CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="id" dir="ltr"> <head> <meta charset="UTF-8"> <title>Sifat asosiatif - Wikipedia bahasa Indonesia, ensiklopedia bebas</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )idwikimwclientpreferences=([^;]+)/);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":["","Januari","Februari","Maret","April","Mei","Juni","Juli","Agustus","September","Oktober","November","Desember"],"wgRequestId":"34cc19c2-2f66-4166-acfc-1924a288d003","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Sifat_asosiatif","wgTitle":"Sifat asosiatif","wgCurRevisionId":26429488,"wgRevisionId":26429488,"wgArticleId":3157674,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Halaman dengan rujukan yang menggunakan parameter yang tidak didukung","Templat webarchive tautan wayback","Operasi biner","Aljabar abstrak","Analisis fungsional","Kaidah inferensi","Aljabar elementer"],"wgPageViewLanguage":"id","wgPageContentLanguage":"id","wgPageContentModel":"wikitext","wgRelevantPageName":"Sifat_asosiatif","wgRelevantArticleId":3157674,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[], "wgRedirectedFrom":"Asosiatif","wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"accuracy":{"levels":2}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"id","pageLanguageDir":"ltr","pageVariantFallbacks":"id"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgInternalRedirectTargetUrl":"/wiki/Sifat_asosiatif","wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q177251","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false, "wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.charinsert-styles":"ready","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"};RLPAGEMODULES=["mediawiki.action.view.redirect","ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.watchlist-notice","ext.gadget.charinsert","ext.gadget.refToolbar","ext.gadget.AdvancedSiteNotices","ext.gadget.switcher", "ext.gadget.Bagikan","ext.gadget.CurIDLink","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","oojs-ui.styles.icons-media","oojs-ui-core.icons"];</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=id&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%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=id&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=id&modules=ext.gadget.charinsert-styles&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=id&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.18"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Sifat asosiatif - Wikipedia bahasa Indonesia, ensiklopedia bebas"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//id.m.wikipedia.org/wiki/Sifat_asosiatif"> <link rel="alternate" type="application/x-wiki" title="Sunting" href="/w/index.php?title=Sifat_asosiatif&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 (id)"> <link rel="EditURI" type="application/rsd+xml" href="//id.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://id.wikipedia.org/wiki/Sifat_asosiatif"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.id"> <link rel="alternate" type="application/atom+xml" title="Umpan Atom Wikipedia" href="/w/index.php?title=Istimewa:Perubahan_terbaru&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-Sifat_asosiatif rootpage-Sifat_asosiatif skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Lompat ke isi</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="Situs"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Menu utama" > <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="Menu utama" > <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">Menu utama</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">Menu utama</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">sembunyikan</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigasi </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage" class="mw-list-item"><a href="/wiki/Halaman_Utama" title="Kunjungi Halaman Utama [z]" accesskey="z"><span>Halaman Utama</span></a></li><li id="n-Daftar-isi" class="mw-list-item"><a href="/wiki/Wikipedia:Isi"><span>Daftar isi</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Istimewa:Perubahan_terbaru" title="Daftar perubahan terbaru dalam wiki. [r]" accesskey="r"><span>Perubahan terbaru</span></a></li><li id="n-Artikel-pilihan" class="mw-list-item"><a href="/wiki/Wikipedia:Artikel_pilihan/Topik"><span>Artikel pilihan</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Peristiwa_terkini" title="Temukan informasi tentang peristiwa terkini"><span>Peristiwa terkini</span></a></li><li id="n-newpage" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_baru"><span>Halaman baru</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_sembarang" title="Tampilkan sembarang halaman [x]" accesskey="x"><span>Halaman sembarang</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_istimewa"><span>Halaman istimewa</span></a></li> </ul> </div> </div> <div id="p-Komunitas" class="vector-menu mw-portlet mw-portlet-Komunitas" > <div class="vector-menu-heading"> Komunitas </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-Warung-Kopi" class="mw-list-item"><a href="/wiki/Wikipedia:Warung_Kopi"><span>Warung Kopi</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Portal:Komunitas" title="Tentang proyek, apa yang dapat Anda lakukan, di mana untuk mencari sesuatu"><span>Portal komunitas</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Bantuan:Isi" title="Tempat mencari bantuan."><span>Bantuan</span></a></li> </ul> </div> </div> <div id="p-Wikipedia" class="vector-menu mw-portlet mw-portlet-Wikipedia" > <div class="vector-menu-heading"> Wikipedia </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:Perihal"><span>Tentang Wikipedia</span></a></li><li id="n-Pancapilar" class="mw-list-item"><a href="/wiki/Wikipedia:Pancapilar"><span>Pancapilar</span></a></li><li id="n-Kebijakan" class="mw-list-item"><a href="/wiki/Wikipedia:Kebijakan_dan_pedoman"><span>Kebijakan</span></a></li><li id="n-Hubungi-kami" class="mw-list-item"><a href="/wiki/Wikipedia:Hubungi_kami"><span>Hubungi kami</span></a></li><li id="n-Bak-pasir" class="mw-list-item"><a href="/wiki/Wikipedia:Bak_pasir"><span>Bak pasir</span></a></li> </ul> </div> </div> <div id="p-Bagikan" class="vector-menu mw-portlet mw-portlet-Bagikan emptyPortlet" > <div class="vector-menu-heading"> Bagikan </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Halaman_Utama" 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="Ensiklopedia Bebas" src="/static/images/mobile/copyright/wikipedia-tagline-id.svg" width="120" height="14" style="width: 7.5em; height: 0.875em;"> </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/Istimewa:Pencarian" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Cari di Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Pencarian</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="Telusuri Wikipedia" aria-label="Telusuri Wikipedia" autocapitalize="sentences" title="Cari di 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="Istimewa:Pencarian"> </div> <button class="cdx-button cdx-search-input__end-button">Cari</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Perkakas pribadi"> <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="Tampilan"> <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="Tampilan" > <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">Tampilan</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=id.wikipedia.org&uselang=id" class=""><span>Menyumbang</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=Istimewa:Buat_akun&returnto=Sifat+asosiatif" title="Anda dianjurkan untuk membuat akun dan masuk log; meskipun, hal itu tidak diwajibkan" class=""><span>Buat akun baru</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=Istimewa:Masuk_log&returnto=Sifat+asosiatif" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o" class=""><span>Masuk log</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="Opsi lainnya" > <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="Perkakas pribadi" > <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">Perkakas pribadi</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menu pengguna" > <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=id.wikipedia.org&uselang=id"><span>Menyumbang</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Buat_akun&returnto=Sifat+asosiatif" title="Anda dianjurkan untuk membuat akun dan masuk log; meskipun, hal itu tidak diwajibkan"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Buat akun baru</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Masuk_log&returnto=Sifat+asosiatif" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Masuk log</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"> Halaman penyunting yang telah keluar log <a href="/wiki/Bantuan:Pengantar" aria-label="Pelajari lebih lanjut tentang menyunting"><span>pelajari lebih lanjut</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/Istimewa:Kontribusi_saya" title="Daftar suntingan yang dibuat dari alamat IP ini [y]" accesskey="y"><span>Kontribusi</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Istimewa:Pembicaraan_saya" title="Pembicaraan tentang suntingan dari alamat IP ini [n]" accesskey="n"><span>Pembicaraan</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="Situs"> <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="Daftar isi" 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">Daftar isi</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sembunyikan</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">Awal</div> </a> </li> <li id="toc-Definisi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definisi"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definisi</span> </div> </a> <ul id="toc-Definisi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hukum_asosiatif_yang_digeneralisasikan" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hukum_asosiatif_yang_digeneralisasikan"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Hukum asosiatif yang digeneralisasikan</span> </div> </a> <ul id="toc-Hukum_asosiatif_yang_digeneralisasikan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Contoh" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Contoh"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Contoh</span> </div> </a> <ul id="toc-Contoh-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logika_proposisional" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Logika_proposisional"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Logika proposisional</span> </div> </a> <button aria-controls="toc-Logika_proposisional-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>Gulingkan subbagian Logika proposisional</span> </button> <ul id="toc-Logika_proposisional-sublist" class="vector-toc-list"> <li id="toc-Aturan_penggantian" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aturan_penggantian"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Aturan penggantian</span> </div> </a> <ul id="toc-Aturan_penggantian-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Penghubung_fungsional_kebenaran" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Penghubung_fungsional_kebenaran"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Penghubung fungsional kebenaran</span> </div> </a> <ul id="toc-Penghubung_fungsional_kebenaran-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Operasi_nonasosiatif" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operasi_nonasosiatif"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Operasi nonasosiatif</span> </div> </a> <button aria-controls="toc-Operasi_nonasosiatif-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>Gulingkan subbagian Operasi nonasosiatif</span> </button> <ul id="toc-Operasi_nonasosiatif-sublist" class="vector-toc-list"> <li id="toc-Nonasosiatif_dari_perhitungan_titik_mengambang" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonasosiatif_dari_perhitungan_titik_mengambang"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Nonasosiatif dari perhitungan titik mengambang</span> </div> </a> <ul id="toc-Nonasosiatif_dari_perhitungan_titik_mengambang-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notasi_untuk_operasi-operasi_nonasosiastif" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notasi_untuk_operasi-operasi_nonasosiastif"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Notasi untuk operasi-operasi nonasosiastif</span> </div> </a> <ul id="toc-Notasi_untuk_operasi-operasi_nonasosiastif-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lihat_pula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lihat_pula"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Lihat pula</span> </div> </a> <ul id="toc-Lihat_pula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referensi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referensi"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Referensi</span> </div> </a> <ul id="toc-Referensi-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="Daftar isi" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Daftar Isi" > <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="Gulingkan daftar isi" > <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">Gulingkan daftar isi</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">Sifat asosiatif</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="Pergi ke artikel dalam bahasa lain. Terdapat 67 bahasa" > <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-67" 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">67 bahasa</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%AA%D8%AC%D9%85%D9%8A%D8%B9%D9%8A%D8%A9" title="عملية تجميعية – Arab" lang="ar" hreflang="ar" data-title="عملية تجميعية" data-language-autonym="العربية" data-language-local-name="Arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Asociativid%C3%A1" title="Asociatividá – Asturia" lang="ast" hreflang="ast" data-title="Asociatividá" data-language-autonym="Asturianu" data-language-local-name="Asturia" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D1%81%D1%81%D0%BE%D1%86%D0%B8%D0%B0%D1%82%D0%B8%D0%B2%D0%BB%D1%8B%D2%A1_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ассоциативлыҡ (математика) – Bashkir" lang="ba" hreflang="ba" data-title="Ассоциативлыҡ (математика)" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D1%81%D0%B0%D1%86%D1%8B%D1%8F%D1%82%D1%8B%D1%9E%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D1%8B%D1%8F" title="Асацыятыўная аперацыя – Belarusia" lang="be" hreflang="be" data-title="Асацыятыўная аперацыя" data-language-autonym="Беларуская" data-language-local-name="Belarusia" 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%90%D1%81%D0%BE%D1%86%D0%B8%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82" title="Асоциативност – Bulgaria" lang="bg" hreflang="bg" data-title="Асоциативност" data-language-autonym="Български" data-language-local-name="Bulgaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Asocijativnost" title="Asocijativnost – Bosnia" lang="bs" hreflang="bs" data-title="Asocijativnost" data-language-autonym="Bosanski" data-language-local-name="Bosnia" 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/Propietat_associativa" title="Propietat associativa – Katalan" lang="ca" hreflang="ca" data-title="Propietat associativa" data-language-autonym="Català" data-language-local-name="Katalan" 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/%DB%8C%DB%95%DA%A9%D8%AA%D8%B1%D8%A8%DB%95%D8%B3%D8%AA%D9%86" title="یەکتربەستن – Kurdi Sorani" lang="ckb" hreflang="ckb" data-title="یەکتربەستن" data-language-autonym="کوردی" data-language-local-name="Kurdi Sorani" 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/Asociativita" title="Asociativita – Ceko" lang="cs" hreflang="cs" data-title="Asociativita" data-language-autonym="Čeština" data-language-local-name="Ceko" 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%90%D1%81%D1%81%D0%B0%D1%86%D0%B8%D0%B0%D1%82%D0%B8%D0%B2%D0%BB%C4%83%D1%85_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ассациативлăх (математика) – Chuvash" lang="cv" hreflang="cv" data-title="Ассациативлăх (математика)" 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/Associativitet" title="Associativitet – Dansk" lang="da" hreflang="da" data-title="Associativitet" data-language-autonym="Dansk" data-language-local-name="Dansk" 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/Assoziativgesetz" title="Assoziativgesetz – Jerman" lang="de" hreflang="de" data-title="Assoziativgesetz" data-language-autonym="Deutsch" data-language-local-name="Jerman" 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%A0%CF%81%CE%BF%CF%83%CE%B5%CF%84%CE%B1%CE%B9%CF%81%CE%B9%CF%83%CF%84%CE%B9%CE%BA%CE%AE_%CE%B9%CE%B4%CE%B9%CF%8C%CF%84%CE%B7%CF%84%CE%B1" title="Προσεταιριστική ιδιότητα – Yunani" lang="el" hreflang="el" data-title="Προσεταιριστική ιδιότητα" data-language-autonym="Ελληνικά" data-language-local-name="Yunani" 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/Associative_property" title="Associative property – Inggris" lang="en" hreflang="en" data-title="Associative property" data-language-autonym="English" data-language-local-name="Inggris" 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/Asocieco" title="Asocieco – Esperanto" lang="eo" hreflang="eo" data-title="Asocieco" 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/Asociatividad_(%C3%A1lgebra)" title="Asociatividad (álgebra) – Spanyol" lang="es" hreflang="es" data-title="Asociatividad (álgebra)" data-language-autonym="Español" data-language-local-name="Spanyol" 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/Assotsiatiivsus" title="Assotsiatiivsus – Estonia" lang="et" hreflang="et" data-title="Assotsiatiivsus" data-language-autonym="Eesti" data-language-local-name="Estonia" 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/Elkarkortasun" title="Elkarkortasun – Basque" lang="eu" hreflang="eu" data-title="Elkarkortasun" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AE%D8%A7%D8%B5%DB%8C%D8%AA_%D8%B4%D8%B1%DA%A9%D8%AA%E2%80%8C%D9%BE%D8%B0%DB%8C%D8%B1%DB%8C" title="خاصیت شرکت‌پذیری – Persia" lang="fa" hreflang="fa" data-title="خاصیت شرکت‌پذیری" data-language-autonym="فارسی" data-language-local-name="Persia" 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/Liit%C3%A4nn%C3%A4isyys" title="Liitännäisyys – Suomi" lang="fi" hreflang="fi" data-title="Liitännäisyys" data-language-autonym="Suomi" data-language-local-name="Suomi" 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/Associativit%C3%A9" title="Associativité – Prancis" lang="fr" hreflang="fr" data-title="Associativité" data-language-autonym="Français" data-language-local-name="Prancis" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Asotsiatiifgesets" title="Asotsiatiifgesets – Frisia Utara" lang="frr" hreflang="frr" data-title="Asotsiatiifgesets" data-language-autonym="Nordfriisk" data-language-local-name="Frisia Utara" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Oibr%C3%ADocht_chomhthiomsaitheach" title="Oibríocht chomhthiomsaitheach – Irlandia" lang="ga" hreflang="ga" data-title="Oibríocht chomhthiomsaitheach" data-language-autonym="Gaeilge" data-language-local-name="Irlandia" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Asociatividade_(%C3%A1lxebra)" title="Asociatividade (álxebra) – Galisia" lang="gl" hreflang="gl" data-title="Asociatividade (álxebra)" data-language-autonym="Galego" data-language-local-name="Galisia" 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%90%D7%A1%D7%95%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%91%D7%99%D7%AA" title="פעולה אסוציאטיבית – Ibrani" lang="he" hreflang="he" data-title="פעולה אסוציאטיבית" data-language-autonym="עברית" data-language-local-name="Ibrani" 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/Asocijativnost" title="Asocijativnost – Kroasia" lang="hr" hreflang="hr" data-title="Asocijativnost" data-language-autonym="Hrvatski" data-language-local-name="Kroasia" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Asszociativit%C3%A1s" title="Asszociativitás – Hungaria" lang="hu" hreflang="hu" data-title="Asszociativitás" data-language-autonym="Magyar" data-language-local-name="Hungaria" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B6%D5%B8%D6%82%D5%A3%D5%B8%D6%80%D5%A4%D5%A1%D5%AF%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Զուգորդականություն – Armenia" lang="hy" hreflang="hy" data-title="Զուգորդականություն" data-language-autonym="Հայերեն" data-language-local-name="Armenia" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Associativitate" title="Associativitate – Interlingua" lang="ia" hreflang="ia" data-title="Associativitate" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tengiregla" title="Tengiregla – Islandia" lang="is" hreflang="is" data-title="Tengiregla" data-language-autonym="Íslenska" data-language-local-name="Islandia" 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/Associativit%C3%A0" title="Associatività – Italia" lang="it" hreflang="it" data-title="Associatività" data-language-autonym="Italiano" data-language-local-name="Italia" 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/%E7%B5%90%E5%90%88%E6%B3%95%E5%89%87" title="結合法則 – Jepang" lang="ja" hreflang="ja" data-title="結合法則" data-language-autonym="日本語" data-language-local-name="Jepang" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%81%D1%81%D0%BE%D1%86%D0%B8%D0%B0%D1%82%D0%B8%D0%B2%D1%82%D1%96%D0%BA_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Ассоциативтік операция – Kazakh" lang="kk" hreflang="kk" data-title="Ассоциативтік операция" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B2%B0%ED%95%A9%EB%B2%95%EC%B9%99" title="결합법칙 – Korea" lang="ko" hreflang="ko" data-title="결합법칙" data-language-autonym="한국어" data-language-local-name="Korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Associativitas_(mathematica)" title="Associativitas (mathematica) – Latin" lang="la" hreflang="la" data-title="Associativitas (mathematica)" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Asociatyvumas" title="Asociatyvumas – Lituania" lang="lt" hreflang="lt" data-title="Asociatyvumas" data-language-autonym="Lietuvių" data-language-local-name="Lituania" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Asociativit%C4%81te" title="Asociativitāte – Latvia" lang="lv" hreflang="lv" data-title="Asociativitāte" data-language-autonym="Latviešu" data-language-local-name="Latvia" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%90%D1%81%D0%BE%D1%86%D0%B8%D1%98%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82" title="Асоцијативност – Makedonia" lang="mk" hreflang="mk" data-title="Асоцијативност" data-language-autonym="Македонски" data-language-local-name="Makedonia" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%BE%E0%B4%B9%E0%B4%9A%E0%B4%B0%E0%B5%8D%E0%B4%AF%E0%B4%A8%E0%B4%BF%E0%B4%AF%E0%B4%AE%E0%B4%82" title="സാഹചര്യനിയമം – Malayalam" lang="ml" hreflang="ml" data-title="സാഹചര്യനിയമം" 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/Kalis_sekutuan" title="Kalis sekutuan – Melayu" lang="ms" hreflang="ms" data-title="Kalis sekutuan" data-language-autonym="Bahasa Melayu" data-language-local-name="Melayu" 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/Associativiteit_(wiskunde)" title="Associativiteit (wiskunde) – Belanda" lang="nl" hreflang="nl" data-title="Associativiteit (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Belanda" 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/Assosiativitet" title="Assosiativitet – Nynorsk Norwegia" lang="nn" hreflang="nn" data-title="Assosiativitet" data-language-autonym="Norsk nynorsk" data-language-local-name="Nynorsk Norwegia" 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/Assosiativ_lov" title="Assosiativ lov – Bokmål Norwegia" lang="nb" hreflang="nb" data-title="Assosiativ lov" data-language-autonym="Norsk bokmål" data-language-local-name="Bokmål Norwegia" 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/Associativitat" title="Associativitat – Ositania" lang="oc" hreflang="oc" data-title="Associativitat" data-language-autonym="Occitan" data-language-local-name="Ositania" 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/%C5%81%C4%85czno%C5%9B%C4%87_(matematyka)" title="Łączność (matematyka) – Polski" lang="pl" hreflang="pl" data-title="Łączność (matematyka)" data-language-autonym="Polski" data-language-local-name="Polski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Associatividade" title="Associatividade – Portugis" lang="pt" hreflang="pt" data-title="Associatividade" data-language-autonym="Português" data-language-local-name="Portugis" 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/Asociativitate" title="Asociativitate – Rumania" lang="ro" hreflang="ro" data-title="Asociativitate" data-language-autonym="Română" data-language-local-name="Rumania" 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%90%D1%81%D1%81%D0%BE%D1%86%D0%B8%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82%D1%8C_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ассоциативность (математика) – Rusia" lang="ru" hreflang="ru" data-title="Ассоциативность (математика)" data-language-autonym="Русский" data-language-local-name="Rusia" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Asocijativnost" title="Asocijativnost – Serbo-Kroasia" lang="sh" hreflang="sh" data-title="Asocijativnost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Kroasia" 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/Associativity" title="Associativity – Simple English" lang="en-simple" hreflang="en-simple" data-title="Associativity" 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/Asociat%C3%ADvnos%C5%A5" title="Asociatívnosť – Slovak" lang="sk" hreflang="sk" data-title="Asociatívnosť" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Asociativnost" title="Asociativnost – Slovenia" lang="sl" hreflang="sl" data-title="Asociativnost" data-language-autonym="Slovenščina" data-language-local-name="Slovenia" 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/Vetia_e_shoq%C3%ABrimit" title="Vetia e shoqërimit – Albania" lang="sq" hreflang="sq" data-title="Vetia e shoqërimit" data-language-autonym="Shqip" data-language-local-name="Albania" 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%90%D1%81%D0%BE%D1%86%D0%B8%D1%98%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82" title="Асоцијативност – Serbia" lang="sr" hreflang="sr" data-title="Асоцијативност" data-language-autonym="Српски / srpski" data-language-local-name="Serbia" 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/Associativitet" title="Associativitet – Swedia" lang="sv" hreflang="sv" data-title="Associativitet" data-language-autonym="Svenska" data-language-local-name="Swedia" 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%9A%E0%AF%87%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AE%A3%E0%AF%8D%E0%AE%AA%E0%AF%81" title="சேர்ப்புப் பண்பு – Tamil" lang="ta" hreflang="ta" data-title="சேர்ப்புப் பண்பு" 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%AA%E0%B8%A1%E0%B8%9A%E0%B8%B1%E0%B8%95%E0%B8%B4%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%9B%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%99%E0%B8%AB%E0%B8%A1%E0%B8%B9%E0%B9%88" title="สมบัติการเปลี่ยนหมู่ – Thai" lang="th" hreflang="th" data-title="สมบัติการเปลี่ยนหมู่" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Birle%C5%9Fme_%C3%B6zelli%C4%9Fi_(ikili_i%C5%9Flemler)" title="Birleşme özelliği (ikili işlemler) – Turki" lang="tr" hreflang="tr" data-title="Birleşme özelliği (ikili işlemler)" data-language-autonym="Türkçe" data-language-local-name="Turki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%90%D1%81%D1%81%D0%BE%D1%86%D0%B8%D0%B0%D1%82%D0%B8%D0%B2%D0%BB%D1%8B%D0%BA" title="Ассоциативлык – Tatar" lang="tt" hreflang="tt" data-title="Ассоциативлык" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D1%81%D0%BE%D1%86%D1%96%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Асоціативність – Ukraina" lang="uk" hreflang="uk" data-title="Асоціативність" data-language-autonym="Українська" data-language-local-name="Ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Assotsiativlik" title="Assotsiativlik – Uzbek" lang="uz" hreflang="uz" data-title="Assotsiativlik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Propiet%C3%A0_asociativa" title="Propietà asociativa – Venesia" lang="vec" hreflang="vec" data-title="Propietà asociativa" data-language-autonym="Vèneto" data-language-local-name="Venesia" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%C3%ADnh_k%E1%BA%BFt_h%E1%BB%A3p" title="Tính kết hợp – Vietnam" lang="vi" hreflang="vi" data-title="Tính kết hợp" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnam" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BB%93%E5%90%88%E5%BE%8B" title="结合律 – Wu Tionghoa" lang="wuu" hreflang="wuu" data-title="结合律" data-language-autonym="吴语" data-language-local-name="Wu Tionghoa" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%A1%D7%90%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%95%D7%95%D7%99%D7%98%D7%A2%D7%98" title="אסאציאטיוויטעט – Yiddish" lang="yi" hreflang="yi" data-title="אסאציאטיוויטעט" 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/%E7%BB%93%E5%90%88%E5%BE%8B" title="结合律 – Tionghoa" lang="zh" hreflang="zh" data-title="结合律" data-language-autonym="中文" data-language-local-name="Tionghoa" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B5%90%E5%90%88%E5%BE%8B" title="結合律 – Kanton" lang="yue" hreflang="yue" data-title="結合律" data-language-autonym="粵語" data-language-local-name="Kanton" 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/Q177251#sitelinks-wikipedia" title="Sunting pranala interwiki" class="wbc-editpage">Sunting pranala</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="Ruang nama"> <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/Sifat_asosiatif" title="Lihat halaman isi [c]" accesskey="c"><span>Halaman</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Pembicaraan:Sifat_asosiatif" rel="discussion" title="Pembicaraan halaman isi [t]" accesskey="t"><span>Pembicaraan</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="Ubah varian bahasa" > <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">Bahasa Indonesia</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="Tampilan"> <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/Sifat_asosiatif"><span>Baca</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit" title="Sunting halaman ini [v]" accesskey="v"><span>Sunting</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sifat_asosiatif&action=edit" title="Sunting kode sumber halaman ini [e]" accesskey="e"><span>Sunting sumber</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sifat_asosiatif&action=history" title="Revisi sebelumnya dari halaman ini. [h]" accesskey="h"><span>Lihat riwayat</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Peralatan halaman"> <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="Perkakas" > <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">Perkakas</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">Perkakas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">sembunyikan</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Opsi lainnya" > <div class="vector-menu-heading"> Tindakan </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/Sifat_asosiatif"><span>Baca</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit" title="Sunting halaman ini [v]" accesskey="v"><span>Sunting</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Sifat_asosiatif&action=edit" title="Sunting kode sumber halaman ini [e]" accesskey="e"><span>Sunting sumber</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Sifat_asosiatif&action=history"><span>Lihat riwayat</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Umum </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Istimewa:Pranala_balik/Sifat_asosiatif" title="Daftar semua halaman wiki yang memiliki pranala ke halaman ini [j]" accesskey="j"><span>Pranala balik</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Istimewa:Perubahan_terkait/Sifat_asosiatif" rel="nofollow" title="Perubahan terbaru halaman-halaman yang memiliki pranala ke halaman ini [k]" accesskey="k"><span>Perubahan terkait</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Sifat_asosiatif&oldid=26429488" title="Pranala permanen untuk revisi halaman ini"><span>Pranala permanen</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Sifat_asosiatif&action=info" title="Informasi lanjut tentang halaman ini"><span>Informasi halaman</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Kutip&page=Sifat_asosiatif&id=26429488&wpFormIdentifier=titleform" title="Informasi tentang bagaimana mengutip halaman ini"><span>Kutip halaman ini</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Istimewa:UrlShortener&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FSifat_asosiatif"><span>Lihat URL pendek</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Istimewa:QrCode&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FSifat_asosiatif"><span>Unduh kode 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"> Cetak/ekspor </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=Istimewa:Buku&bookcmd=book_creator&referer=Sifat+asosiatif"><span>Buat buku</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Istimewa:DownloadAsPdf&page=Sifat_asosiatif&action=show-download-screen"><span>Unduh versi PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Sifat_asosiatif&printable=yes" title="Versi cetak halaman ini [p]" accesskey="p"><span>Versi cetak</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"> Dalam proyek lain </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q177251" title="Pranala untuk menghubungkan butir pada ruang penyimpanan data [g]" accesskey="g"><span>Butir di 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="Peralatan halaman"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Tampilan"> <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">Tampilan</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sembunyikan</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">Dari Wikipedia bahasa Indonesia, ensiklopedia bebas</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Dialihkan dari <a href="/w/index.php?title=Asosiatif&redirect=no" class="mw-redirect" title="Asosiatif">Asosiatif</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="id" dir="ltr"><p><a href="/w/index.php?title=Templat:Kaidah_transformasi&action=edit&redlink=1" class="new" title="Templat:Kaidah transformasi (halaman belum tersedia)">Templat:Kaidah transformasi</a> </p><p>Dalam <a href="/wiki/Matematika" title="Matematika">matematika</a>, <b>sifat asosiatif</b><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> adalah sifat dari beberapa <a href="/wiki/Operasi_biner" title="Operasi biner">operasi biner</a>, yang berarti bahwa mengatur ulang tanda kurung dalam ekspresi yang tidak mengubah hasilnya. Dalam <a href="/w/index.php?title=Logika_proposisional&action=edit&redlink=1" class="new" title="Logika proposisional (halaman belum tersedia)">logika proposisional</a>, <b>asosiativitas</b> adalah <a href="/wiki/Validitas_(logika)" title="Validitas (logika)">valid</a> <a href="/w/index.php?title=Kaidah_penggantian&action=edit&redlink=1" class="new" title="Kaidah penggantian (halaman belum tersedia)">kaidah penggantian</a> untuk <a href="/w/index.php?title=Rumus_bentuk_baik&action=edit&redlink=1" class="new" title="Rumus bentuk baik (halaman belum tersedia)">ekspresi</a> dalam <a href="/w/index.php?title=Bukti_formal&action=edit&redlink=1" class="new" title="Bukti formal (halaman belum tersedia)">bukti logika</a>. </p><p>Dalam ekspresi dengan dua atau lebih dari satu baris dari operasi asosiatif, urutan <a href="/wiki/Operasi_(matematika)" title="Operasi (matematika)">operasi</a> untuk urutan <a href="/wiki/Operand" class="mw-redirect" title="Operand">operand</a> yang tidak berubah. Artinya, menata ulang <a href="/wiki/Tanda_kurung" title="Tanda kurung">tanda kurung</a> dalam ekspresi tersebut tidak akan mengubah nilainya. Perhatikan persamaan berikut: </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 (2+3)+4=2+(3+4)=9\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>4</mn> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>9</mn> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2+3)+4=2+(3+4)=9\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b33314f4fc13b0ee84b3386ca9d0755137b026ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.701ex; height:2.843ex;" alt="{\displaystyle (2+3)+4=2+(3+4)=9\,}" /></span></dd></dl> <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 2\times (3\times 4)=(2\times 3)\times 4=24.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>×</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>×</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mn>4</mn> <mo>=</mo> <mn>24.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times (3\times 4)=(2\times 3)\times 4=24.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4322e3534b2318b8f51a29363c45fed9754c3019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.124ex; height:2.843ex;" alt="{\displaystyle 2\times (3\times 4)=(2\times 3)\times 4=24.}" /></span></dd></dl> <p>Meskipun tanda kurung diatur ulang pada setiap baris, nilai ekspresi tersebut tidak diubah. Karena penjumlahan dan perkalian terdapat pada <a href="/wiki/Bilangan_riil" title="Bilangan riil">bilangan riil</a>, maka dikatakan bahwa "penjumlahan dan perkalian bilangan riil adalah operasi asosiatif". </p><p>Asosiatif berbeda dengan <a href="/wiki/Komutativitas" class="mw-redirect" title="Komutativitas">komutativitas</a>, dengan urutan dua <a href="/wiki/Operan" title="Operan">operan</a> memengaruhi hasil. Misalnya, urutan tidak menjadi masalah dalam perkalian bilangan riil, yaitu <span class="nowrap"><i>a</i> × <i>b</i> = <i>b</i> × <i>a</i></span>, jadi perkalian bilangan riil adalah operasi komutatif. </p><p>Operasi asosiatif dalam matematika; pada kenyataannya, banyak <a href="/wiki/Struktur_aljabar" title="Struktur aljabar">struktur aljabar</a> (yaitu <a href="/w/index.php?title=Semigrup_(matematika)&action=edit&redlink=1" class="new" title="Semigrup (matematika) (halaman belum tersedia)">semigrup</a> dan <a href="/wiki/Kategori_(matematika)" title="Kategori (matematika)">kategori</a>) secara eksplisit membutuhkan operasi biner untuk menjadi asosiatif. </p><p>Namun, terdapat operasi yang bukan asosiatif yaitu nonasosiatif; beberapa contoh termasuk <a href="/wiki/Pengurangan" title="Pengurangan">pengurangan</a>, <a href="/wiki/Eksponen" class="mw-redirect" title="Eksponen">eksponen</a>, dan <a href="/w/index.php?title=Perkalian_silang_vektor&action=edit&redlink=1" class="new" title="Perkalian silang vektor (halaman belum tersedia)">perkalian silang vektor</a>. Berbeda dengan sifat teoritis bilangan riil, penambahan bilangan <a href="/w/index.php?title=Titik_pengambangan&action=edit&redlink=1" class="new" title="Titik pengambangan (halaman belum tersedia)">titik pengambangan</a> dalam <a href="/wiki/Ilmu_komputer" title="Ilmu komputer">ilmu komputer</a> yang tidak bersifat asosiatif, dan pilihan cara mengaitkan ekspresi dapat berpengaruh signifikan pada kesalahan pembulatan. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definisi">Definisi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit&section=1" title="Sunting bagian: Definisi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sifat_asosiatif&action=edit&section=1" title="Sunting kode sumber bagian: Definisi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Semigroup_associative.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Semigroup_associative.svg/220px-Semigroup_associative.svg.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Semigroup_associative.svg/330px-Semigroup_associative.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Semigroup_associative.svg/440px-Semigroup_associative.svg.png 2x" data-file-width="250" data-file-height="125" /></a><figcaption>Sebuah operasi biner ∗ pada himpunan <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> asosiatif ketika <a href="/wiki/Diagram_komutatif" title="Diagram komutatif">diagram ini komutatif</a>. Artinya, ketika dua jalur dari <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\times S\times S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>×</mo> <mi>S</mi> <mo>×</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\times S\times S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ed3b1dad02488be7c62875f85e6567be43794ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.178ex; height:2.176ex;" alt="{\displaystyle S\times S\times S}" /></span> ke komposisi <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> menadi fungsi yang sama dari <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\times S\times S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>×</mo> <mi>S</mi> <mo>×</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\times S\times S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ed3b1dad02488be7c62875f85e6567be43794ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.178ex; height:2.176ex;" alt="{\displaystyle S\times S\times S}" /></span> ke <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span>.