CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Glossary of mathematical symbols - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"508201f1-dde7-4b0a-88c5-ca35ef018673","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Glossary_of_mathematical_symbols","wgTitle":"Glossary of mathematical symbols","wgCurRevisionId":1259267463,"wgRevisionId":1259267463,"wgArticleId":73634,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","All articles with unsourced statements","Articles with unsourced statements from November 2020","Articles containing Latin-language text","Wikipedia glossaries using description lists","Pages that use a deprecated format of the math tags","Mathematical symbols","Mathematical notation","Lists of symbols","Mathematical logic","Mathematical tables", "Glossaries of mathematics"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Glossary_of_mathematical_symbols","wgRelevantArticleId":73634,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":80000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false, "wgWikibaseItemId":"Q180159","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","ext.scribunto.logs","site","mediawiki.page.ready", "jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Glossary of mathematical symbols - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Glossary_of_mathematical_symbols"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbols"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Glossary_of_mathematical_symbols rootpage-Glossary_of_mathematical_symbols skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Main menu </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z">Main page</a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia">Contents</a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events">Current events</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x">Random article</a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works">About Wikipedia</a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia">Contact us</a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia">Help</a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia">Learn to edit</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors">Community portal</a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r">Recent changes</a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia">Upload file</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"> Search </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Appearance </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class="">Donate</a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Glossary+of+mathematical+symbols" title="You are encouraged to create an account and log in; however, it is not mandatory" class="">Create account</a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Glossary+of+mathematical+symbols" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class="">Log in</a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Personal tools </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en">Donate</a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Glossary+of+mathematical+symbols" title="You are encouraged to create an account and log in; however, it is not mandatory"> Create account</a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Glossary+of+mathematical+symbols" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"> Log in</a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing">learn more</a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y">Contributions</a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n">Talk</a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Layout_of_this_article" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Layout_of_this_article"> <div class="vector-toc-text"> 1 Layout of this article </div> </a> <ul id="toc-Layout_of_this_article-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arithmetic_operators" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Arithmetic_operators"> <div class="vector-toc-text"> 2 Arithmetic operators </div> </a> <ul id="toc-Arithmetic_operators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equality,_equivalence_and_similarity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equality,_equivalence_and_similarity"> <div class="vector-toc-text"> 3 Equality, equivalence and similarity </div> </a> <ul id="toc-Equality,_equivalence_and_similarity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparison" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Comparison"> <div class="vector-toc-text"> 4 Comparison </div> </a> <ul id="toc-Comparison-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Set_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Set_theory"> <div class="vector-toc-text"> 5 Set theory </div> </a> <ul id="toc-Set_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basic_logic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Basic_logic"> <div class="vector-toc-text"> 6 Basic logic </div> </a> <ul id="toc-Basic_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Blackboard_bold" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Blackboard_bold"> <div class="vector-toc-text"> 7 Blackboard bold </div> </a> <ul id="toc-Blackboard_bold-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calculus" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Calculus"> <div class="vector-toc-text"> 8 Calculus </div> </a> <ul id="toc-Calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_and_multilinear_algebra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Linear_and_multilinear_algebra"> <div class="vector-toc-text"> 9 Linear and multilinear algebra </div> </a> <ul id="toc-Linear_and_multilinear_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Advanced_group_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Advanced_group_theory"> <div class="vector-toc-text"> 10 Advanced group theory </div> </a> <ul id="toc-Advanced_group_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinite_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Infinite_numbers"> <div class="vector-toc-text"> 11 Infinite numbers </div> </a> <ul id="toc-Infinite_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Brackets" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Brackets"> <div class="vector-toc-text"> 12 Brackets </div> </a> <button aria-controls="toc-Brackets-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Brackets subsection </button> <ul id="toc-Brackets-sublist" class="vector-toc-list"> <li id="toc-Parentheses" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parentheses"> <div class="vector-toc-text"> 12.1 Parentheses </div> </a> <ul id="toc-Parentheses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Square_brackets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Square_brackets"> <div class="vector-toc-text"> 12.2 Square brackets </div> </a> <ul id="toc-Square_brackets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Braces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Braces"> <div class="vector-toc-text"> 12.3 Braces </div> </a> <ul id="toc-Braces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_brackets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_brackets"> <div class="vector-toc-text"> 12.4 Other brackets </div> </a> <ul id="toc-Other_brackets-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Symbols_that_do_not_belong_to_formulas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Symbols_that_do_not_belong_to_formulas"> <div class="vector-toc-text"> 13 Symbols that do not belong to formulas </div> </a> <ul id="toc-Symbols_that_do_not_belong_to_formulas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Miscellaneous" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Miscellaneous"> <div class="vector-toc-text"> 14 Miscellaneous </div> </a> <ul id="toc-Miscellaneous-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 15 See also </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle See also subsection </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Related_articles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Related_articles"> <div class="vector-toc-text"> 15.1 Related articles </div> </a> <ul id="toc-Related_articles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_lists" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Related_lists"> <div class="vector-toc-text"> 15.2 Related lists </div> </a> <ul id="toc-Related_lists-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unicode_symbols" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unicode_symbols"> <div class="vector-toc-text"> 15.3 Unicode symbols </div> </a> <ul id="toc-Unicode_symbols-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 16 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 17 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Glossary of mathematical symbols</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 56 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-56" aria-hidden="true" > 56 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%88%92%E1%88%B3%E1%89%A5_%E1%88%9D%E1%88%8D%E1%8A%AD%E1%89%B6%E1%89%BD" title="የሒሳብ ምልክቶች – Amharic" lang="am" hreflang="am" data-title="የሒሳብ ምልክቶች" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D8%A6%D9%85%D8%A9_%D8%A7%D9%84%D8%B1%D9%85%D9%88%D8%B2_%D8%A7%D9%84%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A9" title="قائمة الرموز الرياضية – Arabic" lang="ar" hreflang="ar" data-title="قائمة الرموز الرياضية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%A4%E0%A7%80%E0%A6%95%E0%A7%87%E0%A6%B0_%E0%A6%A4%E0%A6%BE%E0%A6%B2%E0%A6%BF%E0%A6%95%E0%A6%BE" title="গাণিতিক প্রতীকের তালিকা – Bangla" lang="bn" hreflang="bn" data-title="গাণিতিক প্রতীকের তালিকা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B0%D0%B1%D0%BB%D0%B8%D1%86%D0%B0_%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8_%D1%81%D0%B8%D0%BC%D0%B2%D0%BE%D0%BB%D0%B8" title="Таблица на математически символи – Bulgarian" lang="bg" hreflang="bg" data-title="Таблица на математически символи" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Tabela_matemati%C4%8Dkih_simbola" title="Tabela matematičkih simbola – Bosnian" lang="bs" hreflang="bs" data-title="Tabela matematičkih simbola" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Taula_de_s%C3%ADmbols_matem%C3%A0tics" title="Taula de símbols matemàtics – Catalan" lang="ca" hreflang="ca" data-title="Taula de símbols matemàtics" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Matematick%C3%A9_symboly_a_zna%C4%8Dky" title="Matematické symboly a značky – Czech" lang="cs" hreflang="cs" data-title="Matematické symboly a značky" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhestr_symbolau_mathemategol" title="Rhestr symbolau mathemategol – Welsh" lang="cy" hreflang="cy" data-title="Rhestr symbolau mathemategol" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Liste_mathematischer_Symbole" title="Liste mathematischer Symbole – German" lang="de" hreflang="de" data-title="Liste mathematischer Symbole" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Matemaatiliste_s%C3%BCmbolite_loend" title="Matemaatiliste sümbolite loend – Estonian" lang="et" hreflang="et" data-title="Matemaatiliste sümbolite loend" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Anexo:S%C3%ADmbolos_matem%C3%A1ticos" title="Anexo:Símbolos matemáticos – Spanish" lang="es" hreflang="es" data-title="Anexo:Símbolos matemáticos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zerrenda:Sinbolo_matematikoak" title="Zerrenda:Sinbolo matematikoak – Basque" lang="eu" hreflang="eu" data-title="Zerrenda:Sinbolo matematikoak" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D9%87%D8%B1%D8%B3%D8%AA_%D8%A7%D8%B5%D8%B7%D9%84%D8%A7%D8%AD%D8%A7%D8%AA_%D9%86%D9%85%D8%A7%D8%AF%D9%87%D8%A7%DB%8C_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="فهرست اصطلاحات نمادهای ریاضی – Persian" lang="fa" hreflang="fa" data-title="فهرست اصطلاحات نمادهای ریاضی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Table_de_symboles_math%C3%A9matiques" title="Table de symboles mathématiques – French" lang="fr" hreflang="fr" data-title="Table de symboles mathématiques" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/S%C3%ADmbolos_matem%C3%A1ticos" title="Símbolos matemáticos – Galician" lang="gl" hreflang="gl" data-title="Símbolos matemáticos" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%ED%95%99_%EA%B8%B0%ED%98%B8" title="수학 기호 – Korean" lang="ko" hreflang="ko" data-title="수학 기호" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B6%D5%B7%D5%A1%D5%B6%D5%B6%D5%A5%D6%80" title="Մաթեմատիկական նշաններ – Armenian" lang="hy" hreflang="hy" data-title="Մաթեմատիկական նշաններ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A5%80%E0%A4%95%E0%A5%8B%E0%A4%82_%E0%A4%95%E0%A5%80_%E0%A4%B8%E0%A4%BE%E0%A4%B0%E0%A4%A3%E0%A5%80" title="गणितीय प्रतीकों की सारणी – Hindi" lang="hi" hreflang="hi" data-title="गणितीय प्रतीकों की सारणी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Daftar_simbol_matematika" title="Daftar simbol matematika – Indonesian" lang="id" hreflang="id" data-title="Daftar simbol matematika" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Tabella_de_symbolos_mathematic" title="Tabella de symbolos mathematic – Interlingua" lang="ia" hreflang="ia" data-title="Tabella de symbolos mathematic" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Glossario_della_simbologia_matematica" title="Glossario della simbologia matematica – Italian" lang="it" hreflang="it" data-title="Glossario della simbologia matematica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%9E%D7%AA%D7%9E%D7%98%D7%99" title="סימון מתמטי – Hebrew" lang="he" hreflang="he" data-title="סימון מתמטי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4_%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%A4%E0%B3%80%E0%B2%95%E0%B2%97%E0%B2%B3%E0%B3%81" title="ಗಣಿತ ಪ್ರತೀಕಗಳು – Kannada" lang="kn" hreflang="kn" data-title="ಗಣಿತ ಪ್ರತೀಕಗಳು" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%B1%D0%B5%D0%BB%D0%B3%D1%96%D0%BB%D0%B5%D1%80" title="Математикалық белгілер – Kazakh" lang="kk" hreflang="kk" data-title="Математикалық белгілер" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/N%C3%AE%C5%9Faney%C3%AAn_b%C3%AErkariy%C3%AA" title="Nîşaneyên bîrkariyê – Kurdish" lang="ku" hreflang="ku" data-title="Nîşaneyên bîrkariyê" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Simboles_matematical" title="Simboles matematical – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Simboles matematical" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target">Lingua Franca Nova</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Tavula_de_s%C3%ACmboli_matem%C3%A0tich" title="Tavula de sìmboli matemàtich – Lombard" lang="lmo" hreflang="lmo" data-title="Tavula de sìmboli matemàtich" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Matematikai_szimb%C3%B3lumok_list%C3%A1ja" title="Matematikai szimbólumok listája – Hungarian" lang="hu" hreflang="hu" data-title="Matematikai szimbólumok listája" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%BF%E0%B4%B2%E0%B5%81%E0%B4%AA%E0%B4%AF%E0%B5%8B%E0%B4%97%E0%B4%BF%E0%B4%95%E0%B5%8D%E0%B4%95%E0%B5%81%E0%B4%A8%E0%B5%8D%E0%B4%A8_%E0%B4%9A%E0%B4%BF%E0%B4%B9%E0%B5%8D%E0%B4%A8%E0%B4%99%E0%B5%8D%E0%B4%99%E0%B4%B3%E0%B5%81%E0%B4%9F%E0%B5%86_%E0%B4%AA%E0%B4%9F%E0%B5%8D%E0%B4%9F%E0%B4%BF%E0%B4%95" title="ഗണിതത്തിലുപയോഗിക്കുന്ന ചിഹ്നങ്ങളുടെ പട്ടിക – Malayalam" lang="ml" hreflang="ml" data-title="ഗണിതത്തിലുപയോഗിക്കുന്ന ചിഹ്നങ്ങളുടെ പട്ടിക" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%84%D8%B3%D8%AA%D8%A9_%D8%A7%D9%84%D8%B1%D9%85%D9%88%D8%B2_%D8%A7%D9%84%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D9%87" title="لستة الرموز الرياضيه – Egyptian Arabic" lang="arz" hreflang="arz" data-title="لستة الرموز الرياضيه" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target">مصرى</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Glosari_simbol_matematik" title="Glosari simbol matematik – Malay" lang="ms" hreflang="ms" data-title="Glosari simbol matematik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lijst_van_wiskundige_symbolen" title="Lijst van wiskundige symbolen – Dutch" lang="nl" hreflang="nl" data-title="Lijst van wiskundige symbolen" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%E8%A8%98%E5%8F%B7%E3%81%AE%E8%A1%A8" title="数学記号の表 – Japanese" lang="ja" hreflang="ja" data-title="数学記号の表" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Matematiske_symboler" title="Matematiske symboler – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Matematiske symboler" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Symbol_i_matematikk" title="Symbol i matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Symbol i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%D8%B1%DB%8C%D8%A7%D8%B6%D9%8A_%D8%AF_%D8%B3%D9%85%D8%A8%D9%88%D9%84%D9%88%D9%86%D9%88_%D9%84%DA%93%D9%84%DB%8C%DA%A9" title="د ریاضي د سمبولونو لړلیک – Pashto" lang="ps" hreflang="ps" data-title="د ریاضي د سمبولونو لړلیک" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Lista_symboli_matematycznych" title="Lista symboli matematycznych – Polish" lang="pl" hreflang="pl" data-title="Lista symboli matematycznych" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Lista_de_s%C3%ADmbolos_matem%C3%A1ticos" title="Lista de símbolos matemáticos – Portuguese" lang="pt" hreflang="pt" data-title="Lista de símbolos matemáticos" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Tabel_de_simboluri_matematice" title="Tabel de simboluri matematice – Romanian" lang="ro" hreflang="ro" data-title="Tabel de simboluri matematice" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B0%D0%B1%D0%BB%D0%B8%D1%86%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D1%85_%D1%81%D0%B8%D0%BC%D0%B2%D0%BE%D0%BB%D0%BE%D0%B2" title="Таблица математических символов – Russian" lang="ru" hreflang="ru" data-title="Таблица математических символов" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/List_of_mathematical_symbols" title="List of mathematical symbols – Simple English" lang="en-simple" hreflang="en-simple" data-title="List of mathematical symbols" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Matematick%C3%A9_zna%C4%8Dky" title="Matematické značky – Slovak" lang="sk" hreflang="sk" data-title="Matematické značky" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Seznam_matemati%C4%8Dnih_simbolov" title="Seznam matematičnih simbolov – Slovenian" lang="sl" hreflang="sl" data-title="Seznam matematičnih simbolov" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Sumad_xisaabeed" title="Sumad xisaabeed – Somali" lang="so" hreflang="so" data-title="Sumad xisaabeed" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Tabel_lambang_matematis" title="Tabel lambang matematis – Sundanese" lang="su" hreflang="su" data-title="Tabel lambang matematis" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Luettelo_matemaattisista_merkeist%C3%A4" title="Luettelo matemaattisista merkeistä – Finnish" lang="fi" hreflang="fi" data-title="Luettelo matemaattisista merkeistä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Lista_%C3%B6ver_matematiska_symboler" title="Lista över matematiska symboler – Swedish" lang="sv" hreflang="sv" data-title="Lista över matematiska symboler" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B2%E0%B8%A2%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%AA%E0%B8%B1%E0%B8%8D%E0%B8%A5%E0%B8%B1%E0%B8%81%E0%B8%A9%E0%B8%93%E0%B9%8C%E0%B8%97%E0%B8%B2%E0%B8%87%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C" title="รายการสัญลักษณ์ทางคณิตศาสตร์ – Thai" lang="th" hreflang="th" data-title="รายการสัญลักษณ์ทางคณิตศาสตร์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Matematiksel_sembollerin_listesi" title="Matematiksel sembollerin listesi – Turkish" lang="tr" hreflang="tr" data-title="Matematiksel sembollerin listesi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B0%D0%B1%D0%BB%D0%B8%D1%86%D1%8F_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%B8%D1%85_%D1%81%D0%B8%D0%BC%D0%B2%D0%BE%D0%BB%D1%96%D0%B2" title="Таблиця математичних символів – Ukrainian" lang="uk" hreflang="uk" data-title="Таблиця математичних символів" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Danh_s%C3%A1ch_k%C3%BD_hi%E1%BB%87u_to%C3%A1n_h%E1%BB%8Dc" title="Danh sách ký hiệu toán học – Vietnamese" lang="vi" hreflang="vi" data-title="Danh sách ký hiệu toán học" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Mat%C3%B5maatigat%C3%A4ht" title="Matõmaatigatäht – Võro" lang="vro" hreflang="vro" data-title="Matõmaatigatäht" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E7%AC%A6%E8%A1%A8" title="數學符表 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="數學符表" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%E7%AC%A6%E5%8F%B7%E8%A1%A8" title="数学符号表 – Wu" lang="wuu" hreflang="wuu" data-title="数学符号表" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E7%AC%A6%E8%99%9F" title="數學符號 – Cantonese" lang="yue" hreflang="yue" data-title="數學符號" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%E7%AC%A6%E5%8F%B7%E8%A1%A8" title="数学符号表 – Chinese" lang="zh" hreflang="zh" data-title="数学符号表" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q180159#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Glossary_of_mathematical_symbols" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Glossary_of_mathematical_symbols" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t">Talk</a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >English </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Glossary_of_mathematical_symbols">Read</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=history" title="Past revisions of this page [h]" accesskey="h">View history</a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >Tools </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Glossary_of_mathematical_symbols">Read</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=history">View history</a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Glossary_of_mathematical_symbols" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j">What links here</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Glossary_of_mathematical_symbols" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k">Related changes</a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q">Special pages</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Glossary_of_mathematical_symbols&oldid=1259267463" title="Permanent link to this revision of this page">Permanent link</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=info" title="More information about this page">Page information</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Glossary_of_mathematical_symbols&id=1259267463&wpFormIdentifier=titleform" title="Information on how to cite this page">Cite this page</a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FGlossary_of_mathematical_symbols">Get shortened URL</a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FGlossary_of_mathematical_symbols">Download QR code</a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Glossary_of_mathematical_symbols&action=show-download-screen" title="Download this page as a PDF file">Download as PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Glossary_of_mathematical_symbols&printable=yes" title="Printable version of this page [p]" accesskey="p">Printable version</a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Skyscrapers_in_Europe" hreflang="en">Wikimedia Commons</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q180159" title="Structured data on this page hosted by Wikidata [g]" accesskey="g">Wikidata item</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> A mathematical symbol is a figure or a combination of figures that is used to represent a <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a>, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a <a href="/wiki/Formula" title="Formula">formula</a>. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics. The most basic symbols are the <a href="/wiki/Decimal_digit" class="mw-redirect" title="Decimal digit">decimal digits</a> (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the <a href="/wiki/Latin_alphabet" title="Latin alphabet">Latin alphabet</a>. The decimal digits are used for representing numbers through the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a>. Historically, upper-case letters were used for representing <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> in geometry, and lower-case letters were used for <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> and <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constants</a>. Letters are used for representing many other sorts of <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a>. As the number of these sorts has remarkably increased in modern mathematics, the <a href="/wiki/Greek_alphabet" title="Greek alphabet">Greek alphabet</a> and some <a href="/wiki/Hebrew_alphabet" title="Hebrew alphabet">Hebrew letters</a> are also used. In mathematical <a href="/wiki/Formula" title="Formula">formulas</a>, the standard <a href="/wiki/Typeface" title="Typeface">typeface</a> is <a href="/wiki/Italic_type" title="Italic type">italic type</a> for Latin letters and lower-case Greek letters, and upright type for upper case Greek letters. For having more symbols, other typefaces are also used, mainly <a href="/wiki/Boldface" class="mw-redirect" title="Boldface">boldface</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a,A,b,B} ,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold">A</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold">b</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold">B</mi> </mrow> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a,A,b,B} ,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3142a9c980db3fa2c296c61070f1314b6bf16192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.85ex; height:2.509ex;" alt="{\displaystyle \mathbf {a,A,b,B} ,\ldots }">, <a href="/wiki/Script_typeface" title="Script typeface">script typeface</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A,B}},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> <mo class="MJX-tex-caligraphic" mathvariant="script">,</mo> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A,B}},\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad450ebee43ba59fd5b1637f8db2e5612477f9b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.238ex; height:2.676ex;" alt="{\displaystyle {\mathcal {A,B}},\ldots }"> (the lower-case script face is rarely used because of the possible confusion with the standard face), <a href="/wiki/Fraktur" title="Fraktur">German fraktur</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a,A,b,B}},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> <mo mathvariant="fraktur">,</mo> <mi mathvariant="fraktur">A</mi> <mo mathvariant="fraktur">,</mo> <mi mathvariant="fraktur">b</mi> <mo mathvariant="fraktur">,</mo> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a,A,b,B}},\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a10069edb64682b1e478309c43eea22d6b155e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.937ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a,A,b,B}},\ldots }">, and <a href="/wiki/Blackboard_bold" title="Blackboard bold">blackboard bold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N,Z,Q,R,C,H,F} _{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> <mo mathvariant="double-struck">,</mo> <mi mathvariant="double-struck">Z</mi> <mo mathvariant="double-struck">,</mo> <mi mathvariant="double-struck">Q</mi> <mo mathvariant="double-struck">,</mo> <mi mathvariant="double-struck">R</mi> <mo mathvariant="double-struck">,</mo> <mi mathvariant="double-struck">C</mi> <mo mathvariant="double-struck">,</mo> <mi mathvariant="double-struck">H</mi> <mo mathvariant="double-struck">,</mo> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N,Z,Q,R,C,H,F} _{q}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b0161f396cc2a249d4106fc2364337027dd592" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.813ex; height:3.009ex;" alt="{\displaystyle \mathbb {N,Z,Q,R,C,H,F} _{q}}"> (the other letters are rarely used in this face, or their use is unconventional). The use of Latin and Greek letters as symbols for denoting mathematical objects is not described in this article. For such uses, see <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">Variable (mathematics)</a> and <a href="/wiki/List_of_mathematical_constants" title="List of mathematical constants">List of mathematical constants</a>. However, some symbols that are described here have the same shape as the letter from which they are derived, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod {}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \prod {}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea33ea4f80bcf98a55c46d7b86267fe5fa74df1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.581ex; height:2.843ex;" alt="{\displaystyle \textstyle \prod {}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum {}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum {}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac6f70c9bf4aa9214bd3f9d56bae2d173a3ffda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.841ex; height:2.843ex;" alt="{\displaystyle \textstyle \sum {}}">. These letters alone are not sufficient for the needs of mathematicians, and many other symbols are used. Some take their origin in <a href="/wiki/Punctuation_mark" class="mw-redirect" title="Punctuation mark">punctuation marks</a> and <a href="/wiki/Diacritic" title="Diacritic">diacritics</a> traditionally used in <a href="/wiki/Typography" title="Typography">typography</a>; others by deforming <a href="/wiki/Letter_form" class="mw-redirect" title="Letter form">letter forms</a>, as in the cases of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \in }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \in }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe4d5b0a594c1da89b5e78e7dfbeed90bdcc32f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:1.843ex;" alt="{\displaystyle \in }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc1a1a9c4c0f8d5df989c98aa2773ed657c5937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \forall }">. Others, such as + and =, were specially designed for mathematics. