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>Discrete mathematics - 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":"e7d71356-36a6-48c0-a590-d5351c194b09","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Discrete_mathematics","wgTitle":"Discrete mathematics","wgCurRevisionId":1246834323,"wgRevisionId":1246834323,"wgArticleId":8492,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description matches Wikidata","Pages using sidebar with the child parameter","CS1: long volume value","Commons category link from Wikidata","Webarchive template wayback links","Discrete mathematics"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Discrete_mathematics","wgRelevantArticleId":8492,"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":30000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q121416","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.cite.styles":"ready","ext.math.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","mediawiki.page.media","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.quicksurveys.init","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.4"> <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 property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/5/5b/6n-graf.svg/1200px-6n-graf.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="793"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/5/5b/6n-graf.svg/800px-6n-graf.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="529"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/5/5b/6n-graf.svg/640px-6n-graf.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="423"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Discrete mathematics - 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/Discrete_mathematics"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Discrete_mathematics&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/Discrete_mathematics"> <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-Discrete_mathematics rootpage-Discrete_mathematics 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" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </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"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></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"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></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"><span>Recent changes</span></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"><span>Upload file</span></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"> <span class="mw-logo-container skin-invert"> <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;"> </span> </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"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </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" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </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" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </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=""><span>Donate</span></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=Discrete+mathematics" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></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=Discrete+mathematics" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></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" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </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"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Discrete+mathematics" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Discrete+mathematics" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></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"><span>learn more</span></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"><span>Contributions</span></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"><span>Talk</span></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-Topics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Topics"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Topics</span> </div> </a> <button aria-controls="toc-Topics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Topics subsection</span> </button> <ul id="toc-Topics-sublist" class="vector-toc-list"> <li id="toc-Theoretical_computer_science" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theoretical_computer_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Theoretical computer science</span> </div> </a> <ul id="toc-Theoretical_computer_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Information_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Information_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Information theory</span> </div> </a> <ul id="toc-Information_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Logic</span> </div> </a> <ul id="toc-Logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Set_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Set_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Set theory</span> </div> </a> <ul id="toc-Set_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combinatorics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinatorics"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Combinatorics</span> </div> </a> <ul id="toc-Combinatorics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graph_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graph_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Graph theory</span> </div> </a> <ul id="toc-Graph_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.7</span> <span>Number theory</span> </div> </a> <ul id="toc-Number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_structures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_structures"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.8</span> <span>Algebraic structures</span> </div> </a> <ul id="toc-Algebraic_structures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discrete_analogues_of_continuous_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete_analogues_of_continuous_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.9</span> <span>Discrete analogues of continuous mathematics</span> </div> </a> <ul id="toc-Discrete_analogues_of_continuous_mathematics-sublist" class="vector-toc-list"> <li id="toc-Calculus_of_finite_differences,_discrete_analysis,_and_discrete_calculus" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Calculus_of_finite_differences,_discrete_analysis,_and_discrete_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.9.1</span> <span>Calculus of finite differences, discrete analysis, and discrete calculus</span> </div> </a> <ul id="toc-Calculus_of_finite_differences,_discrete_analysis,_and_discrete_calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discrete_geometry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Discrete_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.9.2</span> <span>Discrete geometry</span> </div> </a> <ul id="toc-Discrete_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discrete_modelling" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Discrete_modelling"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.9.3</span> <span>Discrete modelling</span> </div> </a> <ul id="toc-Discrete_modelling-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Challenges" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Challenges"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Challenges</span> </div> </a> <ul id="toc-Challenges-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"> <span class="vector-toc-numb">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </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"> <span class="vector-toc-numb">4</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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"> <span class="vector-toc-numb">6</span> <span>External links</span> </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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </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"><span class="mw-page-title-main">Discrete mathematics</span></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 74 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-74" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">74 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Diskrete_wiskunde" title="Diskrete wiskunde – Afrikaans" lang="af" hreflang="af" data-title="Diskrete wiskunde" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA_%D9%85%D8%AA%D9%82%D8%B7%D8%B9%D8%A9" title="رياضيات متقطعة – Arabic" lang="ar" hreflang="ar" data-title="رياضيات متقطعة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Matematica_discreta" title="Matematica discreta – Aragonese" lang="an" hreflang="an" data-title="Matematica discreta" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Matem%C3%A1tiques_discretes" title="Matemátiques discretes – Asturian" lang="ast" hreflang="ast" data-title="Matemátiques discretes" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Diskret_riyaziyyat" title="Diskret riyaziyyat – Azerbaijani" lang="az" hreflang="az" data-title="Diskret riyaziyyat" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AF%DB%8C%D8%B3%DA%A9%D8%B1%D8%AA_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA" title="دیسکرت ریاضیات – South Azerbaijani" lang="azb" hreflang="azb" data-title="دیسکرت ریاضیات" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BF%E0%A6%9A%E0%A7%8D%E0%A6%9B%E0%A6%BF%E0%A6%A8%E0%A7%8D%E0%A6%A8_%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="বিচ্ছিন্ন গণিত – Bangla" lang="bn" hreflang="bn" data-title="বিচ্ছিন্ন গণিত" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D1%8B%D1%81%D0%BA%D1%80%D1%8D%D1%82%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0" title="Дыскрэтная матэматыка – Belarusian" lang="be" hreflang="be" data-title="Дыскрэтная матэматыка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%94%D1%8B%D1%81%D0%BA%D1%80%D1%8D%D1%82%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0" title="Дыскрэтная матэматыка – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Дыскрэтная матэматыка" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B5%D1%82%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Дискретна математика – Bulgarian" lang="bg" hreflang="bg" data-title="Дискретна математика" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Diskretna_matematika" title="Diskretna matematika – Bosnian" lang="bs" hreflang="bs" data-title="Diskretna matematika" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matem%C3%A0tica_discreta" title="Matemàtica discreta – Catalan" lang="ca" hreflang="ca" data-title="Matemàtica discreta" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B5%D1%82%D0%BB%C4%83_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Дискретлă математика – Chuvash" lang="cv" hreflang="cv" data-title="Дискретлă математика" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Diskr%C3%A9tn%C3%AD_matematika" title="Diskrétní matematika – Czech" lang="cs" hreflang="cs" data-title="Diskrétní matematika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Diskret_matematik" title="Diskret matematik – Danish" lang="da" hreflang="da" data-title="Diskret matematik" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Diskrete_Mathematik" title="Diskrete Mathematik – German" lang="de" hreflang="de" data-title="Diskrete Mathematik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Diskreetne_matemaatika" title="Diskreetne matemaatika – Estonian" lang="et" hreflang="et" data-title="Diskreetne matemaatika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%B1%CE%BA%CF%81%CE%B9%CF%84%CE%AC_%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC" title="Διακριτά μαθηματικά – Greek" lang="el" hreflang="el" data-title="Διακριτά μαθηματικά" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Matem%C3%A1tica_discreta" title="Matemática discreta – Spanish" lang="es" hreflang="es" data-title="Matemática discreta" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Diskreta_matematiko" title="Diskreta matematiko – Esperanto" lang="eo" hreflang="eo" data-title="Diskreta matematiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Matematika_diskretu" title="Matematika diskretu – Basque" lang="eu" hreflang="eu" data-title="Matematika diskretu" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA_%DA%AF%D8%B3%D8%B3%D8%AA%D9%87" title="ریاضیات گسسته – Persian" lang="fa" hreflang="fa" data-title="ریاضیات گسسته" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Math%C3%A9matiques_discr%C3%A8tes" title="Mathématiques discrètes – French" lang="fr" hreflang="fr" data-title="Mathématiques discrètes" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Matem%C3%A1tica_discreta" title="Matemática discreta – Galician" lang="gl" hreflang="gl" data-title="Matemática discreta" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%B4%EC%82%B0%EC%88%98%ED%95%99" title="이산수학 – Korean" lang="ko" hreflang="ko" data-title="이산수학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%AB%D5%BD%D5%AF%D6%80%D5%A5%D5%BF_%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1" title="Դիսկրետ մաթեմատիկա – Armenian" lang="hy" hreflang="hy" data-title="Դիսկրետ մաթեմատիկա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B5%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%A4_%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="विविक्त गणित – Hindi" lang="hi" hreflang="hi" data-title="विविक्त गणित" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Diskretna_matematika" title="Diskretna matematika – Croatian" lang="hr" hreflang="hr" data-title="Diskretna matematika" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Matematika_diskrit" title="Matematika diskrit – Indonesian" lang="id" hreflang="id" data-title="Matematika diskrit" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Matematica_discreta" title="Matematica discreta – Italian" lang="it" hreflang="it" data-title="Matematica discreta" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94_%D7%91%D7%93%D7%99%D7%93%D7%94" title="מתמטיקה בדידה – Hebrew" lang="he" hreflang="he" data-title="מתמטיקה בדידה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%93%E1%83%98%E1%83%A1%E1%83%99%E1%83%A0%E1%83%94%E1%83%A2%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90" title="დისკრეტული მათემატიკა – Georgian" lang="ka" hreflang="ka" data-title="დისკრეტული მათემატიკა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B5%D1%82%D1%82%D1%96%D0%BA_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Дискреттік математика – Kazakh" lang="kk" hreflang="kk" data-title="Дискреттік математика" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B5%D1%82%D1%82%D0%B8%D0%BA_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Дискреттик математика – Kyrgyz" lang="ky" hreflang="ky" data-title="Дискреттик математика" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Mathematica_discreta" title="Mathematica discreta – Latin" lang="la" hreflang="la" data-title="Mathematica discreta" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Diskr%C4%93t%C4%81_matem%C4%81tika" title="Diskrētā matemātika – Latvian" lang="lv" hreflang="lv" data-title="Diskrētā matemātika" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Diskre%C4%8Dioji_matematika" title="Diskrečioji matematika – Lithuanian" lang="lt" hreflang="lt" data-title="Diskrečioji matematika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Matematega_discreta" title="Matematega discreta – Lombard" lang="lmo" hreflang="lmo" data-title="Matematega discreta" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Diszkr%C3%A9t_matematika" title="Diszkrét matematika – Hungarian" lang="hu" hreflang="hu" data-title="Diszkrét matematika" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BF%E0%B4%B5%E0%B5%87%E0%B4%9A%E0%B4%A8_%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%B6%E0%B4%BE%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82" title="വിവേചന ഗണിതശാസ്ത്രം – Malayalam" lang="ml" hreflang="ml" data-title="വിവേചന ഗണിതശാസ്ത്രം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B5%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%A4_%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="विविक्त गणित – Marathi" lang="mr" hreflang="mr" data-title="विविक्त गणित" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Matematik_diskret" title="Matematik diskret – Malay" lang="ms" hreflang="ms" data-title="Matematik diskret" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%95%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%B8%E1%80%85%E1%80%9E%E1%80%AE%E1%80%B8%E1%80%81%E1%80%BC%E1%80%AC%E1%80%B8%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="ပိုင်းစသီးခြားသင်္ချာ – Burmese" lang="my" hreflang="my" data-title="ပိုင်းစသီးခြားသင်္ချာ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Discrete_wiskunde" title="Discrete wiskunde – Dutch" lang="nl" hreflang="nl" data-title="Discrete wiskunde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%9B%A2%E6%95%A3%E6%95%B0%E5%AD%A6" title="離散数学 – Japanese" lang="ja" hreflang="ja" data-title="離散数学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Diskret_matematikk" title="Diskret matematikk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Diskret matematikk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Diskret_matematikk" title="Diskret matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Diskret matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Matematicas_discr%C3%A8tas" title="Matematicas discrètas – Occitan" lang="oc" hreflang="oc" data-title="Matematicas discrètas" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Chekli_matematika" title="Chekli matematika – Uzbek" lang="uz" hreflang="uz" data-title="Chekli matematika" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Matem%C3%A0tica_discreta" title="Matemàtica discreta – Piedmontese" lang="pms" hreflang="pms" data-title="Matemàtica discreta" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Matematyka_dyskretna" title="Matematyka dyskretna – Polish" lang="pl" hreflang="pl" data-title="Matematyka dyskretna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Matem%C3%A1tica_discreta" title="Matemática discreta – Portuguese" lang="pt" hreflang="pt" data-title="Matemática discreta" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Matematic%C4%83_discret%C4%83" title="Matematică discretă – Romanian" lang="ro" hreflang="ro" data-title="Matematică discretă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B5%D1%82%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Дискретная математика – Russian" lang="ru" hreflang="ru" data-title="Дискретная математика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Matematika_diskrete" title="Matematika diskrete – Albanian" lang="sq" hreflang="sq" data-title="Matematika diskrete" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Discrete_mathematics" title="Discrete mathematics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Discrete mathematics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Diskr%C3%A9tna_matematika" title="Diskrétna matematika – Slovak" lang="sk" hreflang="sk" data-title="Diskrétna matematika" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Diskretna_matematika" title="Diskretna matematika – Slovenian" lang="sl" hreflang="sl" data-title="Diskretna matematika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B5%D1%82%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Дискретна математика – Serbian" lang="sr" hreflang="sr" data-title="Дискретна математика" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Diskretna_matematika" title="Diskretna matematika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Diskretna matematika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Diskreetti_matematiikka" title="Diskreetti matematiikka – Finnish" lang="fi" hreflang="fi" data-title="Diskreetti matematiikka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Diskret_matematik" title="Diskret matematik – Swedish" lang="sv" hreflang="sv" data-title="Diskret matematik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Matematikang_diskreto" title="Matematikang diskreto – Tagalog" lang="tl" hreflang="tl" data-title="Matematikang diskreto" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%B2%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%AE%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="இலக்கமியல் கணிதம் – Tamil" lang="ta" hreflang="ta" data-title="இலக்கமியல் கணிதம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A7%E0%B8%B4%E0%B8%A2%E0%B8%B8%E0%B8%95%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95" title="วิยุตคณิต – Thai" lang="th" hreflang="th" data-title="วิยุตคณิต" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A0%D0%B8%D1%91%D0%B7%D0%B8%D1%91%D1%82%D0%B8_%D0%B3%D1%83%D1%81%D0%B0%D1%81%D1%82%D0%B0" title="Риёзиёти гусаста – Tajik" lang="tg" hreflang="tg" data-title="Риёзиёти гусаста" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Ayr%C4%B1k_matematik" title="Ayrık matematik – Turkish" lang="tr" hreflang="tr" data-title="Ayrık matematik" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B5%D1%82%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Дискретна математика – Ukrainian" lang="uk" hreflang="uk" data-title="Дискретна математика" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AA%D9%81%D8%B1%D8%AF_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="متفرد ریاضی – Urdu" lang="ur" hreflang="ur" data-title="متفرد ریاضی" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/To%C3%A1n_h%E1%BB%8Dc_r%E1%BB%9Di_r%E1%BA%A1c" title="Toán học rời rạc – Vietnamese" lang="vi" hreflang="vi" data-title="Toán học rời rạc" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6" title="离散数学 – Wu" lang="wuu" hreflang="wuu" data-title="离散数学" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%93%D7%99%D7%A1%D7%A7%D7%A8%D7%A2%D7%98%D7%A2_%D7%9E%D7%90%D7%98%D7%A2%D7%9E%D7%90%D7%98%D7%99%D7%A7" title="דיסקרעטע מאטעמאטיק – Yiddish" lang="yi" hreflang="yi" data-title="דיסקרעטע מאטעמאטיק" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9B%A2%E6%95%A3%E6%95%B8%E5%AD%B8" title="離散數學 – Cantonese" lang="yue" hreflang="yue" data-title="離散數學" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6" title="离散数学 – Chinese" lang="zh" hreflang="zh" data-title="离散数学" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q121416#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></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/Discrete_mathematics" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Discrete_mathematics" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></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" ><span class="vector-dropdown-label-text">English</span> </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/Discrete_mathematics"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discrete_mathematics&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discrete_mathematics&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></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" ><span class="vector-dropdown-label-text">Tools</span> </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/Discrete_mathematics"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Discrete_mathematics&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Discrete_mathematics&action=history"><span>View history</span></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/Discrete_mathematics" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Discrete_mathematics" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></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"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Discrete_mathematics&oldid=1246834323" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Discrete_mathematics&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Discrete_mathematics&id=1246834323&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></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%2FDiscrete_mathematics"><span>Get shortened URL</span></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%2FDiscrete_mathematics"><span>Download QR code</span></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=Discrete_mathematics&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Discrete_mathematics&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></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:Discrete_mathematics" hreflang="en"><span>Wikimedia Commons</span></a></li><li class="wb-otherproject-link wb-otherproject-wikibooks mw-list-item"><a href="https://en.wikibooks.org/wiki/Discrete_Mathematics" hreflang="en"><span>Wikibooks</span></a></li><li class="wb-otherproject-link wb-otherproject-wikiversity mw-list-item"><a href="https://en.wikiversity.org/wiki/Discrete_mathematics" hreflang="en"><span>Wikiversity</span></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/Q121416" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></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"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Study of discrete mathematical structures</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the mathematics journal, see <a href="/wiki/Discrete_Mathematics_(journal)" title="Discrete Mathematics (journal)"><i>Discrete Mathematics</i> (journal)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Finite math" redirects here. For the syllabus, see <a href="/wiki/Finite_mathematics" title="Finite mathematics">Finite mathematics</a>.