</figcaption></figure> <p>Secara formal, sebuah <a href="/wiki/Operasi_biner" title="Operasi biner">operasi biner</a> <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span> pada sebuah himpunan <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> disebut <b>asosiatif</b> jika memenuhi <b>hukum asosiatif</b>. </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 (x*y)*z=x*(y*z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x*y)*z=x*(y*z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43534cf00082214688f06873ec9ea32caa7e3305" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.643ex; height:2.843ex;" alt="{\displaystyle (x*y)*z=x*(y*z)}" /></span>, untuk semua <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 x,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}" /></span> dalam <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> </p><p>Disini <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span> digunakan untuk menggantikan simbol operasi, yang mungkin merupakan simbol apapun, dan meskipun ketiadaan dari simbol (<a href="https://en.wiktionary.org/wiki/juxtaposition" class="extiw" title="wiktionary:juxtaposition">penjajaran</a>) sebagai untuk <a href="/wiki/Perkalian" title="Perkalian">perkalian</a>. </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 (xy)z=x(yz)=xyz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mi>y</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xy)z=x(yz)=xyz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73ffabcc29c9c198634297c4f9b58f21d124cb53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.535ex; height:2.843ex;" alt="{\displaystyle (xy)z=x(yz)=xyz}" /></span>, untuk semua <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 x,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}" /></span> dalam <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span>. </p><p>Hukum asosiatif bisa juga diekspresikan dalam notasi fungsional jadiː <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 f(f(x,y),z)=f(x,f(y,z))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(f(x,y),z)=f(x,f(y,z))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b42ae7d31aa2b0eaeb54a409811229114b4e70a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.732ex; height:2.843ex;" alt="{\displaystyle f(f(x,y),z)=f(x,f(y,z))}" /></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Hukum_asosiatif_yang_digeneralisasikan">Hukum asosiatif yang digeneralisasikan</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit&section=2" title="Sunting bagian: Hukum asosiatif yang digeneralisasikan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sifat_asosiatif&action=edit&section=2" title="Sunting kode sumber bagian: Hukum asosiatif yang digeneralisasikan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Berkas:Tamari_lattice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Tamari_lattice.svg/250px-Tamari_lattice.svg.png" decoding="async" width="250" height="348" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Tamari_lattice.svg/375px-Tamari_lattice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Tamari_lattice.svg/500px-Tamari_lattice.svg.png 2x" data-file-width="504" data-file-height="702" /></a><figcaption>Dalam ketiadaan dari sifat asosiatif, kelima faktor <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,b,c,d,e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c,d,e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a298ef3bf401ccd1730dcb2834357e5c61d6cdf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.669ex; height:2.509ex;" alt="{\displaystyle a,b,c,d,e}" /></span> menghasilkan sebuah <a href="/w/index.php?title=Kisi_Tamari&action=edit&redlink=1" class="new" title="Kisi Tamari (halaman belum tersedia)">kisi Tamari</a> urutan keempat, produk yang mungkin berbeda.</figcaption></figure> <p>Jika sebuah operasi biner adalah asosiatif, penerapan berulang dari operasi menghasilkan hasil yang sama terlepas dan bagaimana pasangan tanda kurung yang sah disisipkan dalam ekspresi.<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> Ini disebut <b>hukum asosiatif yang digeneralisasi</b>. Misalnya, sebuah porduk fari empat anggota bisa ditulis bisa ditulis, tanpa menggantikan urutan dari faktor-faktor, dalam lima kemungkinanː </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 ((ab)c)d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((ab)c)d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1483412ecf78ec5c5d17fb5d7bebfa2a2166e107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.069ex; height:2.843ex;" alt="{\displaystyle ((ab)c)d}" /></span></dd> <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 (ab)(cd)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ab)(cd)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3f01186cb5d5152b51841e993d8ae657b701fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.069ex; height:2.843ex;" alt="{\displaystyle (ab)(cd)}" /></span></dd> <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(bc))d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a(bc))d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b76ed950d0e00a64c6526257b8e5f6ed797b8db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.069ex; height:2.843ex;" alt="{\displaystyle (a(bc))d}" /></span></dd> <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((bc)d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a((bc)d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/816d85fb33e74ec781a83bc54c883bcccfbe135b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.069ex; height:2.843ex;" alt="{\displaystyle a((bc)d)}" /></span></dd> <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(b(cd))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(b(cd))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a7009d4cbf69b3e7cb09bd386d39df221551c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.069ex; height:2.843ex;" alt="{\displaystyle a(b(cd))}" /></span></dd></dl> <p>Jika operasi produk adalah asosiatif, hukum asosiatif yang digeneralisasi mengatakan bahwa semua rumus-rumus ini akan menghasilkan hasil yang sama. Jadi kecuali rumus dengan tanda kurung yang dihilangkan sudah memiliki sebuah arti yang berbeda (lihat bawah), tanda kurung bisa dianggap tidak perlu dan produk"nya" bisa ditulis dengan jelas sebagaiː </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 abcd.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle abcd.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ae535cf143a8c140f062ba5630d55d525f621f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.097ex; height:2.176ex;" alt="{\displaystyle abcd.}" /></span></dd></dl> <p>Sebagai bilangan dari anggota-anggota meningkat, <a href="/w/index.php?title=Bilangan_Catalan&action=edit&redlink=1" class="new" title="Bilangan Catalan (halaman belum tersedia)">bilangan dari kemungkinan cara untuk memasukkan tanda kurung</a> tumbuh dengan cepat, tetapi tidak perlu untuk disambiguasi. </p><p>Sebuah contoh di mana tidak bekerja adalah <a href="/w/index.php?title=Bikondisional_logis&action=edit&redlink=1" class="new" title="Bikondisional logis (halaman belum tersedia)">bikondisional logis</a> <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 \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }" /></span>. Ini adalah asosiatif, demikian <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\leftrightarrow (B\leftrightarrow C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">↔</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightarrow (B\leftrightarrow C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3837a80384c25055f7d7cbc862508b0075cddd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.311ex; height:2.843ex;" alt="{\displaystyle A\leftrightarrow (B\leftrightarrow C)}" /></span> ekuivalen dengan <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\leftrightarrow B)\leftrightarrow C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\leftrightarrow B)\leftrightarrow C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8471e31b8aeb7acb8ca9edc8bb4c2c3ebfd931d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.311ex; height:2.843ex;" alt="{\displaystyle (A\leftrightarrow B)\leftrightarrow C}" /></span>, namun <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\leftrightarrow B\leftrightarrow C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> <mo stretchy="false">↔</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightarrow B\leftrightarrow C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f30344c8db3af85ff6df541388514705df96808" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.502ex; height:2.176ex;" alt="{\displaystyle A\leftrightarrow B\leftrightarrow C}" /></span> paling umum mengartikan (<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\leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftrightarrow B}" /></span> dan <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 B\leftrightarrow C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">↔</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\leftrightarrow C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b663a4607d9c9196ccb8a67dce483ba8f2676ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.144ex; height:2.176ex;" alt="{\displaystyle B\leftrightarrow C}" /></span>), yang tidak ekuivalen </p> <div class="mw-heading mw-heading2"><h2 id="Contoh">Contoh</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit&section=3" title="Sunting bagian: Contoh" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sifat_asosiatif&action=edit&section=3" title="Sunting kode sumber bagian: Contoh"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Associativity_of_binary_operations_(without_question_marks).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Associativity_of_binary_operations_%28without_question_marks%29.svg/220px-Associativity_of_binary_operations_%28without_question_marks%29.svg.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Associativity_of_binary_operations_%28without_question_marks%29.svg/330px-Associativity_of_binary_operations_%28without_question_marks%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Associativity_of_binary_operations_%28without_question_marks%29.svg/440px-Associativity_of_binary_operations_%28without_question_marks%29.svg.png 2x" data-file-width="390" data-file-height="283" /></a><figcaption>Dalam operasi-operasi asosiatif adalah <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 (x\circ y)\circ z=x\circ (y\circ z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∘</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∘</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x\circ y)\circ z=x\circ (y\circ z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd29fd7cf89daf385c7625a32d21027ad068d0b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.643ex; height:2.843ex;" alt="{\displaystyle (x\circ y)\circ z=x\circ (y\circ z)}" /></span>.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Associativity_of_real_number_addition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Associativity_of_real_number_addition.svg/220px-Associativity_of_real_number_addition.svg.png" decoding="async" width="220" height="67" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Associativity_of_real_number_addition.svg/330px-Associativity_of_real_number_addition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Associativity_of_real_number_addition.svg/440px-Associativity_of_real_number_addition.svg.png 2x" data-file-width="316" data-file-height="96" /></a><figcaption>Penjumlahan dari bilangan real adalah asosiatif.</figcaption></figure> <p>Beberapa contoh dari operasi-operasi asosiatif termasuk yang berikut ini. </p> <ul><li><a href="/w/index.php?title=Penggabungan_rangkaian&action=edit&redlink=1" class="new" title="Penggabungan rangkaian (halaman belum tersedia)">Penggabungan</a> dari tiga rangkaian <code>"hello"</code>, <code>" "</code>, <code>"world"</code> bisa dihitung oleh penggabungan dua rangkaian pertama (diberikan <code>"hello "</code>) dan menambhakan rangkaian ketiga (<code>"world"</code>), atau dengan menggabungkan rangkaian kedua atau ketiga (diberikan <code>" world"</code>) dan menggabungkan rangkaian pertama (<code>"hello"</code>) dengan hasilnya. Keuda metodenya menghasilkan hasil yang sama, penggabungan rangkaian adalah asosiatig (tetapi bukan komutatif).</li> <li>Dalam <a href="/wiki/Aritmetika" title="Aritmetika">aritmetika</a>, <a href="/wiki/Penjumlahan" class="mw-redirect" title="Penjumlahan">penjumlahan</a> dan <a href="/wiki/Perkalian" title="Perkalian">perkalian</a> dari <a href="/wiki/Bilangan_real" class="mw-redirect" title="Bilangan real">bilangan real</a> adalah asosiatif, yaitu,</li></ul> <dl><dd><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 \left.{\begin{matrix}(x+y)+z=x+(y+z)=x+y+z\quad \\(x\,y)z=x(y\,z)=x\,y\,z\qquad \qquad \qquad \quad \ \ \,\end{matrix}}\right\}{\mbox{untuk semua }}x,y,z\in \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mspace width="1em"></mspace> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo stretchy="false">)</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mi>z</mi> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="1em"></mspace> <mtext> </mtext> <mtext> </mtext> <mspace width="thinmathspace"></mspace> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>untuk semua </mtext> </mstyle> </mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{matrix}(x+y)+z=x+(y+z)=x+y+z\quad \\(x\,y)z=x(y\,z)=x\,y\,z\qquad \qquad \qquad \quad \ \ \,\end{matrix}}\right\}{\mbox{untuk semua }}x,y,z\in \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dffa2146bf4671f3997307df7ea237c51300049" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:67.099ex; height:6.176ex;" alt="{\displaystyle \left.{\begin{matrix}(x+y)+z=x+(y+z)=x+y+z\quad \\(x\,y)z=x(y\,z)=x\,y\,z\qquad \qquad \qquad \quad \ \ \,\end{matrix}}\right\}{\mbox{untuk semua }}x,y,z\in \mathbb {R} .}" /></span></dd></dl></dd> <dd>Karena asosiatif, pengelompokan tanda kurung bisa dihilangkan tanpa kemenduaan.</dd></dl> <ul><li>Operasi biasa <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 x*y=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x*y=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db8978d34a1a4cf3ec3733e5163144d2b4bb3e2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.108ex; height:2.009ex;" alt="{\displaystyle x*y=x}" /></span> (artinya, hasilnya adalah argumen pertama, tidak peduli apa argumen keduanya) adalah asosiatif, tetapi bukan komutatif. Demikian juga, operasi trivial <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 x\circ y=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∘</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\circ y=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/886c4252fc08fb3ea9b05e505b0c402117040a86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.934ex; height:2.009ex;" alt="{\displaystyle x\circ y=y}" /></span> (artinya, hasilnya adalah argumen kedua, tidak peduli apa argumen kepertamanya) adalah asosiatif, tetapi bukan komutatif.</li> <li>Penjumlahan dan peralian dari <a href="/wiki/Bilangan_kompleks" title="Bilangan kompleks">bilangan kompleks</a> dan <a href="/wiki/Kuaternion" title="Kuaternion">kuaternion</a> adalah asosiatif. Penjumlahan dari <a href="/wiki/Oktonion" title="Oktonion">oktonion</a> juga asosiatif, tetapi perkalian dari oktonion adalah tidak asosiatif.</li> <li>Fungsi <a href="/wiki/Faktor_persekutuan_terbesar" title="Faktor persekutuan terbesar">faktor persekutuan terbesar</a> dan <a href="/wiki/Kelipatan_persekutuan_terkecil" title="Kelipatan persekutuan terkecil">kelipatan persekutuan terkecil</a> bersifat secara asosiatif.</li></ul> <dl><dd><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 \left.