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Layout_of_this_article">Layout of this article</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=1" title="Edit section: Layout of this article">edit</a>]</div> <ul><li>Normally, entries of a <a href="/wiki/Glossary" title="Glossary">glossary</a> are structured by topics and sorted alphabetically. This is not possible here, as there is no natural order on symbols, and many symbols are used in different parts of mathematics with different meanings, often completely unrelated. Therefore, some arbitrary choices had to be made, which are summarized below.</li> <li>The article is split into sections that are sorted by an increasing level of technicality. That is, the first sections contain the symbols that are encountered in most mathematical texts, and that are supposed to be known even by beginners. On the other hand, the last sections contain symbols that are specific to some area of mathematics and are ignored outside these areas. However, the long <a href="#Brackets">section on brackets</a> has been placed near to the end, although most of its entries are elementary: this makes it easier to search for a symbol entry by scrolling.</li> <li>Most symbols have multiple meanings that are generally distinguished either by the area of mathematics where they are used or by their syntax, that is, by their position inside a formula and the nature of the other parts of the formula that are close to them.</li> <li>As readers may not be aware of the area of mathematics to which the symbol that they are looking for is related, the different meanings of a symbol are grouped in the section corresponding to their most common meaning.</li> <li>When the meaning depends on the syntax, a symbol may have different entries depending on the syntax. For summarizing the syntax in the entry name, the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"> is used for representing the neighboring parts of a formula that contains the symbol. See <a href="#Brackets">§ Brackets</a> for examples of use.</li> <li>Most symbols have two printed versions. They can be displayed as <a href="/wiki/Unicode" title="Unicode">Unicode</a> characters, or in <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a> format. With the Unicode version, using <a href="/wiki/Search_engine" title="Search engine">search engines</a> and <a href="/wiki/Copy_and_paste" class="mw-redirect" title="Copy and paste">copy-pasting</a> are easier. On the other hand, the LaTeX rendering is often much better (more aesthetic), and is generally considered a standard in mathematics. Therefore, in this article, the Unicode version of the symbols is used (when possible) for labelling their entry, and the LaTeX version is used in their description. So, for finding how to type a symbol in LaTeX, it suffices to look at the source of the article.</li> <li>For most symbols, the entry name is the corresponding Unicode symbol. So, for searching the entry of a symbol, it suffices to type or copy the Unicode symbol into the search textbox. Similarly, when possible, the entry name of a symbol is also an <a href="/wiki/Wikipedia:ANCHOR" class="mw-redirect" title="Wikipedia:ANCHOR">anchor</a>, which allows linking easily from another Wikipedia article. When an entry name contains special characters such as [,], and |, there is also an anchor, but one has to look at the article source to know it.</li> <li>Finally, when there is an article on the symbol itself (not its mathematical meaning), it is linked to in the entry name.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Arithmetic_operators">Arithmetic operators</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=2" title="Edit section: Arithmetic operators">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1228772891">.mw-parser-output .glossary dt{margin-top:0.4em}.mw-parser-output .glossary dt+dt{margin-top:-0.2em}.mw-parser-output .glossary .templatequote{margin-top:0;margin-bottom:-0.5em}</style> <dl class="glossary"> <dt id="+"><dfn>+    (<a href="/wiki/Plus_sign" class="mw-redirect" title="Plus sign">plus sign</a>)</dfn></dt> <dd>1.  Denotes <a href="/wiki/Addition" title="Addition">addition</a> and is read as plus; for example, 3 + 2.</dd> <dd>2.  Denotes that a number is <a href="/wiki/Sign_(mathematics)#Terminology_for_signs" title="Sign (mathematics)">positive</a> and is read as plus. Redundant, but sometimes used for emphasizing that a number is <a href="/wiki/Positive_number" class="mw-redirect" title="Positive number">positive</a>, specially when other numbers in the context are or may be negative; for example, +2.</dd> <dd>3.  Sometimes used instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sqcup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sqcup }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1596aedf354da694149e44ce2bf53ede54eca8cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \sqcup }"> for a <a href="/wiki/Disjoint_union" title="Disjoint union">disjoint union</a> of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a>.</dd> <dt id="−"><dfn>−    (<a href="/wiki/Minus_sign" class="mw-redirect" title="Minus sign">minus sign</a>)</dfn></dt> <dd>1.  Denotes <a href="/wiki/Subtraction" title="Subtraction">subtraction</a> and is read as minus; for example, 3 – 2.</dd> <dd>2.  Denotes the <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a> and is read as minus, the negative of, or the opposite of; for example, –2.</dd> <dd>3.  Also used in place of \ for denoting the <a href="/wiki/Set-theoretic_complement" class="mw-redirect" title="Set-theoretic complement">set-theoretic complement</a>; see <a href="#\">\</a> in <a href="#Set_theory">§ Set theory</a>.</dd> <dt id="×"><dfn>×    (<a href="/wiki/Multiplication_sign" title="Multiplication sign">multiplication sign</a>)</dfn></dt> <dd>1.  In <a href="/wiki/Elementary_arithmetic" title="Elementary arithmetic">elementary arithmetic</a>, denotes <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, and is read as times; for example, 3 × 2.</dd> <dd>2.  In <a href="/wiki/Geometry" title="Geometry">geometry</a> and <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, denotes the <a href="/wiki/Cross_product" title="Cross product">cross product</a>.</dd> <dd>3.  In <a href="/wiki/Set_theory" title="Set theory">set theory</a> and <a href="/wiki/Category_theory" title="Category theory">category theory</a>, denotes the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> and the <a href="/wiki/Direct_product" title="Direct product">direct product</a>. See also <a href="#cartesian">×</a> in <a href="#Set_theory">§ Set theory</a>.</dd> <dt id="·"><dfn>·    (<a href="/wiki/Interpunct" title="Interpunct">dot</a>)</dfn></dt> <dd>1.  Denotes <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> and is read as times; for example, 3 ⋅ 2.</dd> <dd>2.  In <a href="/wiki/Geometry" title="Geometry">geometry</a> and <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, denotes the <a href="/wiki/Dot_product" title="Dot product">dot product</a>.</dd> <dd>3.  Placeholder used for replacing an indeterminate element. For example, saying "the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> is denoted by | · |" is perhaps clearer than saying that it is denoted as | |.</dd> <dt id="±"><dfn>±    (<a href="/wiki/Plus%E2%80%93minus_sign" title="Plus–minus sign">plus–minus sign</a>)</dfn></dt> <dd>1.  Denotes either a plus sign or a minus sign.</dd> <dd>2.  Denotes the range of values that a measured quantity may have; for example, 10 ± 2 denotes an unknown value that lies between 8 and 12.</dd> <dt id="∓"><dfn>∓    (<a href="/wiki/Minus-plus_sign" class="mw-redirect" title="Minus-plus sign">minus-plus sign</a>)</dfn></dt> <dd>Used paired with ±, denotes the opposite sign; that is, + if ± is –, and – if ± is +.</dd> <dt id="÷"><dfn>÷    (<a href="/wiki/Division_sign" title="Division sign">division sign</a>)</dfn></dt> <dd>Widely used for denoting <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> in <a href="/wiki/English-speaking_world" title="English-speaking world"> Anglophone</a> countries, it is no longer in common use in mathematics and its use is "not recommended".<a href="#cite_note-ISO-1">[1]</a> In some countries, it can indicate subtraction.</dd> <dt id="ratio"><dfn>:    (<a href="/wiki/Colon_(punctuation)" title="Colon (punctuation)">colon</a>)</dfn></dt> <dd>1.  Denotes the <a href="/wiki/Ratio" title="Ratio">ratio</a> of two quantities.</dd> <dd>2.  In some countries, may denote <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>.</dd> <dd>3.  In <a href="/wiki/Set-builder_notation" title="Set-builder notation">set-builder notation</a>, it is used as a separator meaning "such that"; see <a href="#b:b">{□ : □}</a>.</dd> <dt id="/"><dfn>/    (<a href="/wiki/Slash_(punctuation)" title="Slash (punctuation)">slash</a>)</dfn></dt> <dd>1.  Denotes <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> and is read as divided by or over. Often replaced by a horizontal bar. For example, 3 / 2 or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4d39a9f7beda4ddbeffafaca691c386d3142cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{2}}}">.</dd> <dd>2.  Denotes a <a href="/wiki/Quotient_(disambiguation)#Mathematics" class="mw-disambig" title="Quotient (disambiguation)">quotient structure</a>. For example, <a href="/wiki/Quotient_set" class="mw-redirect" title="Quotient set">quotient set</a>, <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a>, <a href="/wiki/Quotient_category" title="Quotient category">quotient category</a>, etc.</dd> <dd>3.  In <a href="/wiki/Number_theory" title="Number theory">number theory</a> and <a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F/E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F/E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba04c235083fd37db07fce9ec43901e0cb6d1d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.679ex; height:2.843ex;" alt="{\displaystyle F/E}"> denotes a <a href="/wiki/Field_extension" title="Field extension">field extension</a>, where F is an <a href="/wiki/Extension_field" class="mw-redirect" title="Extension field">extension field</a> of the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> E.</dd> <dd>4.  In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, denotes a <a href="/wiki/Conditional_probability" title="Conditional probability">conditional probability</a>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A/B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A/B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8ea898d35e8bbd6aad8df7328635a699751856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.224ex; height:2.843ex;" alt="{\displaystyle P(A/B)}"> denotes the probability of A, given that B occurs. Usually denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\mid B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∣</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A\mid B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8f30f4da85b53901e0871eb41ed8827f511bb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.999ex; height:2.843ex;" alt="{\displaystyle P(A\mid B)}">: see "<a href="#vbar">|</a>".</dd> <dt id="√"><dfn>√    (<a href="/wiki/Radical_symbol#Encoding" title="Radical symbol">square-root symbol</a>)</dfn></dt> <dd>Denotes <a href="/wiki/Square_root" title="Square root">square root</a> and is read as the square root of. Rarely used in modern mathematics without a horizontal bar delimiting the width of its argument (see the next item). For example, √2.</dd> <dt id="sqrt"><dfn>√     (<a href="/wiki/Radical_symbol" title="Radical symbol">radical symbol</a>)</dfn></dt> <dd>1.  Denotes <a href="/wiki/Square_root" title="Square root">square root</a> and is read as the square root of. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {3+2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> <mo>+</mo> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3+2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc70eb95fd7972cbb4a2ebf5adb1b62577062d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.101ex; height:3.009ex;" alt="{\displaystyle {\sqrt {3+2}}}">.</dd> <dd>2.  With an integer greater than 2 as a left superscript, denotes an <a href="/wiki/Nth_root" title="Nth root">nth root</a>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{7}]{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{7}]{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c775908a306fe7496d4302ab87bfd414e1261c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{7}]{3}}}"> denotes the 7th root of 3.</dd> <dt id="caret"><dfn>^    (<a href="/wiki/Caret" title="Caret">caret</a>)</dfn></dt> <dd>1.  <a href="/wiki/Exponentiation" title="Exponentiation">Exponentiation</a> is normally denoted with a <a href="/wiki/Superscript" class="mw-redirect" title="Superscript">superscript</a>. However, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8561c712e86598255e8434a70affa18ffd7e0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.379ex; height:2.343ex;" alt="{\displaystyle x^{y}}"> is often denoted x^y when superscripts are not easily available, such as in <a href="/wiki/Programming_language" title="Programming language">programming languages</a> (including <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a>) or plain text <a href="/wiki/Email" title="Email">emails</a>.</dd> <dd>2.  Not to be confused with <a href="#∧">∧</a></dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Equality,_equivalence_and_similarity">Equality, equivalence and similarity</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=3" title="Edit section: Equality, equivalence and similarity">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="equal"><dfn>=    (<a href="/wiki/Equals_sign" title="Equals sign">equals sign</a>)</dfn></dt> <dd>1.  Denotes <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equality</a>.</dd> <dd>2.  Used for naming a <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a> in a sentence like "let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c10dd6f167d5c527a1dd2d5b38c2b0acfdc8b13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.204ex; height:2.176ex;" alt="{\displaystyle x=E}"> ", where E is an <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expression</a>. See also ≝, ≜ or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9a320a04a814e22f58952141fd0d92cd5ac402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:1.676ex;" alt="{\displaystyle :=}">. </dd> <dt id="≝"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangleq \quad {\stackrel {\scriptscriptstyle \mathrm {def} }{=}}\quad :=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≜</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mspace width="1em" /> <mo>:=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangleq \quad {\stackrel {\scriptscriptstyle \mathrm {def} }{=}}\quad :=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de0286754a5820d8c399d517ed37823425a43045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.03ex; height:3.009ex;" alt="{\displaystyle \triangleq \quad {\stackrel {\scriptscriptstyle \mathrm {def} }{=}}\quad :=}"></dfn></dt> <dd>Any of these is sometimes used for naming a <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a>. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\triangleq E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≜</mo> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\triangleq E,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84982af2eb248f5a7cfd263fed633e0823f2e710" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.851ex; height:2.843ex;" alt="{\displaystyle x\triangleq E,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mathrel {\stackrel {\scriptscriptstyle \mathrm {def} }{=}} E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mathrel {\stackrel {\scriptscriptstyle \mathrm {def} }{=}} E,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a21ee4e84a4d9c4c434a9ac799f4fc696518e1a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.874ex; height:3.343ex;" alt="{\displaystyle x\mathrel {\stackrel {\scriptscriptstyle \mathrm {def} }{=}} E,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mathrel {:=} E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mrow class="MJX-TeXAtom-REL"> <mo>:=</mo> </mrow> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mathrel {:=} E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bf2585d24c48dcd4df885ddc97fad29fce36aae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.851ex; height:2.176ex;" alt="{\displaystyle x\mathrel {:=} E}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\mathrel {=:} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow class="MJX-TeXAtom-REL"> <mo>=:</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\mathrel {=:} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38955eb5c1108d6449c35e2630bb29b61b39a9e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.851ex; height:2.176ex;" alt="{\displaystyle E\mathrel {=:} x}"> are each an abbreviation of the phrase "let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c10dd6f167d5c527a1dd2d5b38c2b0acfdc8b13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.204ex; height:2.176ex;" alt="{\displaystyle x=E}">", where ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}">⁠ is an <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expression</a> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">⁠ is a <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a>. This is similar to the concept of <a href="/wiki/Assignment_(computer_science)" title="Assignment (computer science)">assignment</a> in computer science, which is variously denoted (depending on the <a href="/wiki/Programming_language" title="Programming language">programming language</a> used) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =,:=,\leftarrow ,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo>,</mo> <mo>:=</mo> <mo>,</mo> <mo stretchy="false">←</mo> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =,:=,\leftarrow ,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61fae5b3fdf4de85acb3ad8370af4763ca969c4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.412ex; height:2.176ex;" alt="{\displaystyle =,:=,\leftarrow ,\ldots }"></dd> <dt id="inequal"><dfn>≠    (<a href="/wiki/Not-equal_sign" class="mw-redirect" title="Not-equal sign">not-equal sign</a>)</dfn></dt> <dd>Denotes <a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">inequality</a> and means "not equal".</dd> <dt id="approx"><dfn>≈</dfn></dt> <dd>The most common symbol for denoting <a href="/wiki/Approximate_equality" class="mw-redirect" title="Approximate equality">approximate equality</a>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \approx 3.14159.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>≈</mo> <mn>3.14159.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \approx 3.14159.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e445b5e559460acd65d84c2333d7ab78fe2833d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.699ex; height:2.176ex;" alt="{\displaystyle \pi \approx 3.14159.}"> </dd> <dt id="tilde"><dfn>~    (<a href="/wiki/Tilde" title="Tilde">tilde</a>)</dfn></dt> <dd>1.  Between two numbers, either it is used instead of ≈ to mean "approximatively equal", or it means "has the same <a href="/wiki/Order_of_magnitude" title="Order of magnitude">order of magnitude</a> as".</dd> <dd>2.  Denotes the <a href="/wiki/Asymptotic_equivalence" class="mw-redirect" title="Asymptotic equivalence">asymptotic equivalence</a> of two functions or sequences.</dd> <dd>3.  Often used for denoting other types of similarity, for example, <a href="/wiki/Matrix_similarity" title="Matrix similarity">matrix similarity</a> or <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similarity of geometric shapes</a>.</dd> <dd>4.  Standard notation for an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>.</dd> <dd>5.  In <a href="/wiki/Probability" title="Probability">probability</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, may specify the <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> of a <a href="/wiki/Random_variable" title="Random variable">random variable</a>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim N(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim N(0,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21715fc79ffc6f31cbbcacb5cda0e82f08d807f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.31ex; height:2.843ex;" alt="{\displaystyle X\sim N(0,1)}"> means that the distribution of the random variable X is <a href="/wiki/Standard_normal_distribution" class="mw-redirect" title="Standard normal distribution">standard normal</a>.<a href="#cite_note-2">[2]</a></dd> <dd>6.  Notation for <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportionality</a>. See also <a href="#∝">∝</a> for a less ambiguous symbol.</dd> <dt id="triple_bar"><dfn>≡    (<a href="/wiki/Triple_bar" title="Triple bar">triple bar</a>)</dfn></dt> <dd>1.  Denotes an <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identity</a>; that is, an equality that is true whichever values are given to the variables occurring in it.</dd> <dd>2.  In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, and more specifically in <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>, denotes the <a href="/wiki/Congruence_modulo_n" class="mw-redirect" title="Congruence modulo n">congruence</a> modulo an integer.</dd> <dd>3.  May denote a <a href="/wiki/Logical_equivalence" title="Logical equivalence">logical equivalence</a>.</dd> <dt id="≅"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cong }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cong }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a725ebc5ab8de11d7b71a8aa5a3706c2ea467885" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.049ex; margin-bottom: -0.22ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \cong }"></dfn></dt> <dd>1.  May denote an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> between two <a href="/wiki/Mathematical_structures" class="mw-redirect" title="Mathematical structures">mathematical structures</a>, and is read as "is isomorphic to".</dd> <dd>2.  In <a href="/wiki/Geometry" title="Geometry">geometry</a>, may denote the <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruence</a> of two <a href="/wiki/Geometric_shape" class="mw-redirect" title="Geometric shape">geometric shapes</a> (that is the equality <a href="/wiki/Up_to" title="Up to">up to</a> a <a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">displacement</a>), and is read "is congruent to".</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Comparison">Comparison</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=4" title="Edit section: Comparison">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="less"><dfn><    (<a href="/wiki/Less-than_sign" title="Less-than sign">less-than sign</a>)</dfn></dt> <dd>1.  <a href="/wiki/Strict_inequality" class="mw-redirect" title="Strict inequality">Strict inequality</a> between two numbers; means and is read as "<a href="/wiki/Less_than" class="mw-redirect" title="Less than">less than</a>".</dd> <dd>2.  Commonly used for denoting any <a href="/wiki/Strict_order" class="mw-redirect" title="Strict order">strict order</a>.</dd> <dd>3.  Between two <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, may mean that the first one is a <a href="/wiki/Proper_subgroup" class="mw-redirect" title="Proper subgroup">proper subgroup</a> of the second one.</dd> <dt id="greater"><dfn>>    (<a href="/wiki/Greater-than_sign" title="Greater-than sign">greater-than sign</a>)</dfn></dt> <dd>1.  <a href="/wiki/Strict_inequality" class="mw-redirect" title="Strict inequality">Strict inequality</a> between two numbers; means and is read as "<a href="/wiki/Greater_than" class="mw-redirect" title="Greater than">greater than</a>".</dd> <dd>2.  Commonly used for denoting any <a href="/wiki/Strict_order" class="mw-redirect" title="Strict order">strict order</a>.</dd> <dd>3.  Between two <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, may mean that the second one is a <a href="/wiki/Proper_subgroup" class="mw-redirect" title="Proper subgroup">proper subgroup</a> of the first one.</dd> <dt id="less-equal_sign"><dfn>≤</dfn></dt> <dd>1.  Means "<a href="/wiki/Less_than_or_equal_to" class="mw-redirect" title="Less than or equal to">less than or equal to</a>". That is, whatever A and B are, A ≤ B is equivalent to A < B or A = B.</dd> <dd>2.  Between two <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, may mean that the first one is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the second one.</dd> <dt id="greater-equal_sign"><dfn>≥</dfn></dt> <dd>1.  Means "<a href="/wiki/Greater_than_or_equal_to" class="mw-redirect" title="Greater than or equal to">greater than or equal to</a>". That is, whatever A and B are, A ≥ B is equivalent to A > B or A = B.</dd> <dd>2.  Between two <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, may mean that the second one is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the first one.</dd> <dt id="much-greater-or-less"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ll {\text{ and }}\gg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≪</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mo>≫</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ll {\text{ and }}\gg }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be1a37d6af9566bc9a3b5e79d5dec111dedacbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.847ex; height:2.176ex;" alt="{\displaystyle \ll {\text{ and }}\gg }"></dfn></dt> <dd>1.  Means "<a href="/wiki/Much_less_than" class="mw-redirect" title="Much less than">much less than</a>" and "<a href="/wiki/Much_greater_than" class="mw-redirect" title="Much greater than">much greater than</a>". Generally, much is not formally defined, but means that the lesser quantity can be neglected with respect to the other. This is generally the case when the lesser quantity is smaller than the other by one or several <a href="/wiki/Orders_of_magnitude" class="mw-redirect" title="Orders of magnitude">orders of magnitude</a>.</dd> <dd>2.  In <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \ll \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>≪</mo> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \ll \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57cb96c52c3ac95056970f5973908ee3e7c134bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.248ex; height:2.343ex;" alt="{\displaystyle \mu \ll \nu }"> means that the measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> is absolutely continuous with respect to the measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }">.</dd> <dt id="less-equal_sign"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leqq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≦</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leqq }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1dd3e3e5dbf6fc91ddef46a531a96e5e3f57143" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.676ex;" alt="{\displaystyle \leqq }"></dfn></dt> <dd>A rarely used symbol, generally a synonym of ≤.