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-disabled skin-theme-clientpref-day vector-toc-available" lang="ast" dir="ltr"> <head> <meta charset="UTF-8"> <title>Operación binaria - 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-disabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )astwikimwclientpreferences=([^;]+)/);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":[",\t."," \t,"],"wgDigitTransformTable":["",""], "wgDefaultDateFormat":"dmy","wgMonthNames":["","xineru","febreru","marzu","abril","mayu","xunu","xunetu","agostu","setiembre","ochobre","payares","avientu"],"wgRequestId":"4e9ad685-238f-4455-bf60-ddacd72389f7","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Operación_binaria","wgTitle":"Operación binaria","wgCurRevisionId":4121027,"wgRevisionId":4121027,"wgArticleId":190731,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Wikipedia:Páxines con etiquetes de Wikidata ensin traducir","Wikipedia:Artículos con plantíes de notes d'encabezamientu enllaciando a páxines que nun esisten","Álxebra elemental","Álxebra astrauta"],"wgPageViewLanguage":"ast","wgPageContentLanguage":"ast","wgPageContentModel":"wikitext","wgRelevantPageName":"Operación_binaria","wgRelevantArticleId":190731,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[], "wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"ast","pageLanguageDir":"ltr","pageVariantFallbacks":"ast"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":10000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q164307","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","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","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns", "ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=ast&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.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=ast&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=ast&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/7/79/Binary_operations_as_black_box.svg/1200px-Binary_operations_as_black_box.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1200"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/7/79/Binary_operations_as_black_box.svg/800px-Binary_operations_as_black_box.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="800"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/7/79/Binary_operations_as_black_box.svg/640px-Binary_operations_as_black_box.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="640"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Operación binaria - 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="//ast.m.wikipedia.org/wiki/Operaci%C3%B3n_binaria"> <link rel="alternate" type="application/x-wiki" title="Editar" href="/w/index.php?title=Operaci%C3%B3n_binaria&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 (ast)"> <link rel="EditURI" type="application/rsd+xml" href="//ast.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://ast.wikipedia.org/wiki/Operaci%C3%B3n_binaria"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.ast"> <link rel="alternate" type="application/atom+xml" title="Canal Atom Wikipedia" href="/w/index.php?title=Especial:CambeosRecientes&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-Operación_binaria rootpage-Operación_binaria skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Saltar al conteníu</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="Sitio"> <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="Menú principal" > <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">Menú principal</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">Menú principal</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">despintar</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navegación </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Portada" title="Visita la portada [z]" accesskey="z"><span>Portada</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Portal_de_la_comunid%C3%A1" title="Tocante al proyeutu, lo qué pues facer, ú s'alcuentren les coses"><span>Portal de la comunidá</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Especial:CambeosRecientes" title="La llista de cambios recientes de la wiki. [r]" accesskey="r"><span>Cambeos recién</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Especial:Aleatoria" title="Carga una páxina al debalu [x]" accesskey="x"><span>Páxina al debalu</span></a></li><li id="n-help" class="mw-list-item"><a href="https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Contents" title="El llugar pa deprender"><span>Ayuda</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Portada" 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="La Enciclopedia Llibre" src="/static/images/mobile/copyright/wikipedia-tagline-ast.svg" width="119" height="13" style="width: 7.4375em; 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/Especial:Gueta" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Busca en Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Buscar</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="Buscar en Wikipedia" aria-label="Buscar en Wikipedia" autocapitalize="sentences" title="Busca en 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="Especial:Gueta"> </div> <button class="cdx-button cdx-search-input__end-button">Guetar</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Ferramientes personales"> <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="Apariencia"> <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="Apariencia" > <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">Apariencia</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/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=ast.wikipedia.org&uselang=ast" class=""><span>Donativos</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=Especial:Crear_una_cuenta&returnto=Operaci%C3%B3n+binaria" title="Encamentámoste que crees una cuenta y qu'anicies sesión; sicasí, nun ye obligatorio" class=""><span>Crear una cuenta</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=Especial:Entrar&returnto=Operaci%C3%B3n+binaria" title="T'encamentamos que t'identifiques, anque nun ye obligatorio [o]" accesskey="o" class=""><span>Entrar</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="Más opciones" > <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="Ferramientes personales" > <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">Ferramientes personales</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menú de usuario" > <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/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=ast.wikipedia.org&uselang=ast"><span>Donativos</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Crear_una_cuenta&returnto=Operaci%C3%B3n+binaria" title="Encamentámoste que crees una cuenta y qu'anicies sesión; sicasí, nun ye obligatorio"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Crear una cuenta</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Entrar&returnto=Operaci%C3%B3n+binaria" title="T'encamentamos que t'identifiques, anque nun ye obligatorio [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Entrar</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"> Páginas para editores desconectados <a href="/wiki/Ayuda:Introducci%C3%B3n" aria-label="Obtenga más información sobre editar"><span>más información</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/Especial:MisContribuciones" title="Una llista d'ediciones feches dende esta dirección IP [y]" accesskey="y"><span>Contribuciones</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:MiDiscusi%C3%B3n" title="Alderique de les ediciones feches con esta direición IP [n]" accesskey="n"><span>Alderique</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="Sitio"> <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="Conteníu" 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">Conteníu</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">despintar</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">Entamu</div> </a> </li> <li id="toc-Notación" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notación"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Notación</span> </div> </a> <ul id="toc-Notación-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exemplu_d'operación_binaria" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Exemplu_d'operación_binaria"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Exemplu d'operación binaria</span> </div> </a> <ul id="toc-Exemplu_d'operación_binaria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tipos" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Tipos"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Tipos</span> </div> </a> <button aria-controls="toc-Tipos-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>Alternar subsección Tipos</span> </button> <ul id="toc-Tipos-sublist" class="vector-toc-list"> <li id="toc-Operación_binaria_interna" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operación_binaria_interna"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Operación binaria interna</span> </div> </a> <ul id="toc-Operación_binaria_interna-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operación_binaria_esterna" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operación_binaria_esterna"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Operación binaria esterna</span> </div> </a> <ul id="toc-Operación_binaria_esterna-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Propiedaes_d'una_operación_binaria" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Propiedaes_d'una_operación_binaria"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Propiedaes d'una operación binaria</span> </div> </a> <button aria-controls="toc-Propiedaes_d'una_operación_binaria-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>Alternar subsección Propiedaes d'una operación binaria</span> </button> <ul id="toc-Propiedaes_d'una_operación_binaria-sublist" class="vector-toc-list"> <li id="toc-Conmutatividá" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conmutatividá"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Conmutatividá</span> </div> </a> <ul id="toc-Conmutatividá-sublist" class="vector-toc-list"> <li id="toc-Anticonmutatividá" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Anticonmutatividá"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Anticonmutatividá</span> </div> </a> <ul id="toc-Anticonmutatividá-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Asociatividá" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Asociatividá"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Asociatividá</span> </div> </a> <ul id="toc-Asociatividá-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Propiedaes_de_dos_operaciones_binaries" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Propiedaes_de_dos_operaciones_binaries"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Propiedaes de dos operaciones binaries</span> </div> </a> <button aria-controls="toc-Propiedaes_de_dos_operaciones_binaries-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>Alternar subsección Propiedaes de dos operaciones binaries</span> </button> <ul id="toc-Propiedaes_de_dos_operaciones_binaries-sublist" class="vector-toc-list"> <li id="toc-Distributividá" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distributividá"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Distributividá</span> </div> </a> <ul id="toc-Distributividá-sublist" class="vector-toc-list"> <li id="toc-Distributividá_pela_esquierda" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Distributividá_pela_esquierda"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.1</span> <span>Distributividá pela esquierda</span> </div> </a> <ul id="toc-Distributividá_pela_esquierda-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distributividá_pela_derecha" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Distributividá_pela_derecha"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.2</span> <span>Distributividá