</div> <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:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></th></tr><tr><td class="sidebar-above" style="padding-bottom:0.35em;"> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Index</a></li></ul></td></tr><tr><td class="sidebar-content-with-subgroup"> <table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)"><a href="/wiki/Areas_of_mathematics" class="mw-redirect" title="Areas of mathematics">Areas</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Number_theory" title="Number theory">Number theory</a></li> <li><a href="/wiki/Geometry" title="Geometry">Geometry</a></li> <li><a href="/wiki/Algebra" title="Algebra">Algebra</a></li> <li><a href="/wiki/Calculus" title="Calculus">Calculus</a> and <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></li> <li><a class="mw-selflink selflink">Discrete mathematics</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Logic</a> and <a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a> and <a href="/wiki/Decision_theory" title="Decision theory">Decision theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)">Relationship with sciences</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Physics</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li> <li><a href="/wiki/Geomathematics" title="Geomathematics">Geosciences</a></li> <li><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computation</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li> <li><a href="/wiki/Computational_linguistics" title="Computational linguistics">Linguistics</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Economics</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Education</a></li></ul></div></div></td> </tr></tbody></table></td> </tr><tr><th class="sidebar-heading"> <span typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/20px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/30px-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/40px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics Portal</a></th></tr><tr><td class="sidebar-navbar"><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:Math_topics_sidebar" title="Template:Math topics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Math_topics_sidebar" title="Template talk:Math topics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Math_topics_sidebar" title="Special:EditPage/Template:Math topics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:6n-graf.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/6n-graf.svg/250px-6n-graf.svg.png" decoding="async" width="250" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/6n-graf.svg/375px-6n-graf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/6n-graf.svg/500px-6n-graf.svg.png 2x" data-file-width="333" data-file-height="220" /></a><figcaption><a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">Graphs</a> such as these are among the objects studied by discrete mathematics, for their interesting <a href="/wiki/Graph_property" title="Graph property">mathematical properties</a>, their usefulness as models of real-world problems, and their importance in developing computer <a href="/wiki/Algorithm" title="Algorithm">algorithms</a>.</figcaption></figure> <p><b>Discrete mathematics</b> is the study of <a href="/wiki/Mathematical_structures" class="mw-redirect" title="Mathematical structures">mathematical structures</a> that can be considered "discrete" (in a way analogous to <a href="/wiki/Discrete_variable" class="mw-redirect" title="Discrete variable">discrete variables</a>, having a <a href="/wiki/Bijection" title="Bijection">bijection</a> with the set of <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>) rather than "continuous" (analogously to <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a>). Objects studied in discrete mathematics include <a href="/wiki/Integer" title="Integer">integers</a>, <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>, and <a href="/wiki/Statement_(logic)" title="Statement (logic)">statements</a> in <a href="/wiki/Mathematical_logic" title="Mathematical logic">logic</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> By contrast, discrete mathematics excludes topics in "continuous mathematics" such as <a href="/wiki/Real_number" title="Real number">real numbers</a>, <a href="/wiki/Calculus" title="Calculus">calculus</a> or <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. Discrete objects can often be <a href="/wiki/Enumeration" title="Enumeration">enumerated</a> by <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a>; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with <a href="/wiki/Countable_set" title="Countable set">countable sets</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> (finite sets or sets with the same <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> as the natural numbers). However, there is no exact definition of the term "discrete mathematics".<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The set of objects studied in discrete mathematics can be finite or infinite. The term <b>finite mathematics</b> is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. </p><p>Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of <a href="/wiki/Digital_computers" class="mw-redirect" title="Digital computers">digital computers</a> which operate in "discrete" steps and store data in "discrete" bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of <a href="/wiki/Computer_science" title="Computer science">computer science</a>, such as <a href="/wiki/Computer_algorithm" class="mw-redirect" title="Computer algorithm">computer algorithms</a>, <a href="/wiki/Programming_language" title="Programming language">programming languages</a>, <a href="/wiki/Cryptography" title="Cryptography">cryptography</a>, <a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">automated theorem proving</a>, and <a href="/wiki/Software_development" title="Software development">software development</a>. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems. </p><p>Although the main objects of study in discrete mathematics are discrete objects, <a href="/wiki/Analysis_(mathematics)" class="mw-redirect" title="Analysis (mathematics)">analytic</a> methods from "continuous" mathematics are often employed as well. </p><p>In university curricula, discrete mathematics appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction with efforts by <a href="/wiki/Association_for_Computing_Machinery" title="Association for Computing Machinery">ACM</a> and <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">MAA</a> into a course that is basically intended to develop <a href="/wiki/Mathematical_maturity" title="Mathematical maturity">mathematical maturity</a> in first-year students; therefore, it is nowadays a prerequisite for mathematics majors in some universities as well.<sup id="cite_ref-LevasseurDoerr_6-0" class="reference"><a href="#cite_note-LevasseurDoerr-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Howson1988_7-0" class="reference"><a href="#cite_note-Howson1988-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Some high-school-level discrete mathematics textbooks have appeared as well.<sup id="cite_ref-Rosenstein_8-0" class="reference"><a href="#cite_note-Rosenstein-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> At this level, discrete mathematics is sometimes seen as a preparatory course, like <a href="/wiki/Precalculus" title="Precalculus">precalculus</a> in this respect.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Fulkerson_Prize" title="Fulkerson Prize">Fulkerson Prize</a> is awarded for outstanding papers in discrete mathematics. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Topics">Topics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=1" title="Edit section: Topics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Outline_of_discrete_mathematics" title="Outline of discrete mathematics">Outline of discrete mathematics</a></div> <div class="mw-heading mw-heading3"><h3 id="Theoretical_computer_science">Theoretical computer science</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=2" title="Edit section: Theoretical computer science"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Sorting_quicksort_anim.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Sorting_quicksort_anim.gif/210px-Sorting_quicksort_anim.gif" decoding="async" width="210" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/6a/Sorting_quicksort_anim.gif 1.5x" data-file-width="280" data-file-height="214" /></a><figcaption><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Complexity</a> studies the time taken by <a href="/wiki/Algorithm" title="Algorithm">algorithms</a>, such as this <a href="/wiki/Quicksort" title="Quicksort">sorting routine</a>.</figcaption></figure> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:SimplexRangeSearching.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/SimplexRangeSearching.svg/150px-SimplexRangeSearching.svg.png" decoding="async" width="150" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/SimplexRangeSearching.svg/225px-SimplexRangeSearching.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/SimplexRangeSearching.svg/300px-SimplexRangeSearching.svg.png 2x" data-file-width="512" data-file-height="552" /></a><figcaption><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a> applies computer <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> to representations of <a href="/wiki/Geometry" title="Geometry">geometrical</a> objects.</figcaption></figure> <p>Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a> and <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>. Included within theoretical computer science is the study of algorithms and data structures. <a href="/wiki/Computability" title="Computability">Computability</a> studies what can be computed in principle, and has close ties to logic, while complexity studies the time, space, and other resources taken by computations. <a href="/wiki/Automata_theory" title="Automata theory">Automata theory</a> and <a href="/wiki/Formal_language" title="Formal language">formal language</a> theory are closely related to computability. <a href="/wiki/Petri_net" title="Petri net">Petri nets</a> and <a href="/wiki/Process_algebra" class="mw-redirect" title="Process algebra">process algebras</a> are used to model computer systems, and methods from discrete mathematics are used in analyzing <a href="/wiki/VLSI" class="mw-redirect" title="VLSI">VLSI</a> electronic circuits. <a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a> applies algorithms to geometrical problems and representations of <a href="/wiki/Geometry" title="Geometry">geometrical</a> objects, while <a href="/wiki/Computer_image_analysis" class="mw-redirect" title="Computer image analysis">computer image analysis</a> applies them to representations of images. Theoretical computer science also includes the study of various continuous computational topics. </p> <div class="mw-heading mw-heading3"><h3 id="Information_theory">Information theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=3" title="Edit section: Information theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Information_theory" title="Information theory">Information theory</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:WikipediaBinary.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/WikipediaBinary.svg/150px-WikipediaBinary.svg.png" decoding="async" width="150" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/WikipediaBinary.svg/225px-WikipediaBinary.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/WikipediaBinary.svg/300px-WikipediaBinary.svg.png 2x" data-file-width="88" data-file-height="106" /></a><figcaption>The <a href="/wiki/ASCII" title="ASCII">ASCII</a> codes for the word "Wikipedia", given here in <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a>, provide a way of representing the word in <a href="/wiki/Information_theory" title="Information theory">information theory</a>, as well as for information-processing <a href="/wiki/Algorithm" title="Algorithm">algorithms</a>.