</dd> <dt id="pred-succ"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prec {\text{ and }}\succ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≺</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mo>≻</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prec {\text{ and }}\succ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8da58a796d9630e18278d55d8dd03e897a24860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.815ex; height:2.176ex;" alt="{\displaystyle \prec {\text{ and }}\succ }"></dfn></dt> <dd>1.  Often used for denoting an <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">order</a> or, more generally, a <a href="/wiki/Preorder" title="Preorder">preorder</a>, when it would be confusing or not convenient to use < and >.</dd> <dd>2.  <a href="/wiki/Sequention" class="mw-redirect" title="Sequention">Sequention</a> in <a href="/wiki/Asynchronous_logic" class="mw-redirect" title="Asynchronous logic">asynchronous logic</a>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Set_theory">Set theory</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=5" title="Edit section: Set theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="∅"><dfn>∅</dfn></dt> <dd>Denotes the <a href="/wiki/Empty_set" title="Empty set">empty set</a>, and is more often written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }">. Using <a href="/wiki/Set-builder_notation" title="Set-builder notation">set-builder notation</a>, it may also be denoted <a href="#bb"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6f1caa524dfcc90158ad69a51b5f9577fe5f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.325ex; height:2.843ex;" alt="{\displaystyle \{\}}"></a>.</dd> <dt id="sharp"><dfn>#    (<a href="/wiki/Number_sign" title="Number sign">number sign</a>)</dfn></dt> <dd id="sharp-defn1">1.  Number of elements: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#{}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">#</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#{}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d090ffa4d8ba010cce1e735ecbe698456e2119d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.435ex; height:2.509ex;" alt="{\displaystyle \#{}S}"> may denote the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> S. An alternative notation is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |S|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |S|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28d901e98a035ff4c0e37fe6dd8e750ece6c1f0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.793ex; height:2.843ex;" alt="{\displaystyle |S|}">; see <a href="#!□!"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\square |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>◻</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\square |}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44a50c3115f8152b08b0b34d3c821933b7a31049" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.102ex; height:2.843ex;" alt="{\displaystyle |\square |}"></a>.</dd> <dd id="sharp-defn2">2.  <a href="/wiki/Primorial" title="Primorial">Primorial</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n{}\#}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi mathvariant="normal">#</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n{}\#}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b4740012a136055c90e15adfb910615d4a0fd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.331ex; height:2.509ex;" alt="{\displaystyle n{}\#}"> denotes the product of the <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> that are not greater than n.</dd> <dd id="sharp-defn3">3.  In <a href="/wiki/Topology" title="Topology">topology</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\#N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi mathvariant="normal">#</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\#N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7b0504c47f566ed4fdc6a6850e4538f20a62df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.442ex; height:2.509ex;" alt="{\displaystyle M\#N}"> denotes the <a href="/wiki/Connected_sum" title="Connected sum">connected sum</a> of two <a href="/wiki/Manifolds" class="mw-redirect" title="Manifolds">manifolds</a> or two <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knots</a>.</dd> <dt id="∈"><dfn>∈</dfn></dt> <dd>Denotes <a href="/wiki/Set_membership" class="mw-redirect" title="Set membership">set membership</a>, and is read "is in", "belongs to", or "is a member of". That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51186ba8afb2067573a9082d55dd383df1ea9214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle x\in S}"> means that x is an element of the set S.</dd> <dt id="∉"><dfn>∉</dfn></dt> <dd>Means "is not in". That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\notin S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\notin S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/142de939258c1dcfad19ba4988d890f05afb769e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.67ex; height:2.676ex;" alt="{\displaystyle x\notin S}"> means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (x\in S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (x\in S)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c906af95a0dfd43c3b116fc88cbc256e44f54e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.029ex; height:2.843ex;" alt="{\displaystyle \neg (x\in S)}">.</dd> <dt id="⊂"><dfn>⊂</dfn></dt> <dd>Denotes <a href="/wiki/Set_inclusion" class="mw-redirect" title="Set inclusion">set inclusion</a>. However two slightly different definitions are common.</dd> <dd id="⊂-defn1">1.  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}"> may mean that A is a <a href="/wiki/Subset" title="Subset">subset</a> of B, and is possibly equal to B; that is, every element of A belongs to B; expressed as a formula, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall {}x,\,x\in A\Rightarrow x\in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>x</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>∈</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall {}x,\,x\in A\Rightarrow x\in B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94a68cd9952be5300dc6a54b57bc059c9cced3e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.505ex; height:2.509ex;" alt="{\displaystyle \forall {}x,\,x\in A\Rightarrow x\in B}">.</dd> <dd id="⊂-defn2">2.  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}"> may mean that A is a <a href="/wiki/Proper_subset" class="mw-redirect" title="Proper subset">proper subset</a> of B, that is the two sets are different, and every element of A belongs to B; expressed as a formula, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\neq B\land \forall {}x,\,x\in A\Rightarrow x\in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≠</mo> <mi>B</mi> <mo>∧</mo> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>x</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>∈</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\neq B\land \forall {}x,\,x\in A\Rightarrow x\in B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c8f0be836f38bd475082d895cc73b3fa1c278a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.693ex; height:2.676ex;" alt="{\displaystyle A\neq B\land \forall {}x,\,x\in A\Rightarrow x\in B}">.</dd> <dt id="⊆"><dfn>⊆</dfn></dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b09068bd2f7ba899aeb883ebe670b2ad07b0c851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle A\subseteq B}"> means that A is a <a href="/wiki/Subset" title="Subset">subset</a> of B. Used for emphasizing that equality is possible, or when <a href="#⊂-defn2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}"></a> means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is a proper subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eccf5bca7cdc1fa4439af2d31831db6bde00473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle B.}"></dd> <dt id="⊊"><dfn>⊊</dfn></dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subsetneq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊊</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subsetneq B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf81e9a4a81df2d596b4db1cde6b9bdf82c73db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.606ex; height:2.676ex;" alt="{\displaystyle A\subsetneq B}"> means that A is a <a href="/wiki/Proper_subset" class="mw-redirect" title="Proper subset">proper subset</a> of B. Used for emphasizing that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\neq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≠</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\neq B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78362703472ea51edc4614b6b7a7bda8e83131c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.606ex; height:2.676ex;" alt="{\displaystyle A\neq B}">, or when <a href="#⊂-defn1"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}"></a> does not imply that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is a proper subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eccf5bca7cdc1fa4439af2d31831db6bde00473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle B.}"></dd> <dt id="⊃"><dfn>⊃, ⊇, ⊋</dfn></dt> <dd>Denote the converse relation of <a href="#⊂"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊂</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subset }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f51f0eeff0c2a9dcb9c856f87ca0359e701ef01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \subset }"></a>, <a href="#⊆"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subseteq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subseteq }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a924f8dcb2847bb8871edfdbf4c6b5cca0669228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \subseteq }"></a>, and <a href="#⊊"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subsetneq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊊</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subsetneq }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccda11a9c5eb10088602771f7c0f05ffea7f41c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \subsetneq }"></a> respectively. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\supset A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊃</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\supset A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/450398271587fcd521f7313ee3ebfdb5023e1c07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle B\supset A}"> is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}">.</dd> <dt id="∪"><dfn>∪</dfn></dt> <dd>Denotes <a href="/wiki/Set-theoretic_union" class="mw-redirect" title="Set-theoretic union">set-theoretic union</a>, that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cup B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cup B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb575990bcfbcdf616aa6fd76e8b30bf7fd2169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\cup B}"> is the set formed by the elements of A and B together. That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cup B=\{x\mid (x\in A)\lor (x\in B)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cup B=\{x\mid (x\in A)\lor (x\in B)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a39dca33012cd10a5c33d39dc72e75d83d3890f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.829ex; height:2.843ex;" alt="{\displaystyle A\cup B=\{x\mid (x\in A)\lor (x\in B)\}}">.</dd> <dt id="∩"><dfn>∩</dfn></dt> <dd>Denotes <a href="/wiki/Set-theoretic_intersection" class="mw-redirect" title="Set-theoretic intersection">set-theoretic intersection</a>, that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb27b38cf9eac6060e67b61f66cd9beec5067f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\cap B}"> is the set formed by the elements of both A and B. That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B=\{x\mid (x\in A)\land (x\in B)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B=\{x\mid (x\in A)\land (x\in B)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc1a3c3c0e6404731c06d2dce0f84ae2dd0a3daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.829ex; height:2.843ex;" alt="{\displaystyle A\cap B=\{x\mid (x\in A)\land (x\in B)\}}">.</dd> <dt id="∖"><dfn>∖    (<a href="/wiki/Backslash" title="Backslash">backslash</a>)</dfn></dt> <dd><a href="/wiki/Set_difference" class="mw-redirect" title="Set difference">Set difference</a>; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\setminus B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef797ed5deb971321592e34281d9fac27c3249d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.702ex; height:2.843ex;" alt="{\displaystyle A\setminus B}"> is the set formed by the elements of A that are not in B. Sometimes, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>−</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A-B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc58c452f31f578fdf98cafc1c53fe98a0c0975" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A-B}"> is used instead; see <a href="#–">–</a> in <a href="#Arithmetic_operators">§ Arithmetic operators</a>.</dd> <dt id="⊖"><dfn>⊖ or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d909fe94e8277a4c44a50853cb7dbbf0fa3148ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle \triangle }"></dfn></dt> <dd><a href="/wiki/Symmetric_difference" title="Symmetric difference">Symmetric difference</a>: that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\ominus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊖</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\ominus B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd5c8d3fdf145dc31b779da31b1d8793d4fc059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\ominus B}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\operatorname {\triangle } B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">△</mi> </mrow> <mo>⁡</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\operatorname {\triangle } B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/981e8f252c4ed3ccf8237db6114ebb00f296621d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.347ex; height:2.176ex;" alt="{\displaystyle A\operatorname {\triangle } B}"> is the set formed by the elements that belong to exactly one of the two sets A and B.</dd> <dt id="∁"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \complement }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>∁</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \complement }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2479e2cdb7ce0c5be60408f111d2354369189f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \complement }"></dfn></dt> <dd>1.  With a subscript, denotes a <a href="/wiki/Set_complement" class="mw-redirect" title="Set complement">set complement</a>: that is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subseteq A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8124cb68686ede7083aa2a5a821f262eb62954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle B\subseteq A}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \complement _{A}B=A\setminus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>∁</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>B</mi> <mo>=</mo> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \complement _{A}B=A\setminus B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2488b249ebb02b1630b89853088b9525da7636a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.192ex; height:3.009ex;" alt="{\displaystyle \complement _{A}B=A\setminus B}">.</dd> <dd>2.  Without a subscript, denotes the <a href="/wiki/Absolute_complement" class="mw-redirect" title="Absolute complement">absolute complement</a>; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \complement A=\complement _{U}A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>∁</mi> <mi>A</mi> <mo>=</mo> <msub> <mi>∁</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \complement A=\complement _{U}A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb4e64079605c31d77038fab45156dd6d80da4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.402ex; height:2.843ex;" alt="{\displaystyle \complement A=\complement _{U}A}">, where U is a set implicitly defined by the context, which contains all sets under consideration. This set U is sometimes called the <a href="/wiki/Universe_of_discourse" class="mw-redirect" title="Universe of discourse">universe of discourse</a>. </dd> <dt id="cartesian"><dfn>×    (<a href="/wiki/Multiplication_sign" title="Multiplication sign">multiplication sign</a>)</dfn></dt> <dd>See also <a href="#×">×</a> in <a href="#Arithmetic_operators">§ Arithmetic operators</a>.</dd> <dd>1.  Denotes the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of two sets. That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f31ae45b0098f06b5d22c38d317eb097a88fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.348ex; height:2.176ex;" alt="{\displaystyle A\times B}"> is the set formed by all <a href="/wiki/Pair_(mathematics)" class="mw-redirect" title="Pair (mathematics)">pairs</a> of an element of A and an element of B.</dd> <dd>2.  Denotes the <a href="/wiki/Direct_product" title="Direct product">direct product</a> of two <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structures</a> of the same type, which is the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of the underlying sets, equipped with a structure of the same type. For example, <a href="/wiki/Direct_product_of_rings" class="mw-redirect" title="Direct product of rings">direct product of rings</a>, <a href="/wiki/Product_topology" title="Product topology">direct product of topological spaces</a>.</dd> <dd>3.  In <a href="/wiki/Category_theory" title="Category theory">category theory</a>, denotes the <a href="/wiki/Product_(category_theory)" title="Product (category theory)">direct product</a> (often called simply product) of two objects, which is a generalization of the preceding concepts of product.</dd> <dt id="⊔"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sqcup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sqcup }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1596aedf354da694149e44ce2bf53ede54eca8cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \sqcup }"></dfn></dt> <dd>Denotes the <a href="/wiki/Disjoint_union" title="Disjoint union">disjoint union</a>. That is, if A and B are sets then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\sqcup B=\left(A\times \{i_{A}\}\right)\cup \left(B\times \{i_{B}\}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊔</mo> <mi>B</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mrow> <mo>)</mo> </mrow> <mo>∪</mo> <mrow> <mo>(</mo> <mrow> <mi>B</mi> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\sqcup B=\left(A\times \{i_{A}\}\right)\cup \left(B\times \{i_{B}\}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cad9e18591b61ca1d7ed9a8fb3a3a7144bd4bb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.776ex; height:2.843ex;" alt="{\displaystyle A\sqcup B=\left(A\times \{i_{A}\}\right)\cup \left(B\times \{i_{B}\}\right)}"> is a set of <a href="/wiki/Ordered_pair" title="Ordered pair">pairs</a> where iA and iB are distinct indices discriminating the members of A and B in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\sqcup B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\sqcup B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30206a16da3623a1244336c395f9f268b69c4aef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\sqcup B}">⁠.</dd> <dt id="∐"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigsqcup {\text{ or }}\coprod }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⨆</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mo>∐</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigsqcup {\text{ or }}\coprod }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f111b54a788f2892a1626068e74ca9d521b7445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.561ex; height:3.843ex;" alt="{\displaystyle \bigsqcup {\text{ or }}\coprod }"></dfn></dt> <dd>1.  Used for the <a href="/wiki/Disjoint_union" title="Disjoint union">disjoint union</a> of a family of sets, such as in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigsqcup _{i\in I}A_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>⨆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigsqcup _{i\in I}A_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae6dbe70b85b840226996445fda38350347c9648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.237ex; height:3.009ex;" alt="{\textstyle \bigsqcup _{i\in I}A_{i}.}"></dd> <dd>2.  Denotes the <a href="/wiki/Coproduct" title="Coproduct">coproduct</a> of <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structures</a> or of objects in a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Basic_logic">Basic logic</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=6" title="Edit section: Basic logic">edit</a>]</div> Several <a href="/wiki/Logical_symbol" class="mw-redirect" title="Logical symbol">logical symbols</a> are widely used in all mathematics, and are listed here. For symbols that are used only in <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, or are rarely used, see <a href="/wiki/List_of_logic_symbols" title="List of logic symbols">List of logic symbols</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="¬"><dfn>¬    (<a href="/wiki/Not_sign" class="mw-redirect" title="Not sign">not sign</a>)</dfn></dt> <dd>Denotes <a href="/wiki/Logical_negation" class="mw-redirect" title="Logical negation">logical negation</a>, and is read as "not". If E is a <a href="/wiki/Logical_predicate" class="mw-redirect" title="Logical predicate">logical predicate</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2771dd525bad1bb656104318ae9d6986459164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.326ex; height:2.176ex;" alt="{\displaystyle \neg E}"> is the predicate that evaluates to true if and only if E evaluates to false. For clarity, it is often replaced by the word "not". In <a href="/wiki/Programming_language" title="Programming language">programming languages</a> and some mathematical texts, it is sometimes replaced by "~" or "!", which are easier to type on some keyboards.</dd> <dt id="∨"><dfn>∨    (<a href="/wiki/Descending_wedge" title="Descending wedge">descending wedge</a>)</dfn></dt> <dd>1.  Denotes the <a href="/wiki/Logical_or" class="mw-redirect" title="Logical or">logical or</a>, and is read as "or". If E and F are <a href="/wiki/Logical_predicate" class="mw-redirect" title="Logical predicate">logical predicates</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\lor F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>∨</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\lor F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf9c188ff01798007e2d2a0ebc39389390537e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.099ex; height:2.176ex;" alt="{\displaystyle E\lor F}"> is true if either E, F, or both are true. It is often replaced by the word "or".</dd> <dd>2.  In <a href="/wiki/Lattice_theory" class="mw-redirect" title="Lattice theory">lattice theory</a>, denotes the <a href="/wiki/Join_(lattice_theory)" class="mw-redirect" title="Join (lattice theory)">join</a> or <a href="/wiki/Least_upper_bound" class="mw-redirect" title="Least upper bound">least upper bound</a> operation.</dd> <dd>3.  In <a href="/wiki/Topology" title="Topology">topology</a>, denotes the <a href="/wiki/Wedge_sum" title="Wedge sum">wedge sum</a> of two <a href="/wiki/Pointed_space" title="Pointed space">pointed spaces</a>.</dd> <dt id="∧"><dfn>∧    (<a href="/wiki/Wedge_(symbol)" title="Wedge (symbol)">wedge</a>)</dfn></dt> <dd>1.  Denotes the <a href="/wiki/Logical_and" class="mw-redirect" title="Logical and">logical and</a>, and is read as "and". If E and F are <a href="/wiki/Logical_predicate" class="mw-redirect" title="Logical predicate">logical predicates</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\land F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>∧</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\land F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4f514d64c7e8ab06025e7341af09430796d9bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.099ex; height:2.176ex;" alt="{\displaystyle E\land F}"> is true if E and F are both true. It is often replaced by the word "and" or the symbol "&".</dd> <dd>2.  In <a href="/wiki/Lattice_theory" class="mw-redirect" title="Lattice theory">lattice theory</a>, denotes the <a href="/wiki/Meet_(lattice_theory)" class="mw-redirect" title="Meet (lattice theory)">meet</a> or <a href="/wiki/Greatest_lower_bound" class="mw-redirect" title="Greatest lower bound">greatest lower bound</a> operation.</dd> <dd>3.  In <a href="/wiki/Multilinear_algebra" title="Multilinear algebra">multilinear algebra</a>, <a href="/wiki/Geometry" title="Geometry">geometry</a>, and <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable calculus</a>, denotes the <a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">wedge product</a> or the <a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">exterior product</a>.</dd> <dt id="⊻"><dfn>⊻</dfn></dt> <dd><a href="/wiki/Exclusive_or" title="Exclusive or">Exclusive or</a>: if E and F are two <a href="/wiki/Boolean_variable" class="mw-redirect" title="Boolean variable">Boolean variables</a> or <a href="/wiki/Predicate_(mathematical_logic)#Simplified_overview" title="Predicate (mathematical logic)">predicates</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\veebar F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊻</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\veebar F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a178813e18a7dde9424964f919f76ffb4b87955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.969ex; height:2.176ex;" alt="{\displaystyle E\veebar F}"> denotes the exclusive or. Notations E XOR F and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\oplus F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊕</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\oplus F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4548df9d548d438b02c47efd6eabca20aa46aabf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.357ex; height:2.343ex;" alt="{\displaystyle E\oplus F}"> are also commonly used; see <a href="#⊕">⊕</a>.</dd> <dt id="∀"><dfn>∀    (<a href="/wiki/Turned_A" title="Turned A">turned A</a>)</dfn></dt> <dd>1.  Denotes <a href="/wiki/Universal_quantification" title="Universal quantification">universal quantification</a> and is read as "for all". If E is a <a href="/wiki/Logical_predicate" class="mw-redirect" title="Logical predicate">logical predicate</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\;E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thickmathspace" /> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\;E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb331871cd0cd60fb9f03082e5ab92050c346de7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.043ex; height:2.176ex;" alt="{\displaystyle \forall x\;E}"> means that E is true for all possible values of the variable x.</dd> <dd>2.  Often used in plain text as an abbreviation of "for all" or "for every".</dd> <dt id="∃"><dfn>∃</dfn></dt> <dd>1.  Denotes <a href="/wiki/Existential_quantification" title="Existential quantification">existential quantification</a> and is read "there exists ... such that". If E is a <a href="/wiki/Logical_predicate" class="mw-redirect" title="Logical predicate">logical predicate</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x\;E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thickmathspace" /> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\;E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b7c99fd6f045db473ead91387550cdee77cf21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.043ex; height:2.176ex;" alt="{\displaystyle \exists x\;E}"> means that there exists at least one value of x for which E is true.</dd> <dd>2.  Often used in plain text as an abbreviation of "there exists".</dd> <dt id="∃!"><dfn>∃!</dfn></dt> <dd>Denotes <a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">uniqueness quantification</a>, that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists !x\;P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mo>!</mo> <mi>x</mi> <mspace width="thickmathspace" /> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists !x\;P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7934d5574713dba21190bf397dd91612d4e9f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.66ex; height:2.176ex;" alt="{\displaystyle \exists !x\;P}"> means "there exists exactly one x such that P (is true)". In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists !x\;P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mo>!