pela derecha</span> </div> </a> <ul id="toc-Distributividá_pela_derecha-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Elementos_distinguidos" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Elementos_distinguidos"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Elementos distinguidos</span> </div> </a> <button aria-controls="toc-Elementos_distinguidos-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>Alternar subsección Elementos distinguidos</span> </button> <ul id="toc-Elementos_distinguidos-sublist" class="vector-toc-list"> <li id="toc-Elementu_neutru" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elementu_neutru"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Elementu neutru</span> </div> </a> <ul id="toc-Elementu_neutru-sublist" class="vector-toc-list"> <li id="toc-Elementu_neutru_pela_derecha" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Elementu_neutru_pela_derecha"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.1</span> <span>Elementu neutru pela derecha</span> </div> </a> <ul id="toc-Elementu_neutru_pela_derecha-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementu_neutru_pela_esquierda" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Elementu_neutru_pela_esquierda"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.2</span> <span>Elementu neutru pela esquierda</span> </div> </a> <ul id="toc-Elementu_neutru_pela_esquierda-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unicidá_del_elementu_neutru" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Unicidá_del_elementu_neutru"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.3</span> <span>Unicidá del elementu neutru</span> </div> </a> <ul id="toc-Unicidá_del_elementu_neutru-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Elementu_simétricu" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elementu_simétricu"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Elementu simétricu</span> </div> </a> <ul id="toc-Elementu_simétricu-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementu_involutivu" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elementu_involutivu"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Elementu involutivu</span> </div> </a> <ul id="toc-Elementu_involutivu-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementu_absorbente" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elementu_absorbente"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Elementu absorbente</span> </div> </a> <ul id="toc-Elementu_absorbente-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Operación_inversa" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Operación_inversa"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Operación inversa</span> </div> </a> <ul id="toc-Operación_inversa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Otres_propiedaes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Otres_propiedaes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Otres propiedaes</span> </div> </a> <button aria-controls="toc-Otres_propiedaes-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>Alternar subsección Otres propiedaes</span> </button> <ul id="toc-Otres_propiedaes-sublist" class="vector-toc-list"> <li id="toc-Simplificación_o_cancelativa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simplificación_o_cancelativa"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Simplificación o cancelativa</span> </div> </a> <ul id="toc-Simplificación_o_cancelativa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Divisores_del_cero" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Divisores_del_cero"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Divisores del cero</span> </div> </a> <ul id="toc-Divisores_del_cero-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ver_tamién" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ver_tamién"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Ver tamién</span> </div> </a> <ul id="toc-Ver_tamién-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referencies" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Referencies"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Referencies</span> </div> </a> <ul id="toc-Referencies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enllaces_esternos" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Enllaces_esternos"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Enllaces esternos</span> </div> </a> <ul id="toc-Enllaces_esternos-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="Conteníu" 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="Cambiar a la tabla de contenidos" > <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">Cambiar a la tabla de contenidos</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">Operación binaria</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="Ir a un artículo en otro idioma. Disponible en 64 idiomas" > <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-64" 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">64 llingües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D9%85%D9%84%D9%8A%D8%A9_%D8%AB%D9%86%D8%A7%D8%A6%D9%8A%D8%A9" title="عملية ثنائية – árabe" lang="ar" hreflang="ar" data-title="عملية ثنائية" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%91%D1%96%D0%BD%D0%B0%D1%80%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D1%8B%D1%8F" title="Бінарная аперацыя – bielorrusu" lang="be" hreflang="be" data-title="Бінарная аперацыя" data-language-autonym="Беларуская" data-language-local-name="bielorrusu" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%B0_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Бинарна операция – búlgaru" lang="bg" hreflang="bg" data-title="Бинарна операция" data-language-autonym="Български" data-language-local-name="búlgaru" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AA%E0%A6%BE%E0%A6%B0%E0%A7%87%E0%A6%B6%E0%A6%A8_(%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4)" title="অপারেশন (গণিত) – bengalín" lang="bn" hreflang="bn" data-title="অপারেশন (গণিত)" data-language-autonym="বাংলা" data-language-local-name="bengalín" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Binarna_operacija" title="Binarna operacija – bosniu" lang="bs" hreflang="bs" data-title="Binarna operacija" data-language-autonym="Bosanski" data-language-local-name="bosniu" 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/Operaci%C3%B3_bin%C3%A0ria" title="Operació binària – catalán" lang="ca" hreflang="ca" data-title="Operació binària" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%A9%D8%B1%D8%AF%D8%A7%D8%B1_(%D8%A8%DB%8C%D8%B1%DA%A9%D8%A7%D8%B1%DB%8C)" title="کردار (بیرکاری) – kurdu central" lang="ckb" hreflang="ckb" data-title="کردار (بیرکاری)" data-language-autonym="کوردی" data-language-local-name="kurdu central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bin%C3%A1rn%C3%AD_operace" title="Binární operace – checu" lang="cs" hreflang="cs" data-title="Binární operace" data-language-autonym="Čeština" data-language-local-name="checu" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BB%C4%83_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8" title="Бинарлă операци – chuvash" lang="cv" hreflang="cv" data-title="Бинарлă операци" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bin%C3%A6r_operator" title="Binær operator – danés" lang="da" hreflang="da" data-title="Binær operator" data-language-autonym="Dansk" data-language-local-name="danés" 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/Zweistellige_Verkn%C3%BCpfung" title="Zweistellige Verknüpfung – alemán" lang="de" hreflang="de" data-title="Zweistellige Verknüpfung" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CF%85%CE%B1%CE%B4%CE%B9%CE%BA%CE%AE_%CF%80%CF%81%CE%AC%CE%BE%CE%B7" title="Δυαδική πράξη – griegu" lang="el" hreflang="el" data-title="Δυαδική πράξη" data-language-autonym="Ελληνικά" data-language-local-name="griegu" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Binary_operation" title="Binary operation – inglés" lang="en" hreflang="en" data-title="Binary operation" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Duvalenta_operacio" title="Duvalenta operacio – esperanto" lang="eo" hreflang="eo" data-title="Duvalenta operacio" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Operaci%C3%B3n_binaria" title="Operación binaria – español" lang="es" hreflang="es" data-title="Operación binaria" data-language-autonym="Español" data-language-local-name="español" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Binaarne_tehe" title="Binaarne tehe – estoniu" lang="et" hreflang="et" data-title="Binaarne tehe" data-language-autonym="Eesti" data-language-local-name="estoniu" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eragiketa_bitar" title="Eragiketa bitar – vascu" lang="eu" hreflang="eu" data-title="Eragiketa bitar" data-language-autonym="Euskara" data-language-local-name="vascu" 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%B9%D9%85%D9%84_%D8%AF%D9%88%D8%AA%D8%A7%DB%8C%DB%8C" title="عمل دوتایی – persa" lang="fa" hreflang="fa" data-title="عمل دوتایی" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bin%C3%A4%C3%A4rioperaatio" title="Binäärioperaatio – finlandés" lang="fi" hreflang="fi" data-title="Binäärioperaatio" data-language-autonym="Suomi" data-language-local-name="finlandés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Op%C3%A9ration_binaire" title="Opération binaire – francés" lang="fr" hreflang="fr" data-title="Opération binaire" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Obrachadh_c%C3%A0raideach" title="Obrachadh càraideach – gaélicu escocés" lang="gd" hreflang="gd" data-title="Obrachadh càraideach" data-language-autonym="Gàidhlig" data-language-local-name="gaélicu escocés" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Operaci%C3%B3n_binaria" title="Operación binaria – gallegu" lang="gl" hreflang="gl" data-title="Operación binaria" data-language-autonym="Galego" data-language-local-name="gallegu" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_%D7%91%D7%99%D7%A0%D7%90%D7%A8%D7%99%D7%AA" title="פעולה בינארית – hebréu" lang="he" hreflang="he" data-title="פעולה בינארית" data-language-autonym="עברית" data-language-local-name="hebréu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Binarna_operacija" title="Binarna operacija – croata" lang="hr" hreflang="hr" data-title="Binarna operacija" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Operation_binari" title="Operation binari – interlingua" lang="ia" hreflang="ia" data-title="Operation binari" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Operasi_biner" title="Operasi biner – indonesiu" lang="id" hreflang="id" data-title="Operasi biner" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiu" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/A%C3%B0ger%C3%B0_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Aðgerð (stærðfræði) – islandés" lang="is" hreflang="is" data-title="Aðgerð (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="islandés" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Operazione_binaria" title="Operazione binaria – italianu" lang="it" hreflang="it" data-title="Operazione binaria" data-language-autonym="Italiano" data-language-local-name="italianu" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%8C%E9%A0%85%E6%BC%94%E7%AE%97" title="二項演算 – xaponés" lang="ja" hreflang="ja" data-title="二項演算" data-language-autonym="日本語" data-language-local-name="xaponés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko badge-Q70893996 mw-list-item" title=""><a href="https://ko.wikipedia.org/wiki/%EC%9D%B4%ED%95%AD_%EC%97%B0%EC%82%B0" title="이항 연산 – coreanu" lang="ko" hreflang="ko" data-title="이항 연산" data-language-autonym="한국어" data-language-local-name="coreanu" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Kiryar_(matemat%C3%AEk)" title="Kiryar (matematîk) – curdu" lang="ku" hreflang="ku" data-title="Kiryar (matematîk)" data-language-autonym="Kurdî" data-language-local-name="curdu" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Bin%C4%81ra_oper%C4%81cija" title="Bināra operācija – letón" lang="lv" hreflang="lv" data-title="Bināra operācija" data-language-autonym="Latviešu" data-language-local-name="letón" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Operasi_biner" title="Operasi biner – minangkabau" lang="min" hreflang="min" data-title="Operasi biner" data-language-autonym="Minangkabau" data-language-local-name="minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A6%E0%B5%8D%E0%B4%B5%E0%B4%AF%E0%B4%BE%E0%B4%99%E0%B5%8D%E0%B4%95%E0%B4%B8%E0%B4%82%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%AF" title="ദ്വയാങ്കസംക്രിയ – malayalam" lang="ml" hreflang="ml" data-title="ദ്വയാങ്കസംക്രിയ" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Operasi_dedua" title="Operasi dedua – malayu" lang="ms" hreflang="ms" data-title="Operasi dedua" data-language-autonym="Bahasa Melayu" data-language-local-name="malayu" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Binaire_operatie" title="Binaire operatie – neerlandés" lang="nl" hreflang="nl" data-title="Binaire operatie" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Bin%C3%A6r_operasjon" title="Binær operasjon – noruegu Nynorsk" lang="nn" hreflang="nn" data-title="Binær operasjon" data-language-autonym="Norsk nynorsk" data-language-local-name="noruegu Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bin%C3%A6r_operasjon" title="Binær operasjon – noruegu Bokmål" lang="nb" hreflang="nb" data-title="Binær operasjon" data-language-autonym="Norsk bokmål" data-language-local-name="noruegu Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/L%C3%A8i_de_composicion_int%C3%A8rna" title="Lèi de composicion intèrna – occitanu" lang="oc" hreflang="oc" data-title="Lèi de composicion intèrna" data-language-autonym="Occitan" data-language-local-name="occitanu" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dzia%C5%82anie_dwuargumentowe" title="Działanie dwuargumentowe – polacu" lang="pl" hreflang="pl" data-title="Działanie dwuargumentowe" data-language-autonym="Polski" data-language-local-name="polacu" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Operassion" title="Operassion – piamontés" lang="pms" hreflang="pms" data-title="Operassion" data-language-autonym="Piemontèis" data-language-local-name="piamontés" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Opera%C3%A7%C3%A3o_bin%C3%A1ria" title="Operação binária – portugués" lang="pt" hreflang="pt" data-title="Operação binária" data-language-autonym="Português" data-language-local-name="portugués" 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/Opera%C8%9Bie_binar%C4%83" title="Operație binară – rumanu" lang="ro" hreflang="ro" data-title="Operație binară" data-language-autonym="Română" data-language-local-name="rumanu" 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%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%B0%D1%8F_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Бинарная операция – rusu" lang="ru" hreflang="ru" data-title="Бинарная операция" data-language-autonym="Русский" data-language-local-name="rusu" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Opirazzioni_binaria" title="Opirazzioni binaria – sicilianu" lang="scn" hreflang="scn" data-title="Opirazzioni binaria" data-language-autonym="Sicilianu" data-language-local-name="sicilianu" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Binarna_operacija" title="Binarna operacija – serbo-croata" lang="sh" hreflang="sh" data-title="Binarna operacija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Binary_operation" title="Binary operation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Binary operation" 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/Bin%C3%A1rna_oper%C3%A1cia" title="Binárna operácia – eslovacu" lang="sk" hreflang="sk" data-title="Binárna operácia" data-language-autonym="Slovenčina" data-language-local-name="eslovacu" 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/Dvo%C4%8Dlena_operacija" title="Dvočlena operacija – eslovenu" lang="sl" hreflang="sl" data-title="Dvočlena operacija" data-language-autonym="Slovenščina" data-language-local-name="eslovenu" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Veprimi_binar" title="Veprimi binar – albanu" lang="sq" hreflang="sq" data-title="Veprimi binar" data-language-autonym="Shqip" data-language-local-name="albanu" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%B0_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Бинарна операција – serbiu" lang="sr" hreflang="sr" data-title="Бинарна операција" data-language-autonym="Српски / srpski" data-language-local-name="serbiu" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bin%C3%A4r_operator" title="Binär operator – suecu" lang="sv" hreflang="sv" data-title="Binär operator" data-language-autonym="Svenska" data-language-local-name="suecu" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%88%E0%AE%B0%E0%AF%81%E0%AE%B1%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AF%86%E0%AE%AF%E0%AE%B2%E0%AE%BF" title="ஈருறுப்புச் செயலி – tamil" lang="ta" hreflang="ta" data-title="ஈருறுப்புச் செயலி" 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%81%E0%B8%B2%E0%B8%A3%E0%B8%94%E0%B8%B3%E0%B9%80%E0%B8%99%E0%B8%B4%E0%B8%99%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%97%E0%B8%A7%E0%B8%B4%E0%B8%A0%E0%B8%B2%E0%B8%84" title="การดำเนินการทวิภาค – tailandés" lang="th" hreflang="th" data-title="การดำเนินการทวิภาค" data-language-autonym="ไทย" data-language-local-name="tailandés" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Operasyong_tambalan" title="Operasyong tambalan – tagalog" lang="tl" hreflang="tl" data-title="Operasyong tambalan" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0kili_i%C5%9Flem" title="İkili işlem – turcu" lang="tr" hreflang="tr" data-title="İkili işlem" data-language-autonym="Türkçe" data-language-local-name="turcu" 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%91%D1%96%D0%BD%D0%B0%D1%80%D0%BD%D0%B0_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D1%96%D1%8F" title="Бінарна операція – ucraín" lang="uk" hreflang="uk" data-title="Бінарна операція" data-language-autonym="Українська" data-language-local-name="ucraín" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_to%C3%A1n_hai_ng%C3%B4i" title="Phép toán hai ngôi – vietnamín" lang="vi" hreflang="vi" data-title="Phép toán hai ngôi" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamín" 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/%E4%BA%8C%E5%85%83%E8%BF%90%E7%AE%97" title="二元运算 – chinu wu" lang="wuu" hreflang="wuu" data-title="二元运算" data-language-autonym="吴语" data-language-local-name="chinu wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BD_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Бинарн операция – calmuco" lang="xal" hreflang="xal" data-title="Бинарн операция" data-language-autonym="Хальмг" data-language-local-name="calmuco" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%91%D7%99%D7%A0%D7%90%D7%A8%D7%99%D7%A9%D7%A2_%D7%90%D7%A4%D7%A2%D7%A8%D7%90%D7%A6%D7%99%D7%A2" title="בינארישע אפעראציע – yiddish" lang="yi" hreflang="yi" data-title="בינארישע אפעראציע" data-language-autonym="ייִדיש" data-language-local-name="yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BA%8C%E5%85%83%E8%BF%90%E7%AE%97" title="二元运算 – chinu" lang="zh" hreflang="zh" data-title="二元运算" data-language-autonym="中文" data-language-local-name="chinu" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E4%BA%8C%E5%85%83%E9%81%8B%E7%AE%97" title="二元運算 – chinu lliterariu" lang="lzh" hreflang="lzh" data-title="二元運算" data-language-autonym="文言" data-language-local-name="chinu lliterariu" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BA%8C%E5%85%83%E9%81%8B%E7%AE%97" title="二元運算 – cantonés" lang="yue" hreflang="yue" data-title="二元運算" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</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/Q164307#sitelinks-wikipedia" title="Editar los enllaces d'interllingua" class="wbc-editpage">Editar los enllaces</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="Espacios de nome"> <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/Operaci%C3%B3n_binaria" title="Ver la páxina de conteníu [c]" accesskey="c"><span>Páxina</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Alderique:Operaci%C3%B3n_binaria&action=edit&redlink=1" rel="discussion" class="new" title="Alderique tocante al conteníu de la páxina (la páxina nun esiste) [t]" accesskey="t"><span>Alderique</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="Cambiar variante de idioma" > <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">asturianu</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="Vistes"> <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/Operaci%C3%B3n_binaria"><span>Lleer</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit" title="Editar esta páxina [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit" title="Editar el códigu fonte d'esta páxina [e]" accesskey="e"><span>Editar la fonte</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=history" title="Versiones antigües d'esta páxina [h]" accesskey="h"><span>Ver historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Ferramientes de páxina"> <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="Ferramientes" > <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">Ferramientes</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">Ferramientes</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">despintar</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Más opciones" > <div class="vector-menu-heading"> Aiciones </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/Operaci%C3%B3n_binaria"><span>Lleer</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit" title="Editar esta páxina [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit" title="Editar el códigu fonte d'esta páxina [e]" accesskey="e"><span>Editar la fonte</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=history"><span>Ver historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Xeneral </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:LoQueEnlazaAqu%C3%AD/Operaci%C3%B3n_binaria" title="Llista de toles páxines wiki qu'enllacien equí [j]" accesskey="j"><span>Lo qu'enllaza equí</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:CambiosEnEnlazadas/Operaci%C3%B3n_binaria" rel="nofollow" title="Cambios recientes nes páxines enllazaes dende esta [k]" accesskey="k"><span>Cambios rellacionaos</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=ast" title="Xubir ficheros [u]" accesskey="u"><span>Xubir ficheru</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginasEspeciales" title="Llista de toles páxines especiales [q]" accesskey="q"><span>Páxines especiales</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Operaci%C3%B3n_binaria&oldid=4121027" title="Enllaz permanente a esta revisión de la páxina"><span>Enllaz permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=info" title="Más información sobro esta páxina"><span>Información de la páxina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=Operaci%C3%B3n_binaria&id=4121027&wpFormIdentifier=titleform" title="Información tocante a cómo citar esta páxina"><span>Citar esta páxina</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:Acortador_de_URL&url=https%3A%2F%2Fast.wikipedia.org%2Fwiki%2FOperaci%25C3%25B3n_binaria"><span>Llograr la URL encurtiada</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fast.wikipedia.org%2Fwiki%2FOperaci%25C3%25B3n_binaria"><span>Xenerar códigu QR</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"> Imprentar/esportar </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Libro&bookcmd=book_creator&referer=Operaci%C3%B3n+binaria"><span>Crear un llibru</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Operaci%C3%B3n_binaria&action=show-download-screen"><span>Descargar como PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Operaci%C3%B3n_binaria&printable=yes" title="Versión imprentable d'esta páxina [p]" accesskey="p"><span>Versión pa imprentar</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"> N'otros proyeutos </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:Binary_operations" hreflang="en"><span>Wikimedia Commons</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/Q164307" title="Enllaz al elementu del depósitu de datos coneutáu [g]" accesskey="g"><span>Elementu de Wikidata</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="Ferramientes de páxina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apariencia"> <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">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">despintar</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">De Wikipedia</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="ast" dir="ltr"><table class="infobox plantia-xenerica" style="font-size:90%;width:25em"><tbody><tr><th colspan="2" style="text-align:center;font-size:125%;font-weight:bold;background-color: #acacac">Operación binaria</th></tr><tr><td colspan="2" style="text-align:center;background-color: #cdcdcd"> función binaria <sup>(es)</sup> <span class="mw-valign-baseline skin-invert" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q3737844?uselang=ast" title="Traducir"><img alt="Traducir" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/10px-Noun_Project_label_icon_1116097_cc_mirror.