</figcaption></figure> <p>Information theory involves the quantification of <a href="/wiki/Information" title="Information">information</a>. Closely related is <a href="/wiki/Coding_theory" title="Coding theory">coding theory</a> which is used to design efficient and reliable data transmission and storage methods. Information theory also includes continuous topics such as: <a href="/wiki/Analog_signal" title="Analog signal">analog signals</a>, <a href="/wiki/Analog_coding" class="mw-redirect" title="Analog coding">analog coding</a>, <a href="/wiki/Analog_encryption" class="mw-redirect" title="Analog encryption">analog encryption</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Logic">Logic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=4" title="Edit section: Logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div> <p>Logic is the study of the principles of valid reasoning and <a href="/wiki/Inference" title="Inference">inference</a>, as well as of <a href="/wiki/Consistency" title="Consistency">consistency</a>, <a href="/wiki/Soundness" title="Soundness">soundness</a>, and <a href="/wiki/Completeness_(logic)" title="Completeness (logic)">completeness</a>. For example, in most systems of logic (but not in <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>) <a href="/wiki/Peirce%27s_law" title="Peirce's law">Peirce's law</a> (((<i>P</i>→<i>Q</i>)→<i>P</i>)→<i>P</i>) is a theorem. For classical logic, it can be easily verified with a <a href="/wiki/Truth_table" title="Truth table">truth table</a>. The study of <a href="/wiki/Mathematical_proof" title="Mathematical proof">mathematical proof</a> is particularly important in logic, and has accumulated to <a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">automated theorem proving</a> and <a href="/wiki/Formal_verification" title="Formal verification">formal verification</a> of software. </p><p><a href="/wiki/Well-formed_formula" title="Well-formed formula">Logical formulas</a> are discrete structures, as are <a href="/wiki/Proof_theory" title="Proof theory">proofs</a>, which form finite <a href="/wiki/Tree_structure" title="Tree structure">trees</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> or, more generally, <a href="/wiki/Directed_acyclic_graph" title="Directed acyclic graph">directed acyclic graph</a> structures<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> (with each <a href="/wiki/Rule_of_inference" title="Rule of inference">inference step</a> combining one or more <a href="/wiki/Premise" title="Premise">premise</a> branches to give a single conclusion). The <a href="/wiki/Truth_value" title="Truth value">truth values</a> of logical formulas usually form a finite set, generally restricted to two values: <i>true</i> and <i>false</i>, but logic can also be continuous-valued, e.g., <a href="/wiki/Fuzzy_logic" title="Fuzzy logic">fuzzy logic</a>. Concepts such as infinite proof trees or infinite derivation trees have also been studied,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> e.g. <a href="/wiki/Infinitary_logic" title="Infinitary logic">infinitary logic</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Set_theory">Set theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=5" title="Edit section: Set theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Set_theory" title="Set theory">Set theory</a></div> <p>Set theory is the branch of mathematics that studies <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a>, which are collections of objects, such as {blue, white, red} or the (infinite) set of all <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>. <a href="/wiki/Partially_ordered_set" title="Partially ordered set">Partially ordered sets</a> and sets with other <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relations</a> have applications in several areas. </p><p>In discrete mathematics, <a href="/wiki/Countable_set" title="Countable set">countable sets</a> (including <a href="/wiki/Finite_set" title="Finite set">finite sets</a>) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>'s work distinguishing between different kinds of <a href="/wiki/Infinite_set" title="Infinite set">infinite set</a>, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outside the scope of discrete mathematics. Indeed, contemporary work in <a href="/wiki/Descriptive_set_theory" title="Descriptive set theory">descriptive set theory</a> makes extensive use of traditional continuous mathematics. </p> <div class="mw-heading mw-heading3"><h3 id="Combinatorics">Combinatorics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=6" title="Edit section: Combinatorics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></div> <p>Combinatorics studies the ways in which discrete structures can be combined or arranged. <a href="/wiki/Enumerative_combinatorics" title="Enumerative combinatorics">Enumerative combinatorics</a> concentrates on counting the number of certain combinatorial objects - e.g. the <a href="/wiki/Twelvefold_way" title="Twelvefold way">twelvefold way</a> provides a unified framework for counting <a href="/wiki/Permutations" class="mw-redirect" title="Permutations">permutations</a>, <a href="/wiki/Combinations" class="mw-redirect" title="Combinations">combinations</a> and <a href="/wiki/Partition_of_a_set" title="Partition of a set">partitions</a>. <a href="/wiki/Analytic_combinatorics" title="Analytic combinatorics">Analytic combinatorics</a> concerns the enumeration (i.e., determining the number) of combinatorial structures using tools from <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> and <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>. In contrast with enumerative combinatorics which uses explicit combinatorial formulae and <a href="/wiki/Generating_functions" class="mw-redirect" title="Generating functions">generating functions</a> to describe the results, analytic combinatorics aims at obtaining <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">asymptotic formulae</a>. <a href="/wiki/Topological_combinatorics" title="Topological combinatorics">Topological combinatorics</a> concerns the use of techniques from <a href="/wiki/Topology" title="Topology">topology</a> and <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>/<a href="/wiki/Combinatorial_topology" title="Combinatorial topology">combinatorial topology</a> in <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>. Design theory is a study of <a href="/wiki/Combinatorial_design" title="Combinatorial design">combinatorial designs</a>, which are collections of subsets with certain <a href="/wiki/Set_intersection" class="mw-redirect" title="Set intersection">intersection</a> properties. <a href="/wiki/Partition_theory" class="mw-redirect" title="Partition theory">Partition theory</a> studies various enumeration and asymptotic problems related to <a href="/wiki/Integer_partition" title="Integer partition">integer partitions</a>, and is closely related to <a href="/wiki/Q-series" class="mw-redirect" title="Q-series">q-series</a>, <a href="/wiki/Special_functions" title="Special functions">special functions</a> and <a href="/wiki/Orthogonal_polynomials" title="Orthogonal polynomials">orthogonal polynomials</a>. Originally a part of <a href="/wiki/Number_theory" title="Number theory">number theory</a> and <a href="/wiki/Analysis" title="Analysis">analysis</a>, partition theory is now considered a part of combinatorics or an independent field. <a href="/wiki/Order_theory" title="Order theory">Order theory</a> is the study of <a href="/wiki/Partially_ordered_sets" class="mw-redirect" title="Partially ordered sets">partially ordered sets</a>, both finite and infinite. </p> <div class="mw-heading mw-heading3"><h3 id="Graph_theory">Graph theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=7" title="Edit section: Graph theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:TruncatedTetrahedron.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/TruncatedTetrahedron.gif/200px-TruncatedTetrahedron.gif" decoding="async" width="200" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/TruncatedTetrahedron.gif/300px-TruncatedTetrahedron.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/TruncatedTetrahedron.gif/400px-TruncatedTetrahedron.gif 2x" data-file-width="640" data-file-height="480" /></a><figcaption><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a> has close links to <a href="/wiki/Group_theory" title="Group theory">group theory</a>. This <a href="/wiki/Truncated_tetrahedron" title="Truncated tetrahedron">truncated tetrahedron</a> graph is related to the <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> <i>A</i><sub>4</sub>.</figcaption></figure> <p>Graph theory, the study of <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a> and <a href="/wiki/Network_theory" title="Network theory">networks</a>, is often considered part of combinatorics, but has grown large enough and distinct enough, with its own kind of problems, to be regarded as a subject in its own right.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Graphs are one of the prime objects of study in discrete mathematics. They are among the most ubiquitous models of both natural and human-made structures. They can model many types of relations and process dynamics in physical, biological and social systems. In computer science, they can represent networks of communication, data organization, computational devices, the flow of computation, etc. In mathematics, they are useful in geometry and certain parts of <a href="/wiki/Topology" title="Topology">topology</a>, e.g. <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a>. <a href="/wiki/Algebraic_graph_theory" title="Algebraic graph theory">Algebraic graph theory</a> has close links with group theory and <a href="/wiki/Topological_graph_theory" title="Topological graph theory">topological graph theory</a> has close links to <a href="/wiki/Topology" title="Topology">topology</a>. There are also <a href="/wiki/Continuous_graph" class="mw-redirect" title="Continuous graph">continuous graphs</a>; however, for the most part, research in graph theory falls within the domain of discrete mathematics. </p> <div class="mw-heading mw-heading3"><h3 id="Number_theory">Number theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=8" title="Edit section: Number theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Ulam_1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Ulam_1.png/200px-Ulam_1.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Ulam_1.png/300px-Ulam_1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/69/Ulam_1.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>The <a href="/wiki/Ulam_spiral" title="Ulam spiral">Ulam spiral</a> of numbers, with black pixels showing <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>. This diagram hints at patterns in the <a href="/wiki/Prime_number#Distribution" title="Prime number">distribution</a> of prime numbers.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Number_theory" title="Number theory">Number theory</a></div> <p>Number theory is concerned with the properties of numbers in general, particularly <a href="/wiki/Integer" title="Integer">integers</a>. It has applications to <a href="/wiki/Cryptography" title="Cryptography">cryptography</a> and <a href="/wiki/Cryptanalysis" title="Cryptanalysis">cryptanalysis</a>, particularly with regard to <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>, <a href="/wiki/Diophantine_equations" class="mw-redirect" title="Diophantine equations">diophantine equations</a>, linear and quadratic congruences, prime numbers and <a href="/wiki/Primality_test" title="Primality test">primality testing</a>. Other discrete aspects of number theory include <a href="/wiki/Geometry_of_numbers" title="Geometry of numbers">geometry of numbers</a>. In <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a>, techniques from continuous mathematics are also used. Topics that go beyond discrete objects include <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental numbers</a>, <a href="/wiki/Diophantine_approximation" title="Diophantine approximation">diophantine approximation</a>, <a href="/wiki/P-adic_analysis" title="P-adic analysis">p-adic analysis</a> and <a href="/wiki/Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function fields</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Algebraic_structures">Algebraic structures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=9" title="Edit section: Algebraic structures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract algebra</a></div> <p><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a> occur as both discrete examples and continuous examples. Discrete algebras include: <a href="/wiki/Boolean_algebra_(logic)" class="mw-redirect" title="Boolean algebra (logic)">Boolean algebra</a> used in <a href="/wiki/Logic_gate" title="Logic gate">logic gates</a> and programming; <a href="/wiki/Relational_algebra" title="Relational algebra">relational algebra</a> used in <a href="/wiki/Databases" class="mw-redirect" title="Databases">databases</a>; discrete and finite versions of <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a> and <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> are important in <a href="/wiki/Algebraic_coding_theory" class="mw-redirect" title="Algebraic coding theory">algebraic coding theory</a>; discrete <a href="/wiki/Semigroup" title="Semigroup">semigroups</a> and <a href="/wiki/Monoid" title="Monoid">monoids</a> appear in the theory of <a href="/wiki/Formal_languages" class="mw-redirect" title="Formal languages">formal languages</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Discrete_analogues_of_continuous_mathematics">Discrete analogues of continuous mathematics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=10" title="Edit section: Discrete analogues of continuous mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are many concepts and theories in continuous mathematics which have discrete versions, such as <a href="/wiki/Discrete_calculus" title="Discrete calculus">discrete calculus</a>, <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transforms</a>, <a href="/wiki/Discrete_geometry" title="Discrete geometry">discrete geometry</a>, <a href="/wiki/Discrete_logarithm" title="Discrete logarithm">discrete logarithms</a>, <a href="/wiki/Discrete_differential_geometry" title="Discrete