</mo> <mi>x</mi> <mspace width="thickmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists !x\;P(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75d7392d43aba3d5c891f26058524ba009d6a19f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.799ex; height:2.843ex;" alt="{\displaystyle \exists !x\;P(x)}"> is an abbreviation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x\,(P(x)\,\wedge \neg \exists y\,(P(y)\wedge y\neq x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">∃</mi> <mi>y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>y</mi> <mo>≠</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\,(P(x)\,\wedge \neg \exists y\,(P(y)\wedge y\neq x))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b0c09aa1715e6cb2a804f56f464f06f68df671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.744ex; height:2.843ex;" alt="{\displaystyle \exists x\,(P(x)\,\wedge \neg \exists y\,(P(y)\wedge y\neq x))}">.</dd> <dt id="⇒"><dfn>⇒</dfn></dt> <dd>1.  Denotes <a href="/wiki/Material_conditional" title="Material conditional">material conditional</a>, and is read as "implies". If P and Q are <a href="/wiki/Logical_predicate" class="mw-redirect" title="Logical predicate">logical predicates</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\Rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">⇒</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\Rightarrow Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a57c9bc077d0b20e4f5ec006f5342cfbb18fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\Rightarrow Q}"> means that if P is true, then Q is also true. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\Rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">⇒</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\Rightarrow Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a57c9bc077d0b20e4f5ec006f5342cfbb18fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\Rightarrow Q}"> is logically equivalent with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q\lor \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q\lor \neg P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e5375ce9a1f158c980e10549dde74f8b8c39677" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.717ex; height:2.509ex;" alt="{\displaystyle Q\lor \neg P}">.</dd> <dd>2.  Often used in plain text as an abbreviation of "implies".</dd> <dt id="⇔"><dfn>⇔</dfn></dt> <dd>1.  Denotes <a href="/wiki/Logical_equivalence" title="Logical equivalence">logical equivalence</a>, and is read "is equivalent to" or "<a href="/wiki/If_and_only_if" title="If and only if">if and only if</a>". If P and Q are <a href="/wiki/Logical_predicate" class="mw-redirect" title="Logical predicate">logical predicates</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\Leftrightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">⇔</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\Leftrightarrow Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe75af42226920bc628ac7bbd53c023928f346ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\Leftrightarrow Q}"> is thus an abbreviation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\Rightarrow Q)\land (Q\Rightarrow P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">⇒</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">⇒</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\Rightarrow Q)\land (Q\Rightarrow P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eec3bd1f9b1c8698baba9e3124f6077575a0ff15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.597ex; height:2.843ex;" alt="{\displaystyle (P\Rightarrow Q)\land (Q\Rightarrow P)}">, or of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\land Q)\lor (\neg P\land \neg Q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>Q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\land Q)\lor (\neg P\land \neg Q)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acba17910384f15b1f928005ea6311ad59028898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.635ex; height:2.843ex;" alt="{\displaystyle (P\land Q)\lor (\neg P\land \neg Q)}">.</dd> <dd>2.  Often used in plain text as an abbreviation of "<a href="/wiki/If_and_only_if" title="If and only if">if and only if</a>".</dd> <dt id="⊤"><dfn>⊤    (<a href="/wiki/Tee_(symbol)" title="Tee (symbol)">tee</a>)</dfn></dt> <dd>1.  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"> denotes the <a href="/wiki/Logical_predicate" class="mw-redirect" title="Logical predicate">logical predicate</a> always true.</dd> <dd>2.  Denotes also the <a href="/wiki/Truth_value" title="Truth value">truth value</a> true.</dd> <dd>3.  Sometimes denotes the <a href="/wiki/Top_element" class="mw-redirect" title="Top element">top element</a> of a <a href="/wiki/Bounded_lattice" class="mw-redirect" title="Bounded lattice">bounded lattice</a> (previous meanings are specific examples).</dd> <dd>4.  For the use as a superscript, see <a href="#□⊤">□⊤</a>.</dd> <dt id="⊥"><dfn>⊥    (<a href="/wiki/Up_tack" title="Up tack">up tack</a>)</dfn></dt> <dd>1.  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"> denotes the <a href="/wiki/Logical_predicate" class="mw-redirect" title="Logical predicate">logical predicate</a> always false.</dd> <dd>2.  Denotes also the <a href="/wiki/Truth_value" title="Truth value">truth value</a> false.</dd> <dd>3.  Sometimes denotes the <a href="/wiki/Bottom_element" class="mw-redirect" title="Bottom element">bottom element</a> of a <a href="/wiki/Bounded_lattice" class="mw-redirect" title="Bounded lattice">bounded lattice</a> (previous meanings are specific examples).</dd> <dd>4.  In <a href="/wiki/Cryptography" title="Cryptography">Cryptography</a> often denotes an error in place of a regular value.</dd> <dd>5.  For the use as a superscript, see <a href="#□⊥">□⊥</a>.</dd> <dd>6.  For the similar symbol, see <a href="#⟂"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \perp }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊥</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \perp }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb90d6db42aa12f9e2f31176a4ed4e741c69eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \perp }"></a>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Blackboard_bold">Blackboard bold</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=7" title="Edit section: Blackboard bold">edit</a>]</div> The <a href="/wiki/Blackboard_bold" title="Blackboard bold">blackboard bold</a> <a href="/wiki/Typeface" title="Typeface">typeface</a> is widely used for denoting the basic <a href="/wiki/Number_system" class="mw-redirect" title="Number system">number systems</a>. These systems are often also denoted by the corresponding uppercase bold letter. A clear advantage of blackboard bold is that these symbols cannot be confused with anything else. This allows using them in any area of mathematics, without having to recall their definition. For example, if one encounters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> in <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, one should immediately know that this denotes the <a href="/wiki/Real_number" title="Real number">real numbers</a>, although combinatorics does not study the real numbers (but it uses them for many proofs). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="ℕ"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></dfn></dt> <dd>Denotes the set of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots \},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots \},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c507fade9c89210bf99dc52b7a6e991add6c7990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.088ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots \},}"> or sometimes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1,2,\ldots \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1,2,\ldots \}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe90224ed80730c05540c56ed9e1644218817ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.284ex; height:2.843ex;" alt="{\displaystyle \{0,1,2,\ldots \}.}"> When the distinction is important and readers might assume either definition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cca915d54ae835781191ae19599e11c7ff3d066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} _{1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ab7e98123f0def29a1cd3df96a0b7a58f4202c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} _{0}}"> are used, respectively, to denote one of them unambiguously. Notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f63b6cd6d63ee9b7be0b7e4d14099d7153bd43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {N} }"> is also commonly used.</dd> <dt id="ℤ"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></dfn></dt> <dd>Denotes the set of <a href="/wiki/Integer" title="Integer">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\ldots ,-2,-1,0,1,2,\ldots \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>…</mo> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\ldots ,-2,-1,0,1,2,\ldots \}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/958d9f8708cd5376adc925553b8f3dc66ed29e3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.438ex; height:2.843ex;" alt="{\displaystyle \{\ldots ,-2,-1,0,1,2,\ldots \}.}"> It is often denoted also by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ddd2dc6678fbf5b336535dd6f25bb5681bc5f20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.281ex; height:2.176ex;" alt="{\displaystyle \mathbf {Z} .}"></dd> <dt id="ℤp"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"></dfn></dt> <dd>1.  Denotes the set of <a href="/wiki/P-adic_integer" class="mw-redirect" title="P-adic integer">p-adic integers</a>, where p is a <a href="/wiki/Prime_number" title="Prime number">prime number</a>.</dd> <dd>2.  Sometimes, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b729c334a9863c47f0b7e3ad61342c2f0881bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{n}}"> denotes the <a href="/wiki/Integers_modulo_n" class="mw-redirect" title="Integers modulo n">integers modulo n</a>, where n is an <a href="/wiki/Integer" title="Integer">integer</a> greater than 0. The notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /n\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /n\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2120ebbc85f91df66c6de5446367bf9fd620844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.658ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /n\mathbb {Z} }"> is also used, and is less ambiguous.</dd> <dt id="ℚ"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></dfn></dt> <dd>Denotes the set of <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> (fractions of two integers). It is often denoted also by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4001d22c846e1ec4b5cf71650a58e9fed5df4f76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.655ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} .}"></dd> <dt id="ℚp"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"></dfn></dt> <dd>Denotes the set of <a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a>, where p is a <a href="/wiki/Prime_number" title="Prime number">prime number</a>.</dd> <dt id="ℝ"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></dfn></dt> <dd>Denotes the set of <a href="/wiki/Real_number" title="Real number">real numbers</a>. It is often denoted also by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a067bb21dcf0642bdce48f05a55e218efab3b85e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.65ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} .}"></dd> <dt id="ℂ"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></dfn></dt> <dd>Denotes the set of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. It is often denoted also by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8decdc57398394307879276418d46bc88b01de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.578ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} .}"></dd> <dt id="ℍ"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></dfn></dt> <dd>Denotes the set of <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>. It is often denoted also by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/012ed6cd8c75bd4227713d2bc3db339ef30d8aaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.738ex; height:2.176ex;" alt="{\displaystyle \mathbf {H} .}"></dd> <dt id="fq"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb96e056c071d13fc7702013f9273e7f5cd88a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}"></dfn></dt> <dd>Denotes the <a href="/wiki/Finite_field" title="Finite field">finite field</a> with q elements, where q is a <a href="/wiki/Prime_power" title="Prime power">prime power</a> (including <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>). It is denoted also by GF(q).</dd> <dt id="o"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></dfn></dt> <dd>Used on rare occasions to denote the set of <a href="/wiki/Octonion" title="Octonion">octonions</a>. It is often denoted also by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {O} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">O</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {O} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77bb2976419493443d2fc0b0be6325d211e6a48b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.655ex; height:2.176ex;" alt="{\displaystyle \mathbf {O} .}"></dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Calculus">Calculus</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=8" title="Edit section: Calculus">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="□'"><dfn>□'</dfn></dt> <dd><a href="/wiki/Lagrange%27s_notation" class="mw-redirect" title="Lagrange's notation">Lagrange's notation</a> for the <a href="/wiki/Derivative" title="Derivative">derivative</a>: If f is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of a single variable, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}">, read as "f <a href="/wiki/Prime_(symbol)#Use_in_mathematics,_statistics,_and_science" title="Prime (symbol)">prime</a>", is the derivative of f with respect to this variable. The <a href="/wiki/Second_derivative" title="Second derivative">second derivative</a> is the derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}">, and is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbdf186092f4353b7630fa8dda903e493cbbdc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.458ex; height:2.843ex;" alt="{\displaystyle f''}">.</dd> <dt id="\dot"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\Box }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>◻</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\Box }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957f5d3dffde5c82bd1098e67013f25f9b0eb78d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.676ex;" alt="{\displaystyle {\dot {\Box }}}"></dfn></dt> <dd><a href="/wiki/Newton%27s_notation" class="mw-redirect" title="Newton's notation">Newton's notation</a>, most commonly used for the <a href="/wiki/Derivative" title="Derivative">derivative</a> with respect to time. If x is a variable depending on time, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/447dd24107959e59e1389e419eafcff61f25bcfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.509ex;" alt="{\displaystyle {\dot {x}},}"> read as "x dot", is its derivative with respect to time. In particular, if x represents a moving point, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82c85f33714da82ab42d6b69eae07ab7e5e234b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\dot {x}}}"> is its <a href="/wiki/Velocity" title="Velocity">velocity</a>.</dd> <dt id="\ddot"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {\Box }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>◻</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {\Box }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b2d7f9eb79e6460dc0a8ec8cd2534d79193125d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.676ex;" alt="{\displaystyle {\ddot {\Box }}}"></dfn></dt> <dd><a href="/wiki/Newton%27s_notation" class="mw-redirect" title="Newton's notation">Newton's notation</a>, for the <a href="/wiki/Second_derivative" title="Second derivative">second derivative</a>: If x is a variable that represents a moving point, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06e0e705ddda28c6cd06cdc6e18be9abf88bb395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\ddot {x}}}"> is its <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>.</dd> <dt id="leibnitz"><dfn><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠d □/d □⁠</dfn></dt> <dd><a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a> for the <a href="/wiki/Derivative" title="Derivative">derivative</a>, which is used in several slightly different ways.</dd> <dd>1.  If y is a variable that <a href="/wiki/Dependent_variable" class="mw-redirect" title="Dependent variable">depends</a> on x, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\mathrm {d} y}{\mathrm {d} x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\mathrm {d} y}{\mathrm {d} x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc7db6fcc647d0372d744d4880ea45bf7e3cadb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.69ex; height:4.176ex;" alt="{\displaystyle \textstyle {\frac {\mathrm {d} y}{\mathrm {d} x}}}">, read as "d y over d x" (commonly shortened to "d y d x"), is the derivative of y with respect to x.</dd> <dd>2.  If f is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of a single variable x, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\mathrm {d} f}{\mathrm {d} x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>f</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\mathrm {d} f}{\mathrm {d} x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b6b600ec47290dd76692d146af0908122667fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.69ex; height:4.176ex;" alt="{\displaystyle \textstyle {\frac {\mathrm {d} f}{\mathrm {d} x}}}"> is the derivative of f, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\mathrm {d} f}{\mathrm {d} x}}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>f</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\mathrm {d} f}{\mathrm {d} x}}(a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d280d766623d30f31fc42fa4a07d4a7b9bd9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.729ex; height:4.176ex;" alt="{\displaystyle \textstyle {\frac {\mathrm {d} f}{\mathrm {d} x}}(a)}"> is the value of the derivative at a.</dd> <dd>3.  <a href="/wiki/Total_derivative" title="Total derivative">Total derivative</a>: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{1},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{1},\ldots ,x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61d6311d8c66acc1a5755c4c7cb688d3b1fa0fcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.198ex; height:2.843ex;" alt="{\displaystyle f(x_{1},\ldots ,x_{n})}"> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of several variables that <a href="/wiki/Dependent_variable" class="mw-redirect" title="Dependent variable">depend</a> on x, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\mathrm {d} f}{\mathrm {d} x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>f</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\mathrm {d} f}{\mathrm {d} x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b6b600ec47290dd76692d146af0908122667fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.69ex; height:4.176ex;" alt="{\displaystyle \textstyle {\frac {\mathrm {d} f}{\mathrm {d} x}}}"> is the derivative of f considered as a function of x. That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\mathrm {d} f}{dx}}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}\,{\frac {\mathrm {d} x_{i}}{\mathrm {d} x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\mathrm {d} f}{dx}}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}\,{\frac {\mathrm {d} x_{i}}{\mathrm {d} x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470f0cde8d93adb73922e79c1c327a64826a79f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:18.529ex; height:4.509ex;" alt="{\displaystyle \textstyle {\frac {\mathrm {d} f}{dx}}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}\,{\frac {\mathrm {d} x_{i}}{\mathrm {d} x}}}">.</dd> <dt id="partial"><dfn><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠∂ □/∂ □⁠</dfn></dt> <dd><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a>: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{1},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{1},\ldots ,x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61d6311d8c66acc1a5755c4c7cb688d3b1fa0fcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.198ex; height:2.843ex;" alt="{\displaystyle f(x_{1},\ldots ,x_{n})}"> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of several variables, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\partial f}{\partial x_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\partial f}{\partial x_{i}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa89b17348526def30043c1a4f2e6b4284748bb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:3.333ex; height:4.509ex;" alt="{\displaystyle \textstyle {\frac {\partial f}{\partial x_{i}}}}"> is the derivative with respect to the ith variable considered as an <a href="/wiki/Independent_variable" class="mw-redirect" title="Independent variable">independent variable</a>, the other variables being considered as constants.</dd> <dt id="functional"><dfn><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝛿 □/𝛿 □⁠</dfn></dt> <dd><a href="/wiki/Functional_derivative" title="Functional derivative">Functional derivative</a>: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y_{1},\ldots ,y_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(y_{1},\ldots ,y_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41b7345c44e8c45c9128819e3bd755210a2ec478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.817ex; height:2.843ex;" alt="{\displaystyle f(y_{1},\ldots ,y_{n})}"> is a <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functional</a> of several <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\delta f}{\delta y_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>f</mi> </mrow> <mrow> <mi>δ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\delta f}{\delta y_{i}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9935d7dda8a2fd9993a1979decf2502d5f51d1b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:3.008ex; height:4.509ex;" alt="{\displaystyle \textstyle {\frac {\delta f}{\delta y_{i}}}}"> is the functional derivative with respect to the nth function considered as an <a href="/wiki/Independent_variable" class="mw-redirect" title="Independent variable">independent variable</a>, the other functions being considered constant.</dd> <dt id="overline"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\Box }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>◻</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\Box }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2af38ccf16894144a32ec5fa63b9690339fc831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.923ex; height:3.009ex;" alt="{\displaystyle {\overline {\Box }}}"></dfn></dt> <dd>1.  <a href="/wiki/Complex_conjugate" title="Complex conjugate">Complex conjugate</a>: If z is a <a href="/wiki/Complex_number" title="Complex number">complex number</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64281d029a1d4bef9545644f01821c713f876f76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.208ex; height:2.343ex;" alt="{\displaystyle {\overline {z}}}"> is its complex conjugate. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {a+bi}}=a-bi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {a+bi}}=a-bi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f3fe032bfaad1f9bea098118677ccab943176f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.954ex; height:3.176ex;" alt="{\displaystyle {\overline {a+bi}}=a-bi}">.</dd> <dd>2.  <a href="/wiki/Topological_closure" class="mw-redirect" title="Topological closure">Topological closure</a>: If S is a <a href="/wiki/Subset" title="Subset">subset</a> of a <a href="/wiki/Topological_space" title="Topological space">topological space</a> T, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0353b71f671221a0796d94febf9079b11dcca124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.663ex; height:3.009ex;" alt="{\displaystyle {\overline {S}}}"> is its topological closure, that is, the smallest <a href="/wiki/Closed_subset" class="mw-redirect" title="Closed subset">closed subset</a> of T that contains S.</dd> <dd>3.  <a href="/wiki/Algebraic_closure" title="Algebraic closure">Algebraic closure</a>: If F is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d122dfa2be8a341b1c30e7b3405af6ac2c157105" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.016ex; height:3.009ex;" alt="{\displaystyle {\overline {F}}}"> is its algebraic closure, that is, the smallest <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a> that contains F. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {Q} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbb {Q} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377a8814b1ca454c488e409813988dd5dd906148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.923ex; height:3.343ex;" alt="{\displaystyle {\overline {\mathbb {Q} }}}"> is the field of all <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>.</dd> <dd>4.  <a href="/wiki/Mean_value" class="mw-redirect" title="Mean value">Mean value</a>: If x is a <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a> that takes its values in some sequence of numbers S, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4039bbc2a0048c3a3c02e5fd24390cab0dc97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.445ex; height:2.343ex;" alt="{\displaystyle {\overline {x}}}"> may denote the mean of the elements of S.</dd> <dd>5.  <a href="/wiki/Negation" title="Negation">Negation</a>: Sometimes used to denote negation of the entire expression under the bar, particularly when dealing with <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a>. For example, one of <a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a> says that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A\land B}}={\overline {A}}\lor {\overline {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A\land B}}={\overline {A}}\lor {\overline {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed12be97d3f36682b9322c53247b927a3d254d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.623ex; height:3.009ex;" alt="{\displaystyle {\overline {A\land B}}={\overline {A}}\lor {\overline {B}}}"> .</dd> <dt id="→"><dfn>→</dfn></dt> <dd>1.  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"> denotes a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> with <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> A and <a href="/wiki/Codomain" title="Codomain">codomain</a> B. For naming such a function, one writes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20040a52d9391f2fe271f0aaa300bf7887a0c7b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:2.509ex;" alt="{\displaystyle f:A\to B}">, which is read as "f from A to B".</dd> <dd>2.  More generally, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"> denotes a <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a> or a <a href="/wiki/Morphism" title="Morphism">morphism</a> from A to B.</dd> <dd>3.  May denote a <a href="/wiki/Logical_implication" class="mw-redirect" title="Logical implication">logical implication</a>. For the <a href="/wiki/Material_conditional" title="Material conditional">material implication</a> that is widely used in mathematics reasoning, it is nowadays generally replaced by <a href="#⇒">⇒</a>. In <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, it remains used for denoting implication, but its exact meaning depends on the specific theory that is studied.</dd> <dd>4.  Over a <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable name</a>, means that the variable represents a <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a>, in a context where ordinary variables represent <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a>; for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdbb1a68c861cbd0cfda4f71510f67eed27c7cb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:3.009ex;" alt="{\displaystyle {\overrightarrow {v}}}">. Boldface (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }">) or a <a href="/wiki/Circumflex" title="Circumflex">circumflex</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2790b6cafab4cc0f98a9ae8beb550947e60062f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.176ex;" alt="{\displaystyle {\hat {v}}}">) are often used for the same purpose.