Associative property

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<span class="vector-toc-numb">2</span> <span>Generalized associative law</span> </div> </a> <ul id="toc-Generalized_associative_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propositional_logic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propositional_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Propositional logic</span> </div> </a> <button aria-controls="toc-Propositional_logic-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Propositional logic subsection</span> </button> <ul id="toc-Propositional_logic-sublist" class="vector-toc-list"> <li id="toc-Rule_of_replacement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rule_of_replacement"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Rule of replacement</span> </div> </a> <ul id="toc-Rule_of_replacement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Truth_functional_connectives" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Truth_functional_connectives"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Truth functional connectives</span> </div> </a> <ul id="toc-Truth_functional_connectives-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Non-associative_operation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non-associative_operation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Non-associative operation</span> </div> </a> <button aria-controls="toc-Non-associative_operation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Non-associative operation subsection</span> </button> <ul id="toc-Non-associative_operation-sublist" class="vector-toc-list"> <li id="toc-Nonassociativity_of_floating_point_calculation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonassociativity_of_floating_point_calculation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Nonassociativity of floating point calculation</span> </div> </a> <ul id="toc-Nonassociativity_of_floating_point_calculation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notation_for_non-associative_operations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notation_for_non-associative_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Notation for non-associative operations</span> </div> </a> <ul id="toc-Notation_for_non-associative_operations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Associative property</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 67 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-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 languages</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 – Arabic" lang="ar" hreflang="ar" data-title="عملية تجميعية" data-language-autonym="العربية" data-language-local-name="Arabic" 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 – Asturian" lang="ast" hreflang="ast" data-title="Asociatividá" data-language-autonym="Asturianu" data-language-local-name="Asturian" 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 – Belarusian" lang="be" hreflang="be" data-title="Асацыятыўная аперацыя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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 – Bulgarian" lang="bg" hreflang="bg" data-title="Асоциативност" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Asocijativnost" title="Asocijativnost – Bosnian" lang="bs" hreflang="bs" data-title="Asocijativnost" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Propietat_associativa" title="Propietat associativa – Catalan" lang="ca" hreflang="ca" data-title="Propietat associativa" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%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-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Asociativita" title="Asociativita – Czech" lang="cs" hreflang="cs" data-title="Asociativita" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Associativitet" title="Associativitet – Danish" lang="da" hreflang="da" data-title="Associativitet" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Assoziativgesetz" title="Assoziativgesetz – German" lang="de" hreflang="de" data-title="Assoziativgesetz" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Assotsiatiivsus" title="Assotsiatiivsus – Estonian" lang="et" hreflang="et" data-title="Assotsiatiivsus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%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 – Greek" lang="el" hreflang="el" data-title="Προσεταιριστική ιδιότητα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Asociatividad_(%C3%A1lgebra)" title="Asociatividad (álgebra) – Spanish" lang="es" hreflang="es" data-title="Asociatividad (álgebra)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/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-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 – Persian" lang="fa" hreflang="fa" data-title="خاصیت شرکت‌پذیری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Associativit%C3%A9" title="Associativité – French" lang="fr" hreflang="fr" data-title="Associativité" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</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 – Irish" lang="ga" hreflang="ga" data-title="Oibríocht chomhthiomsaitheach" data-language-autonym="Gaeilge" data-language-local-name="Irish" 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) – Galician" lang="gl" hreflang="gl" data-title="Asociatividade (álxebra)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B2%B0%ED%95%A9%EB%B2%95%EC%B9%99" title="결합법칙 – Korean" lang="ko" hreflang="ko" data-title="결합법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%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 – Armenian" lang="hy" hreflang="hy" data-title="Զուգորդականություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Asocijativnost" title="Asocijativnost – Croatian" lang="hr" hreflang="hr" data-title="Asocijativnost" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sifat_asosiatif" title="Sifat asosiatif – Indonesian" lang="id" hreflang="id" data-title="Sifat asosiatif" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-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 – Icelandic" lang="is" hreflang="is" data-title="Tengiregla" data-language-autonym="Íslenska" data-language-local-name="Icelandic" 