differential geometry">discrete differential geometry</a>, <a href="/wiki/Discrete_exterior_calculus" title="Discrete exterior calculus">discrete exterior calculus</a>, <a href="/wiki/Discrete_Morse_theory" title="Discrete Morse theory">discrete Morse theory</a>, <a href="/wiki/Discrete_optimization" title="Discrete optimization">discrete optimization</a>, <a href="/wiki/Discrete_probability_theory" class="mw-redirect" title="Discrete probability theory">discrete probability theory</a>, <a href="/wiki/Discrete_probability_distribution" class="mw-redirect" title="Discrete probability distribution">discrete probability distribution</a>, <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">difference equations</a>, <a href="/wiki/Discrete_dynamical_system" class="mw-redirect" title="Discrete dynamical system">discrete dynamical systems</a>, and <a href="/wiki/Shapley%E2%80%93Folkman_lemma#Probability_and_measure_theory" title="Shapley–Folkman lemma">discrete vector measures</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Calculus_of_finite_differences,_discrete_analysis,_and_discrete_calculus"><span id="Calculus_of_finite_differences.2C_discrete_analysis.2C_and_discrete_calculus"></span>Calculus of finite differences, discrete analysis, and discrete calculus</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=11" title="Edit section: Calculus of finite differences, discrete analysis, and discrete calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Discrete_calculus" title="Discrete calculus">discrete calculus</a> and the <a href="/wiki/Calculus_of_finite_differences" class="mw-redirect" title="Calculus of finite differences">calculus of finite differences</a>, a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> defined on an interval of the <a href="/wiki/Integer" title="Integer">integers</a> is usually called a <a href="/wiki/Sequence" title="Sequence">sequence</a>. A sequence could be a finite sequence from a data source or an infinite sequence from a <a href="/wiki/Discrete_dynamical_system" class="mw-redirect" title="Discrete dynamical system">discrete dynamical system</a>. Such a discrete function could be defined explicitly by a list (if its domain is finite), or by a formula for its general term, or it could be given implicitly by a <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a> or <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">difference equation</a>. Difference equations are similar to <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>, but replace <a href="/wiki/Derivative" title="Derivative">differentiation</a> by taking the difference between adjacent terms; they can be used to approximate differential equations or (more often) studied in their own right. Many questions and methods concerning differential equations have counterparts for difference equations. For instance, where there are <a href="/wiki/Integral_transforms" class="mw-redirect" title="Integral transforms">integral transforms</a> in <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a> for studying continuous functions or analogue signals, there are <a href="/wiki/Discrete_transform" title="Discrete transform">discrete transforms</a> for discrete functions or digital signals. As well as <a href="/wiki/Discrete_metric_space" class="mw-redirect" title="Discrete metric space">discrete metric spaces</a>, there are more general <a href="/wiki/Discrete_topological_space" class="mw-redirect" title="Discrete topological space">discrete topological spaces</a>, <a href="/wiki/Finite_metric_space" class="mw-redirect" title="Finite metric space">finite metric spaces</a>, <a href="/wiki/Finite_topological_space" title="Finite topological space">finite topological spaces</a>. </p><p>The <a href="/wiki/Time_scale_calculus" class="mw-redirect" title="Time scale calculus">time scale calculus</a> is a unification of the theory of <a href="/wiki/Difference_equations" class="mw-redirect" title="Difference equations">difference equations</a> with that of <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a>, which has applications to fields requiring simultaneous modelling of discrete and continuous data. Another way of modeling such a situation is the notion of <a href="/wiki/Hybrid_system" title="Hybrid system">hybrid dynamical systems</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Discrete_geometry">Discrete geometry</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=12" title="Edit section: Discrete geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete geometry</a> and combinatorial geometry are about combinatorial properties of <i>discrete collections</i> of geometrical objects. A long-standing topic in discrete geometry is <a href="/wiki/Tessellation" title="Tessellation">tiling of the plane</a>. </p><p>In <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, the concept of a curve can be extended to discrete geometries by taking the <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectra</a> of <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial rings</a> over <a href="/wiki/Finite_field" title="Finite field">finite fields</a> to be models of the <a href="/wiki/Affine_space" title="Affine space">affine spaces</a> over that field, and letting <a href="/wiki/Algebraic_variety" title="Algebraic variety">subvarieties</a> or spectra of other rings provide the curves that lie in that space. Although the space in which the curves appear has a finite number of points, the curves are not so much sets of points as analogues of curves in continuous settings. For example, every point of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x-c)\subset \operatorname {Spec} K[x]=\mathbb {A} ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <mi>Spec</mi> <mo>⁡</mo> <mi>K</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(x-c)\subset \operatorname {Spec} K[x]=\mathbb {A} ^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/147df15a1780a002602cd26438adbec315699e2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.429ex; height:3.176ex;" alt="{\displaystyle V(x-c)\subset \operatorname {Spec} K[x]=\mathbb {A} ^{1}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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 K}"></span> a field can be studied either as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} K[x]/(x-c)\cong \operatorname {Spec} K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mi>K</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <mi>Spec</mi> <mo>⁡</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} K[x]/(x-c)\cong \operatorname {Spec} K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80cc3220b7c86e6d0862f1bcf3fbac3ffc0191a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.076ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spec} K[x]/(x-c)\cong \operatorname {Spec} K}"></span>, a point, or as the spectrum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} K[x]_{(x-c)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mi>K</mi> <mo stretchy="false">[</mo> <mi>x</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} K[x]_{(x-c)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76e6e66e203e1805ec5b90fa25d9f0c817f28dd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.169ex; height:3.176ex;" alt="{\displaystyle \operatorname {Spec} K[x]_{(x-c)}}"></span> of the <a href="/wiki/Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">local ring at (x-c)</a>, a point together with a neighborhood around it. Algebraic varieties also have a well-defined notion of <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> called the <a href="/wiki/Zariski_tangent_space" title="Zariski tangent space">Zariski tangent space</a>, making many features of calculus applicable even in finite settings. </p> <div class="mw-heading mw-heading4"><h4 id="Discrete_modelling">Discrete modelling</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=13" title="Edit section: Discrete modelling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>, <a href="/wiki/Discrete_modelling" title="Discrete modelling">discrete modelling</a> is the discrete analogue of <a href="/wiki/Continuous_modelling" title="Continuous modelling">continuous modelling</a>. In discrete modelling, discrete formulae are fit to <a href="/wiki/Data" title="Data">data</a>. A common method in this form of modelling is to use <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a>. <a href="/wiki/Discretization" title="Discretization">Discretization</a> concerns the process of transferring continuous models and equations into discrete counterparts, often for the purposes of making calculations easier by using approximations. <a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a> provides an important example. </p> <div class="mw-heading mw-heading2"><h2 id="Challenges">Challenges</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=14" title="Edit section: Challenges"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Four_Colour_Map_Example.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Four_Colour_Map_Example.svg/180px-Four_Colour_Map_Example.svg.png" decoding="async" width="180" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Four_Colour_Map_Example.svg/270px-Four_Colour_Map_Example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Four_Colour_Map_Example.svg/360px-Four_Colour_Map_Example.svg.png 2x" data-file-width="300" data-file-height="400" /></a><figcaption>Much research in <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a> was motivated by attempts to prove that all maps, like this one, can be <a href="/wiki/Graph_coloring" title="Graph coloring">colored</a> using <a href="/wiki/Four_color_theorem" title="Four color theorem">only four colors</a> so that no areas of the same color share an edge. <a href="/wiki/Kenneth_Appel" title="Kenneth Appel">Kenneth Appel</a> and <a href="/wiki/Wolfgang_Haken" title="Wolfgang Haken">Wolfgang Haken</a> proved this in 1976.<sup id="cite_ref-4colors_15-0" class="reference"><a href="#cite_note-4colors-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a>, first stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance).<sup id="cite_ref-4colors_15-1" class="reference"><a href="#cite_note-4colors-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Mathematical_logic" title="Mathematical logic">logic</a>, the <a href="/wiki/Hilbert%27s_second_problem" title="Hilbert's second problem">second problem</a> on <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>'s list of open <a href="/wiki/Hilbert%27s_problems" title="Hilbert's problems">problems</a> presented in 1900 was to prove that the <a href="/wiki/Axioms" class="mw-redirect" title="Axioms">axioms</a> of <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> are <a href="/wiki/Consistent" class="mw-redirect" title="Consistent">consistent</a>. <a href="/wiki/G%C3%B6del%27s_second_incompleteness_theorem" class="mw-redirect" title="Gödel's second incompleteness theorem">Gödel's second incompleteness theorem</a>, proved in 1931, showed that this was not possible – at least not within arithmetic itself. <a href="/wiki/Hilbert%27s_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a> was to determine whether a given polynomial <a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equation</a> with integer coefficients has an integer solution. In 1970, <a href="/wiki/Yuri_Matiyasevich" title="Yuri Matiyasevich">Yuri Matiyasevich</a> proved that this <a href="/wiki/Matiyasevich%27s_theorem" class="mw-redirect" title="Matiyasevich's theorem">could not be done</a>. </p><p>The need to <a href="/wiki/Cryptanalysis" title="Cryptanalysis">break</a> German codes in <a href="/wiki/World_War_II" title="World War II">World War II</a> led to advances in <a href="/wiki/Cryptography" title="Cryptography">cryptography</a> and <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a>, with the <a href="/wiki/Colossus_computer" title="Colossus computer">first programmable digital electronic computer</a> being developed at England's <a href="/wiki/Bletchley_Park" title="Bletchley Park">Bletchley Park</a> with the guidance of <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a> and his seminal work, On Computable Numbers.