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/15px-Noun_Project_label_icon_1116097_cc_mirror.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/20px-Noun_Project_label_icon_1116097_cc_mirror.svg.png 2x" data-file-width="158" data-file-height="161" /></a></span> y partial binary operation <sup>(en)</sup> <span class="mw-valign-baseline skin-invert" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q97152408?uselang=ast" title="Traducir"><img alt="Traducir" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/10px-Noun_Project_label_icon_1116097_cc_mirror.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/15px-Noun_Project_label_icon_1116097_cc_mirror.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/20px-Noun_Project_label_icon_1116097_cc_mirror.svg.png 2x" data-file-width="158" data-file-height="161" /></a></span></td></tr><tr><td colspan="2" style="text-align:center"> <span typeof="mw:File"><a href="/wiki/Ficheru:Binary_operations_as_black_box.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Binary_operations_as_black_box.svg/260px-Binary_operations_as_black_box.svg.png" decoding="async" width="260" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Binary_operations_as_black_box.svg/390px-Binary_operations_as_black_box.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Binary_operations_as_black_box.svg/520px-Binary_operations_as_black_box.svg.png 2x" data-file-width="142" data-file-height="142" /></a></span></td></tr><tr><td colspan="2" style="text-align:center"> <table style="width:100%; text-align:center; background-color:transparent; border:0; margin:0; padding:0;"> <tbody><tr><td style="width:33%; padding:0.2em 0.1em 0.2em 0;vertical-align: middle;"><span style="font-style:;"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q657596" class="extiw" title="d:Special:EntityPage/Q657596">operación unaria</a> <sup>(es)</sup> <span class="mw-valign-baseline skin-invert" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q657596?uselang=ast" title="Traducir"><img alt="Traducir" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/10px-Noun_Project_label_icon_1116097_cc_mirror.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/15px-Noun_Project_label_icon_1116097_cc_mirror.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/20px-Noun_Project_label_icon_1116097_cc_mirror.svg.png 2x" data-file-width="158" data-file-height="161" /></a></span></span></td> <td style="padding:0.2em 0.1em; background-color:#ABD2D0; vertical-align: middle;"><span style="font-weight:bold; font-style:;">Operación binaria</span></td> <td style="width:33%; padding:0.2em 0 0.2em 0.1em;vertical-align: middle;"><span style="font-style:;"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1524945" class="extiw" title="d:Special:EntityPage/Q1524945">operación ternaria</a> <sup>(es)</sup> <span class="mw-valign-baseline skin-invert" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q1524945?uselang=ast" title="Traducir"><img alt="Traducir" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/10px-Noun_Project_label_icon_1116097_cc_mirror.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/15px-Noun_Project_label_icon_1116097_cc_mirror.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Noun_Project_label_icon_1116097_cc_mirror.svg/20px-Noun_Project_label_icon_1116097_cc_mirror.svg.png 2x" data-file-width="158" data-file-height="161" /></a></span></span></td> </tr></tbody></table></td></tr><tr><td colspan="2" style="text-align:right"><span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q164307" title="Cambiar los datos en Wikidata"><img alt="Cambiar los datos en Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/12px-Arbcom_ru_editing.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/18px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/24px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheru:Operaci%C3%B3n_binaria_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Operaci%C3%B3n_binaria_1.svg/220px-Operaci%C3%B3n_binaria_1.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Operaci%C3%B3n_binaria_1.svg/330px-Operaci%C3%B3n_binaria_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Operaci%C3%B3n_binaria_1.svg/440px-Operaci%C3%B3n_binaria_1.svg.png 2x" data-file-width="500" data-file-height="500" /></a><figcaption>Esquema d'una operación binaria.</figcaption></figure> <p>Defínese como <b>operación binaria</b> (o llei de composición) aquella <a href="/w/index.php?title=Operaci%C3%B3n_matem%C3%A1tica&action=edit&redlink=1" class="new" title="Operación matemática (la páxina nun esiste)">operación matemática</a>, que precisa de l'operador y dos operandos (argumentos) por el que calculese un valor.<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> </p><p>Formalmente, daos trés conxuntos <i>A</i>, <i>B</i> y <i>C</i> una operación binaria <a href="/wiki/Multiplicaci%C3%B3n" title="Multiplicación">productu</a>, representando la operación pol signu ∘ , ye una aplicación qu'asigna a cada par de valores <i>a</i> de <i>A</i> y <i>b</i> de <i>B</i> un solu valor <i>c</i> de <i>C</i>, que podemos representar: </p><p><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 {\begin{array}{rccl}\circ :&A\times B&\longrightarrow &C\\&(a,b)&\longmapsto &c\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>∘</mo> <mo>:</mo> </mtd> <mtd> <mi>A</mi> <mo>×</mo> <mi>B</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>C</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">⟼</mo> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rccl}\circ :&A\times B&\longrightarrow &C\\&(a,b)&\longmapsto &c\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e183b798f9e06de125ab5ffe6912c6266ea0324" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.093ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{rccl}\circ :&A\times B&\longrightarrow &C\\&(a,b)&\longmapsto &c\end{array}}}"></span> </p><p>En particular, <i>A</i>, <i>B</i> y <i>C</i> podríen ser el mesmu <a href="/wiki/Conxuntu" title="Conxuntu">conxuntu</a>, que denotamos <i>A</i>. Polo tanto, una operación binaria nel conxuntu <i>A</i> ye una <a href="/wiki/Funci%C3%B3n_matem%C3%A1tica" title="Función matemática">aplicación</a> d'elementos del <a href="/w/index.php?title=Productu_cartesianu&action=edit&redlink=1" class="new" title="Productu cartesianu (la páxina nun esiste)">productu cartesianu</a> <i>A×A</i> na <i>A</i>. </p><p>Esisten dos tipos d'operaciones binaries, les operaciones binaries internes y les operaciones binaries esternes. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notación"><span id="Notaci.C3.B3n"></span>Notación</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=1" title="Editar seición: Notación" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=1" title="Editar el código fuente de la sección: Notación"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una operación binaria ∘ ente dos elementos, <i>a</i> y <i>b</i>, de dos conxuntos, <i>A</i> y <i>B</i>, puede denotase por: </p><p><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 a\circ b=c\;,\quad \circ (a,b)=c\;,\quad (a,b)\xrightarrow {\circ } c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∘</mo> <mi>b</mi> <mo>=</mo> <mi>c</mi> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="1em" /> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mo>∘</mo> </mpadded> </mover> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\circ b=c\;,\quad \circ (a,b)=c\;,\quad (a,b)\xrightarrow {\circ } c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912b36cefa259e96d7a59a491f2dc00ba365a257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-top: -0.31ex; width:36.561ex; height:3.676ex;" alt="{\displaystyle a\circ b=c\;,\quad \circ (a,b)=c\;,\quad (a,b)\xrightarrow {\circ } c}"></span> </p><p>siendo la primera la más común. </p> <div class="mw-heading mw-heading2"><h2 id="Exemplu_d'operación_binaria"><span id="Exemplu_d.27operaci.C3.B3n_binaria"></span>Exemplu d'operación binaria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=2" title="Editar seición: Exemplu d'operación binaria" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=2" title="Editar el código fuente de la sección: Exemplu d'operación binaria"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Adici%C3%B3n_(matem%C3%A1tica)" title="Adición (matemática)">suma</a> (+) de <a href="/wiki/N%C3%BAmberu_natural" title="Númberu natural">númberos naturales</a> ye un exemplu d'operación binaria interna nel conxunto <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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span>. </p><p><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 {\begin{array}{rccl}+:&N\times N&\longrightarrow &N\\&(a,b)&\longmapsto &c=a+b\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>+</mo> <mo>:</mo> </mtd> <mtd> <mi>N</mi> <mo>×</mo> <mi>N</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>N</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">⟼</mo> </mtd> <mtd> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rccl}+:&N\times N&\longrightarrow &N\\&(a,b)&\longmapsto &c=a+b\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571c20321e2e30cb5cb6632ead465d6dcb50ba7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.766ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{rccl}+:&N\times N&\longrightarrow &N\\&(a,b)&\longmapsto &c=a+b\end{array}}}"></span> </p><p>y tenemos que: </p><p><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 2+3=5\;,\quad +(2,3)=5\;,\quad (2,3)\xrightarrow {+} 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>5</mn> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="1em" /> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mo>+</mo> </mpadded> </mover> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+3=5\;,\quad +(2,3)=5\;,\quad (2,3)\xrightarrow {+} 5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da66fe1996c8ff3a562190bb5be241dcc7fb1760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-top: -0.326ex; width:38.89ex; height:4.009ex;" alt="{\displaystyle 2+3=5\;,\quad +(2,3)=5\;,\quad (2,3)\xrightarrow {+} 5}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Tipos">Tipos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=3" title="Editar seición: Tipos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=3" title="Editar el código fuente de la sección: Tipos"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Según los conxuntos A, B y C podemos estremar dos tipos d'operaciones, les internes nes qu'A = B = C, y les esternes que son toles demás. Denomínase <a href="/w/index.php?title=Llei_de_Composici%C3%B3n&action=edit&redlink=1" class="new" title="Llei de Composición (la páxina nun esiste)">Llei de Composición</a> a un subtipo d'operación binaria. </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/Ficheru:AL_Operaci%C3%B3n_binaria.