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 – Italian" lang="it" hreflang="it" data-title="Associatività" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%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 – Hebrew" lang="he" hreflang="he" data-title="פעולה אסוציאטיבית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</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-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-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Asociativit%C4%81te" title="Asociativitāte – Latvian" lang="lv" hreflang="lv" data-title="Asociativitāte" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Asociatyvumas" title="Asociatyvumas – Lithuanian" lang="lt" hreflang="lt" data-title="Asociatyvumas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Asszociativit%C3%A1s" title="Asszociativitás – Hungarian" lang="hu" hreflang="hu" data-title="Asszociativitás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</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 – Macedonian" lang="mk" hreflang="mk" data-title="Асоцијативност" data-language-autonym="Македонски" data-language-local-name="Macedonian" 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 – Malay" lang="ms" hreflang="ms" data-title="Kalis sekutuan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Associativiteit_(wiskunde)" title="Associativiteit (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Associativiteit (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B5%90%E5%90%88%E6%B3%95%E5%89%87" title="結合法則 – Japanese" lang="ja" hreflang="ja" data-title="結合法則" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Asotsiatiifgesets" title="Asotsiatiifgesets – Northern Frisian" lang="frr" hreflang="frr" data-title="Asotsiatiifgesets" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Assosiativ_lov" title="Assosiativ lov – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Assosiativ lov" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Assosiativitet" title="Assosiativitet – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Assosiativitet" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Associativitat" title="Associativitat – Occitan" lang="oc" hreflang="oc" data-title="Associativitat" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/%C5%81%C4%85czno%C5%9B%C4%87_(matematyka)" title="Łączność (matematyka) – Polish" lang="pl" hreflang="pl" data-title="Łączność (matematyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Associatividade" title="Associatividade – Portuguese" lang="pt" hreflang="pt" data-title="Associatividade" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Asociativitate" title="Asociativitate – Romanian" lang="ro" hreflang="ro" data-title="Asociativitate" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%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) – Russian" lang="ru" hreflang="ru" data-title="Ассоциативность (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Vetia_e_shoq%C3%ABrimit" title="Vetia e shoqërimit – Albanian" lang="sq" hreflang="sq" data-title="Vetia e shoqërimit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/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 – Slovenian" lang="sl" hreflang="sl" data-title="Asociativnost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</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 – Central Kurdish" lang="ckb" hreflang="ckb" data-title="یەکتربەستن" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</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 – Serbian" lang="sr" hreflang="sr" data-title="Асоцијативност" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Asocijativnost" title="Asocijativnost – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Asocijativnost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Liit%C3%A4nn%C3%A4isyys" title="Liitännäisyys – Finnish" lang="fi" hreflang="fi" data-title="Liitännäisyys" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Associativitet" title="Associativitet – Swedish" lang="sv" hreflang="sv" data-title="Associativitet" data-language-autonym="Svenska" data-language-local-name="Swedish" 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-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-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) – Turkish" lang="tr" hreflang="tr" data-title="Birleşme özelliği (ikili işlemler)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%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 – Ukrainian" lang="uk" hreflang="uk" data-title="Асоціативність" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</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 – Venetian" lang="vec" hreflang="vec" data-title="Propietà asociativa" data-language-autonym="Vèneto" data-language-local-name="Venetian" 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 – Vietnamese" lang="vi" hreflang="vi" data-title="Tính kết hợp" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BB%93%E5%90%88%E5%BE%8B" title="结合律 – Wu" lang="wuu" hreflang="wuu" data-title="结合律" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%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" 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class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Property of a mathematical operation</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the associative property in mathematics. For associativity in the central processing unit memory cache, see <a href="/wiki/CPU_cache#Associativity" title="CPU cache">CPU cache § Associativity</a>. For associativity in programming languages, see <a href="/wiki/Operator_associativity" title="Operator associativity">operator associativity</a>. For the meaning of an associated group of people in linguistics, see <a href="/wiki/Associativity_(linguistics)" class="mw-redirect" title="Associativity (linguistics)">Associativity (linguistics)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Associative" and "non-associative" redirect here. 