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Cold_War" title="Cold War">Cold War</a> meant that cryptography remained important, with fundamental advances such as <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a> being developed in the following decades. The <a href="/wiki/Telecommunications_industry" title="Telecommunications industry">telecommunications industry</a> has also motivated advances in discrete mathematics, particularly in graph theory and <a href="/wiki/Information_theory" title="Information theory">information theory</a>. <a href="/wiki/Formal_verification" title="Formal verification">Formal verification</a> of statements in logic has been necessary for <a href="/wiki/Software_development" title="Software development">software development</a> of <a href="/wiki/Safety-critical_system" title="Safety-critical system">safety-critical systems</a>, and advances in <a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">automated theorem proving</a> have been driven by this need. </p><p><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a> has been an important part of the <a href="/wiki/Computer_graphics_(computer_science)" title="Computer graphics (computer science)">computer graphics</a> incorporated into modern <a href="/wiki/Video_game" title="Video game">video games</a> and <a href="/wiki/Computer-aided_design" title="Computer-aided design">computer-aided design</a> tools. </p><p>Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, are important in addressing the challenging <a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a> problems associated with understanding the <a href="/wiki/Phylogenetic_tree" title="Phylogenetic tree">tree of life</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Currently, one of the most famous open problems in theoretical computer science is the <a href="/wiki/P_%3D_NP_problem" class="mw-redirect" title="P = NP problem">P = NP problem</a>, which involves the relationship between the <a href="/wiki/Complexity_class" title="Complexity class">complexity classes</a> <a href="/wiki/P_(complexity)" title="P (complexity)">P</a> and <a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a>. The <a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a> has offered a $1 million <a href="/wiki/USD" class="mw-redirect" title="USD">USD</a> prize for the first correct proof, along with prizes for <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">six other mathematical problems</a>.<sup id="cite_ref-CMI_Millennium_Prize_Problems_18-0" class="reference"><a href="#cite_note-CMI_Millennium_Prize_Problems-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><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/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-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/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <ul><li><a href="/wiki/Outline_of_discrete_mathematics" title="Outline of discrete mathematics">Outline of discrete mathematics</a></li> <li><a href="/wiki/Cyberchase" title="Cyberchase">Cyberchase</a>, a show that teaches discrete mathematics to children</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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 mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="/wiki/Richard_Johnsonbaugh" title="Richard Johnsonbaugh">Richard Johnsonbaugh</a>, <i>Discrete Mathematics</i>, Prentice Hall, 2008.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><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 id="CITEREFFranklin2017" class="citation journal cs1"><a href="/wiki/James_Franklin_(philosopher)" title="James Franklin (philosopher)">Franklin, James</a> (2017). <a rel="nofollow" class="external text" href="https://scholarship.claremont.edu/jhm/vol7/iss2/18/">"Discrete and continuous: a fundamental dichotomy in mathematics"</a>. <i>Journal of Humanistic Mathematics</i>. <b>7</b> (2): 355–378. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.5642%2Fjhummath.201702.18">10.5642/jhummath.201702.18</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6945363">6945363</a><span class="reference-accessdate">. Retrieved <span class="nowrap">30 June</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Humanistic+Mathematics&rft.atitle=Discrete+and+continuous%3A+a+fundamental+dichotomy+in+mathematics&rft.volume=7&rft.issue=2&rft.pages=355-378&rft.date=2017&rft_id=info%3Adoi%2F10.5642%2Fjhummath.201702.18&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6945363%23id-name%3DS2CID&rft.aulast=Franklin&rft.aufirst=James&rft_id=https%3A%2F%2Fscholarship.claremont.edu%2Fjhm%2Fvol7%2Fiss2%2F18%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://cse.buffalo.edu/~rapaport/191/S09/whatisdiscmath.html">"Discrete Structures: What is Discrete Math?"</a>. <i>cse.buffalo.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">16 November</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=cse.buffalo.edu&rft.atitle=Discrete+Structures%3A+What+is+Discrete+Math%3F&rft_id=https%3A%2F%2Fcse.buffalo.edu%2F~rapaport%2F191%2FS09%2Fwhatisdiscmath.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBiggs2002" class="citation cs2"><a href="/wiki/Norman_L._Biggs" title="Norman L. Biggs">Biggs, Norman L.</a> (2002), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Mj9gzZMrXDIC&pg=PA89"><i>Discrete mathematics</i></a>, Oxford Science Publications (2nd ed.), The Clarendon Press Oxford University Press, p. 89, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198507178" title="Special:BookSources/9780198507178"><bdi>9780198507178</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1078626">1078626</a>, <q>Discrete Mathematics is the branch of Mathematics in which we deal with questions involving finite or countably infinite sets.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+mathematics&rft.series=Oxford+Science+Publications&rft.pages=89&rft.edition=2nd&rft.pub=The+Clarendon+Press+Oxford+University+Press&rft.date=2002&rft.isbn=9780198507178&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1078626%23id-name%3DMR&rft.aulast=Biggs&rft.aufirst=Norman+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMj9gzZMrXDIC%26pg%3DPA89&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHopkins2009" class="citation book cs1">Hopkins, Brian, ed. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=05DEJ8Kh67AC&pg=PR11"><i>Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles</i></a>. Mathematical Association of America. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-184-5" title="Special:BookSources/978-0-88385-184-5"><bdi>978-0-88385-184-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Resources+for+Teaching+Discrete+Mathematics%3A+Classroom+Projects%2C+History+Modules%2C+and+Articles&rft.pub=Mathematical+Association+of+America&rft.date=2009&rft.isbn=978-0-88385-184-5&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D05DEJ8Kh67AC%26pg%3DPR11&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-LevasseurDoerr-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-LevasseurDoerr_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevasseurDoerr" class="citation book cs1">Levasseur, Ken; Doerr, Al. <a rel="nofollow" class="external text" href="https://discretemath.org/ads/index-ads.html"><i>Applied Discrete Structures</i></a>. p. 8.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Discrete+Structures&rft.pages=8&rft.aulast=Levasseur&rft.aufirst=Ken&rft.au=Doerr%2C+Al&rft_id=https%3A%2F%2Fdiscretemath.org%2Fads%2Findex-ads.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-Howson1988-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Howson1988_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeoffrey_Howson1988" class="citation book cs1">Geoffrey Howson, Albert, ed. (1988). <i>Mathematics as a Service Subject</i>. Cambridge University Press. pp. 77–78. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-35395-3" title="Special:BookSources/978-0-521-35395-3"><bdi>978-0-521-35395-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+as+a+Service+Subject&rft.pages=77-78&rft.pub=Cambridge+University+Press&rft.date=1988&rft.isbn=978-0-521-35395-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-Rosenstein-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rosenstein_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosenstein" class="citation book cs1">Rosenstein, Joseph G. <i>Discrete Mathematics in the Schools</i>. American Mathematical Society. p. 323. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-8578-9" title="Special:BookSources/978-0-8218-8578-9"><bdi>978-0-8218-8578-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics+in+the+Schools&rft.pages=323&rft.pub=American+Mathematical+Society&rft.isbn=978-0-8218-8578-9&rft.aulast=Rosenstein&rft.aufirst=Joseph+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://ucsmp.uchicago.edu/secondary/curriculum/precalculus-discrete/">"UCSMP"</a>. <i>uchicago.edu</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=uchicago.edu&rft.atitle=UCSMP&rft_id=http%3A%2F%2Fucsmp.uchicago.edu%2Fsecondary%2Fcurriculum%2Fprecalculus-discrete%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTroelstraSchwichtenberg2000" class="citation book cs1">Troelstra, A.S.; Schwichtenberg, H. (2000-07-27). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x9x6F_4mUPgC&pg=PA186"><i>Basic Proof Theory</i></a>. Cambridge University Press. p. 186. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-77911-1" title="Special:BookSources/978-0-521-77911-1"><bdi>978-0-521-77911-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Proof+Theory&rft.pages=186&rft.pub=Cambridge+University+Press&rft.date=2000-07-27&rft.isbn=978-0-521-77911-1&rft.aulast=Troelstra&rft.aufirst=A.S.&rft.au=Schwichtenberg%2C+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dx9x6F_4mUPgC%26pg%3DPA186&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuss1998" class="citation book cs1">Buss, Samuel R. (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MfTMDeCq7ukC&pg=PA13"><i>Handbook of Proof Theory</i></a>. Elsevier. p. 13. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-89840-1" title="Special:BookSources/978-0-444-89840-1"><bdi>978-0-444-89840-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Proof+Theory&rft.pages=13&rft.pub=Elsevier&rft.date=1998&rft.isbn=978-0-444-89840-1&rft.aulast=Buss&rft.aufirst=Samuel+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMfTMDeCq7ukC%26pg%3DPA13&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaaderBrewkaEiter2001" class="citation book cs1">Baader, Franz; Brewka, Gerhard; Eiter, Thomas (2001-10-16). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=27A2XJPYwIkC&pg=PA325"><i>KI 2001: Advances in Artificial Intelligence: Joint German/Austrian Conference on AI, Vienna, Austria, September 19-21, 2001. Proceedings</i></a>. Springer. p. 325. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-42612-7" title="Special:BookSources/978-3-540-42612-7"><bdi>978-3-540-42612-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=KI+2001%3A+Advances+in+Artificial+Intelligence%3A+Joint+German%2FAustrian+Conference+on+AI%2C+Vienna%2C+Austria%2C+September+19-21%2C+2001.+Proceedings&rft.pages=325&rft.pub=Springer&rft.date=2001-10-16&rft.isbn=978-3-540-42612-7&rft.aulast=Baader&rft.aufirst=Franz&rft.au=Brewka%2C+Gerhard&rft.au=Eiter%2C+Thomas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D27A2XJPYwIkC%26pg%3DPA325&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrotherstonBornatCalcagno2008" class="citation journal cs1">Brotherston, J.; Bornat, R.; Calcagno, C. (January 2008). "Cyclic proofs of program termination in separation logic". <i>ACM SIGPLAN Notices</i>. <b>43</b> (1): 101–112. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1328897.1328453">10.1145/1328897.1328453</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+SIGPLAN+Notices&rft.atitle=Cyclic+proofs+of+program+termination+in+separation+logic&rft.volume=43&rft.issue=1&rft.pages=101-112&rft.date=2008-01&rft_id=info%3Adoi%2F10.1145%2F1328897.1328453&rft.aulast=Brotherston&rft.aufirst=J.&rft.au=Bornat%2C+R.&rft.au=Calcagno%2C+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoharThomassen2001" class="citation book cs1"><a href="/wiki/Bojan_Mohar" title="Bojan Mohar">Mohar, Bojan</a>; <a href="/wiki/Carsten_Thomassen_(mathematician)" title="Carsten Thomassen (mathematician)">Thomassen, Carsten</a> (2001). <a rel="nofollow" class="external text" href="https://www.press.jhu.edu/books/title/1675/graphs-surfaces"><i>Graphs on Surfaces</i></a>. Johns Hopkins University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-6689-0" title="Special:BookSources/978-0-8018-6689-0"><bdi>978-0-8018-6689-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/45102952">45102952</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Graphs+on+Surfaces&rft.pub=Johns+Hopkins+University+Press&rft.date=2001&rft_id=info%3Aoclcnum%2F45102952&rft.isbn=978-0-8018-6689-0&rft.aulast=Mohar&rft.aufirst=Bojan&rft.au=Thomassen%2C+Carsten&rft_id=https%3A%2F%2Fwww.press.jhu.edu%2Fbooks%2Ftitle%2F1675%2Fgraphs-surfaces&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-4colors-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-4colors_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-4colors_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilson2002" class="citation book cs1"><a href="/wiki/Robin_Wilson_(mathematician)" title="Robin Wilson (mathematician)">Wilson, Robin</a> (2002). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/fourcolorssuffic00wils"><i>Four Colors Suffice</i></a></span>. London: Penguin Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11533-7" title="Special:BookSources/978-0-691-11533-7"><bdi>978-0-691-11533-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Four+Colors+Suffice&rft.place=London&rft.pub=Penguin+Books&rft.date=2002&rft.isbn=978-0-691-11533-7&rft.aulast=Wilson&rft.aufirst=Robin&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffourcolorssuffic00wils&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodges1992" class="citation book cs1"><a href="/wiki/Andrew_Hodges" title="Andrew Hodges">Hodges, Andrew</a> (1992). <i><a href="/wiki/Alan_Turing:_The_Enigma" title="Alan Turing: The Enigma">Alan Turing: The Enigma</a></i>. <a href="/wiki/Random_House" title="Random House">Random House</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alan+Turing%3A+The+Enigma&rft.pub=Random+House&rft.date=1992&rft.aulast=Hodges&rft.aufirst=Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodkinsonParnell2007" class="citation book cs1">Hodkinson, Trevor R.; Parnell, John A. N. (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7GKkbJ4yOKAC&pg=PA97"><i>Reconstruction the Tree of Life: Taxonomy And Systematics of Large And Species Rich Taxa</i></a>. CRC Press. p. 97. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-9579-6" title="Special:BookSources/978-0-8493-9579-6"><bdi>978-0-8493-9579-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reconstruction+the+Tree+of+Life%3A+Taxonomy+And+Systematics+of+Large+And+Species+Rich+Taxa&rft.pages=97&rft.pub=CRC+Press&rft.date=2007&rft.isbn=978-0-8493-9579-6&rft.aulast=Hodkinson&rft.aufirst=Trevor+R.&rft.au=Parnell%2C+John+A.+N.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7GKkbJ4yOKAC%26pg%3DPA97&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> <li id="cite_note-CMI_Millennium_Prize_Problems-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-CMI_Millennium_Prize_Problems_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.claymath.org/millennium/">"Millennium Prize Problems"</a>. 2000-05-24<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-01-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Millennium+Prize+Problems&rft.date=2000-05-24&rft_id=http%3A%2F%2Fwww.claymath.org%2Fmillennium%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=17" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBiggs2002" class="citation book cs1"><a href="/wiki/Norman_L._Biggs" title="Norman L. Biggs">Biggs, Norman L.</a> (2002). <i>Discrete Mathematics</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850717-8" title="Special:BookSources/978-0-19-850717-8"><bdi>978-0-19-850717-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics&rft.pub=Oxford+University+Press&rft.date=2002&rft.isbn=978-0-19-850717-8&rft.aulast=Biggs&rft.aufirst=Norman+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDwyer2010" class="citation book cs1">Dwyer, John (2010). <i>An Introduction to Discrete Mathematics for Business & Computing</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-907934-00-1" title="Special:BookSources/978-1-907934-00-1"><bdi>978-1-907934-00-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Discrete+Mathematics+for+Business+%26+Computing&rft.date=2010&rft.isbn=978-1-907934-00-1&rft.aulast=Dwyer&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEpp2010" class="citation book cs1"><a href="/wiki/Susanna_S._Epp" title="Susanna S. Epp">Epp, Susanna S.</a> (2010-08-04). <i>Discrete Mathematics With Applications</i>. Thomson Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-39132-6" title="Special:BookSources/978-0-495-39132-6"><bdi>978-0-495-39132-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics+With+Applications&rft.pub=Thomson+Brooks%2FCole&rft.date=2010-08-04&rft.isbn=978-0-495-39132-6&rft.aulast=Epp&rft.aufirst=Susanna+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrahamKnuthPatashnik1994" class="citation book cs1"><a href="/wiki/Ronald_Graham" title="Ronald Graham">Graham, Ronald</a>; <a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a>; <a href="/wiki/Oren_Patashnik" title="Oren Patashnik">Patashnik, Oren</a> (1994). <a href="/wiki/Concrete_Mathematics" title="Concrete Mathematics"><i>Concrete Mathematics</i></a> (2nd ed.). Addison–Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-55802-5" title="Special:BookSources/0-201-55802-5"><bdi>0-201-55802-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concrete+Mathematics&rft.edition=2nd&rft.pub=Addison%E2%80%93Wesley&rft.date=1994&rft.isbn=0-201-55802-5&rft.aulast=Graham&rft.aufirst=Ronald&rft.au=Knuth%2C+Donald+E.&rft.au=Patashnik%2C+Oren&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrimaldi2004" class="citation book cs1"><a href="/wiki/Ralph_P._Grimaldi" class="mw-redirect" title="Ralph P. Grimaldi">Grimaldi, Ralph P.</a> (2004). <i>Discrete and Combinatorial Mathematics: An Applied Introduction</i>. Addison Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-72634-3" title="Special:BookSources/978-0-201-72634-3"><bdi>978-0-201-72634-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+and+Combinatorial+Mathematics%3A+An+Applied+Introduction&rft.pub=Addison+Wesley&rft.date=2004&rft.isbn=978-0-201-72634-3&rft.aulast=Grimaldi&rft.aufirst=Ralph+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth2011" class="citation book cs1 cs1-prop-long-vol">Knuth, Donald E. (2011). <a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming"><i>The Art of Computer Programming</i></a>. Vol. 1–4a Boxed Set. Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-75104-1" title="Special:BookSources/978-0-321-75104-1"><bdi>978-0-321-75104-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Art+of+Computer+Programming&rft.pub=Addison-Wesley&rft.date=2011&rft.isbn=978-0-321-75104-1&rft.aulast=Knuth&rft.aufirst=Donald+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatoušekNešetřil1998" class="citation book cs1"><a href="/wiki/Ji%C5%99%C3%AD_Matou%C5%A1ek_(mathematician)" title="Jiří Matoušek (mathematician)">Matoušek, Jiří</a>; <a href="/wiki/Jaroslav_Ne%C5%A1et%C5%99il" title="Jaroslav Nešetřil">Nešetřil, Jaroslav</a> (1998). <i>Discrete Mathematics</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850208-1" title="Special:BookSources/978-0-19-850208-1"><bdi>978-0-19-850208-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics&rft.pub=Oxford+University+Press&rft.date=1998&rft.isbn=978-0-19-850208-1&rft.aulast=Matou%C5%A1ek&rft.aufirst=Ji%C5%99%C3%AD&rft.au=Ne%C5%A1et%C5%99il%2C+Jaroslav&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFObrenic2003" class="citation book cs1">Obrenic, Bojana (2003). <i>Practice Problems in Discrete Mathematics</i>. Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-045803-2" title="Special:BookSources/978-0-13-045803-2"><bdi>978-0-13-045803-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Practice+Problems+in+Discrete+Mathematics&rft.pub=Prentice+Hall&rft.date=2003&rft.isbn=978-0-13-045803-2&rft.aulast=Obrenic&rft.aufirst=Bojana&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosenMichaels2000" class="citation book cs1">Rosen, Kenneth H.; Michaels, John G. (2000). <i>Hand Book of Discrete and Combinatorial Mathematics</i>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-0149-0" title="Special:BookSources/978-0-8493-0149-0"><bdi>978-0-8493-0149-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hand+Book+of+Discrete+and+Combinatorial+Mathematics&rft.pub=CRC+Press&rft.date=2000&rft.isbn=978-0-8493-0149-0&rft.aulast=Rosen&rft.aufirst=Kenneth+H.&rft.au=Michaels%2C+John+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosen2007" class="citation book cs1">Rosen, Kenneth H. (2007). <i>Discrete Mathematics: And Its Applications</i>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-288008-3" title="Special:BookSources/978-0-07-288008-3"><bdi>978-0-07-288008-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics%3A+And+Its+Applications&rft.pub=McGraw-Hill&rft.date=2007&rft.isbn=978-0-07-288008-3&rft.aulast=Rosen&rft.aufirst=Kenneth+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimpson2002" class="citation book cs1"><a href="/wiki/Andrew_Clive_Simpson" title="Andrew Clive Simpson">Simpson, Andrew</a> (2002). <i>Discrete Mathematics by Example</i>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-709840-7" title="Special:BookSources/978-0-07-709840-7"><bdi>978-0-07-709840-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics+by+Example&rft.pub=McGraw-Hill&rft.date=2002&rft.isbn=978-0-07-709840-7&rft.aulast=Simpson&rft.aufirst=Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+mathematics" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_mathematics&action=edit&section=18" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></span></span></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Discrete_Mathematics" class="extiw" title="wikibooks:Discrete Mathematics">Discrete Mathematics</a></b></i></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Discrete_mathematics" class="extiw" title="commons:Category:Discrete mathematics">Discrete mathematics</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://archives.math.utk.edu/topics/discreteMath.html">Discrete mathematics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110829184228/http://archives.math.utk.edu/topics/discreteMath.html">Archived</a> 2011-08-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> at the utk.edu Mathematics Archives, providing links to syllabi, tutorials, programs, etc.</li> <li><a rel="nofollow" class="external text" href="http://www.iowacentral.edu/industrial_technology/electrical_technologies/index.asp">Iowa Central: Electrical Technologies Program</a> Discrete mathematics for <a href="/wiki/Electrical_engineering" title="Electrical engineering">Electrical engineering</a>.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><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"><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: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 href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">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 class="mw-selflink selflink">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><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><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></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Category"><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" /></span></span> <b><a href="/wiki/Category:Fields_of_mathematics" title="Category:Fields of mathematics">Category</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><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" /></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw" title="commons:Category:Mathematics">Commons</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="WikiProject"><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" /></span></span> <b><a href="/wiki/Wikipedia:WikiProject_Mathematics" title="Wikipedia:WikiProject Mathematics">WikiProject</a></b></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="Industrial_and_applied_mathematics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed 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:Industrial_and_applied_mathematics" title="Template:Industrial and applied mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Industrial_and_applied_mathematics" title="Template talk:Industrial and applied mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Industrial_and_applied_mathematics" title="Special:EditPage/Template:Industrial and applied mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Industrial_and_applied_mathematics" style="font-size:114%;margin:0 4em"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Industrial and applied mathematics</a></div></th></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/Algorithm" title="Algorithm">Algorithms</a> <ul><li><a href="/wiki/Algorithm_design" class="mw-redirect" title="Algorithm design">design</a></li> <li><a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">analysis</a></li></ul></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata theory</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Coding_theory" title="Coding theory">Coding theory</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a></li> <li><a href="/wiki/Constraint_satisfaction_problem" title="Constraint satisfaction problem">Constraint satisfaction</a> <ul><li><a href="/wiki/Constraint_programming" title="Constraint programming">Constraint programming</a></li></ul></li> <li><a href="/wiki/Logic_in_computer_science" title="Logic in computer science">Computational logic</a></li> <li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Computational_statistics" title="Computational statistics">Statistics</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Mathematical_software" scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Mathematical_software" title="Mathematical software">Mathematical software</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/List_of_arbitrary-precision_arithmetic_software" title="List of arbitrary-precision arithmetic software">Arbitrary-precision arithmetic</a></li> <li><a href="/wiki/List_of_finite_element_software_packages" title="List of finite element software packages">Finite element