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/AL_Operaci%C3%B3n_binaria.svg/495px-AL_Operaci%C3%B3n_binaria.svg.png" decoding="async" width="495" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/AL_Operaci%C3%B3n_binaria.svg/743px-AL_Operaci%C3%B3n_binaria.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/AL_Operaci%C3%B3n_binaria.svg/990px-AL_Operaci%C3%B3n_binaria.svg.png 2x" data-file-width="1200" data-file-height="600" /></a><figcaption>Esquema de les tipos d'operaciones binaries en castellanu</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Operación_binaria_interna"><span id="Operaci.C3.B3n_binaria_interna"></span>Operación binaria interna</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=4" title="Editar seición: Operación binaria interna" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=4" title="Editar el código fuente de la sección: Operación binaria interna"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si a cada par de valores (<b>a</b>, <b>b</b>) de <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> la operación correspuéndelu a un valor <b>c</b> de <i>A</i>: </p> <dl><dd><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 {\begin{array}{rccl}\circledast :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\circledast b\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>⊛</mo> <mo>:</mo> </mtd> <mtd> <mi>A</mi> <mo>×</mo> <mi>A</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>A</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">⟼</mo> </mtd> <mtd> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⊛</mo> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rccl}\circledast :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\circledast b\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5113674d72cdf5c079c136dabc4b6f4842d4801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.125ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{rccl}\circledast :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\circledast b\end{array}}}"></span></dd></dl> <p>dizse qu'esta operación ye interna, tamién se llama <b>llei de composición interna</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Operación_binaria_esterna"><span id="Operaci.C3.B3n_binaria_esterna"></span>Operación binaria esterna</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=5" title="Editar seición: Operación binaria esterna" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=5" title="Editar el código fuente de la sección: Operación binaria esterna"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si la operación nun ye interna entós ye esterna, pudiéndose presentar los siguientes casos: </p> <ul><li>Si a cada par de valores <b>a</b> de <b>A</b> y <b>b</b> de <b>B</b>, asígnase-y un valor <b>c</b> de <b>A</b>,</li></ul> <p><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 {\begin{array}{rccl}\star :&A\times B&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\star b\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>⋆</mo> <mo>:</mo> </mtd> <mtd> <mi>A</mi> <mo>×</mo> <mi>B</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>A</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">⟼</mo> </mtd> <mtd> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⋆</mo> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rccl}\star :&A\times B&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\star b\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62c0a0a91974a2f66f8ad6f989cab00431f31fa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.854ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{rccl}\star :&A\times B&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\star b\end{array}}}"></span> </p><p>a esta operación tamién se denomina <b>llei de composición esterna</b>. </p> <ul><li>Si la operación ye de la forma:</li></ul> <p><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 {\begin{array}{rccl}\star :&A\times A&\longrightarrow &B\\&(a,b)&\longmapsto &c=a\star b\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>⋆</mo> <mo>:</mo> </mtd> <mtd> <mi>A</mi> <mo>×</mo> <mi>A</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">⟼</mo> </mtd> <mtd> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⋆</mo> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rccl}\star :&A\times A&\longrightarrow &B\\&(a,b)&\longmapsto &c=a\star b\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cdcbc8e2f518da05a6e26f5cbcd122e6ba14836" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.833ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{rccl}\star :&A\times A&\longrightarrow &B\\&(a,b)&\longmapsto &c=a\star b\end{array}}}"></span> </p><p>na qu'a cada par de valores <b>a</b>, <b>b</b> de <b>A</b> asígnase-y un <b>c</b> de <b>B</b>, esta operación nun se denomina llei de composición. </p> <ul><li>Si la operación asigna a cada par de valores <b>a</b> de <b>A</b> y <b>b</b> de <b>B</b> un <b>c</b> de <b>C</b>, siendo <b>A</b>, <b>B</b> y <b>C</b> conxuntos distintos:</li></ul> <p><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 {\begin{array}{rccl}\star :&A\times B&\longrightarrow &C\\&(a,b)&\longmapsto &c=a\star b\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>⋆</mo> <mo>:</mo> </mtd> <mtd> <mi>A</mi> <mo>×</mo> <mi>B</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>C</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">⟼</mo> </mtd> <mtd> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⋆</mo> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rccl}\star :&A\times B&\longrightarrow &C\\&(a,b)&\longmapsto &c=a\star b\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f78b0ef42c69a0d00a43888a0f91f7e86aac9d49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.854ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{rccl}\star :&A\times B&\longrightarrow &C\\&(a,b)&\longmapsto &c=a\star b\end{array}}}"></span> </p><p>ye'l casu más xeneral, y tampoco se denomina llei de composición. </p> <div class="mw-heading mw-heading2"><h2 id="Propiedaes_d'una_operación_binaria"><span id="Propiedaes_d.27una_operaci.C3.B3n_binaria"></span>Propiedaes d'una operación binaria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=6" title="Editar seición: Propiedaes d'una operación binaria" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=6" title="Editar el código fuente de la sección: Propiedaes d'una operación binaria"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dáu un conxuntu <b>A</b> non vacíu y definida una aplicación de <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 A\times A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91e587030076802eec026dc75906339cf1f61b70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.327ex; height:2.176ex;" alt="{\displaystyle A\times A}"></span> sobre <b>A</b>, onde a cada par ordenáu <b>(a,b)</b> asígnase-y un valor <b>c</b> de <b>A</b>, que representamos: <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 (A,\odot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>⊙</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\odot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8e9f9a88778a58cf3b1b6f417a2d9332b767783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" alt="{\displaystyle (A,\odot )}"></span> </p> <dl><dd><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 {\begin{array}{rccl}\odot :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\odot b\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>⊙</mo> <mo>:</mo> </mtd> <mtd> <mi>A</mi> <mo>×</mo> <mi>A</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>A</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">⟼</mo> </mtd> <mtd> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rccl}\odot :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\odot b\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5561c227f9d78bdc583690c28ebaf9b3aea1ddd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.125ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{rccl}\odot :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\odot b\end{array}}}"></span></dd></dl> <p>Puede tener les siguientes propiedaes: </p> <div class="mw-heading mw-heading3"><h3 id="Conmutatividá"><span id="Conmutativid.C3.A1"></span>Conmutatividá</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=7" title="Editar seición: Conmutatividá" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=7" title="Editar el código fuente de la sección: Conmutatividá"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r4219085">.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">Artículu principal: <a href="/wiki/Conmutativid%C3%A1" title="Conmutatividá">Conmutatividá</a></div> <p>Dizse que <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 \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"></span> tien la propiedá conmutativa na A si cumplese: </p> <dl><dd><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 \forall a,b\in A\;:\quad a\odot b=b\odot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>⊙</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a,b\in A\;:\quad a\odot b=b\odot a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9a25632943922b608dd6ffcb8b5439535995ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.276ex; height:2.509ex;" alt="{\displaystyle \forall a,b\in A\;:\quad a\odot b=b\odot a}"></span></dd></dl> <p>Para tou a, b de A, cumplese que la resultancia d'operar a con b ye igual al d'operar b con a. </p><p>De la mesma podemos dicir que la llei de composición interna <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 \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"></span>, <b>nun ye conmutativa</b> na A si: </p> <dl><dd><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 \exists a,b\in A\;:\quad a\odot b\neq b\odot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> <mo>≠</mo> <mi>b</mi> <mo>⊙</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists a,b\in A\;:\quad a\odot b\neq b\odot a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46656bc1e5b6373be3dfea207b818e8294079ae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.276ex; height:2.676ex;" alt="{\displaystyle \exists a,b\in A\;:\quad a\odot b\neq b\odot a}"></span></dd></dl> <p>Si esiste dalgún a, b na A, que cumple que la resultancia d'operar a con b ye distintu d'operar b con a. </p> <div class="mw-heading mw-heading4"><h4 id="Anticonmutatividá"><span id="Anticonmutativid.C3.A1"></span>Anticonmutatividá</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=8" title="Editar seición: Anticonmutatividá" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=8" title="Editar el código fuente de la sección: Anticonmutatividá"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La operación <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 \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"></span> en <b>A</b> ye anticonmutativa si: </p> <dl><dd><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 \forall a,b\in A\;:\quad a\odot b=-(b\odot a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⊙</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a,b\in A\;:\quad a\odot b=-(b\odot a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e4eceb8df53819c25be7972f4d29e58d211e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.894ex; height:2.843ex;" alt="{\displaystyle \forall a,b\in A\;:\quad a\odot b=-(b\odot a)}"></span></dd></dl> <p>Para tou a, b de A, cúmplese que la resultancia d'operar a con b ye igual al opuestu d'operar b con a. </p> <div class="mw-heading mw-heading3"><h3 id="Asociatividá"><span id="Asociativid.C3.A1"></span>Asociatividá</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=9" title="Editar seición: Asociatividá" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=9" title="Editar