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Associative+property%22">"Associative property"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Associative+property%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Associative+property%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Associative+property%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Associative+property%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Associative+property%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">June 2009</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox vcard"><caption class="infobox-title fn" style="padding-bottom:0.2em;">Associative property</caption><tbody><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File: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></span><div class="infobox-caption">A visual graph representing associative operations; <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></div></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Principle" title="Principle">Law</a>, <a href="/wiki/Rule_of_replacement" title="Rule of replacement">rule of replacement</a></td></tr><tr><th scope="row" class="infobox-label">Field</th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"> <ul><li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary algebra</a></li> <li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="infobox-label">Symbolic statement</th><td class="infobox-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"> <ol><li>Elementary algebra <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)\forall x,y,z\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mo>∗</mo> <mspace width="thinmathspace" /> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>∗</mo> <mspace width="thinmathspace" /> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mo>∗</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>y</mi> <mspace width="thinmathspace" /> <mo>∗</mo> <mspace width="thinmathspace" /> <mi>z</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∀</mi> <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 (x\,*\,y)\,*\,z=x\,*\,(y\,*\,z)\forall x,y,z\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c347b6353379114f9d70f61e1a608aed9f356d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.013ex; height:2.843ex;" alt="{\displaystyle (x\,*\,y)\,*\,z=x\,*\,(y\,*\,z)\forall x,y,z\in S}"></span></dd></dl></li> <li>Propositional calculus <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> <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></li></ol> </div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>associative property</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> is a property of some <a href="/wiki/Binary_operation" title="Binary operation">binary operations</a> that means that rearranging the <a href="/wiki/Parentheses" class="mw-redirect" title="Parentheses">parentheses</a> in an expression will not change the result. In <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>, <b>associativity</b> is a <a href="/wiki/Validity_(logic)" title="Validity (logic)">valid</a> <a href="/wiki/Rule_of_replacement" title="Rule of replacement">rule of replacement</a> for <a href="/wiki/Well-formed_formula" title="Well-formed formula">expressions</a> in <a href="/wiki/Formal_proof" title="Formal proof">logical proofs</a>. </p><p>Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> are performed does not matter as long as the sequence of the <a href="/wiki/Operand" title="Operand">operands</a> is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(2+3)+4&=2+(3+4)=9\,\\2\times (3\times 4)&=(2\times 3)\times 4=24.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>4</mn> </mtd> <mtd> <mi></mi> <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" /> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>×</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <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> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(2+3)+4&=2+(3+4)=9\,\\2\times (3\times 4)&=(2\times 3)\times 4=24.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb094798b35a217beb6a0a79b6e72772cd585cfb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.875ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(2+3)+4&=2+(3+4)=9\,\\2\times (3\times 4)&=(2\times 3)\times 4=24.\end{aligned}}}"></span> </p><p>Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any <a href="/wiki/Real_number" title="Real number">real numbers</a>, it can be said that "addition and multiplication of real numbers are associative operations". </p><p>Associativity is not the same as <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a>, which addresses whether the order of two operands affects the result. For example, the order does not matter in the multiplication of real numbers, that is, <span class="texhtml"><var style="padding-right: 1px;">a</var> × <var style="padding-right: 1px;">b</var> = <var style="padding-right: 1px;">b</var> × <var style="padding-right: 1px;">a</var></span>, so we say that the multiplication of real numbers is a commutative operation. However, operations such as <a href="/wiki/Function_composition" title="Function composition">function composition</a> and <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> are associative, but not (generally) commutative. </p><p>Associative operations are abundant in mathematics; in fact, many <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a> (such as <a href="/wiki/Semigroup_(mathematics)" class="mw-redirect" title="Semigroup (mathematics)">semigroups</a> and <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">categories</a>) explicitly require their binary operations to be associative. </p><p>However, many important and interesting operations are non-associative; some examples include <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>, and the <a href="/wiki/Vector_cross_product" class="mw-redirect" title="Vector cross product">vector cross product</a>. In contrast to the theoretical properties of real numbers, the addition of <a href="/wiki/Floating_point" class="mw-redirect" title="Floating point">floating point</a> numbers in computer science is not associative, and the choice of how to associate an expression can have a significant effect on rounding error. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associative_property&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File: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>A binary operation ∗ on the set <i>S</i> is associative when <a href="/wiki/Commutative_diagram" title="Commutative diagram">this diagram commutes</a>. That is, when the two paths from <span class="texhtml"><var style="padding-right: 1px;">S</var>×<var style="padding-right: 1px;">S</var>×<var style="padding-right: 1px;">S</var></span> to <span class="texhtml mvar" style="font-style:italic;">S</span> <a href="/wiki/Function_composition" title="Function composition">compose</a> to the same function from <span class="texhtml"><var style="padding-right: 1px;">S</var>×<var style="padding-right: 1px;">S</var>×<var style="padding-right: 1px;">S</var></span> to <span class="texhtml mvar" style="font-style:italic;">S</span>.</figcaption></figure> <p>Formally, a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</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 \ast }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ast }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1858484bef51b1435c2b986c728a81788051803" 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 \ast }"></span> on a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <span class="texhtml mvar" style="font-style:italic;">S</span> is called <b>associative</b> if it satisfies the <b>associative law</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 (x\ast y)\ast z=x\ast (y\ast 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\ast y)\ast z=x\ast (y\ast z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ef6c71d78dc8cc3e1ae35676e5dcf94787201f" 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\ast y)\ast z=x\ast (y\ast z)}"></span>, for all <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> in <span class="texhtml mvar" style="font-style:italic;">S</span>.}}</dd></dl> <p>Here, ∗ is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbol (<a href="/wiki/Juxtaposition#Mathematics" title="Juxtaposition">juxtaposition</a>) as for <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>. </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 (xy)z=x(yz)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xy)z=x(yz)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efb185761d3d71a9a59cf8ed17b9a40c518e08ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.864ex; height:2.843ex;" alt="{\displaystyle (xy)z=x(yz)}"></span>, for all <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> in <span class="texhtml mvar" style="font-style:italic;">S</span>.</dd></dl> <p>The associative law can also be expressed in functional notation thus: <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))(x)=((f\circ g)\circ h)(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\circ (g\circ h))(x)=((f\circ g)\circ h)(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6195ade61d73243f355a0de9fc7a3334c1a91b9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.86ex; height:2.843ex;" alt="{\displaystyle (f\circ (g\circ h))(x)=((f\circ g)\circ h)(x)}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Generalized_associative_law">Generalized associative law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associative_property&action=edit&section=2" title="Edit section: Generalized associative law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tamari_lattice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Tamari_lattice.svg/220px-Tamari_lattice.svg.png" decoding="async" width="220" height="306" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Tamari_lattice.svg/330px-Tamari_lattice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Tamari_lattice.svg/440px-Tamari_lattice.svg.png 2x" data-file-width="504" data-file-height="702" /></a><figcaption>In the absence of the associative property, five factors <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>,<span class="texhtml mvar" style="font-style:italic;">c</span>, <span class="texhtml mvar" style="font-style:italic;">d</span>, <span class="texhtml mvar" style="font-style:italic;">e</span> result in a <a href="/wiki/Tamari_lattice" title="Tamari lattice">Tamari lattice</a> of order four, possibly different products.</figcaption></figure> <p>If a binary operation is associative, repeated application of the operation produces the same result regardless of how valid pairs of parentheses are inserted in the expression.<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> This is called the <b>generalized associative law</b>. </p><p>The number of possible bracketings is just the <a href="/wiki/Catalan_number" title="Catalan number">Catalan number</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 C_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0301812adb392070d834ca2df4ed97f1cf132f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle C_{n}}"></span> , for <i>n</i> operations on <i>n+1</i> values. For instance, a product of 3 operations on 4 elements may be written (ignoring permutations of the arguments), in <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_{3}=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{3}=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75450a359cca8f4b99600ec68e656c55f35d0b1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.977ex; height:2.509ex;" alt="{\displaystyle C_{3}=5}"></span> possible ways: </p> <ul><li><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></li> <li><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></li> <li><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></li> <li><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"> <mo stretchy="false">(</mo> <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/03b9e8c4e23e874496ba5b0adaedb47c9c6c41ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.973ex; height:2.843ex;" alt="{\displaystyle (a(b(cd))}"></span></li> <li><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></li></ul> <p>If the product operation is associative, the generalized associative law says that all these expressions will yield the same result. So unless the expression with omitted parentheses already has a different meaning (see below), the parentheses can be