</dd> <dd>5.  In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> and more generally in <a href="/wiki/Affine_geometry" title="Affine geometry">affine geometry</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {PQ}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>Q</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {PQ}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/675febeea8e91072fb11994af206714d0bc598a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.4ex; width:3.714ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {PQ}}}"> denotes the <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> defined by the two points P and Q, which can be identified with the <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> that maps P to Q. The same vector can be denoted also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q-P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>−</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q-P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0eb4781c90e13f7a69c3762ca5a29b1aa54c566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.424ex; height:2.509ex;" alt="{\displaystyle Q-P}">; see <a href="/wiki/Affine_space" title="Affine space">Affine space</a>.</dd> <dt id="↦"><dfn>↦</dfn></dt> <dd>"<a href="/wiki/Maps_to" title="Maps to">Maps to</a>": Used for defining a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> without having to name it. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b49de3850ec5b2be3acb8db45514958c5e80ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.328ex; height:2.676ex;" alt="{\displaystyle x\mapsto x^{2}}"> is the <a href="/wiki/Square_function" class="mw-redirect" title="Square function">square function</a>.</dd> <dt id="∘"><dfn>○<a href="#cite_note-3">[3]</a></dfn></dt> <dd>1.  <a href="/wiki/Function_composition" title="Function composition">Function composition</a>: If f and g are two functions, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"> is the function such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g\circ f)(x)=g(f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g\circ f)(x)=g(f(x))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8a73f8d834a602ee506ac323b8a36ce17ac2b9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.979ex; height:2.843ex;" alt="{\displaystyle (g\circ f)(x)=g(f(x))}"> for every value of x.</dd> <dd>2.  <a href="/wiki/Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product of matrices</a>: If A and B are two matrices of the same size, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\circ B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∘</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\circ B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c341d3106d2763836b32f992b74e73f4cef0d24d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.702ex; height:2.176ex;" alt="{\displaystyle A\circ B}"> is the matrix such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\circ B)_{i,j}=(A)_{i,j}(B)_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∘</mo> <mi>B</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\circ B)_{i,j}=(A)_{i,j}(B)_{i,j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c463ca925b4af07b1ac93b1c9a56d562ba1c2dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.539ex; height:3.009ex;" alt="{\displaystyle (A\circ B)_{i,j}=(A)_{i,j}(B)_{i,j}}">. Possibly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"> is also used instead of <a href="#⊙">⊙</a> for the <a href="/wiki/Hadamard_product_(series)" class="mw-redirect" title="Hadamard product (series)">Hadamard product of power series</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]</dd> <dt id="∂"><dfn>∂</dfn></dt> <dd>1.  <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">Boundary</a> of a <a href="/wiki/Topological_subspace" class="mw-redirect" title="Topological subspace">topological subspace</a>: If S is a subspace of a topological space, then its boundary, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="{\displaystyle \partial S}">, is the <a href="/wiki/Set_difference" class="mw-redirect" title="Set difference">set difference</a> between the <a href="/wiki/Closure_(topology)" title="Closure (topology)">closure</a> and the <a href="/wiki/Interior_(topology)" title="Interior (topology)">interior</a> of S.</dd> <dd>2.  <a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a>: see <a href="#partial"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠∂□/∂□⁠</a>.</dd> <dt id="integral"><dfn>∫</dfn></dt> <dd>1.  Without a subscript, denotes an <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int x^{2}dx={\frac {x^{3}}{3}}+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mi>C</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int x^{2}dx={\frac {x^{3}}{3}}+C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/784a38f079bd887e8e311be5c4aa8da4f38053f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.048ex; height:4.176ex;" alt="{\displaystyle \textstyle \int x^{2}dx={\frac {x^{3}}{3}}+C}">.</dd> <dd>2.  With a subscript and a superscript, or expressions placed below and above it, denotes a <a href="/wiki/Definite_integral" class="mw-redirect" title="Definite integral">definite integral</a>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{a}^{b}x^{2}dx={\frac {b^{3}-a^{3}}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>3</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{a}^{b}x^{2}dx={\frac {b^{3}-a^{3}}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788d4bfdfabd75b9fa2ca28745ac4ff6aa525976" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.22ex; height:4.343ex;" alt="{\displaystyle \textstyle \int _{a}^{b}x^{2}dx={\frac {b^{3}-a^{3}}{3}}}">.</dd> <dd>3.  With a subscript that denotes a curve, denotes a <a href="/wiki/Line_integral" title="Line integral">line integral</a>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{C}f=\int _{a}^{b}f(r(t))r'(t)\operatorname {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mi>f</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>t</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{C}f=\int _{a}^{b}f(r(t))r'(t)\operatorname {d} t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ac62efef15ab1a68ad52ce999a551f49d885dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.256ex; height:3.676ex;" alt="{\displaystyle \textstyle \int _{C}f=\int _{a}^{b}f(r(t))r'(t)\operatorname {d} t}">, if r is a parametrization of the curve C, from a to b.</dd> <dt id="oint"><dfn>∮</dfn></dt> <dd>Often used, typically in physics, instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20aefa830c82b373910d5d569b67f11fe5acf74c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.418ex; height:3.176ex;" alt="{\displaystyle \textstyle \int }"> for <a href="/wiki/Line_integral" title="Line integral">line integrals</a> over a <a href="/wiki/Closed_curve" class="mw-redirect" title="Closed curve">closed curve</a>.</dd> <dt id="iint"><dfn>∬, ∯</dfn></dt> <dd>Similar to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20aefa830c82b373910d5d569b67f11fe5acf74c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.418ex; height:3.176ex;" alt="{\displaystyle \textstyle \int }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \oint }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∮</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \oint }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc0c73b2f6624a2aa31528983c8f0a53b450a476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.418ex; height:3.176ex;" alt="{\displaystyle \textstyle \oint }"> for <a href="/wiki/Surface_integral" title="Surface integral">surface integrals</a>.</dd> <dt id="∇"><dfn><a href="/wiki/Nabla_symbol" title="Nabla symbol"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\nabla }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86517a50dbbbc3ab9b4e9071f9c3627594dd0040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.226ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\nabla }}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\nabla }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65eeb56775c5bb9a9645ca773c8b8f744ca6ce0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.843ex;" alt="{\displaystyle {\vec {\nabla }}}"></a></dfn></dt> <dd><a href="/wiki/Del" title="Del">Nabla</a>, the <a href="/wiki/Gradient" title="Gradient">gradient</a>, vector derivative operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d230ab1c50528c0640da5e9a78827504eb5ed4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.675ex; height:4.843ex;" alt="{\displaystyle \textstyle \left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)}">, also called del or grad,</dd> or the <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a>. <dt id="laplacian"><dfn>∇2 or ∇⋅∇</dfn></dt> <dd><a href="/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a> or Laplacian: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738a3f7fd623c3ccda4bb14047a7d2d6dfcf34f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.012ex; height:4.676ex;" alt="{\displaystyle \textstyle {\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}}">. The forms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4be87ad083e5ead48d92b0c82f2d4e719cb34a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.99ex; height:2.676ex;" alt="{\displaystyle \nabla ^{2}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d48fadc100d258f78fc0ec2f8d33eba7c7759dfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.132ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }}}"> represent the dot product of the <a href="#∇">gradient</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\nabla }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86517a50dbbbc3ab9b4e9071f9c3627594dd0040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.226ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\nabla }}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\nabla }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65eeb56775c5bb9a9645ca773c8b8f744ca6ce0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.843ex;" alt="{\displaystyle {\vec {\nabla }}}">) with itself. Also notated Δ (next item).</dd> <dt id="delta"><dfn>Δ</dfn></dt> (Capital Greek letter <a href="/wiki/Delta_(letter)" title="Delta (letter)">delta</a>—not to be confused with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d909fe94e8277a4c44a50853cb7dbbf0fa3148ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle \triangle }">, which may denote a geometric <a href="/wiki/Triangle" title="Triangle">triangle</a> or, alternatively, the <a href="/wiki/Symmetric_difference" title="Symmetric difference">symmetric difference</a> of two sets.) <dd>1.  Another notation for the <a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a> (see above).</dd> <dd>2.  Operator of <a href="/wiki/Finite_difference" title="Finite difference">finite difference</a>.</dd> <dt id="four-gradient"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\partial }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∂</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\partial }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5eea7d641ff8530d15afea312051dfe8ab8716" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.527ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\partial }}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be3e5cf12940b60f8569a9927ad28823277177a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.458ex; height:2.843ex;" alt="{\displaystyle \partial _{\mu }}"></dfn></dt> (Note: the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"> is not recommended for the four-gradient since both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Box }^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>◻</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Box }^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc359533bf8adb4d1b6d94d147732aeabd22f2f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.862ex; height:2.676ex;" alt="{\displaystyle {\Box }^{2}}"> are used to denote the <a href="/wiki/D%27Alembertian" class="mw-redirect" title="D'Alembertian">d'Alembertian</a>; see below.) <dd><a href="/wiki/Four-gradient" title="Four-gradient">Quad</a>, the <a href="/wiki/Four-gradient" title="Four-gradient">4-vector gradient operator</a> or <a href="/wiki/Four-gradient" title="Four-gradient">four-gradient</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \left({\frac {\partial }{\partial t}},{\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \left({\frac {\partial }{\partial t}},{\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ffe07900a2c97ce093a2a12eab95ad79df2906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.07ex; height:4.843ex;" alt="{\displaystyle \textstyle \left({\frac {\partial }{\partial t}},{\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)}">.</dd> <dt id="d'alembertian"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Box }^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>◻</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Box }^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc359533bf8adb4d1b6d94d147732aeabd22f2f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.862ex; height:2.676ex;" alt="{\displaystyle {\Box }^{2}}"></dfn></dt> (here an actual box, not a placeholder) <dd>Denotes the <a href="/wiki/D%27Alembertian" class="mw-redirect" title="D'Alembertian">d'Alembertian</a> or squared <a href="/wiki/Four-gradient" title="Four-gradient">four-gradient</a>, which is a generalization of the <a href="/wiki/Laplacian" class="mw-redirect" title="Laplacian">Laplacian</a> to four-dimensional spacetime. In flat spacetime with Euclidean coordinates, this may mean either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\textstyle -{\frac {\partial ^{2}}{\partial t^{2}}}+{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}~\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mspace width="thickmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\textstyle -{\frac {\partial ^{2}}{\partial t^{2}}}+{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}~\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6688cd8387ff1bc95836f7b032745037757b6fcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.692ex; height:4.676ex;" alt="{\displaystyle ~\textstyle -{\frac {\partial ^{2}}{\partial t^{2}}}+{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}~\;}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;~\textstyle +{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}~\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mtext> </mtext> <mstyle displaystyle="false" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mspace width="thickmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;~\textstyle +{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}~\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3032338a76ce05ba71206ee0aa6ebef1759c7c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.338ex; height:4.676ex;" alt="{\displaystyle \;~\textstyle +{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}~\;}">; the sign convention must be specified. In curved spacetime (or flat spacetime with non-Euclidean coordinates), the definition is more complicated. Also called box or quabla.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Linear_and_multilinear_algebra">Linear and multilinear algebra</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=9" title="Edit section: Linear and multilinear algebra">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="\sum"><dfn>∑    (<a href="/wiki/Capital-sigma_notation" class="mw-redirect" title="Capital-sigma notation">capital-sigma notation</a>)</dfn></dt> <dd>1.  Denotes the <a href="/wiki/Summation" title="Summation">sum</a> of a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{i=1}^{n}i^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{i=1}^{n}i^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903324fc6af69cc735307ebf24a7e54776913611" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.598ex; height:3.176ex;" alt="{\displaystyle \textstyle \sum _{i=1}^{n}i^{2}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{0<i<j<n}j-i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo><</mo> <mi>n</mi> </mrow> </munder> <mi>j</mi> <mo>−</mo> <mi>i</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{0<i<j<n}j-i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d985dbdbd790e940dbf9dd8b3fdb3441970766c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.563ex; height:3.343ex;" alt="{\displaystyle \textstyle \sum _{0<i<j<n}j-i}">.</dd> <dd>2.  Denotes a <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> and, if the series is <a href="/wiki/Convergent_series" title="Convergent series">convergent</a>, the <a href="/wiki/Sum_of_series" class="mw-redirect" title="Sum of series">sum of the series</a>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow> <mi>i</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c30bcd697ab79c49d0a611ea1b2e8923a85cae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.497ex; height:4.176ex;" alt="{\displaystyle \textstyle \sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{x}}">.</dd> <dt id="\prod"><dfn>∏    (<a href="/wiki/Capital-pi_notation" class="mw-redirect" title="Capital-pi notation">capital-pi notation</a>)</dfn></dt> <dd>1.  Denotes the <a href="/wiki/Product_(mathematics)#Product_of_sequences" title="Product (mathematics)">product</a> of a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod _{i=1}^{n}i^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \prod _{i=1}^{n}i^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54656df0ea189908b2e2188af40ce6af6a36f6ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.338ex; height:3.176ex;" alt="{\displaystyle \textstyle \prod _{i=1}^{n}i^{2}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod _{0<i<j<n}j-i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo><</mo> <mi>n</mi> </mrow> </munder> <mi>j</mi> <mo>−</mo> <mi>i</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \prod _{0<i<j<n}j-i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1c0d7064292f977663a3ef1f932add5e0c896f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.303ex; height:3.343ex;" alt="{\displaystyle \textstyle \prod _{0<i<j<n}j-i}">.</dd> <dd>2.  Denotes an <a href="/wiki/Infinite_product" title="Infinite product">infinite product</a>. For example, the <a href="/wiki/Riemann_zeta_function#Euler's_product_formula" title="Riemann zeta function">Euler product formula for the Riemann zeta function</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \zeta (z)=\prod _{n=1}^{\infty }{\frac {1}{1-p_{n}^{-z}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>z</mi> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \zeta (z)=\prod _{n=1}^{\infty }{\frac {1}{1-p_{n}^{-z}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98b60bb66bee039b6a047bd4adec1938b2eb7244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.581ex; height:4.176ex;" alt="{\displaystyle \textstyle \zeta (z)=\prod _{n=1}^{\infty }{\frac {1}{1-p_{n}^{-z}}}}">.</dd> <dd>3.  Also used for the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of any number of sets and the <a href="/wiki/Direct_product" title="Direct product">direct product</a> of any number of <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structures</a>.</dd> <dt id="⊕"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></dfn></dt> <dd>1.  Internal <a href="/wiki/Direct_sum" title="Direct sum">direct sum</a>: if E and F are <a href="/wiki/Abelian_subgroup" class="mw-redirect" title="Abelian subgroup">abelian subgroups</a> of an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> V, notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=E\oplus F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>E</mi> <mo>⊕</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=E\oplus F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c0580aad131758ec88f37027fe55582ffa8bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.242ex; height:2.343ex;" alt="{\displaystyle V=E\oplus F}"> means that V is the direct sum of E and F; that is, every element of V can be written in a unique way as the sum of an element of E and an element of F. This applies also when E and F are <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspaces</a> or <a href="/wiki/Submodule" class="mw-redirect" title="Submodule">submodules</a> of the <a href="/wiki/Vector_space" title="Vector space">vector space</a> or <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> V.</dd> <dd>2.  <a href="/wiki/Direct_sum" title="Direct sum">Direct sum</a>: if E and F are two <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a>, <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>, or <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a>, then their direct sum, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\oplus F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊕</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\oplus F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4548df9d548d438b02c47efd6eabca20aa46aabf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.357ex; height:2.343ex;" alt="{\displaystyle E\oplus F}"> is an abelian group, vector space, or module (respectively) equipped with two <a href="/wiki/Monomorphism" title="Monomorphism">monomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:E\to E\oplus F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>E</mi> <mo>⊕</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:E\to E\oplus F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25f5f810a85f547fd4945be3236d90b4a0c0df51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.962ex; height:2.509ex;" alt="{\displaystyle f:E\to E\oplus F}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:F\to E\oplus F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>F</mi> <mo stretchy="false">→</mo> <mi>E</mi> <mo>⊕</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:F\to E\oplus F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4083080da88c12812252429b6aed0ed72ea491d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.765ex; height:2.509ex;" alt="{\displaystyle g:F\to E\oplus F}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\oplus F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊕</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\oplus F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4548df9d548d438b02c47efd6eabca20aa46aabf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.357ex; height:2.343ex;" alt="{\displaystyle E\oplus F}"> is the internal direct sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3465496148d29e6543003de4cdbac8e9ce6beda7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.864ex; height:2.843ex;" alt="{\displaystyle f(E)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(F)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d8a54002fd4a644d4960aa5e495152da0d37f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.666ex; height:2.843ex;" alt="{\displaystyle g(F)}">. This definition makes sense because this direct sum is unique up to a unique <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>.</dd> <dd>3.  <a href="/wiki/Exclusive_or" title="Exclusive or">Exclusive or</a>: if E and F are two <a href="/wiki/Boolean_variable" class="mw-redirect" title="Boolean variable">Boolean variables</a> or <a href="/wiki/Predicate_(mathematical_logic)#Simplified_overview" title="Predicate (mathematical logic)">predicates</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\oplus F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊕</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\oplus F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4548df9d548d438b02c47efd6eabca20aa46aabf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.357ex; height:2.343ex;" alt="{\displaystyle E\oplus F}"> may denote the exclusive or. Notations E XOR F and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\veebar F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊻</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\veebar F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a178813e18a7dde9424964f919f76ffb4b87955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.969ex; height:2.176ex;" alt="{\displaystyle E\veebar F}"> are also commonly used; see <a href="#⊻">⊻</a>.</dd> <dt id="⊗"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \otimes }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }"></dfn></dt> <dd>1.  Denotes the <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> of <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a>, <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>, <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a>, or other mathematical structures, such as in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\otimes F,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊗</mo> <mi>F</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\otimes F,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a865a1e952cdeb9987b1fa68879770c8d95a87ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.004ex; height:2.509ex;" alt="{\displaystyle E\otimes F,}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\otimes _{K}F.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mi>F</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\otimes _{K}F.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/871ed2f2f257298c9067ca65e735b1d31a6ef985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.697ex; height:2.509ex;" alt="{\displaystyle E\otimes _{K}F.}"></dd> <dd>2.  Denotes the <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> of elements: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b1971b01bc31d5b816f03cc7e1d9215d6c2ad8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.946ex; height:2.176ex;" alt="{\displaystyle x\in E}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in F,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in F,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280cbc261f07373469be0454ff54d4c0d237e845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.384ex; height:2.509ex;" alt="{\displaystyle y\in F,}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\otimes y\in E\otimes F.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⊗</mo> <mi>y</mi> <mo>∈</mo> <mi>E</mi> <mo>⊗</mo> <mi>F</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\otimes y\in E\otimes F.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00cd6c71b2b6b1fe7bfc82bdc396ce9cb617f2a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.17ex; height:2.509ex;" alt="{\displaystyle x\otimes y\in E\otimes F.}"></dd> <dt id="□⊤"><dfn>□⊤</dfn></dt> <dd>1.  <a href="/wiki/Transpose" title="Transpose">Transpose</a>: if A is a matrix, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e39088f8137b6309ed7585fc60f3bfe4f62bb0d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.254ex; height:2.676ex;" alt="{\displaystyle A^{\top }}"> denotes the transpose of A, that is, the matrix obtained by exchanging rows and columns of A. Notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{\top }\!\!A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ^{\top }\!\!A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e958a0dc305dc5b424fee9df3e53fd593fdedd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.48ex; height:2.676ex;" alt="{\displaystyle ^{\top }\!\!A}"> is also used. The symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"> is often replaced by the letter T or t.</dd> <dd>2.  For inline uses of the symbol, see <a href="#⊤">⊤</a>.</dd> <dt id="□⊥"><dfn>□⊥</dfn></dt> <dd>1.  <a href="/wiki/Orthogonal_complement" title="Orthogonal complement">Orthogonal complement</a>: If W is a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of an <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a> V, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{\bot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊥</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W^{\bot }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f522b1d57d040199a6a66c9b60a7121625693e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.019ex; height:2.676ex;" alt="{\displaystyle W^{\bot }}"> denotes its orthogonal complement, that is, the linear space of the elements of V whose inner products with the elements of W are all zero.