el código fuente de la sección: Asociatividá"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219085"><div role="note" class="hatnote navigation-not-searchable">Artículu principal: <a href="/wiki/Propied%C3%A1_asociativa" class="mw-redirect" title="Propiedá asociativa">Propiedá asociativa</a></div> <p>Dizse que <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 (A,\odot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>⊙</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\odot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8e9f9a88778a58cf3b1b6f417a2d9332b767783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" alt="{\displaystyle (A,\odot )}"></span> ye asociativa si, solu si: </p> <dl><dd><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 \forall a,b,c\in A\;:\quad (a\odot b)\odot c=a\odot (b\odot c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⊙</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⊙</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⊙</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a,b,c\in A\;:\quad (a\odot b)\odot c=a\odot (b\odot c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/687fcc9e0664b79672782cf14b1a3d06a2c4974b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.63ex; height:2.843ex;" alt="{\displaystyle \forall a,b,c\in A\;:\quad (a\odot b)\odot c=a\odot (b\odot c)}"></span></dd></dl> <p>Para tou <i>a, b, c</i> de A cumplese qu'operando <i>a</i> con <i>b</i> y la resultancia con <i>c</i> ye igual a operar <i>a</i> cola resultancia d'operar <i>b</i> con <i>c</i>. </p><p>Tamién puede dicise que la operación <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 \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"></span> nun ye asociativa si cumplese: </p> <dl><dd><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 \exists a,b,c\in A\;:\quad (a\odot b)\odot c\neq a\odot (b\odot c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⊙</mo> <mi>c</mi> <mo>≠</mo> <mi>a</mi> <mo>⊙</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⊙</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists a,b,c\in A\;:\quad (a\odot b)\odot c\neq a\odot (b\odot c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c4ffbf85c6e8cda1852be8237a9962c74be4417" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.63ex; height:2.843ex;" alt="{\displaystyle \exists a,b,c\in A\;:\quad (a\odot b)\odot c\neq a\odot (b\odot c)}"></span></dd></dl> <p>Esisten <i>a, b, c</i> na A que cumplen qu'operando <i>a</i> con <i>b</i> y la resultancia con <i>c</i> ye distintu d'operar <i>a</i> cola resultancia d'operar <i>b</i> con <i>c.</i> </p> <div class="mw-heading mw-heading2"><h2 id="Propiedaes_de_dos_operaciones_binaries">Propiedaes de dos operaciones binaries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=10" title="Editar seición: Propiedaes de dos operaciones binaries" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=10" title="Editar el código fuente de la sección: Propiedaes de dos operaciones binaries"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dáu un conxuntu A non vacíu y definíes dos aplicación d'A por A sobre A, onde a cada par ordenáu (a,b) asígnase-y cola operación <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 \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"></span> un valor c de A y con la operación <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 \circledcirc }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circledcirc }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf70a53592b87a725eabcbb2dffc880e9aa9b66c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \circledcirc }"></span> el valor d de A que representamos: <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 (A,\odot ,\circledcirc )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>⊙</mo> <mo>,</mo> <mo>⊚</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\odot ,\circledcirc )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c2ab81c20733faf2a9750e4ffe6f846fce721a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.237ex; height:2.843ex;" alt="{\displaystyle (A,\odot ,\circledcirc )}"></span>. </p> <dl><dd><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 {\begin{array}{rccl}\odot :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\odot b\end{array}}\quad \quad {\begin{array}{rccl}\circledcirc :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &d=a\circledcirc b\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>⊙</mo> <mo>:</mo> </mtd> <mtd> <mi>A</mi> <mo>×</mo> <mi>A</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>A</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">⟼</mo> </mtd> <mtd> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>⊚</mo> <mo>:</mo> </mtd> <mtd> <mi>A</mi> <mo>×</mo> <mi>A</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>A</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">⟼</mo> </mtd> <mtd> <mi>d</mi> <mo>=</mo> <mi>a</mi> <mo>⊚</mo> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rccl}\odot :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\odot b\end{array}}\quad \quad {\begin{array}{rccl}\circledcirc :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &d=a\circledcirc b\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e2f623fbfc54d4ccef3c450f4bf846f09128c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.103ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{rccl}\odot :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\odot b\end{array}}\quad \quad {\begin{array}{rccl}\circledcirc :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &d=a\circledcirc b\end{array}}}"></span></dd></dl> <p>Pueden tener les siguientes propiedaes: </p> <div class="mw-heading mw-heading3"><h3 id="Distributividá"><span id="Distributivid.C3.A1"></span>Distributividá</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=11" title="Editar seición: Distributividá" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=11" title="Editar el código fuente de la sección: Distributividá"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219085"><div role="note" class="hatnote navigation-not-searchable">Artículu principal: <a href="/wiki/Distributivid%C3%A1" title="Distributividá">Distributividá</a></div> <p>Dizse qu'una operación binaria ye distributiva si y solu si ye distributiva pela esquierda y pela derecha. </p> <div class="mw-heading mw-heading4"><h4 id="Distributividá_pela_esquierda"><span id="Distributivid.C3.A1_pela_esquierda"></span>Distributividá pela esquierda</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=12" title="Editar seición: Distributividá pela esquierda" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=12" title="Editar el código fuente de la sección: Distributividá pela esquierda"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dizse que la operación <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 \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"></span> ye distributiva pela esquierda de <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 \circledcirc }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circledcirc }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf70a53592b87a725eabcbb2dffc880e9aa9b66c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \circledcirc }"></span> si cumplese: </p> <dl><dd><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 \forall a,b,c\in A\;:\quad a\odot (b\circledcirc c)=(a\odot b)\circledcirc (a\odot c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mi>a</mi> <mo>⊙</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⊚</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⊚</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⊙</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a,b,c\in A\;:\quad a\odot (b\circledcirc c)=(a\odot b)\circledcirc (a\odot c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/463a15bd6132197f7a40a429217bc676a47d98ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.51ex; height:2.843ex;" alt="{\displaystyle \forall a,b,c\in A\;:\quad a\odot (b\circledcirc c)=(a\odot b)\circledcirc (a\odot c)}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Distributividá_pela_derecha"><span id="Distributivid.C3.A1_pela_derecha"></span>Distributividá pela derecha</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=13" title="Editar seición: Distributividá pela derecha" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=13" title="Editar el código fuente de la sección: Distributividá pela derecha"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dizse que la operación <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 \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"></span> ye distributiva pela derecha de <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 \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"></span> si cumplse: </p> <dl><dd><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 \forall a,b,c\in A\;:\quad (a\circledcirc b)\odot c=(a\odot c)\circledcirc (b\odot c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⊚</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⊙</mo> <mi>c</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⊙</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>⊚</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⊙</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a,b,c\in A\;:\quad (a\circledcirc b)\odot c=(a\odot c)\circledcirc (b\odot c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9676264d6dbe07f969413a73063c925884789247" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.287ex; height:2.843ex;" alt="{\displaystyle \forall a,b,c\in A\;:\quad (a\circledcirc b)\odot c=(a\odot c)\circledcirc (b\odot c)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Elementos_distinguidos">Elementos distinguidos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=14" title="Editar seición: Elementos distinguidos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=14" title="Editar el código fuente de la sección: Elementos distinguidos"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Elementu_neutru">Elementu neutru</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=15" title="Editar seición: Elementu neutru" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=15" title="Editar el código fuente de la sección: Elementu neutru"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219085"><div role="note" class="hatnote navigation-not-searchable">Artículu principal: <a href="/wiki/Elementu_neutru" title="Elementu neutru">Elementu neutru</a></div> <p>Un elemento <b>e</b> ye elementu neutru en <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 (A,\odot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>⊙</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\odot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8e9f9a88778a58cf3b1b6f417a2d9332b767783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" alt="{\displaystyle (A,\odot )}"></span> si ye elementu neutru pela derecha y pela esquierda. </p> <dl><dd><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 \forall a\in A\;,\quad \exists e\in A\;:\quad e\odot a=a\odot e=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∃</mi> <mi>e</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mi>e</mi> <mo>⊙</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>⊙</mo> <mi>e</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a\in A\;,\quad \exists e\in A\;:\quad e\odot a=a\odot e=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49d87cfcd8f4cb9788affcd76d8e4178e183a7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.707ex; height:2.509ex;" alt="{\displaystyle \forall a\in A\;,\quad \exists e\in A\;:\quad e\odot a=a\odot e=a}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Elementu_neutru_pela_derecha">Elementu neutru pela derecha</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=16" title="Editar seición: Elementu neutru pela derecha" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=16" title="Editar