{\begin{matrix}\operatorname {gcd} (\operatorname {gcd} (x,y),z)=\operatorname {gcd} (x,\operatorname {gcd} (y,z))=\operatorname {gcd} (x,y,z)\ \quad \\\operatorname {lcm} (\operatorname {lcm} (x,y),z)=\operatorname {lcm} (x,\operatorname {lcm} (y,z))=\operatorname {lcm} (x,y,z)\quad \end{matrix}}\right\}{\mbox{ for all }}x,y,z\in \mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>gcd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>gcd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>gcd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>gcd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>gcd</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mspace width="1em"></mspace> </mtd> </mtr> <mtr> <mtd> <mi>lcm</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>lcm</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>lcm</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>lcm</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>lcm</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="1em"></mspace> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for all </mtext> </mstyle> </mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{matrix}\operatorname {gcd} (\operatorname {gcd} (x,y),z)=\operatorname {gcd} (x,\operatorname {gcd} (y,z))=\operatorname {gcd} (x,y,z)\ \quad \\\operatorname {lcm} (\operatorname {lcm} (x,y),z)=\operatorname {lcm} (x,\operatorname {lcm} (y,z))=\operatorname {lcm} (x,y,z)\quad \end{matrix}}\right\}{\mbox{ for all }}x,y,z\in \mathbb {Z} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071295ba6d997ee1d20db02a03491740853c3b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:73.11ex; height:6.176ex;" alt="{\displaystyle \left.{\begin{matrix}\operatorname {gcd} (\operatorname {gcd} (x,y),z)=\operatorname {gcd} (x,\operatorname {gcd} (y,z))=\operatorname {gcd} (x,y,z)\ \quad \\\operatorname {lcm} (\operatorname {lcm} (x,y),z)=\operatorname {lcm} (x,\operatorname {lcm} (y,z))=\operatorname {lcm} (x,y,z)\quad \end{matrix}}\right\}{\mbox{ for all }}x,y,z\in \mathbb {Z} .}" /></span></dd></dl></dd></dl> <ul><li>Mengambil <a href="/wiki/Irisan_(teori_himpunan)" title="Irisan (teori himpunan)">irisan</a> atau <a href="/wiki/Gabungan_(teori_himpunan)" title="Gabungan (teori himpunan)">gabungan</a> dari <a href="/wiki/Himpunan_(matematika)" title="Himpunan (matematika)">himpunan-himpunan</a>ː</li></ul> <dl><dd><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 \left.{\begin{matrix}(A\cap B)\cap C=A\cap (B\cap C)=A\cap B\cap C\quad \\(A\cup B)\cup C=A\cup (B\cup C)=A\cup B\cup C\quad \end{matrix}}\right\}{\mbox{for all sets }}A,B,C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <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> <mo>=</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo>∩</mo> <mi>C</mi> <mspace width="1em"></mspace> </mtd> </mtr> <mtr> <mtd> <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> <mo>=</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo>∪</mo> <mi>C</mi> <mspace width="1em"></mspace> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for all sets </mtext> </mstyle> </mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{matrix}(A\cap B)\cap C=A\cap (B\cap C)=A\cap B\cap C\quad \\(A\cup B)\cup C=A\cup (B\cup C)=A\cup B\cup C\quad \end{matrix}}\right\}{\mbox{for all sets }}A,B,C.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9378029ca082af451f78ddeb60fbe05c99057b4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.077ex; height:6.176ex;" alt="{\displaystyle \left.{\begin{matrix}(A\cap B)\cap C=A\cap (B\cap C)=A\cap B\cap C\quad \\(A\cup B)\cup C=A\cup (B\cup C)=A\cup B\cup C\quad \end{matrix}}\right\}{\mbox{for all sets }}A,B,C.}" /></span></dd></dl></dd></dl> <ul><li>Jika <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> adalah beberapa himpunan dan <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> melambangkan himpunan dari semua fungsi dari <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> ke <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span>, maka operasi dari <a href="/wiki/Komposisi_fungsi" title="Komposisi fungsi">komposisi fungsi</a> pada <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> adalah asosiatifː</li></ul> <dl><dd><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 (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h\qquad {\mbox{for all }}f,g,h\in S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>h</mi> <mo>=</mo> <mi>f</mi> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for all </mtext> </mstyle> </mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>,</mo> <mi>h</mi> <mo>∈</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h\qquad {\mbox{for all }}f,g,h\in S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f2d14b385cab65c99113b45b2b7972e24cc7e2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.021ex; height:2.843ex;" alt="{\displaystyle (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h\qquad {\mbox{for all }}f,g,h\in S.}" /></span></dd></dl></dd></dl> <ul><li>Sedikit lebih umum, diberikan empat himpunan <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span>, <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>, <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span>, dan <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" /></span>, dengan <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" /></span>ː <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> ke <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>,<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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" /></span>ː <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> ke <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span>, dan <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span>ː <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> ke <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" /></span>, makaː</li></ul> <dl><dd><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 (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>h</mi> <mo>=</mo> <mi>f</mi> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac9ff2f14721c0ad9348126aad6a77a0a014b8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.185ex; height:2.843ex;" alt="{\displaystyle (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h}" /></span></dd></dl></dd></dl> <dl><dd>seperti sebelumnya. Pendeknya, komposisi dari peta selalu asosiatif.</dd></dl> <ul><li>Tinjaulah sebuah himpunan dengan tiga anggota, <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>, <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span>, dan <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span>. Operasi berikut iniː</li></ul> <table class="wikitable"> <caption> </caption> <tbody><tr> <th>× </th> <th>A </th> <th>B </th> <th>C </th></tr> <tr> <td><b>A</b> </td> <td>A </td> <td>A </td> <td>A </td></tr> <tr> <td><b>B</b> </td> <td>A </td> <td>B </td> <td>C </td></tr> <tr> <td><b>C</b> </td> <td>A </td> <td>A </td> <td>A </td></tr></tbody></table> <dl><dd>asosiatif. Demikian, sebagai contoh, <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(BC)=(AB)C=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mi>C</mi> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(BC)=(AB)C=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab5a0cf7f785addf048eaa6607fd51074d2c1957" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.105ex; height:2.843ex;" alt="{\displaystyle A(BC)=(AB)C=A}" /></span>. Operasi ini tidak komutatif.</dd></dl> <ul><li>Karena <a href="/wiki/Matriks_(matematika)" title="Matriks (matematika)">matriks</a> mewakili <a href="/wiki/Peta_linear" title="Peta linear">fungsi linear</a>, dan <a href="/wiki/Perkalian_matriks" title="Perkalian matriks">perkalian matriks</a> mewakili <a href="/wiki/Komposisi_fungsi" title="Komposisi fungsi">komposisi fungsi</a>, salah satunya bisa secepatnya menyimpulkan bahwa perkalian matriks adalah asosiatif.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Logika_proposisional">Logika proposisional</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit&section=4" title="Sunting bagian: Logika proposisional" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sifat_asosiatif&action=edit&section=4" title="Sunting kode sumber bagian: Logika proposisional"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Aturan_penggantian">Aturan penggantian</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit&section=5" title="Sunting bagian: Aturan penggantian" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sifat_asosiatif&action=edit&section=5" title="Sunting kode sumber bagian: Aturan penggantian"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dalam logika proposisional kebenaran fungsional standar, <i>asosiasi</i>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> atau <i>asosiatif<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></i> adalah dua <a href="/w/index.php?title=Aturan_penggantian&action=edit&redlink=1" class="new" title="Aturan penggantian (halaman belum tersedia)">aturan penggantian yang sah</a>. Peraturannya memungkinkan salah satunya untuk memindahkan tanda kurung dalam <a href="/w/index.php?title=Rumus_yang_dibentuk_dengan_baik&action=edit&redlink=1" class="new" title="Rumus yang dibentuk dengan baik (halaman belum tersedia)">ekspresi logis</a> dalam <a href="/w/index.php?title=Bukti_formal&action=edit&redlink=1" class="new" title="Bukti formal (halaman belum tersedia)">bukti logis</a>. Aturan (menggunakan notasi <a href="/wiki/Operator_logika#Dalam_bahasa" title="Operator logika">penghubung logis</a> adalahː </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 (P\lor (Q\lor R))\Leftrightarrow ((P\lor Q)\lor R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∨</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\lor (Q\lor R))\Leftrightarrow ((P\lor Q)\lor R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a8ef30fef20438fb657bb3aff2dfeddb1d1200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.877ex; height:2.843ex;" alt="{\displaystyle (P\lor (Q\lor R))\Leftrightarrow ((P\lor Q)\lor R)}" /></span></dd></dl> <p>dan </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 (P\land (Q\land R))\Leftrightarrow ((P\land Q)\land R),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∧</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\land (Q\land R))\Leftrightarrow ((P\land Q)\land R),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47fcbce6c375735bfea1e21e9c810531bdd4a9c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.524ex; height:2.843ex;" alt="{\displaystyle (P\land (Q\land R))\Leftrightarrow ((P\land Q)\land R),}" /></span></dd></dl> <p>dimana "<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 \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }" /></span>" adalah <a href="/w/index.php?title=Simbol_(formal)&action=edit&redlink=1" class="new" title="Simbol (formal) (halaman belum tersedia)">simbol</a> <a href="/wiki/Metalogika" title="Metalogika">metalogis</a> mewakili "bisa menggantikan dalam sebuah <a href="/w/index.php?title=Bukti_formal&action=edit&redlink=1" class="new" title="Bukti formal (halaman belum tersedia)">bukti</a> dengan." </p> <div class="mw-heading mw-heading3"><h3 id="Penghubung_fungsional_kebenaran">Penghubung fungsional kebenaran</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit&section=6" title="Sunting bagian: Penghubung fungsional kebenaran" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sifat_asosiatif&action=edit&section=6" title="Sunting kode sumber bagian: Penghubung fungsional kebenaran"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Asosiatif</i> adalah sebuah sifat dari beberapa <a href="/wiki/Operator_logika" title="Operator logika">penghubung logis</a>. <a href="/w/index.php?title=Kesetaraan_logis&action=edit&redlink=1" class="new" title="Kesetaraan logis (halaman belum tersedia)">Kesetaraan logis</a> berikut mendemonstrasikan bahwa asosiatif adalah sebuah sifat dari penghubung tertentu. Berikut ini adalah <a href="/wiki/Tautologi_(logika)" title="Tautologi (logika)">tautologi</a> fungsional kebenaran.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p><b>Asosiatif dari disjungsi</b> </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 ((P\lor Q)\lor R)\leftrightarrow (P\lor (Q\lor R))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∨</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((P\lor Q)\lor R)\leftrightarrow (P\lor (Q\lor R))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83022ecdfaf711fdaf67c742e94d33fdabad38bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.877ex; height:2.843ex;" alt="{\displaystyle ((P\lor Q)\lor R)\leftrightarrow (P\lor (Q\lor R))}" /></span></dd> <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 (P\lor (Q\lor R))\leftrightarrow ((P\lor Q)\lor R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∨</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\lor (Q\lor R))\leftrightarrow ((P\lor Q)\lor R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe6682d32332d4ff07effc195b334e497b308f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.877ex; height:2.843ex;" alt="{\displaystyle (P\lor (Q\lor R))\leftrightarrow ((P\lor Q)\lor R)}" /></span></dd></dl> <p><b>Asosatif dari konjungsi</b> </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 ((P\land Q)\land R)\leftrightarrow (P\land (Q\land R))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∧</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((P\land Q)\land R)\leftrightarrow (P\land (Q\land R))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/599ddb9e2cfe377e714700cc2045f57f0d546d05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.877ex; height:2.843ex;" alt="{\displaystyle ((P\land Q)\land R)\leftrightarrow (P\land (Q\land R))}" /></span></dd> <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 (P\land (Q\land R))\leftrightarrow ((P\land Q)\land R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∧</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\land (Q\land R))\leftrightarrow ((P\land Q)\land R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7042ca691d07414dd554984602225f064b1f66ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.877ex; height:2.843ex;" alt="{\displaystyle (P\land (Q\land R))\leftrightarrow ((P\land Q)\land R)}" /></span></dd></dl> <p><b>Asosatif dari kesetaraan</b> </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 ((P\leftrightarrow Q)\leftrightarrow R)\leftrightarrow (P\leftrightarrow (Q\leftrightarrow R))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">↔</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">↔</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((P\leftrightarrow