</dd> <dd>2.  <a href="/wiki/Orthogonal_subspace" class="mw-redirect" title="Orthogonal subspace">Orthogonal subspace</a> in the <a href="/wiki/Dual_space" title="Dual space">dual space</a>: If W is a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> (or a <a href="/wiki/Submodule" class="mw-redirect" title="Submodule">submodule</a>) of a <a href="/wiki/Vector_space" title="Vector space">vector space</a> (or of a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a>) V, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{\bot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊥</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W^{\bot }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f522b1d57d040199a6a66c9b60a7121625693e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.019ex; height:2.676ex;" alt="{\displaystyle W^{\bot }}"> may denote the orthogonal subspace of W, that is, the set of all <a href="/wiki/Linear_forms" class="mw-redirect" title="Linear forms">linear forms</a> that map W to zero.</dd> <dd>3.  For inline uses of the symbol, see <a href="#⊥">⊥</a>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Advanced_group_theory">Advanced group theory</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=10" title="Edit section: Advanced group theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="⋉"><dfn>⋉ ⋊</dfn></dt> <dd>1.  Inner <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a>: if N and H are subgroups of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> G, such that N is a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of G, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=N\rtimes H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mi>N</mi> <mo>⋊</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=N\rtimes H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e88cf0169ffa6973fdcea1a8fc8c0ffe9b6ced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.893ex; height:2.176ex;" alt="{\displaystyle G=N\rtimes H}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=H\ltimes N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mi>H</mi> <mo>⋉</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=H\ltimes N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b942f1e1938f7d9942d5b0dee0544e2c3e6e9087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.893ex; height:2.176ex;" alt="{\displaystyle G=H\ltimes N}"> mean that G is the semidirect product of N and H, that is, that every element of G can be uniquely decomposed as the product of an element of N and an element of H. (Unlike for the <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product of groups</a>, the element of H may change if the order of the factors is changed.)</dd> <dd>2.  Outer <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a>: if N and H are two <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> is a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> from N to the <a href="/wiki/Automorphism_group" title="Automorphism group">automorphism group</a> of H, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\rtimes _{\varphi }H=H\ltimes _{\varphi }N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <msub> <mo>⋊</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mi>H</mi> <mo>=</mo> <mi>H</mi> <msub> <mo>⋉</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\rtimes _{\varphi }H=H\ltimes _{\varphi }N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eeb0a8988d823df9bc7ba5b81e4ce5d73609bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.648ex; height:2.843ex;" alt="{\displaystyle N\rtimes _{\varphi }H=H\ltimes _{\varphi }N}"> denotes a group G, unique up to a <a href="/wiki/Group_isomorphism" title="Group isomorphism">group isomorphism</a>, which is a semidirect product of N and H, with the commutation of elements of N and H defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }">.</dd> <dt id="≀"><dfn>≀</dfn></dt> <dd>In <a href="/wiki/Group_theory" title="Group theory">group theory</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\wr H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≀</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\wr H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36d84e640a7afb56fa8262747d2d135138a12153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.569ex; height:2.343ex;" alt="{\displaystyle G\wr H}"> denotes the <a href="/wiki/Wreath_product" title="Wreath product">wreath product</a> of the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> G and H. It is also denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\operatorname {wr} H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>wr</mi> <mo>⁡</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\operatorname {wr} H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9700045f97e530e4fa1a1a11a0ba0ce527d1e2a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.254ex; height:2.176ex;" alt="{\displaystyle G\operatorname {wr} H}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\operatorname {Wr} H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>Wr</mi> <mo>⁡</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\operatorname {Wr} H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccfd038aa24a7d563602cd3d7d8f8b57bc7157a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.965ex; height:2.176ex;" alt="{\displaystyle G\operatorname {Wr} H}">; see <a href="/wiki/Wreath_product#Notation_and_conventions" title="Wreath product">Wreath product § Notation and conventions</a> for several notation variants.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Infinite_numbers">Infinite numbers</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=11" title="Edit section: Infinite numbers">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="infinity"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }">    (<a href="/wiki/Infinity_symbol" title="Infinity symbol">infinity symbol</a>)</dfn></dt> <dd>1.  The symbol is read as <a href="/wiki/Infinity_(mathematics)" class="mw-redirect" title="Infinity (mathematics)">infinity</a>. As an upper bound of a <a href="/wiki/Summation" title="Summation">summation</a>, an <a href="/wiki/Infinite_product" title="Infinite product">infinite product</a>, an <a href="/wiki/Integral" title="Integral">integral</a>, etc., means that the computation is unlimited. Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"> in a lower bound means that the computation is not limited toward negative values.</dd> <dd>2.  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }"> are the generalized numbers that are added to the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> to form the <a href="/wiki/Extended_real_line" class="mw-redirect" title="Extended real line">extended real line</a>.</dd> <dd>3.  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"> is the generalized number that is added to the real line to form the <a href="/wiki/Projectively_extended_real_line" title="Projectively extended real line">projectively extended real line</a>.</dd> <dt id="𝔠"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {c}}}">   (<a href="/wiki/Fraktur" title="Fraktur">fraktur</a> 𝔠)</dfn></dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {c}}}"> denotes the <a href="/wiki/Cardinality_of_the_continuum" title="Cardinality of the continuum">cardinality of the continuum</a>, which is the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of the set of <a href="/wiki/Real_number" title="Real number">real numbers</a>.</dd> <dt id="ℵ"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">ℵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/306c55e6bc96d94db729ff5821c8f45a34c72bce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \aleph }">   (<a href="/wiki/Aleph" title="Aleph">aleph</a>)</dfn></dt> <dd>With an <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal</a> i as a subscript, denotes the ith <a href="/wiki/Aleph_number" title="Aleph number">aleph number</a>, that is the ith infinite <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal</a>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"> is the smallest infinite cardinal, that is, the cardinal of the natural numbers.</dd> <dt id="ℶ"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℶ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beth }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ca52be7fa2a32db700e775beb85f6dec33cc4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.05ex; width:1.647ex; height:2.343ex;" alt="{\displaystyle \beth }">   (<a href="/wiki/Bet_(letter)" title="Bet (letter)">bet (letter)</a>)</dfn></dt> <dd>With an <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal</a> i as a subscript, denotes the ith <a href="/wiki/Beth_number" title="Beth number">beth number</a>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℶ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beth _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3054002c3a3306a18292717a9c955d761210783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.05ex; width:2.655ex; height:2.676ex;" alt="{\displaystyle \beth _{0}}"> is the <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal</a> of the natural numbers, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beth _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℶ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beth _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb3462a86543187911778e6ff64ed1dc27b19f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.05ex; width:2.655ex; height:2.676ex;" alt="{\displaystyle \beth _{1}}"> is the <a href="/wiki/Cardinal_of_the_continuum" class="mw-redirect" title="Cardinal of the continuum">cardinal of the continuum</a>.</dd> <dt id="omega"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }">   (<a href="/wiki/Omega" title="Omega">omega</a>)</dfn></dt> <dd>1.  Denotes the first <a href="/wiki/Limit_ordinal" title="Limit ordinal">limit ordinal</a>. It is also denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a713d16c489051d4f515e12b1f86061c6be799b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{0}}"> and can be identified with the <a href="/wiki/Ordered_set" class="mw-redirect" title="Ordered set">ordered set</a> of the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>.</dd> <dd>2.  With an <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal</a> i as a subscript, denotes the ith <a href="/wiki/Limit_ordinal" title="Limit ordinal">limit ordinal</a> that has a <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> greater than that of all preceding ordinals.</dd> <dd>3.  In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, denotes the (unknown) greatest lower bound for the exponent of the <a href="/wiki/Computational_complexity" title="Computational complexity">computational complexity</a> of <a href="/wiki/Matrix_multiplication#Complexity" title="Matrix multiplication">matrix multiplication</a>.</dd> <dd>4.  Written as a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of another function, it is used for comparing the <a href="/wiki/Asymptotic_growth" class="mw-redirect" title="Asymptotic growth">asymptotic growth</a> of two functions. See <a href="/wiki/Big_O_notation#Related_asymptotic_notations" title="Big O notation">Big O notation § Related asymptotic notations</a>.</dd> <dd>5.  In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, may denote the <a href="/wiki/Prime_omega_function" title="Prime omega function">prime omega function</a>. That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e1e190faf1ba7931e4e98594df6097d297bed1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.65ex; height:2.843ex;" alt="{\displaystyle \omega (n)}"> is the number of distinct prime factors of the integer n.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Brackets">Brackets</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=12" title="Edit section: Brackets">edit</a>]</div> Many sorts of <a href="/wiki/Bracket_(mathematics)" title="Bracket (mathematics)">brackets</a> are used in mathematics. Their meanings depend not only on their shapes, but also on the nature and the arrangement of what is delimited by them, and sometimes what appears between or before them. For this reason, in the entry titles, the symbol □ is used as a placeholder for schematizing the syntax that underlies the meaning. <div class="mw-heading mw-heading3"><h3 id="Parentheses">Parentheses</h3>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=13" title="Edit section: Parentheses">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="()"><dfn>(□)</dfn></dt> <dd>Used in an <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expression</a> for specifying that the sub-expression between the parentheses has to be considered as a single entity; typically used for specifying the <a href="/wiki/Order_of_operations" title="Order of operations">order of operations</a>.</dd> <dt id="functional"><dfn>□(□) □(□, □) □(□, ..., □)</dfn></dt> <dd>1.  <a href="/wiki/Functional_notation" class="mw-redirect" title="Functional notation">Functional notation</a>: if the first <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"> is the name (symbol) of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, denotes the value of the function applied to the expression between the parentheses; for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x+y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x+y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af84037b516d93ba4a9ff505fb4d6d187fba9e24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.99ex; height:2.843ex;" alt="{\displaystyle \sin(x+y)}">. In the case of a <a href="/wiki/Multivariate_function" class="mw-redirect" title="Multivariate function">multivariate function</a>, the parentheses contain several expressions separated by commas, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29473ed0c4e838ac9dbe074535e507166c0e9101" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.843ex;" alt="{\displaystyle f(x,y)}">.</dd> <dd>2.  May also denote a product, such as in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(b+c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(b+c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/859c3dfc96fbd18909140e34574e4347fd3459f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.884ex; height:2.843ex;" alt="{\displaystyle a(b+c)}">. When the confusion is possible, the context must distinguish which symbols denote functions, and which ones denote <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>.</dd> <dt id="pair"><dfn>(□, □)</dfn></dt> <dd>1.  Denotes an <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a> of <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a>, for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\pi ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>π</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\pi ,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed72c324f3867ace349afdfd89b028dc9e79b80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.338ex; height:2.843ex;" alt="{\displaystyle (\pi ,0)}">.</dd> <dd>2.  If a and b are <a href="/wiki/Real_number" title="Real number">real numbers</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }">, and a < b, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"> denotes the <a href="/wiki/Open_interval" class="mw-redirect" title="Open interval">open interval</a> delimited by a and b. See <a href="#open_interval">]□, □[</a> for an alternative notation.</dd> <dd>3.  If a and b are <a href="/wiki/Integer" title="Integer">integers</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"> may denote the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of a and b. Notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(a,b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b490efc5104ccce7d1908b5cf921f7142a56316" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.558ex; height:2.843ex;" alt="{\displaystyle \gcd(a,b)}"> is often used instead.</dd> <dt id="(□,□,□)"><dfn>(□, □, □)</dfn></dt> <dd>If x, y, z are vectors in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"> may denote the <a href="/wiki/Scalar_triple_product" class="mw-redirect" title="Scalar triple product">scalar triple product</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] See also <a href="#sqb3">[□,□,□]</a> in <a href="#Square_brackets">§ Square brackets</a>.</dd> <dt id="tuple"><dfn>(□, ..., □)</dfn></dt> <dd>Denotes a <a href="/wiki/Tuple" title="Tuple">tuple</a>. If there are n objects separated by commas, it is an n-tuple.</dd> <dt id="sequence"><dfn>(□, □, ...) (□, ..., □, ...)</dfn></dt> <dd>Denotes an <a href="/wiki/Infinite_sequence" class="mw-redirect" title="Infinite sequence">infinite sequence</a>.</dd> <dt id="pmatrix"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>◻</mi> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>◻</mi> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>◻</mi> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>◻</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8604d0d0e8ff7da39de440448683e2743a0fb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:16.058ex; height:11.009ex;" alt="{\displaystyle {\begin{pmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{pmatrix}}}"></dfn></dt> <dd>Denotes a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. Often denoted with <a href="#bmatrix">square brackets</a>.</dd> <dt id="binomial"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {\Box }{\Box }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>◻</mi> <mi>◻</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {\Box }{\Box }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa9ca5c00f150e9758881f85b7373f7a183aeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.229ex; height:6.176ex;" alt="{\displaystyle {\binom {\Box }{\Box }}}"></dfn></dt> <dd>Denotes a <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>: Given two <a href="/wiki/Nonnegative_integer" class="mw-redirect" title="Nonnegative integer">nonnegative integers</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/963a810ba39e3e0725c523d0c98b18f39786ebb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:4.816ex; height:6.176ex;" alt="{\displaystyle {\binom {n}{k}}}"> is read as "n choose k", and is defined as the integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n(n-1)\cdots (n-k+1)}{1\cdot 2\cdots k}}={\frac {n!}{k!\,(n-k)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋯</mo> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n(n-1)\cdots (n-k+1)}{1\cdot 2\cdots k}}={\frac {n!}{k!\,(n-k)!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/444a8adee8acae618036298a0c175c10cd4d77bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.276ex; height:6.509ex;" alt="{\displaystyle {\frac {n(n-1)\cdots (n-k+1)}{1\cdot 2\cdots k}}={\frac {n!}{k!\,(n-k)!}}}"> (if k = 0, its value is conventionally 1). Using the left-hand-side expression, it denotes a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> in n, and is thus defined and used for any <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> value of n.</dd> <dt id="legendre"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\Box }{\Box }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>◻</mi> <mi>◻</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\Box }{\Box }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a18069a10c948f5e78675d85e9b528db6351ad3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.065ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\Box }{\Box }}\right)}"></dfn></dt> <dd><a href="/wiki/Legendre_symbol" title="Legendre symbol">Legendre symbol</a>: If p is an odd <a href="/wiki/Prime_number" title="Prime number">prime number</a> and a is an <a href="/wiki/Integer" title="Integer">integer</a>, the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {a}{p}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {a}{p}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aad4dd123bb7bc0da377dec595d91761753ba294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.487ex; height:6.176ex;" alt="{\displaystyle \left({\frac {a}{p}}\right)}"> is 1 if a is a <a href="/wiki/Quadratic_residue" title="Quadratic residue">quadratic residue</a> modulo p; it is –1 if a is a <a href="/wiki/Quadratic_non-residue" class="mw-redirect" title="Quadratic non-residue">quadratic non-residue</a> modulo p; it is 0 if p divides a. The same notation is used for the <a href="/wiki/Jacobi_symbol" title="Jacobi symbol">Jacobi symbol</a> and <a href="/wiki/Kronecker_symbol" title="Kronecker symbol">Kronecker symbol</a>, which are generalizations where p is respectively any odd positive integer, or any integer.</dd> </dl> <div class="mw-heading mw-heading3"><h3 id="Square_brackets">Square brackets</h3>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=14" title="Edit section: Square brackets">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="sqb1"><dfn>[□]</dfn></dt> <dd>1.  Sometimes used as a synonym of <a href="#()">(□)</a> for avoiding nested parentheses.</dd> <dd>2.  <a href="/wiki/Equivalence_class" title="Equivalence class">Equivalence class</a>: given an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07548563c21e128890501e14eb7c80ee2d6fda4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.623ex; height:2.843ex;" alt="{\displaystyle [x]}"> often denotes the equivalence class of the element x.</dd> <dd>3.  <a href="/wiki/Integral_part" class="mw-redirect" title="Integral part">Integral part</a>: if x is a <a href="/wiki/Real_number" title="Real number">real number</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07548563c21e128890501e14eb7c80ee2d6fda4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.623ex; height:2.843ex;" alt="{\displaystyle [x]}"> often denotes the integral part or <a href="/wiki/Truncation" title="Truncation">truncation</a> of x, that is, the integer obtained by removing all digits after the <a href="/wiki/Decimal_mark" class="mw-redirect" title="Decimal mark">decimal mark</a>. This notation has also been used for other variants of <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">floor and ceiling functions</a>.</dd> <dd>4.  <a href="/wiki/Iverson_bracket" title="Iverson bracket">Iverson bracket</a>: if P is a <a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">predicate</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [P]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>P</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [P]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d78ad4ad13872df07ac9b02a2574250a0e54fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.039ex; height:2.843ex;" alt="{\displaystyle [P]}"> may denote the Iverson bracket, that is the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> that takes the value 1 for the values of the <a href="/wiki/Free_variable" class="mw-redirect" title="Free variable">free variables</a> in P for which P is true, and takes the value 0 otherwise. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x=y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x=y]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8def35f48e2c2bc951587c405ab300430749af9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.877ex; height:2.843ex;" alt="{\displaystyle [x=y]}"> is the <a href="/wiki/Kronecker_delta_function" class="mw-redirect" title="Kronecker delta function">Kronecker delta function</a>, which equals one if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/409a91214d63eabe46ec10ff3cbba689ab687366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}">, and zero otherwise.</dd> <dd>5.  In combinatorics or computer science, sometimes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a26847bfc29bbeb4d6ef62ac3fd076378c0fd1db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.688ex; height:2.843ex;" alt="{\displaystyle [n]}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"> denotes the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3,\ldots ,n\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd810c6a53156f56190667aebf3c8db25d49d2e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.453ex; height:2.843ex;" alt="{\displaystyle \{1,2,3,\ldots ,n\}}"> of positive integers up to n, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0]=\emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0]=\emptyset }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df7f7c88970c268ccf8699cddb116455663ff35a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.717ex; height:2.843ex;" alt="{\displaystyle [0]=\emptyset }">.</dd> <dt id="sqbf"><dfn>□[□]</dfn></dt> <dd><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">Image of a subset</a>: if S is a <a href="/wiki/Subset" title="Subset">subset</a> of the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain of the function</a> f, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[S]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>S</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[S]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc136e58f23572a4001b784eea645e59b735ce49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.071ex; height:2.843ex;" alt="{\displaystyle f[S]}"> is sometimes used for denoting the image of S. When no confusion is possible, notation <a href="#functional">f(S)</a> is commonly used.</dd> <dt id="sqb2"><dfn>[□, □]</dfn></dt> <dd>1.  <a href="/wiki/Closed_interval" class="mw-redirect" title="Closed interval">Closed interval</a>: if a and b are <a href="/wiki/Real_number" title="Real number">real numbers</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41558abc50886fdf38817495b243958d7b3dd39b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leq b}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"> denotes the closed interval defined by them.</dd> <dd>2.  <a href="/wiki/Commutator_(group_theory)" class="mw-redirect" title="Commutator (group theory)">Commutator (group theory)</a>: if a and b belong to a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]=a^{-1}b^{-1}ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]=a^{-1}b^{-1}ab}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eee1488a4b71310ca17e1c49f541be2e548731f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.774ex; height:3.176ex;" alt="{\displaystyle [a,b]=a^{-1}b^{-1}ab}">.</dd> <dd>3.  <a href="/wiki/Commutator_(ring_theory)" class="mw-redirect" title="Commutator (ring theory)">Commutator (ring theory)</a>: if a and b belong to a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]=ab-ba}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mo>−</mo> <mi>b</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]=ab-ba}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23bf78c808a8f58515d59d520bdfac43cf80114f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.949ex; height:2.843ex;" alt="{\displaystyle [a,b]=ab-ba}">.</dd> <dd>4.  Denotes the <a href="/wiki/Lie_bracket" class="mw-redirect" title="Lie bracket">Lie bracket</a>, the operation of a <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a>.</dd> <dt id="sqb2:"><dfn>[□ : □]</dfn></dt> <dd>1.  <a href="/wiki/Degree_of_a_field_extension" title="Degree of a field extension">Degree of a field extension</a>: if F is an <a href="/wiki/Extension_field" class="mw-redirect" title="Extension field">extension</a> of a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> E, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [F:E]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>F</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [F:E]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a21f70810a14f4262ae495020e16fa79b62f93a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.747ex; height:2.843ex;" alt="{\displaystyle [F:E]}"> denotes the degree of the <a href="/wiki/Field_extension" title="Field extension">field extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F/E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F/E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba04c235083fd37db07fce9ec43901e0cb6d1d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.679ex; height:2.843ex;" alt="{\displaystyle F/E}">. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbb {C} :\mathbb {R} ]=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mathbb {C} :\mathbb {R} ]=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca0d230d77b8b6b954c39a9bd67d9ccb31e5547f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.848ex; height:2.843ex;" alt="{\displaystyle [\mathbb {C} :\mathbb {R} ]=2}">.</dd> <dd>2.  <a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">Index of a subgroup</a>: if H is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> E, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [G:H]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>G</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [G:H]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c19ed6f18e6db133b5a0257ecde8026808fd1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.121ex; height:2.843ex;" alt="{\displaystyle [G:H]}"> denotes the index of H in G. The notation <a href="#!:!">|G:H|</a> is also used</dd> <dt id="sqb3"><dfn>[□, □, □]</dfn></dt> <dd>If x, y, z are vectors in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,y,z]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,y,z]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0bfedf9a4341d42ead3affff487ec9debd8365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.935ex; height:2.843ex;" alt="{\displaystyle [x,y,z]}"> may denote the <a href="/wiki/Scalar_triple_product" class="mw-redirect" title="Scalar triple product">scalar triple product</a>.<a href="#cite_note-4">[4]</a> See also <a href="#(□,□,□)">(□,□,□)</a> in <a href="#Parentheses">§ Parentheses</a>.</dd> <dt id="bmatrix"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>◻</mi> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>◻</mi> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>◻</mi> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>◻</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bcf199536acfd9f1424bdaeb0899584ab3b3c03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:15.092ex; height:11.009ex;" alt="{\displaystyle {\begin{bmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{bmatrix}}}"></dfn></dt> <dd>Denotes a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. Often denoted with <a href="#pmatrix">parentheses</a>.</dd> </dl> <div class="mw-heading mw-heading3"><h3 id="Braces">Braces</h3>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=15" title="Edit section: Braces">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="bb"><dfn>{ }</dfn></dt> <dd><a href="/wiki/Set-builder_notation" title="Set-builder notation">Set-builder notation</a> for the <a href="/wiki/Empty_set" title="Empty set">empty set</a>, also denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"> or <a href="#∅">∅</a>.</dd> <dt id="b□b"><dfn>{□} </dfn></dt> <dd>1.  Sometimes used as a synonym of <a href="#()">(□)</a> and <a href="#sqb1">[□]</a> for avoiding nested parentheses.</dd> <dd>2.  <a href="/wiki/Set-builder_notation" title="Set-builder notation">Set-builder notation</a> for a <a href="/wiki/Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a120eeb8a091b516595765bd08b306f2394e7721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}"> denotes the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> that has x as a single element.</dd> <dt id="b,...,b"><dfn>{□, ..., □} </dfn></dt> <dd><a href="/wiki/Set-builder_notation" title="Set-builder notation">Set-builder notation</a>: denotes the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> whose elements are listed between the braces, separated by commas.</dd> <dt id="b:b"><dfn>{□ : □} {□ | □} </dfn></dt> <dd><a href="/wiki/Set-builder_notation" title="Set-builder notation">Set-builder notation</a>: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}"> is a <a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">predicate</a> depending on a <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a> x, then both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x:P(x)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>:</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x:P(x)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb31656e1c3e70f24620da1b97d708eb7d089d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.476ex; height:2.843ex;" alt="{\displaystyle \{x:P(x)\}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\mid P(x)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\mid P(x)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50d836860f856fa5d612f0d9c2a727ba498bfe32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.476ex; height:2.843ex;" alt="{\displaystyle \{x\mid P(x)\}}"> denote the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> formed by the values of x for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}"> is true.</dd> <dt id="single_brace"><dfn>Single brace</dfn></dt> <dd>1.  Used for emphasizing that several <a href="/wiki/Equation_(mathematics)" class="mw-redirect" title="Equation (mathematics)">equations</a> have to be considered as <a href="/wiki/Simultaneous_equations" class="mw-redirect" title="Simultaneous equations">simultaneous equations</a>; for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\begin{cases}2x+y=1\\3x-y=1\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\begin{cases}2x+y=1\\3x-y=1\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70cc6206a4a4e23beedb84b41f81846b7480e5f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.244ex; height:6.176ex;" alt="{\displaystyle \textstyle {\begin{cases}2x+y=1\\3x-y=1\end{cases}}}">.</dd> <dd>2.  <a href="/wiki/Piecewise" class="mw-redirect" title="Piecewise">Piecewise</a> definition; for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle |x|={\begin{cases}x&{\text{if }}x\geq 0\\-x&{\text{if }}x<0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>x</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle |x|={\begin{cases}x&{\text{if }}x\geq 0\\-x&{\text{if }}x<0\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c8d1ba2fd2fd75cb66e6d257f5fee4f734bc43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.207ex; height:6.176ex;" alt="{\displaystyle \textstyle |x|={\begin{cases}x&{\text{if }}x\geq 0\\-x&{\text{if }}x<0\end{cases}}}">.</dd> <dd>3.  Used for grouped annotation of elements in a formula; for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \underbrace {(a,b,\ldots ,z)} _{26}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>26</mn> </mrow> </munder> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \underbrace {(a,b,\ldots ,z)} _{26}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c463f3857fda1f3fbaf69d8f1e900a97b6808bcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; margin-right: -0.028ex; width:11.365ex; height:6.343ex;" alt="{\displaystyle \textstyle \underbrace {(a,b,\ldots ,z)} _{26}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \overbrace {1+2+\cdots +100} ^{=5050}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mn>100</mn> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> <mn>5050</mn> </mrow> </mover> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \overbrace {1+2+\cdots +100} ^{=5050}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d5686d332a08b864f56d86bafe4d1af52dfcfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.028ex; width:17.085ex; height:5.676ex;" alt="{\displaystyle \textstyle \overbrace {1+2+\cdots +100} ^{=5050}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \left.{\begin{bmatrix}A\\B\end{bmatrix}}\right\}m+n{\text{ rows}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> rows</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \left.{\begin{bmatrix}A\\B\end{bmatrix}}\right\}m+n{\text{ rows}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef0be0b8ac969fc1332501681b91f5660a9f445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.625ex; height:6.176ex;" alt="{\displaystyle \textstyle \left.{\begin{bmatrix}A\\B\end{bmatrix}}\right\}m+n{\text{ rows}}}"></dd> </dl> <div class="mw-heading mw-heading3"><h3 id="Other_brackets">Other brackets</h3>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=16" title="Edit section: Other brackets">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="!□!"><dfn>|□|</dfn></dt> <dd>1.  <a href="/wiki/Absolute_value" title="Absolute value">Absolute value</a>: if x is a <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> number, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb41e5fd5dc37eaa1718dfbf4bc082edb991936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.623ex; height:2.843ex;" alt="{\displaystyle |x|}"> denotes its absolute value.</dd> <dd>2.  Number of elements: If S is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |S|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |S|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28d901e98a035ff4c0e37fe6dd8e750ece6c1f0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.793ex; height:2.843ex;" alt="{\displaystyle |S|}"> may denote its <a href="/wiki/Cardinality" title="Cardinality">cardinality</a>, that is, its number of elements. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">#</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d640f54650fa5701b7fa00bffadb192aa43131b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.435ex; height:2.509ex;" alt="{\displaystyle \#S}"> is also often used, see <a href="#sharp">#</a>.</dd> <dd>3.  Length of a <a href="/wiki/Line_segment" title="Line segment">line segment</a>: If P and Q are two points in a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |PQ|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>P</mi> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |PQ|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bce9b5b9a3507edea5dd51e5fe92e446776e4c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.877ex; height:2.843ex;" alt="{\displaystyle |PQ|}"> often denotes the length of the line segment that they define, which is the <a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">distance</a> from P to Q, and is often denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P,Q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P,Q)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b0da2776d38a8b78a8eea2fbb9a10cf8cce485" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.643ex; height:2.843ex;" alt="{\displaystyle d(P,Q)}">.</dd> <dd>4.  For a similar-looking operator, see <a href="#vbar">|</a>.</dd> <dt id="!:!"><dfn>|□:□|</dfn></dt> <dd><a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">Index of a subgroup</a>: if H is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> G, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |G:H|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mo>:</mo> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |G:H|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5114d15a8b2d77dae8c98f1c523d7fdd1a03d814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.121ex; height:2.843ex;" alt="{\displaystyle |G:H|}"> denotes the index of H in G. The notation <a href="#sqb2:">[G:H]</a> is also used</dd> <dt id="determinant"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\begin{vmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{vmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>◻</mi> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>◻</mi> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>◻</mi> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>◻</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\begin{vmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{vmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f922ebc9d04e6f7ac34b3c8e4f538abf4efaa9cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:13.285ex; height:11.009ex;" alt="{\displaystyle \textstyle {\begin{vmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{vmatrix}}}"></dfn></dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}x_{1,1}&\cdots &x_{1,n}\\\vdots &\ddots &\vdots \\x_{n,1}&\cdots &x_{n,n}\end{vmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}x_{1,1}&\cdots &x_{1,n}\\\vdots &\ddots &\vdots \\x_{n,1}&\cdots &x_{n,n}\end{vmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7d1fd8783d4c95105b7d4c7ff59a3492ef5968" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:17.488ex; height:11.509ex;" alt="{\displaystyle {\begin{vmatrix}x_{1,1}&\cdots &x_{1,n}\\\vdots &\ddots &\vdots \\x_{n,1}&\cdots &x_{n,n}\end{vmatrix}}}"> denotes the <a href="/wiki/Determinant" title="Determinant">determinant</a> of the <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}x_{1,1}&\cdots &x_{1,n}\\\vdots &\ddots &\vdots \\x_{n,1}&\cdots &x_{n,n}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}x_{1,1}&\cdots &x_{1,n}\\\vdots &\ddots &\vdots \\x_{n,1}&\cdots &x_{n,n}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17f140a02f2b7ca739780b9f2e8c4a20c616575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:19.295ex; height:11.509ex;" alt="{\displaystyle {\begin{bmatrix}x_{1,1}&\cdots &x_{1,n}\\\vdots &\ddots &\vdots \\x_{n,1}&\cdots &x_{n,n}\end{bmatrix}}}">.</dd> <dt id="norm"><dfn>||□||</dfn></dt> <dd>1.  Denotes the <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> of an element of a <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a>.</dd> <dd>2.  For the similar-looking operator named parallel, see <a href="#∥">∥</a>.</dd> <dt id="⌊⌋"><dfn>⌊□⌋</dfn></dt> <dd><a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">Floor function</a>: if x is a real number, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor x\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor x\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738c94c88678dd08a289f90a47a609ce44eedf14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.394ex; height:2.843ex;" alt="{\displaystyle \lfloor x\rfloor }"> is the greatest <a href="/wiki/Integer" title="Integer">integer</a> that is not greater than x.</dd> <dt id="⌈⌉"><dfn>⌈□⌉</dfn></dt> <dd><a href="/wiki/Ceiling_function" class="mw-redirect" title="Ceiling function">Ceiling function</a>: if x is a real number, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lceil x\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌈</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌉</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lceil x\rceil }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac7f37c8288700904b4a22a2f7c94d45ba917de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.394ex; height:2.843ex;" alt="{\displaystyle \lceil x\rceil }"> is the lowest <a href="/wiki/Integer" title="Integer">integer</a> that is not lesser than x.</dd> <dt id="⌊⌉"><dfn>⌊□⌉</dfn></dt> <dd><a href="/wiki/Nearest_integer_function" class="mw-redirect" title="Nearest integer function">Nearest integer function</a>: if x is a real number, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor x\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌉</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor x\rceil }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d85f3c359bd3e3aee2a21e06fe49b7bf847dcaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.394ex; height:2.843ex;" alt="{\displaystyle \lfloor x\rceil }"> is the <a href="/wiki/Integer" title="Integer">integer</a> that is the closest to x.</dd> <dt id="open_interval"><dfn>]□, □[</dfn></dt> <dd><a href="/wiki/Open_interval" class="mw-redirect" title="Open interval">Open interval</a>: If a and b are real numbers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ]a,b[}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">]</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">[</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ]a,b[}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b51ec208e9582e11a4f340a42d4f17fb4748fcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle ]a,b[}"> denotes the open interval delimited by a and b. See <a href="#pair">(□, □)</a> for an alternative notation.</dd> <dt id="left-open"><dfn>(□, □] ]□, □]</dfn></dt> <dd>Both notations are used for a <a href="/wiki/Half-open_interval" class="mw-redirect" title="Half-open interval">left-open interval</a>.</dd> <dt id="right-open"><dfn>[□, □) [□, □[</dfn></dt> <dd>Both notations are used for a <a href="/wiki/Half-open_interval" class="mw-redirect" title="Half-open interval">right-open interval</a>.</dd> <dt id="⟨⟩"><dfn>⟨□⟩</dfn></dt> <dd>1.  <a href="/wiki/Generating_set" class="mw-redirect" title="Generating set">Generated object</a>: if S is a set of elements in an algebraic structure, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle S\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>S</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle S\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e09a95f17a6c1836a61f42b133a066fd2edd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.309ex; height:2.843ex;" alt="{\displaystyle \langle S\rangle }"> denotes often the object generated by S. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{s_{1},\ldots ,s_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\{s_{1},\ldots ,s_{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a903a482227b7d87cf54bc3f76e3edf55e40861" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.554ex; height:2.843ex;" alt="{\displaystyle S=\{s_{1},\ldots ,s_{n}\}}">, one writes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle s_{1},\ldots ,s_{n}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle s_{1},\ldots ,s_{n}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57a73bba913cf67bd07d9a82582c03069aa9460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.441ex; height:2.843ex;" alt="{\displaystyle \langle s_{1},\ldots ,s_{n}\rangle }"> (that is, braces are omitted). In particular, this may denote <ul><li>the <a href="/wiki/Linear_span" title="Linear span">linear span</a> in a <a href="/wiki/Vector_space" title="Vector space">vector space</a> (also often denoted Span(S)),</li> <li>the generated <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> in a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>,</li> <li>the generated <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> in a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>,</li> <li>the generated <a href="/wiki/Submodule" class="mw-redirect" title="Submodule">submodule</a> in a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a>.</li></ul></dd> <dd>2.  Often used, mainly in physics, for denoting an <a href="/wiki/Expected_value" title="Expected value">expected value</a>. In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2f694d1a54e7ca685e042c3fe5a94a4cfb97317" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.565ex; height:2.843ex;" alt="{\displaystyle E(X)}"> is generally used instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle S\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>S</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle S\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e09a95f17a6c1836a61f42b133a066fd2edd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.309ex; height:2.843ex;" alt="{\displaystyle \langle S\rangle }">.</dd> <dt id="⟨,⟩"><dfn>⟨□, □⟩ ⟨□ | □⟩</dfn></dt> <dd>Both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df1806ebe1fed1a728b18aed82c30be8b2a0acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x\mid y\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>∣</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x\mid y\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5074ba53554d9c746e5ab98e1a336e5d709b5a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.232ex; height:2.843ex;" alt="{\displaystyle \langle x\mid y\rangle }"> are commonly used for denoting the <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> in an <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a>.</dd> <dt id="bra–ket"><dfn><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Box |{\text{ and }}|\Box \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>◻</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>◻</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Box |{\text{ and }}|\Box \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f66e0a5be8336fbbc34e2be273ed3e600ce41d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.628ex; height:2.843ex;" alt="{\displaystyle \langle \Box |{\text{ and }}|\Box \rangle }"></dfn></dt> <dd><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a> or Dirac notation: if x and y are elements of an <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48004887d8f9dfc489bd2bc793780b7f1d8039ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.881ex; height:2.843ex;" alt="{\displaystyle |x\rangle }"> is the vector defined by x, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle y|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle y|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec16e25803f8f4c6fc2cf4d868b77ac1fedd4024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.707ex; height:2.843ex;" alt="{\displaystyle \langle y|}"> is the <a href="/wiki/Covector" class="mw-redirect" title="Covector">covector</a> defined by y; their inner product is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle y\mid x\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mo>∣</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle y\mid x\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e02551ab20798736dc0ab81cf07e3d7f47cb790b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.232ex; height:2.843ex;" alt="{\displaystyle \langle y\mid x\rangle }">.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Symbols_that_do_not_belong_to_formulas">Symbols that do not belong to formulas</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=17" title="Edit section: Symbols that do not belong to formulas">edit</a>]</div> In this section, the symbols that are listed are used as some sorts of punctuation marks in mathematical reasoning, or as abbreviations of natural language phrases. They are generally not used inside a formula. Some were used in <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a> for indicating the logical dependence between sentences written in plain language. Except for the first two, they are normally not used in printed mathematical texts since, for readability, it is generally recommended to have at least one word between two formulas. However, they are still used on a <a href="/wiki/Black_board" class="mw-redirect" title="Black board">black board</a> for indicating relationships between formulas. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="qed"><dfn><a href="/wiki/Tombstone_(typography)" title="Tombstone (typography)">■ , □</a></dfn></dt> <dd>Used for marking the end of a proof and separating it from the current text. The <a href="/wiki/Initialism" class="mw-redirect" title="Initialism">initialism</a> <a href="/wiki/Q.E.D." title="Q.E.D.">Q.E.D. or QED</a> (<a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a>: quod erat demonstrandum, "as was to be shown") is often used for the same purpose, either in its upper-case form or in lower case.</dd> <dt id="☡"><dfn><a href="/wiki/Bourbaki_dangerous_bend_symbol" title="Bourbaki dangerous bend symbol">☡</a></dfn></dt> <dd><a href="/wiki/Bourbaki_dangerous_bend_symbol" title="Bourbaki dangerous bend symbol">Bourbaki dangerous bend symbol</a>: Sometimes used in the margin to forewarn readers against serious errors, where they risk falling, or to mark a passage that is tricky on a first reading because of an especially subtle argument.</dd> <dt id="therefore"><dfn><a href="/wiki/Therefore_sign" title="Therefore sign">∴</a></dfn></dt> <dd>Abbreviation of "therefore". Placed between two assertions, it means that the first one implies the second one. For example: "All humans are mortal, and Socrates is a human. ∴ Socrates is mortal."</dd> <dt id="because"><dfn><a href="/wiki/Because_sign" class="mw-redirect" title="Because sign">∵</a></dfn></dt> <dd>Abbreviation of "because" or "since". Placed between two assertions, it means that the first one is implied by the second one. For example: "11 is <a href="/wiki/Prime_number" title="Prime number">prime</a> ∵ it has no positive integer factors other than itself and one."</dd> <dt id="∋"><dfn>∋</dfn></dt> <dd>1.  Abbreviation of "such that". For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\ni x>3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∋</mo> <mi>x</mi> <mo>></mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\ni x>3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c42487372db57d603493dcc4f7309824394350f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.761ex; height:2.176ex;" alt="{\displaystyle x\ni x>3}"> is normally printed "x such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca13c1461fe5c28b6ba92af1e60b99cde4a53648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>3}">".</dd> <dd>2.  Sometimes used for reversing the operands of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \in }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \in }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe4d5b0a594c1da89b5e78e7dfbeed90bdcc32f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:1.843ex;" alt="{\displaystyle \in }">; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\ni x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>∋</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\ni x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/230a647e153a48c2c3490d2fe0c389de01aea3c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle S\ni x}"> has the same meaning as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51186ba8afb2067573a9082d55dd383df1ea9214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle x\in S}">. See <a href="#∈">∈</a> in <a href="#Set_theory">§ Set theory</a>.</dd> <dt id="∝"><dfn>∝</dfn></dt> <dd>Abbreviation of "is proportional to".</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Miscellaneous">Miscellaneous</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=18" title="Edit section: Miscellaneous">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="!"><dfn><a href="/wiki/!" class="mw-redirect" title="!">!</a></dfn></dt> <dd>1.  <a href="/wiki/Factorial" title="Factorial">Factorial</a>: if n is a <a href="/wiki/Positive_integer" class="mw-redirect" title="Positive integer">positive integer</a>, n! is the product of the first n positive integers, and is read as "n factorial".</dd> <dd>2.  <a href="/wiki/Double_factorial" title="Double factorial">Double factorial</a>: if n is a <a href="/wiki/Positive_integer" class="mw-redirect" title="Positive integer">positive integer</a>, n!! is the product of all positive integers up to n with the same parity as n, and is read as "the double factorial of n".</dd> <dd>3.  <a href="/wiki/Subfactorial" class="mw-redirect" title="Subfactorial">Subfactorial</a>: if n is a positive integer, !n is the number of <a href="/wiki/Derangements" class="mw-redirect" title="Derangements">derangements</a> of a set of n elements, and is read as "the subfactorial of n".</dd> <dt id="*"><dfn><a href="/wiki/*" class="mw-redirect" title="*">*</a></dfn></dt> <dd>Many different uses in mathematics; see <a href="/wiki/Asterisk#Mathematics" title="Asterisk">Asterisk § Mathematics</a>.</dd> <dt id="vbar"><dfn><a href="/wiki/Vertical_bar" title="Vertical bar">|</a></dfn></dt> <dd>1.  <a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">Divisibility</a>: if m and n are two integers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\mid n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∣</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\mid n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8fbcfe5176123851439b522038d471a49ff06be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.372ex; height:2.843ex;" alt="{\displaystyle m\mid n}"> means that m divides n evenly.</dd> <dd>2.  In <a href="/wiki/Set-builder_notation" title="Set-builder notation">set-builder notation</a>, it is used as a separator meaning "such that"; see <a href="#b:b">{□ | □}</a>.</dd> <dd>3.  <a href="/wiki/Restriction_(mathematics)" title="Restriction (mathematics)">Restriction of a function</a>: if f is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, and S is a <a href="/wiki/Subset" title="Subset">subset</a> of its <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f|_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f|_{S}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd8f943a75adb4c382a271541c010eb0d60b3b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.218ex; height:3.009ex;" alt="{\displaystyle f|_{S}}"> is the function with S as a domain that equals f on S.</dd> <dd>4.  <a href="/wiki/Conditional_probability" title="Conditional probability">Conditional probability</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X\mid E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>∣</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X\mid E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/365b05cf3ead2f9a495913beb1a6dc2900b12f47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.248ex; height:2.843ex;" alt="{\displaystyle P(X\mid E)}"> denotes the probability of X given that the event E occurs. Also denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X/E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X/E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab015fda39060fc34f931e8a49685bb202b01ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.473ex; height:2.843ex;" alt="{\displaystyle P(X/E)}">; see "<a href="#/">/</a>".