analysis</a></li> <li><a href="/wiki/Tensor_software" title="Tensor software">Tensor software</a></li> <li><a href="/wiki/List_of_interactive_geometry_software" title="List of interactive geometry software">Interactive geometry software</a></li> <li><a href="/wiki/List_of_optimization_software" title="List of optimization software">Optimization software</a></li> <li><a href="/wiki/List_of_statistical_software" title="List of statistical software">Statistical software</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical-analysis software</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical libraries</a></li> <li><a href="/wiki/Solver" title="Solver">Solvers</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Discrete</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_algebra" title="Computer algebra">Computer algebra</a></li> <li><a href="/wiki/Computational_number_theory" title="Computational number theory">Computational number theory</a></li> <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/Discrete_geometry" title="Discrete geometry">Discrete geometry</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-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_theory" title="Approximation theory">Approximation theory</a></li> <li><a href="/wiki/Clifford_analysis" title="Clifford analysis">Clifford analysis</a> <ul><li><a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equations</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equations</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equations</a></li></ul></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a> <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/Geometric_analysis" title="Geometric analysis">Geometric analysis</a></li></ul></li> <li><a href="/wiki/Dynamical_system" title="Dynamical system">Dynamical systems</a> <ul><li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li></ul></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> <ul><li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Operator_theory" title="Operator theory">Operator theory</a></li></ul></li> <li><a href="/wiki/Harmonic_analysis_(mathematics)" class="mw-redirect" title="Harmonic analysis (mathematics)">Harmonic analysis</a> <ul><li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li></ul></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a> <ul><li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Vector_calculus#Vector_algebra" title="Vector calculus">Vector</a></li></ul></li> <li><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a> <ul><li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li> <li><a href="/wiki/Vector_calculus" title="Vector calculus">Vector</a></li></ul></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a> <ul><li><a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">Numerical linear algebra</a></li> <li><a href="/wiki/Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></li> <li><a href="/wiki/Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Variational calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</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/Probability_distribution" title="Probability distribution">Distributions</a> (<a href="/wiki/Random_variable" title="Random variable">random variables</a>)</li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic processes</a> / <a href="/wiki/Stochastic_calculus" title="Stochastic calculus">analysis</a></li> <li><a href="/wiki/Functional_integration" title="Functional integration">Path integral</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Stochastic variational calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical<br />physics</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/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a> <ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a></li> <li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a></li></ul></li> <li><a href="/wiki/Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">Field theory</a> <ul><li><a href="/wiki/Classical_field_theory" title="Classical field theory">Classical</a></li> <li><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal</a></li> <li><a href="/wiki/Effective_field_theory" title="Effective field theory">Effective</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum</a></li> <li><a href="/wiki/Statistical_field_theory" title="Statistical field theory">Statistical</a></li> <li><a href="/wiki/Topological_field_theory" class="mw-redirect" title="Topological field theory">Topological</a></li></ul></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a> <ul><li><a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">in quantum mechanics</a></li></ul></li> <li><a href="/wiki/Potential_theory" title="Potential theory">Potential theory</a></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a> <ul><li><a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic</a></li> <li><a href="/wiki/Topological_string_theory" title="Topological string theory">Topological</a></li></ul></li> <li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a> <ul><li><a href="/wiki/Supersymmetric_quantum_mechanics" title="Supersymmetric quantum mechanics">Supersymmetric quantum mechanics</a></li> <li><a href="/wiki/Supersymmetric_theory_of_stochastic_dynamics" title="Supersymmetric theory of stochastic dynamics">Supersymmetric theory of stochastic dynamics</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Algebraic_structures" scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Algebraic_structures" class="mw-redirect" title="Algebraic structures">Algebraic structures</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/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Feynman integral</a></li> <li><a href="/wiki/Poisson_algebra" title="Poisson algebra">Poisson algebra</a></li> <li><a href="/wiki/Quantum_group" title="Quantum group">Quantum group</a></li> <li><a href="/wiki/Renormalization_group" title="Renormalization group">Renormalization group</a></li> <li><a href="/wiki/Particle_physics_and_representation_theory" title="Particle physics and representation theory">Representation theory</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a href="/wiki/Superalgebra" title="Superalgebra">Superalgebra</a></li> <li><a href="/wiki/Supersymmetry_algebra" title="Supersymmetry algebra">Supersymmetry algebra</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Decision_theory" title="Decision theory">Decision sciences</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/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice theory</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</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></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other applications</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_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Sociology</a></li> <li>"<a href="/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" title="The Unreasonable Effectiveness of Mathematics in the Natural Sciences">The Unreasonable Effectiveness of Mathematics in the Natural Sciences</a>"</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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/Mathematics" title="Mathematics">Mathematics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</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/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a> <ul><li><a href="/wiki/Japan_Society_for_Industrial_and_Applied_Mathematics" title="Japan Society for Industrial and Applied Mathematics">Japan Society for Industrial and Applied Mathematics</a></li></ul></li> <li><a href="/wiki/Soci%C3%A9t%C3%A9_de_Math%C3%A9matiques_Appliqu%C3%A9es_et_Industrielles" title="Société de Mathématiques Appliquées et Industrielles">Société de Mathématiques Appliquées et Industrielles</a></li> <li><a href="/wiki/International_Council_for_Industrial_and_Applied_Mathematics" title="International Council for Industrial and Applied Mathematics">International Council for Industrial and Applied Mathematics</a></li> <li><a href="/w/index.php?title=European_Community_on_Computational_Methods_in_Applied_Sciences&action=edit&redlink=1" class="new" title="European Community on Computational Methods in Applied Sciences (page does not exist)">European Community on Computational Methods in Applied Sciences</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><a href="/wiki/Category:Mathematics" title="Category:Mathematics">Category</a></b></li> <li><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a> / <a href="/wiki/Topic_outline_of_mathematics" class="mw-redirect" title="Topic outline of mathematics">outline</a> / <a href="/wiki/List_of_mathematics_topics" class="mw-redirect" title="List of mathematics topics">topics list</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"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q121416#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" 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"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q121416#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="font-size:114%;margin:0 4em"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q121416#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">International</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Discrete mathematics"><a rel="nofollow" class="external text" href="http://id.worldcat.org/fast/2009030/">FAST</a></span></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4129143-8">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Discrete mathematics"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh2019000551">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="離散数学"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/001333819">Japan</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="diskrétní matematika"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph119484&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007538304705171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="http://esu.com.ua/search_articles.php?id=24371">Encyclopedia of Modern Ukraine</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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=Discrete_mathematics&oldid=1246834323">https://en.wikipedia.org/w/index.php?title=Discrete_mathematics&oldid=1246834323</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">Category</a>: <ul><li><a href="/wiki/Category:Discrete_mathematics" title="Category:Discrete mathematics">Discrete 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_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Pages_using_sidebar_with_the_child_parameter" title="Category:Pages using sidebar with the child parameter">Pages using sidebar with the child parameter</a></li><li><a href="/wiki/Category:CS1:_long_volume_value" title="Category:CS1: long volume value">CS1: long volume value</a></li><li><a href="/wiki/Category:Commons_category_link_from_Wikidata" title="Category:Commons category link from Wikidata">Commons category link from Wikidata</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</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 21 September 2024, at 10:43<span class="anonymous-show"> (UTC)</span>.</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=Discrete_mathematics&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-669b4ddb54-2jzm8","wgBackendResponseTime":192,"wgPageParseReport":{"limitreport":{"cputime":"0.670","walltime":"0.893","ppvisitednodes":{"value":3012,"limit":1000000},"postexpandincludesize":{"value":120724,"limit":2097152},"templateargumentsize":{"value":2016,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":14,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":130921,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 696.647 1 -total"," 28.71% 199.981 1 Template:Reflist"," 15.30% 106.580 22 Template:Cite_book"," 13.74% 95.702 1 Template:Math_topics_TOC"," 13.53% 94.260 2 Template:Cite_journal"," 13.11% 91.348 1 Template:Sidebar"," 10.57% 73.656 2 Template:Sister_project"," 10.24% 71.354 2 Template:Side_box"," 9.10% 63.429 1 Template:Short_description"," 9.06% 63.091 1 Template:Sidebar_with_collapsible_lists"]},"scribunto":{"limitreport-timeusage":{"value":"0.416","limit":"10.000"},"limitreport-memusage":{"value":6450568,"limit":52428800},"limitreport-logs":"table#1 {\n [\"size\"] = \"tiny\",\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-669b4ddb54-k4q8d","timestamp":"20241127101204","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Discrete mathematics","url":"https:\/\/en.wikipedia.org\/wiki\/Discrete_mathematics","sameAs":"http:\/\/www.wikidata.org\/entity\/Q121416","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q121416","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":"2001-09-23T02:30:52Z","dateModified":"2024-09-21T10:43:12Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/5\/5b\/6n-graf.svg","headline":"study of discrete mathematical structures"}</script> </body> </html>