Q)\leftrightarrow R)\leftrightarrow (P\leftrightarrow (Q\leftrightarrow R))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1488e55751da92bcd3467470cfc8e8a7ce5819d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.003ex; height:2.843ex;" alt="{\displaystyle ((P\leftrightarrow Q)\leftrightarrow R)\leftrightarrow (P\leftrightarrow (Q\leftrightarrow R))}" /></span></dd> <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 (P\leftrightarrow (Q\leftrightarrow R))\leftrightarrow ((P\leftrightarrow Q)\leftrightarrow R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">↔</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">↔</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\leftrightarrow (Q\leftrightarrow R))\leftrightarrow ((P\leftrightarrow Q)\leftrightarrow R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6a49f0d06f9b7d211d1fd49c9d31b1a842cf696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.003ex; height:2.843ex;" alt="{\displaystyle (P\leftrightarrow (Q\leftrightarrow R))\leftrightarrow ((P\leftrightarrow Q)\leftrightarrow R)}" /></span></dd></dl> <p>Penolakan bersama adalah sebuah contoh dari sebuah penghubung fungsional kebenaran yang bukan asosiatif. </p> <div class="mw-heading mw-heading2"><h2 id="Operasi_nonasosiatif">Operasi nonasosiatif</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit&section=7" title="Sunting bagian: Operasi nonasosiatif" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sifat_asosiatif&action=edit&section=7" title="Sunting kode sumber bagian: Operasi nonasosiatif"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sebuah operasi biner <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span> pada sebuah himpunan <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> yang tidak memenuhi hukum asosiatif disebut <b>nonasosiatif</b>. Secara simbolis, </p><p>Untuk sebuah operasi, urutan dari evaluasi itu <i>penting</i>. Sebagai contohː </p> <ul><li><a href="/wiki/Pengurangan" title="Pengurangan">Pengurangan</a></li></ul> <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 (5-3)-2\,\neq \,5-(3-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>5</mn> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mo>≠</mo> <mspace width="thinmathspace"></mspace> <mn>5</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (5-3)-2\,\neq \,5-(3-2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3f6faa396ac65513dabe9782babb3accaf387d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.828ex; height:2.843ex;" alt="{\displaystyle (5-3)-2\,\neq \,5-(3-2)}" /></span></dd></dl> <ul><li><a href="/wiki/Pembagian" title="Pembagian">Pembagian</a></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 4\div (2\div 2)\neq (4\div 2)\div 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>÷</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>÷</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>≠</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>÷</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>÷</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\div (2\div 2)\neq (4\div 2)\div 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f47b04e5678c445003080328b816be546444ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.053ex; height:2.843ex;" alt="{\displaystyle 4\div (2\div 2)\neq (4\div 2)\div 2}" /></span> </p> <ul><li><a href="/wiki/Eksponensiasi" title="Eksponensiasi">Eksponensiasi</a>/Eksponen</li></ul> <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 2^{(1^{2})}\,\neq \,(2^{1})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mo>≠</mo> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{(1^{2})}\,\neq \,(2^{1})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a40b98aac13ac468277c5041735ed8d0763c7adb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.281ex; height:3.509ex;" alt="{\displaystyle 2^{(1^{2})}\,\neq \,(2^{1})^{2}}" /></span></dd> <dd></dd></dl> <p>Studi tentang struktur-struktur nonasosiatif muncul dari alasan-alasan agak berbeda dari arus utama dari aljabar klasik. Satu area dalam <a href="/wiki/Aljabar_nonasosiatif" title="Aljabar nonasosiatif">aljabar nonasosiatif</a> yang tumbuh sangat besar adalah <a href="/wiki/Aljabar_Lie" title="Aljabar Lie">aljabar Lie</a>. Disana hukum asosiatif dignatikan oleh <a href="/wiki/Identitas_Jacobi" title="Identitas Jacobi">identitas Jacobi</a>. Aljabar Lie meringkaskan alami esensial dari <a href="/w/index.php?title=Transformasi_infinitesimal&action=edit&redlink=1" class="new" title="Transformasi infinitesimal (halaman belum tersedia)">transformasi infinitesimal</a>, dan telah menjadi di mana-mana dalam matematika. </p><p>Terdapat jenis-jenis tertentu lainnya yang telah dipelajari secara mendalam; ini cenderung berasal dari beberapa penerapan yang spesifik atau bidang-bidang seperti <a href="/wiki/Kombinatorika" title="Kombinatorika">matematika kombinatorial</a>. Contoh lainnya adalah <a href="/wiki/Kuasigrup" title="Kuasigrup">kuasigrup</a>, <a href="/w/index.php?title=Kuasibidang&action=edit&redlink=1" class="new" title="Kuasibidang (halaman belum tersedia)">kuasibidang</a>, <a href="/w/index.php?title=Gelanggang_nonasosiatif&action=edit&redlink=1" class="new" title="Gelanggang nonasosiatif (halaman belum tersedia)">gelanggang nonasosiatif</a>, <a href="/wiki/Aljabar_nonasosiatif" title="Aljabar nonasosiatif">aljabar nonasosiatif</a> dan <a href="/w/index.php?title=Magma_nonasosiatif_komutatif&action=edit&redlink=1" class="new" title="Magma nonasosiatif komutatif (halaman belum tersedia)">magma nonasosiatif komutatif</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Nonasosiatif_dari_perhitungan_titik_mengambang">Nonasosiatif dari perhitungan titik mengambang</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit&section=8" title="Sunting bagian: Nonasosiatif dari perhitungan titik mengambang" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sifat_asosiatif&action=edit&section=8" title="Sunting kode sumber bagian: Nonasosiatif dari perhitungan titik mengambang"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dalam matematika, penjumlahan dan perkalian dari bilangan real adalah asosiatif. Sebaliknya, dalam ilmu komputer, penjumlahan dan perkalian dari bilangan <a href="/wiki/Floating-point" class="mw-redirect" title="Floating-point">titik mengambang</a> tidak asosiatif, sebagai galat pembulatan diperkenalkan ketika nilai-nilai berukuran berbeda digabungkan berbeda.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Untuk mengilustrasikan ini, tinjaulah sebuah representasi titik mengambang dengan sebuah <a href="/w/index.php?title=Signifikan&action=edit&redlink=1" class="new" title="Signifikan (halaman belum tersedia)">mantissa</a> 4-bit. </p><p>(1.000<sub>2</sub>×2<sup>0</sup> + 1.000<sub>2</sub>×2<sup>0</sup>) + 1.000<sub>2</sub>×2<sup>4</sup> = 1.000<sub>2</sub>×2<sup>1</sup> + 1.000<sub>2</sub>×2<sup>4</sup> = 1.00<span style="color:red;">1</span><sub>2</sub>×2<sup>4</sup> </p><p>1.000<sub>2</sub>×2<sup>0</sup> + (1.000<sub>2</sub>×2<sup>0</sup> + 1.000<sub>2</sub>×2<sup>4</sup>) = 1.000<sub>2</sub>×2<sup>0</sup> + 1.00<span style="color:red;">0</span><sub>2</sub>×2<sup>4</sup> = 1.00<span style="color:red;">0</span><sub>2</sub>×2<sup>4</sup> </p><p>Meskipun sebagian besar komputer-komputer menghitung dengan 24 atau 53 bit mantissa,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> ini adalah sumber yang penting dari galat pembulatan, dan mendekati seperti <a href="/w/index.php?title=Algoritma_penjumlahan_Kahan&action=edit&redlink=1" class="new" title="Algoritma penjumlahan Kahan (halaman belum tersedia)">algoritma penjumlahan Kahan</a> adalah cara untuk memperkecil galat-galatnya. Itu bisa sangat berpengalaman dlam komputer paralel.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Goldberg_1991_11-0" class="reference"><a href="#cite_note-Goldberg_1991-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Notasi_untuk_operasi-operasi_nonasosiastif">Notasi untuk operasi-operasi nonasosiastif</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit&section=9" title="Sunting bagian: Notasi untuk operasi-operasi nonasosiastif" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sifat_asosiatif&action=edit&section=9" title="Sunting kode sumber bagian: Notasi untuk operasi-operasi nonasosiastif"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Secara umum, tanda kurung pasti digunakan untuk menunjukkan <a href="/wiki/Urutan_operasi" title="Urutan operasi">urutan evaluasi</a> jika sebuah operasi nonasosiatif muncul lebih dari satu dalam sebuah ekspresi (kecuali notasinya menentukan urutannya dengan cara lain, seperti <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 {\frac {2}{3/4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3/4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f2608996791bb678dba547004019b5b312ad16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:4.323ex; height:6.009ex;" alt="{\displaystyle {\frac {2}{3/4}}}" /></span>). Namun, <a href="/wiki/Matematikawan" title="Matematikawan">matematikawan</a> setuju pada sebuah urutan evaluasi tertentu untuk beberapa umum operasi nonasosiatif. Ini meyederhanakan sebuah konvensi notasi untuk menghindari tanda kurung. </p><p>Sebuah operasi <b>asosiatif kiri</b> adalah operasi nonasosiatif yang secara konvensional dievaluasikan dari kiri ke kanan, yaitu, </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 \left.{\begin{matrix}x*y*z=(x*y)*z\qquad \qquad \quad \,\\w*x*y*z=((w*x)*y)*z\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{for all }}w,x,y,z\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>z</mi> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="1em"></mspace> <mspace width="thinmathspace"></mspace> </mtd> </mtr> <mtr> <mtd> <mi>w</mi> <mo>∗</mo> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo>∗</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>z</mi> <mspace width="1em"></mspace> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>etc.</mtext> </mstyle> </mrow> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mtext> </mtext> <mtext> </mtext> <mspace width="thinmathspace"></mspace> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for all </mtext> </mstyle> </mrow> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{matrix}x*y*z=(x*y)*z\qquad \qquad \quad \,\\w*x*y*z=((w*x)*y)*z\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{for all }}w,x,y,z\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d398097103855acb5e364c46a902d8e472b701a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:55.323ex; height:9.509ex;" alt="{\displaystyle \left.{\begin{matrix}x*y*z=(x*y)*z\qquad \qquad \quad \,\\w*x*y*z=((w*x)*y)*z\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{for all }}w,x,y,z\in S}" /></span></dd></dl> <p>sedangkan sebuah operasi <b>asosiatif kanan</b> secara konvensional dievaluasikan dari kanan ke kiri. </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 \left.{\begin{matrix}x*y*z=x*(y*z)\qquad \qquad \quad \,\\w*x*y*z=w*(x*(y*z))\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{for all }}w,x,y,z\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="1em"></mspace> <mspace width="thinmathspace"></mspace> </mtd> </mtr> <mtr> <mtd> <mi>w</mi> <mo>∗</mo> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo>=</mo> <mi>w</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="1em"></mspace> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>etc.</mtext> </mstyle> </mrow> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mtext> </mtext> <mtext> </mtext> <mspace width="thinmathspace"></mspace> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for all </mtext> </mstyle> </mrow> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{matrix}x*y*z=x*(y*z)\qquad \qquad \quad \,\\w*x*y*z=w*(x*(y*z))\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{for all }}w,x,y,z\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04dd3ba41a708682179bad22ee14162a221cf3e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:55.323ex; height:9.509ex;" alt="{\displaystyle \left.{\begin{matrix}x*y*z=x*(y*z)\qquad \qquad \quad \,\\w*x*y*z=w*(x*(y*z))\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{for all }}w,x,y,z\in S}" /></span></dd></dl> <p>Kedua operasi asosiatif kiri dan asosiatif kanan terjadi. Operasi asosiatif kiri termasuk yang berikut ini. </p> <ul><li>Pengurangan dan pembagian dari bilangan realː<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Bronstein_1987_16-0" class="reference"><a href="#cite_note-Bronstein_1987-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></li> <li>Penerapanː fungsi</li></ul> <dl><dd><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 (f\,x\,y)=((f\,x)\,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\,x\,y)=((f\,x)\,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b389e63479531bb728bbef3b2b5d0e3bc4324244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.602ex; height:2.843ex;" alt="{\displaystyle (f\,x\,y)=((f\,x)\,y)}" /></span></dd></dl></dd> <dd>Notasi ini bisa dimotivasi dengan <a href="/w/index.php?title=Currying&action=edit&redlink=1" class="new" title="Currying (halaman belum tersedia)">currying</a> <a href="/wiki/Isomorfisme" title="Isomorfisme">isomorfisme</a>.