el código fuente de la sección: Elementu neutru pela derecha"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vamos dicir que l'elementu <b>e</b>, ye l'elementu neutru pela derecha si: </p> <dl><dd><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 \forall a\in A\;,\quad \exists e\in A\;:\quad e\odot a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∃</mi> <mi>e</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mi>e</mi> <mo>⊙</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a\in A\;,\quad \exists e\in A\;:\quad e\odot a=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c3e00027ca320d7eb02e2f7b7812d4552dab2b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.454ex; height:2.509ex;" alt="{\displaystyle \forall a\in A\;,\quad \exists e\in A\;:\quad e\odot a=a}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Elementu_neutru_pela_esquierda">Elementu neutru pela esquierda</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=17" title="Editar seición: Elementu neutru pela esquierda" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=17" title="Editar el código fuente de la sección: Elementu neutru pela esquierda"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vamos dicir que l'elementu <b>e</b>, ye l'elementu neutru pela esquierda si: </p><p><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 \forall a\in A\;,\quad \exists e\in A\;:\quad a\odot e=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∃</mi> <mi>e</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mi>a</mi> <mo>⊙</mo> <mi>e</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a\in A\;,\quad \exists e\in A\;:\quad a\odot e=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb647e60fb02f51d072066b118b208abdb142910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.454ex; height:2.509ex;" alt="{\displaystyle \forall a\in A\;,\quad \exists e\in A\;:\quad a\odot e=a}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Unicidá_del_elementu_neutru"><span id="Unicid.C3.A1_del_elementu_neutru"></span>Unicidá del elementu neutru</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=18" title="Editar seición: Unicidá del elementu neutru" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=18" title="Editar el código fuente de la sección: Unicidá del elementu neutru"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El elementu neutru ye únicu. Demuestrase por reducción al absurdo. Vamos suponer que esisten dos elementos neutros, e y e'. </p> <ul><li>Por ser e l'elementu neutro, pa to tou a cumplese que e<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 \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"></span>a=a.</li> <li>Por ser e' l'elemtu neutru, pa tou a cumplese que e'<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 \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"></span>a=a.</li></ul> <p>Polo tanto, e<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 \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"></span>a=e'<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 \odot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e009eb8a8839c82aa5c76c15e9f2d67006276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }"></span>a y ye claru que e=e'. </p> <div class="mw-heading mw-heading3"><h3 id="Elementu_simétricu"><span id="Elementu_sim.C3.A9tricu"></span>Elementu simétricu</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=19" title="Editar seición: Elementu simétricu" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=19" title="Editar el código fuente de la sección: Elementu simétricu"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219085"><div role="note" class="hatnote navigation-not-searchable">Artículu principal: <a href="/w/index.php?title=Elementu_sim%C3%A9tricu&action=edit&redlink=1" class="new" title="Elementu simétricu (la páxina nun esiste)">Elementu simétricu</a></div> <p>Dizse que <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 {\overline {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/032e261791bd07a59cf1419352fc66f7901d4b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.345ex; height:2.343ex;" alt="{\displaystyle {\overline {a}}}"></span> ye simétricu de <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> si: </p> <dl><dd><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 {\overline {a}}\odot a=a\odot {\overline {a}}=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>⊙</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>⊙</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {a}}\odot a=a\odot {\overline {a}}=e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c80ff3c76fb0441d1d42631fd672c4215fe2eb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.11ex; height:2.509ex;" alt="{\displaystyle {\overline {a}}\odot a=a\odot {\overline {a}}=e}"></span></dd></dl> <p>onde e ye l'elementu neutru. </p> <div class="mw-heading mw-heading3"><h3 id="Elementu_involutivu">Elementu involutivu</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=20" title="Editar seición: Elementu involutivu" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=20" title="Editar el código fuente de la sección: Elementu involutivu"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dizse que <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 d\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19e83d4b82fcad587a69d1be593cc2c501e5dac0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.8ex; height:2.176ex;" alt="{\displaystyle d\in A}"></span> ye elementu involutivu si: </p> <dl><dd><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 d\odot d=d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>⊙</mo> <mi>d</mi> <mo>=</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\odot d=d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2113d6536c12aeb650b375a95e1ba14d1824d1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.586ex; height:2.343ex;" alt="{\displaystyle d\odot d=d}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Elementu_absorbente">Elementu absorbente</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=21" title="Editar seición: Elementu absorbente" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=21" title="Editar el código fuente de la sección: Elementu absorbente"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219085"><div role="note" class="hatnote navigation-not-searchable">Artículu principal: <a href="/wiki/Elementu_absorbente" title="Elementu absorbente">Elementu absorbente</a></div> <p>Dizse que <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 s\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e110ff17c1db9960eae6b79be3e25bb1e09c174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.674ex; height:2.176ex;" alt="{\displaystyle s\in A}"></span> ye elementu absorbente si: </p> <dl><dd><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 \forall a\in A\;:\quad s\odot a=a\odot s=s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mi>s</mi> <mo>⊙</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>⊙</mo> <mi>s</mi> <mo>=</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a\in A\;:\quad s\odot a=a\odot s=s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8aebc89c9c7f4aad13c1fb737893f196fd6751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:29.62ex; height:2.343ex;" alt="{\displaystyle \forall a\in A\;:\quad s\odot a=a\odot s=s}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Operación_inversa"><span id="Operaci.C3.B3n_inversa"></span>Operación inversa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=22" title="Editar seición: Operación inversa" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=22" title="Editar el código fuente de la sección: Operación inversa"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sía A un conxuntu con una operación binaria <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 (a,\odot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mo>⊙</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,\odot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4adac7ff81c9508176d210c7d92509184365076c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.881ex; height:2.843ex;" alt="{\displaystyle (a,\odot )}"></span>: </p> <dl><dd><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 {\begin{array}{rccl}\odot :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\odot b\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>⊙</mo> <mo>:</mo> </mtd> <mtd> <mi>A</mi> <mo>×</mo> <mi>A</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>A</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">⟼</mo> </mtd> <mtd> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rccl}\odot :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\odot b\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5561c227f9d78bdc583690c28ebaf9b3aea1ddd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.125ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{rccl}\odot :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\odot b\end{array}}}"></span></dd></dl> <p>polo que quepe la ecuación: </p> <dl><dd><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 \forall a,b\in A\,,\;\exists c\in A\;:\quad a\odot b=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mi>c</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a,b\in A\,,\;\exists c\in A\;:\quad a\odot b=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92ca26911d4fce8c720cea46b23d01537a2f7325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.165ex; height:2.509ex;" alt="{\displaystyle \forall a,b\in A\,,\;\exists c\in A\;:\quad a\odot b=c}"></span></dd></dl> <p>Si: </p> <dl><dd><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 a\odot b=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\odot b=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78bd91c2311dd16d83c8df3309a12b4be1ff45c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.173ex; height:2.343ex;" alt="{\displaystyle a\odot b=c}"></span></dd></dl> <p>Si A almite elementos simétricos, defínese: </p> <dl><dd><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 a\odot b\odot {\overline {b}}=c\odot {\overline {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⊙</mo> <mi>b</mi> <mo>⊙</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>c</mi> <mo>⊙</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\odot b\odot {\overline {b}}=c\odot {\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddae1079b6b22f2b9654c1309186102495daead4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.079ex; height:3.176ex;" alt="{\displaystyle a\odot b\odot {\overline {b}}=c\odot {\overline {b}}}"></span></dd></dl> <p>Arrexuntando: </p> <dl><dd><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 a\odot (b\odot {\overline {b}})=c\odot {\overline {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⊙</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⊙</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mo>⊙</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\odot (b\odot {\overline {b}})=c\odot {\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb8ce89670491d5910f4558bbe407c2435c8b8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.888ex; height:3.509ex;" alt="{\displaystyle a\odot (b\odot {\overline {b}})=c\odot {\overline {b}}}"></span></dd></dl> <p>onde e ye l'elementu neutru: </p> <dl><dd><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 a\odot e=c\odot {\overline {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⊙</mo> <mi>e</mi> <mo>=</mo> <mi>c</mi> <mo>⊙</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\odot e=c\odot {\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63733c21a046dbcf60d076441ec6431beb7d93a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.212ex; height:3.176ex;" alt="{\displaystyle a\odot e=c\odot {\overline {b}}}"></span></dd></dl> <p>simplificando: </p> <dl><dd><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 a=c\odot {\overline {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>c</mi> <mo>⊙</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=c\odot {\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/890a13334ca2673c39169155334ea221214a98cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.288ex; height:3.176ex;" alt="{\displaystyle a=c\odot {\overline {b}}}"></span></dd></dl> <p>La operación inversa seria <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 {\overline {\odot }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>⊙</mo> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\odot }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be697a9fca55770ca8c846d57784445ae8d0623f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.923ex; height:2.843ex;" alt="{\displaystyle {\overline {\odot }}}"></span> </p> <dl><dd><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 a=c\,{\overline {\odot }}\,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>c</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>⊙</mo> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=c\,{\overline {\odot }}\,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f219cc1c21733287230d07f97764e5d6bad60f59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.03ex; height:2.843ex;" alt="{\displaystyle a=c\,{\overline {\odot }}\,b}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Otres_propiedaes">Otres propiedaes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=23" title="Editar seición: Otres propiedaes" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=23" title="Editar el código fuente de la sección: Otres propiedaes"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Simplificación_o_cancelativa"><span id="Simplificaci.C3.B3n_o_cancelativa"></span>Simplificación o cancelativa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=24" title="Editar seición: Simplificación o cancelativa" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=24" title="Editar el código fuente de la sección: Simplificación o cancelativa"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sía A cola operación  si ab =ac implica que b=c, dizse que se simplificó a pela esquierda. Y si de ba =ca deduzse b=c y dizse que se simplificó pela derecha. Si puede simplificase per dambos llaos falase de <b>simplificación o cancelación</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Divisores_del_cero">Divisores del cero</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=25" title="Editar seición: Divisores del cero" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=25" title="Editar el código fuente de la sección: Divisores del cero"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sía'l conxuntu A y l'operación * , siendo a ≠ 0, b≠ 0 deduzse qu'ab = 0 , dizse qu'a y b son <b>divisores del 0</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Ver_tamién"><span id="Ver_tami.C3.A9n"></span>Ver tamién</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=26" title="Editar seición: Ver tamién" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=26" title="Editar el código fuente de la sección: Ver tamién"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Otres operaciones: <ul><li><a href="/w/index.php?title=Operaci%C3%B3n_nularia&action=edit&redlink=1" class="new" title="Operación nularia (la páxina nun esiste)">Operación nularia</a></li> <li><a href="/w/index.php?title=Operaci%C3%B3n_unaria&action=edit&redlink=1" class="new" title="Operación unaria (la páxina nun esiste)">Operación unaria</a></li> <li><a href="/w/index.php?title=Operaci%C3%B3n_ternaria&action=edit&redlink=1" class="new" title="Operación ternaria (la páxina nun esiste)">Operación ternaria</a></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencies">Referencies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=27" title="Editar seición: Referencies" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=27" title="Editar el código fuente de la sección: Referencies"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r3503771">@media only screen and (max-width:600px){.mw-parser-output .llistaref{column-count:1!important}}</style><div class="llistaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Brainard Braimah. Definitions of Some Mathematical Terms for 11-18 Year Olds. Xulon Press, novembre 2007, p. 23–. ISBN 978-1-60477-357-6.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><i>A Text Book of Mathematics XII Vol. 1</i>. Rastogi Publications, p. 3–. ISBN 978-81-7133-897-9.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Enllaces_esternos">Enllaces esternos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&veaction=edit&section=28" title="Editar seición: Enllaces esternos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operaci%C3%B3n_binaria&action=edit&section=28" title="Editar el código fuente de la sección: Enllaces esternos"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://wmatem.eis.uva.es/~matpag/CONTENIDOS/Operaciones/operaciones/node1.html">Operaciones binarias</a> por Jose Ignacio Farran Martin (<a href="/wiki/Universid%C3%A1_de_Valladolid" title="Universidá de Valladolid">Universidá de Valladolid</a>)</li> <li><a rel="nofollow" class="external text" href="https://www.uv.mx/personal/aherrera/files/2014/08/20a.-ESTRUCTURAS-ALGEBRAICAS-21-pp.pdf">Estructuras Algebraicas</a> por Aldo Jiménez Arteaga (Universidá Veracruzana)</li></ul> <style data-mw-deduplicate="TemplateStyles:r2260362">.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid #c8ccd1;background-color:#eaecf0;color:#222222}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}</style><div class="mw-authority-control navigation-not-searchable"><div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r4075543">.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:r4182964">.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}</style></div><div role="navigation" class="navbox" aria-labelledby="Control_d&#039;autoridaes" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="Control_d&#039;autoridaes" scope="row" class="navbox-group" style="width:1%;width: 12%; text-align:center;"><a href="/wiki/Ayuda:Control_d%27autoridaes" title="Ayuda:Control d'autoridaes">Control d'autoridaes</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><b>Proyeutos Wikimedia</b></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikidata" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span> Datos:</span> <span class="uid"><a href="https://www.wikidata.org/wiki/Q164307" class="extiw" title="wikidata:Q164307">Q164307</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Multimedia:</span> <span class="uid"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Binary_operations">Binary operations</a></span></span></li></ul> <hr /> <ul><li><b>Identificadores</b></li> <li><span style="white-space:nowrap;"><a href="/wiki/National_Library_of_the_Czech_Republic" class="mw-redirect" title="National Library of the Czech Republic">NKC</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph118879">ph118879</a></span></li> <li><b>Diccionarios y enciclopedies</b></li> <li><span style="white-space:nowrap;"><a href="/wiki/Enciclopedia_Brit%C3%A1nica" class="mw-redirect" title="Enciclopedia Británica">Britannica</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/dyadic-operator">url</a></span></li></ul> </div></td></tr></tbody></table></div><div class="mw-mf-linked-projects hlist"> <ul><li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikidata" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span> Datos:</span> <span class="uid"><a href="https://www.wikidata.org/wiki/Q164307" class="extiw" title="wikidata:Q164307">Q164307</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Multimedia:</span> <span class="uid"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Binary_operations">Binary operations</a></span></span></li></ul> </div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Sacáu de «<a dir="ltr" href="https://ast.wikipedia.org/w/index.php?title=Operación_binaria&oldid=4121027">https://ast.wikipedia.org/w/index.php?title=Operación_binaria&oldid=4121027</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categor%C3%ADas" title="Especial:Categorías">Categoríes</a>: <ul><li><a href="/wiki/Categor%C3%ADa:Wikipedia:Art%C3%ADculos_con_plant%C3%ADes_de_notes_d%27encabezamientu_enllaciando_a_p%C3%A1xines_que_nun_esisten" title="Categoría:Wikipedia:Artículos con plantíes de notes d'encabezamientu enllaciando a páxines que nun esisten">Wikipedia:Artículos con plantíes de notes d'encabezamientu enllaciando a páxines que nun esisten</a></li><li><a href="/wiki/Categor%C3%ADa:%C3%81lxebra_elemental" title="Categoría:Álxebra elemental">Álxebra elemental</a></li><li><a href="/wiki/Categor%C3%ADa:%C3%81lxebra_astrauta" title="Categoría:Álxebra astrauta">Álxebra astrauta</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categoría anubrida: <ul><li><a href="/wiki/Categor%C3%ADa:Wikipedia:P%C3%A1xines_con_etiquetes_de_Wikidata_ensin_traducir" title="Categoría:Wikipedia:Páxines con etiquetes de Wikidata ensin traducir">Wikipedia:Páxines con etiquetes de Wikidata ensin traducir</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"> La última edición d'esta páxina foi'l 17 xin 2024 a les 15:36.</li> <li id="footer-info-copyright">El testu ta disponible baxo la <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.es">Llicencia Creative Commons Reconocimientu/CompartirIgual 4.0</a>; puen aplicase términos adicionales. Llei <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">les condiciones d'usu</a> pa más detalles.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy/es">Política d'intimidá</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:Tocante_a">Tocante a Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Avisu_xeneral">Avisu llegal</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Códigu de conducta</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Desendolcadores</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/ast.wikipedia.org">Estadístiques</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement/es">Declaración de cookies</a></li> <li id="footer-places-mobileview"><a href="//ast.m.wikipedia.org/w/index.php?title=Operaci%C3%B3n_binaria&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Vista pa móvil</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5c59558b9d-8crrn","wgBackendResponseTime":158,"wgPageParseReport":{"limitreport":{"cputime":"0.294","walltime":"0.573","ppvisitednodes":{"value":795,"limit":1000000},"postexpandincludesize":{"value":17442,"limit":2097152},"templateargumentsize":{"value":1189,"limit":2097152},"expansiondepth":{"value":9,"limit":100},"expensivefunctioncount":{"value":9,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":10501,"limit":5000000},"entityaccesscount":{"value":4,"limit":400},"timingprofile":["100.00% 284.228 1 -total"," 43.72% 124.273 1 Plantía:Ficha_xenérica"," 42.93% 122.021 1 Plantía:Infobox"," 38.51% 109.447 1 Plantía:Control_d'autoridaes"," 13.32% 37.856 6 Plantía:AP"," 8.22% 23.350 1 Plantía:Ficha/Socesión"," 2.46% 7.004 1 Plantía:Llistaref"," 1.92% 5.456 1 Plantía:Títulu_ensin_añadíos"]},"scribunto":{"limitreport-timeusage":{"value":"0.154","limit":"10.000"},"limitreport-memusage":{"value":2745754,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-694cf4987f-cj7gs","timestamp":"20241126064106","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Operaci\u00f3n binaria","url":"https:\/\/ast.wikipedia.org\/wiki\/Operaci%C3%B3n_binaria","sameAs":"http:\/\/www.wikidata.org\/entity\/Q164307","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q164307","author":{"@type":"Organization","name":"Collaboradores de los proyeutos de Wikimedia"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2020-09-10T15:29:37Z","dateModified":"2024-01-17T15:36:43Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/7\/79\/Binary_operations_as_black_box.svg","headline":"operaci\u00f3n en matem\u00e1tiques que combina dos elementos pa producir otru elementu"}</script> </body> </html>