</dd> <dd>5.  For several uses as <a href="/wiki/Brackets" class="mw-redirect" title="Brackets">brackets</a> (in pairs or with ⟨ and ⟩) see <a href="#Other_brackets">§ Other brackets</a>.</dd> <dt id="∤"><dfn>∤</dfn></dt> <dd><a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">Non-divisibility</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\nmid m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∤</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\nmid m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d0785ea353a817ae656619ab03595b1215a7c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.372ex; height:2.843ex;" alt="{\displaystyle n\nmid m}"> means that n is not a divisor of m.</dd> <dt id="∥"><dfn><a href="/wiki/%E2%88%A5" class="mw-redirect" title="∥">∥</a></dfn></dt> <dd>1.  Denotes <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallelism</a> in <a href="/wiki/Elementary_geometry" class="mw-redirect" title="Elementary geometry">elementary geometry</a>: if PQ and RS are two <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PQ\parallel RS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>Q</mi> <mo>∥</mo> <mi>R</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PQ\parallel RS}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0894c6ea87a071ed039dd4aa99f125ec2a642a27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.3ex; height:2.843ex;" alt="{\displaystyle PQ\parallel RS}"> means that they are parallel.</dd> <dd>2.  <a href="/wiki/Parallel_(operator)" title="Parallel (operator)">Parallel</a>, an <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">arithmetical operation</a> used in <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a> for modeling <a href="/wiki/Parallel_resistors" class="mw-redirect" title="Parallel resistors">parallel resistors</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\parallel y={\frac {1}{{\frac {1}{x}}+{\frac {1}{y}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∥</mo> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>y</mi> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\parallel y={\frac {1}{{\frac {1}{x}}+{\frac {1}{y}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16c5c4704cbbd80b4a7459db7ade123892d338d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.147ex; height:6.843ex;" alt="{\displaystyle x\parallel y={\frac {1}{{\frac {1}{x}}+{\frac {1}{y}}}}}">.</dd> <dd>3.  Used in pairs as brackets, denotes a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>; see <a href="#norm">||□||</a>.</dd> <dd>4.  <a href="/wiki/Concatenation" title="Concatenation">Concatenation</a>: Typically used in computer science, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mathbin {\vert \vert } y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mrow class="MJX-TeXAtom-BIN"> <mo fence="false" stretchy="false">|</mo> <mo fence="false" stretchy="false">|</mo> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mathbin {\vert \vert } y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e991203247ebb9195c292e7ce91410f6beb7740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle x\mathbin {\vert \vert } y}"> is said to represent the value resulting from appending the digits of y to the end of x.</dd> <dd>5.  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\displaystyle D_{\text{KL}}(P\parallel Q)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>KL</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∥</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle D_{\text{KL}}(P\parallel Q)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b184fd4d30dfc8c0ec5a3c7e16c06e717f2837d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.308ex; height:2.843ex;" alt="{\displaystyle {\displaystyle D_{\text{KL}}(P\parallel Q)}}">, denotes a <a href="/wiki/Statistical_distance" title="Statistical distance">statistical distance</a> or measure of how one <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> P is different from a second, reference probability distribution Q.</dd> <dt id="∦"><dfn><a href="/wiki/%E2%88%A5" class="mw-redirect" title="∥">∦</a></dfn></dt> <dd>Sometimes used for denoting that two <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a> are not parallel; for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PQ\not \parallel RS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>Q</mi> <mo>∦</mo> <mi>R</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PQ\not \parallel RS}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b0f4defb90031982f0e583c036d8f724f1c399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.3ex; height:2.843ex;" alt="{\displaystyle PQ\not \parallel RS}">.</dd> <dt id="⟂"><dfn><a href="/wiki/%E2%9F%82" class="mw-redirect" title="⟂"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \perp }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊥</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \perp }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb90d6db42aa12f9e2f31176a4ed4e741c69eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \perp }"></a></dfn></dt> <dd>1.  Denotes <a href="/wiki/Perpendicularity" class="mw-redirect" title="Perpendicularity">perpendicularity</a> and <a href="/wiki/Orthogonality" title="Orthogonality">orthogonality</a>. For example, if A, B, C are three points in a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB\perp AC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mo>⊥</mo> <mi>A</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB\perp AC}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0aa6b30c66732c5a1344357c8569b798abad5b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.115ex; height:2.176ex;" alt="{\displaystyle AB\perp AC}"> means that the <a href="/wiki/Line_segment" title="Line segment">line segments</a> AB and AC are <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a>, and form a <a href="/wiki/Right_angle" title="Right angle">right angle</a>.</dd> <dd>2.  For the similar symbol, see <a href="#⊥"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"></a>.</dd> <dt id="⊙"><dfn>⊙</dfn></dt> <dd><a href="/wiki/Hadamard_product_(series)" class="mw-redirect" title="Hadamard product (series)">Hadamard product of power series</a>: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle S=\sum _{i=0}^{\infty }s_{i}x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle S=\sum _{i=0}^{\infty }s_{i}x^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2548e46b947b2f2f4d469a690d9dd99d25f87adc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.358ex; height:3.176ex;" alt="{\displaystyle \textstyle S=\sum _{i=0}^{\infty }s_{i}x^{i}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle T=\sum _{i=0}^{\infty }t_{i}x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle T=\sum _{i=0}^{\infty }t_{i}x^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5156e31bb2835c9cfa48acdd9aab87bb4e56c11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.245ex; height:3.176ex;" alt="{\displaystyle \textstyle T=\sum _{i=0}^{\infty }t_{i}x^{i}}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle S\odot T=\sum _{i=0}^{\infty }s_{i}t_{i}x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>S</mi> <mo>⊙</mo> <mi>T</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle S\odot T=\sum _{i=0}^{\infty }s_{i}t_{i}x^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd1144bd156b86fb23ca493d63a16554b2f57af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.474ex; height:3.176ex;" alt="{\displaystyle \textstyle S\odot T=\sum _{i=0}^{\infty }s_{i}t_{i}x^{i}}">. Possibly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"> is also used instead of <a href="#∘">○</a> for the <a href="/wiki/Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product of matrices</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=19" title="Edit section: See also">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Related_articles">Related articles</h3>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=20" title="Edit section: Related articles">edit</a>]</div> <ul><li><a href="/wiki/Language_of_mathematics" title="Language of mathematics">Language of mathematics</a></li> <li><a href="/wiki/Mathematical_notation" title="Mathematical notation">Mathematical notation</a></li> <li><a href="/wiki/Notation_in_probability_and_statistics" title="Notation in probability and statistics">Notation in probability and statistics</a></li> <li><a href="/wiki/Physical_constant" title="Physical constant">Physical constants</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Related_lists">Related lists</h3>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=21" title="Edit section: Related lists">edit</a>]</div> <ul><li><a href="/wiki/List_of_logic_symbols" title="List of logic symbols">List of logic symbols</a></li> <li><a href="/wiki/List_of_mathematical_constants" title="List of mathematical constants">List of mathematical constants</a></li> <li><a href="/wiki/Table_of_mathematical_symbols_by_introduction_date" title="Table of mathematical symbols by introduction date">Table of mathematical symbols by introduction date</a></li> <li><a href="/wiki/Blackboard_bold" title="Blackboard bold">Blackboard bold</a></li> <li><a href="/wiki/Greek_letters_used_in_mathematics,_science,_and_engineering" title="Greek letters used in mathematics, science, and engineering">Greek letters used in mathematics, science, and engineering</a></li> <li><a href="/wiki/Latin_letters_used_in_mathematics,_science,_and_engineering" title="Latin letters used in mathematics, science, and engineering">Latin letters used in mathematics, science, and engineering</a></li> <li><a href="/wiki/List_of_common_physics_notations" title="List of common physics notations">List of common physics notations</a></li> <li><a href="/wiki/List_of_letters_used_in_mathematics,_science,_and_engineering" title="List of letters used in mathematics, science, and engineering">List of letters used in mathematics, science, and engineering</a></li> <li><a href="/wiki/List_of_mathematical_abbreviations" title="List of mathematical abbreviations">List of mathematical abbreviations</a></li> <li><a href="/wiki/List_of_typographical_symbols_and_punctuation_marks" title="List of typographical symbols and punctuation marks">List of typographical symbols and punctuation marks</a></li> <li><a href="/wiki/ISO_31-11" title="ISO 31-11">ISO 31-11</a> (Mathematical signs and symbols for use in physical sciences and technology)</li> <li><a href="/wiki/APL_syntax_and_symbols#Monadic_functions" title="APL syntax and symbols">List of APL functions</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Unicode_symbols"><a href="/wiki/Unicode" title="Unicode">Unicode</a> symbols</h3>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=22" title="Edit section: Unicode symbols">edit</a>]</div> <ul><li><a href="/wiki/Unicode_block" title="Unicode block">Unicode block</a></li> <li><a href="/wiki/Mathematical_Alphanumeric_Symbols" title="Mathematical Alphanumeric Symbols">Mathematical Alphanumeric Symbols (Unicode block)</a></li> <li><a href="/wiki/List_of_Unicode_characters" title="List of Unicode characters">List of Unicode characters</a></li> <li><a href="/wiki/Letterlike_Symbols" title="Letterlike Symbols">Letterlike Symbols</a></li> <li><a href="/wiki/Mathematical_operators_and_symbols_in_Unicode" title="Mathematical operators and symbols in Unicode">Mathematical operators and symbols in Unicode</a></li> <li>Miscellaneous Mathematical Symbols: <a href="/wiki/Miscellaneous_Mathematical_Symbols-A" title="Miscellaneous Mathematical Symbols-A">A</a>, <a href="/wiki/Miscellaneous_Mathematical_Symbols-B" title="Miscellaneous Mathematical Symbols-B">B</a>, <a href="/wiki/Miscellaneous_Technical" title="Miscellaneous Technical">Technical</a></li> <li><a href="/wiki/Arrow_(symbol)" title="Arrow (symbol)">Arrow (symbol)</a> and <a href="/wiki/Miscellaneous_Symbols_and_Arrows" title="Miscellaneous Symbols and Arrows">Miscellaneous Symbols and Arrows</a></li> <li><a href="/wiki/Number_Forms" title="Number Forms">Number Forms</a></li> <li><a href="/wiki/Geometric_Shapes_(Unicode_block)" title="Geometric Shapes (Unicode block)">Geometric Shapes</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=23" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-ISO-1"><a href="#cite_ref-ISO_1-0">^</a> <a href="/wiki/ISO_80000-2" class="mw-redirect" title="ISO 80000-2">ISO 80000-2</a>, Section 9 "Operations", 2-9.6 </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.pearson.com/us/higher-education/program/Tamhane-Statistics-and-Data-Analysis-From-Elementary-to-Intermediate/PGM227786.html">"Statistics and Data Analysis: From Elementary to Intermediate"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Statistics+and+Data+Analysis%3A+From+Elementary+to+Intermediate&rft_id=https%3A%2F%2Fwww.pearson.com%2Fus%2Fhigher-education%2Fprogram%2FTamhane-Statistics-and-Data-Analysis-From-Elementary-to-Intermediate%2FPGM227786.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+mathematical+symbols" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> The <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a> equivalent to both <a href="/wiki/Unicode" title="Unicode">Unicode</a> symbols ∘ and ○ is \circ. The Unicode symbol that has the same size as \circ depends on the browser and its implementation. In some cases ∘ is so small that it can be confused with an <a href="/wiki/Interpoint" class="mw-redirect" title="Interpoint">interpoint</a>, and ○ looks similar as \circ. In other cases, ○ is too large for denoting a binary operation, and it is ∘ that looks like \circ. As LaTeX is commonly considered as the standard for mathematical typography, and it does not distinguish these two Unicode symbols, they are considered here as having the same mathematical meaning. </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRutherford1965" class="citation book cs1">Rutherford, D. E. (1965). Vector Methods. University Mathematical Texts. Oliver and Boyd Ltd., Edinburgh.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector+Methods&rft.series=University+Mathematical+Texts&rft.pub=Oliver+and+Boyd+Ltd.%2C+Edinburgh&rft.date=1965&rft.aulast=Rutherford&rft.aufirst=D.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+mathematical+symbols" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Glossary_of_mathematical_symbols&action=edit&section=24" title="Edit section: External links">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathsym.html">Jeff Miller: Earliest Uses of Various Mathematical Symbols</a></li> <li><a rel="nofollow" class="external text" href="http://www.numericana.com/answer/symbol.htm">Numericana: Scientific Symbols and Icons</a></li> <li><a rel="nofollow" class="external text" href="http://us.metamath.org/symbols/symbols.html">GIF and PNG Images for Math Symbols</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070117015443/http://tlt.psu.edu/suggestions/international/bylanguage/math.html">Mathematical Symbols in Unicode</a></li> <li><a rel="nofollow" class="external text" href="https://detexify.kirelabs.org/classify.html">Detexify: LaTeX Handwriting Recognition Tool</a></li></ul> <dl><dt>Some Unicode charts of mathematical operators and symbols:</dt></dl> <ul><li><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/#symbols">Index of Unicode symbols</a></li> <li><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2100.pdf">Range 2100–214F: Unicode Letterlike Symbols</a></li> <li><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2190.pdf">Range 2190–21FF: Unicode Arrows</a></li> <li><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2200.pdf">Range 2200–22FF: Unicode Mathematical Operators</a></li> <li><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U27C0.pdf">Range 27C0–27EF: Unicode Miscellaneous Mathematical Symbols–A</a></li> <li><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2980.pdf">Range 2980–29FF: Unicode Miscellaneous Mathematical Symbols–B</a></li> <li><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2A00.pdf">Range 2A00–2AFF: Unicode Supplementary Mathematical Operators</a></li></ul> <dl><dt>Some Unicode cross-references:</dt></dl> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20141105143723/http://www.artofproblemsolving.com/Wiki/index.php/LaTeX:Symbols">Short list of commonly used LaTeX symbols</a> and <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090323063515/http://mirrors.med.harvard.edu/ctan/info/symbols/comprehensive/">Comprehensive LaTeX Symbol List</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140222144828/http://www.robinlionheart.com/stds/html4/entities-mathml">MathML Characters</a> - sorts out Unicode, HTML and MathML/TeX names on one page</li> <li><a rel="nofollow" class="external text" href="http://www.w3.org/TR/REC-MathML/chap6/bycodes.html">Unicode values and MathML names</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20141126074509/http://svn.ghostscript.com/ghostscript/branches/gs-db/Resource/Decoding/Unicode">Unicode values and Postscript names</a> from the source code for <a href="/wiki/Ghostscript" title="Ghostscript">Ghostscript</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a class="mw-selflink selflink">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Systems_science" title="Systems science">Systems science</a> <ul><li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_science" title="Computer science">Computer science</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a> <ul><li><a href="/wiki/List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li> <li><a href="/wiki/Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li> <li><a href="/wiki/List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li> <li><a href="/wiki/Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Fields_of_mathematics" title="Category:Fields of mathematics">Category</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /> <a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw" title="commons:Category:Mathematics">Commons</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/32px-People_icon.svg.png 2x" data-file-width="100" data-file-height="100" /> <a href="/wiki/Wikipedia:WikiProject_Mathematics" title="Wikipedia:WikiProject Mathematics">WikiProject</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Common_mathematical_notation,_symbols,_and_formulas" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematical_symbols_notation_language" title="Template:Mathematical symbols notation language"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_symbols_notation_language" title="Template talk:Mathematical symbols notation language"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_symbols_notation_language" title="Special:EditPage/Template:Mathematical symbols notation language"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Common_mathematical_notation,_symbols,_and_formulas" style="font-size:114%;margin:0 4em">Common <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> <a href="/wiki/Mathematical_notation" title="Mathematical notation">notation</a>, <a href="/wiki/List_of_mathematical_symbols_by_subject" class="mw-redirect" title="List of mathematical symbols by subject">symbols</a>, and <a href="/wiki/Help:Displaying_a_formula" title="Help:Displaying a formula">formulas</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Lists_of_Unicode_and_LaTeX_mathematical_symbols" style="font-size:114%;margin:0 4em">Lists of <a href="/wiki/Unicode" title="Unicode">Unicode</a> and <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a> mathematical symbols</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_mathematical_symbols_by_subject" class="mw-redirect" title="List of mathematical symbols by subject">List of mathematical symbols by subject</a></li> <li><a class="mw-selflink selflink">Glossary of mathematical symbols</a></li> <li><a href="/wiki/List_of_logic_symbols" title="List of logic symbols">List of logic symbols</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Lists_of_Unicode_symbols" style="font-size:114%;margin:0 4em">Lists of <a href="/wiki/Unicode" title="Unicode">Unicode</a> symbols</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_Unicode_characters" title="List of Unicode characters">List of Unicode characters</a></li> <li><a href="/wiki/Unicode_block" title="Unicode block">Unicode block</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Alphanumeric</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_Alphanumeric_Symbols" title="Mathematical Alphanumeric Symbols">Mathematical Alphanumeric Symbols</a></li> <li><a href="/wiki/Blackboard_bold#Usage" title="Blackboard bold">Blackboard bold</a></li> <li><a href="/wiki/Letterlike_Symbols" title="Letterlike Symbols">Letterlike Symbols</a></li> <li><a href="/wiki/Symbols_for_zero" title="Symbols for zero">Symbols for zero</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Arrow_(symbol)" title="Arrow (symbol)">Arrows</a> and <a href="/wiki/Geometric_Shapes_(Unicode_block)" title="Geometric Shapes (Unicode block)">Geometric Shapes</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arrow_(symbol)" title="Arrow (symbol)">Arrows</a></li> <li><a href="/wiki/Miscellaneous_Symbols_and_Arrows" title="Miscellaneous Symbols and Arrows">Miscellaneous Symbols and Arrows</a></li> <li><a href="/wiki/Geometric_Shapes_(Unicode_block)" title="Geometric Shapes (Unicode block)">Geometric Shapes (Unicode block)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_operators_and_symbols_in_Unicode" title="Mathematical operators and symbols in Unicode">Mathematical operators and symbols</a></li> <li><a href="/wiki/Mathematical_Operators_(Unicode_block)" title="Mathematical Operators (Unicode block)">Mathematical Operators (Unicode block)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Supplemental_Mathematical_Operators" title="Supplemental Mathematical Operators">Supplemental Math Operators</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supplemental_Mathematical_Operators" title="Supplemental Mathematical Operators">Supplemental Mathematical Operators</a></li> <li><a href="/wiki/Number_Forms" title="Number Forms">Number Forms</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Miscellaneous_Mathematical_Symbols-A" title="Miscellaneous Mathematical Symbols-A">A</a></li> <li><a href="/wiki/Miscellaneous_Mathematical_Symbols-B" title="Miscellaneous Mathematical Symbols-B">B</a></li> <li><a href="/wiki/Miscellaneous_Technical" title="Miscellaneous Technical">Technical</a></li> <li><a href="/wiki/ISO_31-11" title="ISO 31-11">ISO 31-11</a> (Mathematical signs and symbols for use in physical sciences and technology)</li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Typographical_conventions_and_notations" style="font-size:114%;margin:0 4em">Typographical conventions and notations</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Language_of_mathematics" title="Language of mathematics">Language</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/APL_syntax_and_symbols#Monadic_functions" title="APL syntax and symbols">APL syntax and symbols</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Letters</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Diacritic" title="Diacritic">Diacritic</a></li> <li><a href="/wiki/List_of_letters_used_in_mathematics,_science,_and_engineering" title="List of letters used in mathematics, science, and engineering">Letters in STEM</a> <ul><li><a href="/wiki/Greek_letters_used_in_mathematics,_science,_and_engineering" title="Greek letters used in mathematics, science, and engineering">Greek letters in STEM</a></li> <li><a href="/wiki/Latin_letters_used_in_mathematics,_science,_and_engineering" title="Latin letters used in mathematics, science, and engineering">Latin letters in STEM</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_notation" title="Mathematical notation">Mathematical notation</a></li> <li><a href="/wiki/List_of_mathematical_abbreviations" title="List of mathematical abbreviations">Abbreviations</a></li> <li><a href="/wiki/Notation_in_probability_and_statistics" title="Notation in probability and statistics">Notation in probability and statistics</a></li> <li><a href="/wiki/List_of_common_physics_notations" title="List of common physics notations">List of common physics notations</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Meanings_of_symbols" style="font-size:114%;margin:0 4em">Meanings of symbols</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Glossary of mathematical symbols</a></li> <li><a href="/wiki/List_of_mathematical_constants" title="List of mathematical constants">List of mathematical constants</a></li> <li><a href="/wiki/Physical_constant" title="Physical constant">Physical constants</a></li> <li><a href="/wiki/Table_of_mathematical_symbols_by_introduction_date" title="Table of mathematical symbols by introduction date">Table of mathematical symbols by introduction date</a></li> <li><a href="/wiki/List_of_typographical_symbols_and_punctuation_marks" title="List of typographical symbols and punctuation marks">List of typographical symbols and punctuation marks</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Glossary_of_mathematical_symbols&oldid=1259267463">https://en.wikipedia.org/w/index.php?title=Glossary_of_mathematical_symbols&oldid=1259267463</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Mathematical_symbols" title="Category:Mathematical symbols">Mathematical symbols</a></li><li><a href="/wiki/Category:Mathematical_notation" title="Category:Mathematical notation">Mathematical notation</a></li><li><a href="/wiki/Category:Lists_of_symbols" title="Category:Lists of symbols">Lists of symbols</a></li><li><a href="/wiki/Category:Mathematical_logic" title="Category:Mathematical logic">Mathematical logic</a></li><li><a href="/wiki/Category:Mathematical_tables" title="Category:Mathematical tables">Mathematical tables</a></li><li><a href="/wiki/Category:Glossaries_of_mathematics" title="Category:Glossaries of mathematics">Glossaries of mathematics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_November_2020" title="Category:Articles with unsourced statements from November 2020">Articles with unsourced statements from November 2020</a></li><li><a href="/wiki/Category:Articles_containing_Latin-language_text" title="Category:Articles containing Latin-language text">Articles containing Latin-language text</a></li><li><a href="/wiki/Category:Wikipedia_glossaries_using_description_lists" title="Category:Wikipedia glossaries using description lists">Wikipedia glossaries using description lists</a></li><li><a href="/wiki/Category:Pages_that_use_a_deprecated_format_of_the_math_tags" title="Category:Pages that use a deprecated format of the math tags">Pages that use a deprecated format of the math tags</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 24 November 2024, at 07:22 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Glossary_of_mathematical_symbols&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5c59558b9d-wt7qm","wgBackendResponseTime":180,"wgPageParseReport":{"limitreport":{"cputime":"1.347","walltime":"1.737","ppvisitednodes":{"value":26021,"limit":1000000},"postexpandincludesize":{"value":205840,"limit":2097152},"templateargumentsize":{"value":96829,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":3,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":44623,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 1103.450 1 -total"," 32.11% 354.369 142 Template:Term"," 30.65% 338.182 264 Template:Defn"," 17.91% 197.595 193 Template:Math"," 14.05% 155.007 1 Template:Areas_of_mathematics"," 13.91% 153.541 1 Template:Navbox"," 9.04% 99.753 1 Template:Short_description"," 8.80% 97.066 1 Template:Langx"," 8.40% 92.711 1 Template:Reflist"," 6.92% 76.359 1 Template:Cite_web"]},"scribunto":{"limitreport-timeusage":{"value":"0.521","limit":"10.000"},"limitreport-memusage":{"value":18945333,"limit":52428800},"limitreport-logs":"table#1 {\n [\"size\"] = \"tiny\",\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-5c59558b9d-5fqwl","timestamp":"20241130105956","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Glossary of mathematical symbols","url":"https:\/\/en.wikipedia.org\/wiki\/Glossary_of_mathematical_symbols","sameAs":"http:\/\/www.wikidata.org\/entity\/Q180159","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q180159","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-08-20T07:47:24Z","dateModified":"2024-11-24T07:22:40Z","headline":"meanings of symbols used in mathematics"}</script> </body> </html>