</dd></dl> <p>Operasi asosiatif kanan termasuk yang berikut ini. </p> <ul><li><a href="/wiki/Eksponensiasi" title="Eksponensiasi">Eksponensiasi</a> atau bilangan real dalam notasi superskripː</li></ul> <dl><dd><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 x^{y^{z}}=x^{(y^{z})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{y^{z}}=x^{(y^{z})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571ed1b1dee0de3134a507fcb430ae2898365e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.721ex; height:2.843ex;" alt="{\displaystyle x^{y^{z}}=x^{(y^{z})}}" /></span></dd></dl></dd></dl> <dl><dd>Eksponensiasi biasanya digunakan dengan tanda kurung atau asosatif kanan karena sebuah operasi eksponensiasi asosiatif kiri yang berulang tidak banyak digunakan. Pangkat berulang sering ditulis ulang dengan perkalian</dd></dl> <dl><dd><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 (x^{y})^{z}=x^{(yz)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{y})^{z}=x^{(yz)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f029ff59aa603e3f8b660abd3a1b6af99a3be8e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.716ex; height:3.343ex;" alt="{\displaystyle (x^{y})^{z}=x^{(yz)}}" /></span></dd></dl></dd></dl> <dl><dd>Diformat dengan benar, supeskrip secara inheren berperilaku sebagai sebuah himpunan dari tanda kurung; misalnya, dalam ekspresi <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^{x+3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{x+3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15b9caae3cbba62c80da961b02d99d6dffe397ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.435ex; height:2.676ex;" alt="{\displaystyle 2^{x+3}}" /></span>, penjumlahan dilkaukan <a href="/wiki/Urutan_operasi" title="Urutan operasi">sebelum</a> eksponensiasi meskipun tidak ada tanda kurung eksplisit <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^{(x+3)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{(x+3)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b75e5618a0dd67c7bef4bd3b260a5832c58bfa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.715ex; height:2.843ex;" alt="{\displaystyle 2^{(x+3)}}" /></span> melilitnya. Demikian diberikan sebuah ekspresi seperti <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 x^{y^{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{y^{z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565240fe524b85973863acdab998c0e296eab57e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.171ex; height:2.676ex;" alt="{\displaystyle x^{y^{z}}}" /></span>, eksponen penuh <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 y^{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8c2cd4410cc6b1d1dd0e34c746bbdb98bf65cf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.162ex; height:2.676ex;" alt="{\displaystyle y^{z}}" /></span> dari dasar <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span> dievaluasikan pertama. Namun, dalam beberapa konteks, termasuk tulis tangan, perbedaan antara <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 {x^{y}}^{z}=(x^{y})^{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x^{y}}^{z}=(x^{y})^{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbe5c78738d252abcbb48b47247b4c5157332e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.669ex; height:2.843ex;" alt="{\displaystyle {x^{y}}^{z}=(x^{y})^{z}}" /></span>, <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 x^{yz}=x^{(yz)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{yz}=x^{(yz)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15777e10957ca62daf347ce016af88d57bce9039" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.675ex; height:2.843ex;" alt="{\displaystyle x^{yz}=x^{(yz)}}" /></span> dan <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 x^{y^{z}}=x^{(y^{z})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{y^{z}}=x^{(y^{z})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571ed1b1dee0de3134a507fcb430ae2898365e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.721ex; height:2.843ex;" alt="{\displaystyle x^{y^{z}}=x^{(y^{z})}}" /></span> bisa jadi sulit untuk dilihat. Dalam kasus seperti itu, asosiatif kanan biasanya tersirat.</dd></dl> <ul><li><a href="/wiki/Fungsi_(matematika)" title="Fungsi (matematika)">Definisi fungsi</a></li></ul> <dl><dd>Menggunakan notasi asosiatif kanan untuk operasi-operasi ini bisa dimotivasi oleh <a href="/w/index.php?title=Korespondensi_Curry-Howard&action=edit&redlink=1" class="new" title="Korespondensi Curry-Howard (halaman belum tersedia)">korespondensi Curry-Howard</a> dan dengan currying isomorfisme.</dd></dl> <p>Operasi nonasosiatif untuk yang urutan evaluasi yang tidak konvensional didefinisikan termasuk sebagai berikut. </p> <ul><li>Eksponensiasi dari bilangan real dalam notasi infiks.<sup id="cite_ref-Codeplea_2016_17-0" class="reference"><a href="#cite_note-Codeplea_2016-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li></ul> <dl><dd><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 (x^{\wedge }y)^{\wedge }z\neq x^{\wedge }(y^{\wedge }z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∧</mo> </mrow> </msup> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∧</mo> </mrow> </msup> <mi>z</mi> <mo>≠</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∧</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∧</mo> </mrow> </msup> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{\wedge }y)^{\wedge }z\neq x^{\wedge }(y^{\wedge }z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4397540767f8b269ed2573338d88ccaaee9ad08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.183ex; height:3.009ex;" alt="{\displaystyle (x^{\wedge }y)^{\wedge }z\neq x^{\wedge }(y^{\wedge }z)}" /></span></dd></dl></dd></dl> <ul><li><a href="/wiki/Notasi_anak_panah_Knuth" class="mw-redirect" title="Notasi anak panah Knuth">Operator panah atas Knuth</a></li></ul> <dl><dd><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\uparrow \uparrow (b\uparrow \uparrow c)\neq (a\uparrow \uparrow b)\uparrow \uparrow c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">↑↑</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">↑↑</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">↑↑</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↑↑</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\uparrow \uparrow (b\uparrow \uparrow c)\neq (a\uparrow \uparrow b)\uparrow \uparrow c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4547cce5cdd90e59a5c3e75f26b026ba5d8ce95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.646ex; height:2.843ex;" alt="{\displaystyle a\uparrow \uparrow (b\uparrow \uparrow c)\neq (a\uparrow \uparrow b)\uparrow \uparrow c}" /></span></dd> <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\uparrow \uparrow \uparrow (b\uparrow \uparrow \uparrow c)\neq (a\uparrow \uparrow \uparrow b)\uparrow \uparrow \uparrow c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">↑↑↑</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">↑↑↑</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">↑↑↑</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↑↑↑</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\uparrow \uparrow \uparrow (b\uparrow \uparrow \uparrow c)\neq (a\uparrow \uparrow \uparrow b)\uparrow \uparrow \uparrow c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85df10c2d070c0816647a0272af330562e62eaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.296ex; height:2.843ex;" alt="{\displaystyle a\uparrow \uparrow \uparrow (b\uparrow \uparrow \uparrow c)\neq (a\uparrow \uparrow \uparrow b)\uparrow \uparrow \uparrow c}" /></span></dd></dl></dd> <dd></dd></dl> <ul><li>Mengambil <a href="/wiki/Perkalian_vektor" title="Perkalian vektor">produk silang</a></li> <li>Mengambil <a href="/wiki/Rata-rata" title="Rata-rata">rata-rata</a> berpasangan dari bilangan realː</li></ul> <dl><dd><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 {(x+y)/2+z \over 2}\neq {x+(y+z)/2 \over 2}\qquad {\mbox{for all }}x,y,z\in \mathbb {R} {\mbox{ with }}x\neq z.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mi>z</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for all </mtext> </mstyle> </mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> with </mtext> </mstyle> </mrow> <mi>x</mi> <mo>≠</mo> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {(x+y)/2+z \over 2}\neq {x+(y+z)/2 \over 2}\qquad {\mbox{for all }}x,y,z\in \mathbb {R} {\mbox{ with }}x\neq z.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2e2b38f210b6a3cc70092f139bbbde968b6ae25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:64.602ex; height:5.676ex;" alt="{\displaystyle {(x+y)/2+z \over 2}\neq {x+(y+z)/2 \over 2}\qquad {\mbox{for all }}x,y,z\in \mathbb {R} {\mbox{ with }}x\neq z.}" /></span></dd></dl></dd></dl> <ul><li>Mengambil <a href="/wiki/Komplemen_(teori_himpunan)" title="Komplemen (teori himpunan)">komplemen relatif</a> dari himpunan <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\backslash B)\backslash C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\backslash B)\backslash C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06365b4731309c65b8d328990f01ca9cfe7d0755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.408ex; height:2.843ex;" alt="{\displaystyle (A\backslash B)\backslash C}" /></span> tidak sama dengan <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\backslash (B\backslash C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\backslash (B\backslash C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1c6146fe14c4242176394843eb8dfb5d616235c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.408ex; height:2.843ex;" alt="{\displaystyle A\backslash (B\backslash C)}" /></span>. (Membandingkan <a href="/w/index.php?title=Nonimplikasi_material&action=edit&redlink=1" class="new" title="Nonimplikasi material (halaman belum tersedia)">nonimplikasi material</a> dalam logika.)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Lihat_pula">Lihat pula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit&section=10" title="Sunting bagian: Lihat pula" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sifat_asosiatif&action=edit&section=10" title="Sunting kode sumber bagian: Lihat pula"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Uji_asosiatif_Light&action=edit&redlink=1" class="new" title="Uji asosiatif Light (halaman belum tersedia)">Uji asosiatif Light</a></li> <li><a href="/w/index.php?title=Deret_teleskopik&action=edit&redlink=1" class="new" title="Deret teleskopik (halaman belum tersedia)">Deret teleskopik</a>, penggunaan dari asosatif penjumlahan untuk membatalkan istilah dalam sebuah <a href="/wiki/Deret_(matematika)" title="Deret (matematika)">deret</a> tak terhingga</li> <li>Sebuah <a href="/wiki/Semigrup" title="Semigrup">semigrup</a> adalah sebuah himpunan dengan operasi biner asosiatif.</li> <li><a href="/wiki/Sifat_komutatif" title="Sifat komutatif">Komutatif</a> dan <a href="/wiki/Properti_distributif" class="mw-redirect" title="Properti distributif">distributif</a> adalah dua lainnya yang sering dibahas sifat-sifat dari operasi-operasi biner.</li> <li><a href="/w/index.php?title=Asosiatif_pangkat&action=edit&redlink=1" class="new" title="Asosiatif pangkat (halaman belum tersedia)">Asosiatif pangkat</a>, <a href="/w/index.php?title=Alternatif_(aljabar_abstrak)&action=edit&redlink=1" class="new" title="Alternatif (aljabar abstrak) (halaman belum tersedia)">alternatif</a>, <a href="/w/index.php?title=Aljabar_fleksibilitas&action=edit&redlink=1" class="new" title="Aljabar fleksibilitas (halaman belum tersedia)">fleksibilitas</a>, dan <a href="/w/index.php?title=Asosiatif_N-ari&action=edit&redlink=1" class="new" title="Asosiatif N-ari (halaman belum tersedia)">asosiatif N-ari</a> adalah bentuk-bentuk yang lemah dari asosiatif.</li> <li><a href="/w/index.php?title=Loop_Moufang&action=edit&redlink=1" class="new" title="Loop Moufang (halaman belum tersedia)">Identitas Moufang</a> juga memberikan bentuk yang lemah dari asosiatif.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Referensi">Referensi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sifat_asosiatif&veaction=edit&section=11" title="Sunting bagian: Referensi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sifat_asosiatif&action=edit&section=11" title="Sunting kode sumber bagian: Referensi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r18833634">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"> <cite class="citation book">Hungerford, Thomas W. (1974). <a rel="nofollow" class="external text" href="https://archive.org/details/algebra0000hung_f8t3"><i>Algebra</i></a> (edisi ke-1st). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. hlm. <a rel="nofollow" class="external text" href="https://archive.org/details/algebra0000hung_f8t3/page/24">24</a>. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0387905181" title="Istimewa:Sumber buku/978-0387905181">978-0387905181</a>. <q>Definisi 1.1 (i)a (bc) = (ab) c untuk semua a, b, c dalam G.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pages=24&rft.edition=1st&rft.pub=Springer&rft.date=1974&rft.isbn=978-0387905181&rft.aulast=Hungerford&rft.aufirst=Thomas+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgebra0000hung_f8t3&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASifat+asosiatif" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation book">Durbin, John R. (1992). <a rel="nofollow" class="external text" href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000258.html"><i>Modern Algebra: an Introduction</i></a> (edisi ke-3rd). New York: Wiley. hlm. 78. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-471-51001-7" title="Istimewa:Sumber buku/978-0-471-51001-7">978-0-471-51001-7</a>. <q>If <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_{1},a_{2},\dots ,a_{n}\,\,(n\geq 2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mi>n</mi> <mo>≥</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\dots ,a_{n}\,\,(n\geq 2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcd535b3f34f8f8b227f20cebc10244cc8850f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.468ex; height:2.843ex;" alt="{\displaystyle a_{1},a_{2},\dots ,a_{n}\,\,(n\geq 2)}" /></span> are elements of a set with an associative operation, then the product <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_{1}a_{2}\dots a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}a_{2}\dots a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261c24ca10b9e43dc5fcf686e138636e8a3749b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.514ex; height:2.009ex;" alt="{\displaystyle a_{1}a_{2}\dots a_{n}}" /></span> is unambiguous; this is, the same element will be obtained regardless of how parentheses are inserted in the product</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Algebra%3A+an+Introduction&rft.place=New+York&rft.pages=78&rft.edition=3rd&rft.pub=Wiley&rft.date=1992&rft.isbn=978-0-471-51001-7&rft.aulast=Durbin&rft.aufirst=John+R.&rft_id=http%3A%2F%2Fwww.wiley.com%2FWileyCDA%2FWileyTitle%2FproductCd-EHEP000258.html&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASifat+asosiatif" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.khanacademy.org/math/linear-algebra/matrix-transformations/composition-of-transformations/v/matrix-product-associativity">"Matrix product associativity"</a>. Khan Academy<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">5 June</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Matrix+product+associativity&rft.pub=Khan+Academy&rft_id=http%3A%2F%2Fwww.khanacademy.org%2Fmath%2Flinear-algebra%2Fmatrix-transformations%2Fcomposition-of-transformations%2Fv%2Fmatrix-product-associativity&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASifat+asosiatif" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation book">Moore, Brooke Noel; Parker, Richard (2017). <i>Critical Thinking (12th edition)</i>. New York: McGraw-Hill Education. hlm. 321. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/9781259690877" title="Istimewa:Sumber buku/9781259690877">9781259690877</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Critical+Thinking+%2812th+edition%29&rft.place=New+York&rft.pages=321&rft.pub=McGraw-Hill+Education&rft.date=2017&rft.isbn=9781259690877&rft.aulast=Moore&rft.aufirst=Brooke+Noel&rft.au=Parker%2C+Richard&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASifat+asosiatif" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation book">Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2014). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontolo0014copi"><i>Introduction to Logic (14th edition)</i></a>. Essex: Pearson Education. hlm. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontolo0014copi/page/387">387</a>. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/9781292024820" title="Istimewa:Sumber buku/9781292024820">9781292024820</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Logic+%2814th+edition%29&rft.place=Essex&rft.pages=387&rft.pub=Pearson+Education&rft.date=2014&rft.isbn=9781292024820&rft.aulast=Copi&rft.aufirst=Irving+M.&rft.au=Cohen%2C+Carl&rft.au=McMahon%2C+Kenneth&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontolo0014copi&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASifat+asosiatif" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite class="citation book">Hurley, Patrick J.; Watson, Lori (2016). <i>A Concise Introduction to Logic (13th edition)</i>. Boston: Cengage Learning. hlm. 427. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/9781305958098" title="Istimewa:Sumber buku/9781305958098">9781305958098</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Concise+Introduction+to+Logic+%2813th+edition%29&rft.place=Boston&rft.pages=427&rft.pub=Cengage+Learning&rft.date=2016&rft.isbn=9781305958098&rft.aulast=Hurley&rft.aufirst=Patrick+J.&rft.au=Watson%2C+Lori&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASifat+asosiatif" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/2197480">"Symbolic Logic Proof of Associativity"</a>. <i>Math.stackexchange.com</i>. 22 March 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math.stackexchange.com&rft.atitle=Symbolic+Logic+Proof+of+Associativity&rft.date=2017-03-22&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fq%2F2197480&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASifat+asosiatif" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Knuth, Donald, <a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>, Volume 3, section 4.2.2</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFIEEE_7542008" class="citation book">IEEE Computer Society (29 August 2008). <i>IEEE Standard for Floating-Point Arithmetic</i>. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FIEEESTD.2008.4610935">10.1109/IEEESTD.2008.4610935</a>. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-7381-5753-5" title="Istimewa:Sumber buku/978-0-7381-5753-5">978-0-7381-5753-5</a>. IEEE Std 754-2008.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=IEEE+Standard+for+Floating-Point+Arithmetic&rft.date=2008-08-29&rft_id=info%3Adoi%2F10.1109%2FIEEESTD.2008.4610935&rft.isbn=978-0-7381-5753-5&rft.au=IEEE+Computer+Society&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASifat+asosiatif" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFVillaChavarría-mirGurumoorthiMárquez" class="citation">Villa, Oreste; Chavarría-mir, Daniel; Gurumoorthi, Vidhya; Márquez, Andrés; Krishnamoorthy, Sriram, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130215171724/http://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdf"><i>Effects of Floating-Point nonassociativity on Numerical Computations on Massively Multithreaded Systems</i></a> <span style="font-size:85%;">(PDF)</span>, diarsipkan dari <a rel="nofollow" class="external text" href="https://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdf">versi asli</a> <span style="font-size:85%;">(PDF)</span> tanggal 15 February 2013<span class="reference-accessdate">, diakses tanggal <span class="nowrap">8 April</span> 2014</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Effects+of+Floating-Point+nonassociativity+on+Numerical+Computations+on+Massively+Multithreaded+Systems&rft.aulast=Villa&rft.aufirst=Oreste&rft.au=Chavarr%C3%ADa-mir%2C+Daniel&rft.au=Gurumoorthi%2C+Vidhya&rft.au=M%C3%A1rquez%2C+Andr%C3%A9s&rft.au=Krishnamoorthy%2C+Sriram&rft_id=http%3A%2F%2Fcass-mt.pnnl.gov%2Fdocs%2Fpubs%2Fpnnleffects_of_floating-pointpaper.pdf&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASifat+asosiatif" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|url-status=</code> yang tidak diketahui akan diabaikan (<a href="/wiki/Bantuan:Galat_CS1#parameter_ignored" title="Bantuan:Galat CS1">bantuan</a>)</span></span> </li> <li id="cite_note-Goldberg_1991-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Goldberg_1991_11-0">^</a></b></span> <span class="reference-text"><cite class="citation journal"><a href="/w/index.php?title=David_Goldberg_(PARC)&action=edit&redlink=1" class="new" title="David Goldberg (PARC) (halaman belum tersedia)">Goldberg, David</a> (March 1991). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160406101256/http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html">"What Every Computer Scientist Should Know About Floating-Point Arithmetic"</a>. <i><a href="/w/index.php?title=ACM_Computing_Surveys&action=edit&redlink=1" class="new" title="ACM Computing Surveys (halaman belum tersedia)">ACM Computing Surveys</a></i>. <b>23</b> (1): 5–48. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F103162.103163">10.1145/103162.103163</a>. Diarsipkan dari <a rel="nofollow" class="external text" href="http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf">versi asli</a> <span style="font-size:85%;">(PDF)</span> tanggal 2016-04-06<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">20 January</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+Computing+Surveys&rft.atitle=What+Every+Computer+Scientist+Should+Know+About+Floating-Point+Arithmetic&rft.volume=23&rft.issue=1&rft.pages=5-48&rft.date=1991-03&rft_id=info%3Adoi%2F10.1145%2F103162.103163&rft.aulast=Goldberg&rft.aufirst=David&rft_id=http%3A%2F%2Fperso.ens-lyon.fr%2Fjean-michel.muller%2Fgoldberg.pdf&rfr_id=info%3Asid%2Fid.wikipedia.org%3ASifat+asosiatif" class="Z3988"><span style="display:none;"> </span></span>(,)</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">George Mark Bergman: <a rel="nofollow" class="external text" href="https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html">Order of arithmetic operations</a></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Education Place: <a rel="nofollow" class="external text" href="http://eduplace.com/math/mathsteps/4/a/index.html">The Order of Operations</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170608144614/http://eduplace.com/math/mathsteps/4/a/index.html">Diarsipkan</a> 2017-06-08 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="/wiki/Khan_Academy" title="Khan Academy">Khan Academy</a>: <a rel="nofollow" class="external text" href="https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-order-of-operations/v/introduction-to-order-of-operations">The Order of Operations</a>, timestamp <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=ClYdw4d4OmA&t=5m40s">5m40s</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Virginia Department of Education: <a rel="nofollow" class="external text" href="http://www.doe.virginia.gov/instruction/mathematics/middle/algebra_readiness/curriculum_companion/order-operations.pdf#page=3">Using Order of Operations and Exploring Properties</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220716062834/http://www.doe.virginia.gov/instruction/mathematics/middle/algebra_readiness/curriculum_companion/order-operations.pdf#page=3">Diarsipkan</a> 2022-07-16 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>., section 9</span> </li> <li id="cite_note-Bronstein_1987-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bronstein_1987_16-0">^</a></b></span> <span class="reference-text">Bronstein: <a href="https://de.wikipedia.org/wiki/Taschenbuch_der_Mathematik" class="extiw" title="de:Taschenbuch der Mathematik">de:Taschenbuch der Mathematik</a>, pages 115-120, chapter: 2.4.1.1, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-3-8085-5673-3" title="Istimewa:Sumber buku/978-3-8085-5673-3">978-3-8085-5673-3</a></span> </li> <li id="cite_note-Codeplea_2016-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Codeplea_2016_17-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://codeplea.com/exponentiationassociativity-options">Exponentiation Associativity and Standard Math Notation</a> Codeplea. 23 August 2016. Retrieved 20 September 2016.</span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Diperoleh dari "<a dir="ltr" href="https://id.wikipedia.org/w/index.php?title=Sifat_asosiatif&oldid=26429488">https://id.wikipedia.org/w/index.php?title=Sifat_asosiatif&oldid=26429488</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Istimewa:Daftar_kategori" title="Istimewa:Daftar kategori">Kategori</a>: <ul><li><a href="/wiki/Kategori:Operasi_biner" title="Kategori:Operasi biner">Operasi biner</a></li><li><a href="/wiki/Kategori:Aljabar_abstrak" title="Kategori:Aljabar abstrak">Aljabar abstrak</a></li><li><a href="/wiki/Kategori:Analisis_fungsional" title="Kategori:Analisis fungsional">Analisis fungsional</a></li><li><a href="/wiki/Kategori:Kaidah_inferensi" title="Kategori:Kaidah inferensi">Kaidah inferensi</a></li><li><a href="/wiki/Kategori:Aljabar_elementer" title="Kategori:Aljabar elementer">Aljabar elementer</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Kategori tersembunyi: <ul><li><a href="/wiki/Kategori:Halaman_dengan_rujukan_yang_menggunakan_parameter_yang_tidak_didukung" title="Kategori:Halaman dengan rujukan yang menggunakan parameter yang tidak didukung">Halaman dengan rujukan yang menggunakan parameter yang tidak didukung</a></li><li><a href="/wiki/Kategori:Templat_webarchive_tautan_wayback" title="Kategori:Templat webarchive tautan wayback">Templat webarchive tautan wayback</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"> Halaman ini terakhir diubah pada 17 Oktober 2024, pukul 14.31.</li> <li id="footer-info-copyright">Teks tersedia di bawah <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Lisensi Atribusi-BerbagiSerupa Creative Commons</a>; ketentuan tambahan mungkin berlaku. Lihat <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">Ketentuan Penggunaan</a> untuk rincian lebih lanjut.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Kebijakan privasi</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:Tentang">Tentang Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Penyangkalan_umum">Penyangkalan</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Kode Etik</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Pengembang</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/id.wikipedia.org">Statistik</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Pernyataan kuki</a></li> <li id="footer-places-mobileview"><a href="//id.m.wikipedia.org/w/index.php?title=Sifat_asosiatif&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Tampilan seluler</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"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></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"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" lang="en" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Pencarian</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div 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="Telusuri Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Istimewa:Pencarian"> </div> <button class="cdx-button cdx-search-input__end-button">Cari</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Daftar isi" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Gulingkan daftar isi" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-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">Gulingkan daftar isi</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Sifat asosiatif</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>67 bahasa</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Bagian baru</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-59986f9cc4-5v75b","wgBackendResponseTime":197,"wgPageParseReport":{"limitreport":{"cputime":"0.214","walltime":"0.401","ppvisitednodes":{"value":1342,"limit":1000000},"postexpandincludesize":{"value":21844,"limit":2097152},"templateargumentsize":{"value":232,"limit":2097152},"expansiondepth":{"value":10,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":21763,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 132.149 1 -total"," 85.48% 112.960 1 Templat:Reflist"," 41.73% 55.141 6 Templat:Cite_book"," 7.17% 9.475 1 Templat:Citation"," 5.23% 6.912 3 Templat:Fontcolor"," 4.84% 6.390 2 Templat:Cite_web"," 4.78% 6.313 1 Templat:ISBN"," 4.33% 5.716 2 Templat:Webarchive"," 3.57% 4.721 1 Templat:Cite_journal"," 1.63% 2.160 6 Templat:Trim"]},"scribunto":{"limitreport-timeusage":{"value":"0.043","limit":"10.000"},"limitreport-memusage":{"value":2462006,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-7f88f964d6-r76c6","timestamp":"20250219113216","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Sifat asosiatif","url":"https:\/\/id.wikipedia.org\/wiki\/Sifat_asosiatif","sameAs":"http:\/\/www.wikidata.org\/entity\/Q177251","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q177251","author":{"@type":"Organization","name":"Kontributor dari proyek Wikimedia."},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2020-10-17T03:58:05Z","dateModified":"2024-10-17T14:31:41Z","headline":"properti operasi biner yang memungkinkan urutan operasi untuk dikelompokkan kembali tanpa mengubah nilainya"}</script> </body> </html>