considered unnecessary and "the" product can be written unambiguously as </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> </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/51b4d1adf9d72170e223b3e92b904eeebe836f84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.45ex; height:2.176ex;" alt="{\displaystyle abcd}"></span></dd></dl> <p>As the number of elements increases, the <a href="/wiki/Catalan_number#Applications_in_combinatorics" title="Catalan number">number of possible ways to insert parentheses</a> grows quickly, but they remain unnecessary for disambiguation. </p><p>An example where this does not work is the <a href="/wiki/Logical_biconditional" title="Logical biconditional">logical biconditional</a> <span class="texhtml">↔</span>. It is associative; thus, <span class="texhtml"><var style="padding-right: 1px;">A</var> ↔ (<var style="padding-right: 1px;">B</var> ↔ <var style="padding-right: 1px;">C</var>)</span> is equivalent to <span class="texhtml">(<var style="padding-right: 1px;">A</var> ↔ <var style="padding-right: 1px;">B</var>) ↔ <var style="padding-right: 1px;">C</var></span>, but <span class="texhtml"><var style="padding-right: 1px;">A</var> ↔ <var style="padding-right: 1px;">B</var> ↔ <var style="padding-right: 1px;">C</var></span> most commonly means <span class="texhtml">(<var style="padding-right: 1px;">A</var> ↔ <var style="padding-right: 1px;">B</var>) and (<var style="padding-right: 1px;">B</var> ↔ <var style="padding-right: 1px;">C</var>)</span>, which is not equivalent. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associative_property&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File: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>The addition of real numbers is associative.</figcaption></figure> <p>Some examples of associative operations include the following. </p> <div><ul><li>The <a href="/wiki/String_concatenation" class="mw-redirect" title="String concatenation">concatenation</a> of the three strings <code>"hello"</code>, <code>" "</code>, <code>"world"</code> can be computed by concatenating the first two strings (giving <code>"hello "</code>) and appending the third string (<code>"world"</code>), or by joining the second and third string (giving <code>" world"</code>) and concatenating the first string (<code>"hello"</code>) with the result. The two methods produce the same result; string concatenation is associative (but not commutative).</li><li>In <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, <a href="/wiki/Addition" title="Addition">addition</a> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> of <a href="/wiki/Real_number" title="Real number">real numbers</a> are associative; i.e., <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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{for all }}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" /> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mspace width="thinmathspace" /> <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" /> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>y</mi> <mspace width="thinmathspace" /> <mi>z</mi> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="1em" /> <mtext> </mtext> <mtext> </mtext> <mspace width="thinmathspace" /> </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">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{for all }}x,y,z\in \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0373f6dd20f2cbcc2106b82d294fda046623cc8e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.992ex; 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{for all }}x,y,z\in \mathbb {R} .}"></span> </p> Because of associativity, the grouping parentheses can be omitted without ambiguity.</li><li>The trivial operation <span class="texhtml"><var style="padding-right: 1px;">x</var> ∗ <var style="padding-right: 1px;">y</var> = <var style="padding-right: 1px;">x</var></span> (that is, the result is the first argument, no matter what the second argument is) is associative but not commutative. Likewise, the trivial operation <span class="texhtml"><var style="padding-right: 1px;">x</var> ∘ <var style="padding-right: 1px;">y</var> = <var style="padding-right: 1px;">y</var></span> (that is, the result is the second argument, no matter what the first argument is) is associative but not commutative.</li><li>Addition and multiplication of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> and <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> are associative. Addition of <a href="/wiki/Octonion" title="Octonion">octonions</a> is also associative, but multiplication of octonions is non-associative.</li><li>The <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> and <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a> functions act associatively. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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" /> </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" /> </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-display 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></li><li>Taking the <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> or the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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" /> </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" /> </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-display 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></li><li>If <span class="texhtml mvar" style="font-style:italic;">M</span> is some set and <span class="texhtml mvar" style="font-style:italic;">S</span> denotes the set of all functions from <span class="texhtml mvar" style="font-style:italic;">M</span> to <span class="texhtml mvar" style="font-style:italic;">M</span>, then the operation of <a href="/wiki/Function_composition" title="Function composition">function composition</a> on <span class="texhtml mvar" style="font-style:italic;">S</span> is associative:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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" /> <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-display 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></li><li>Slightly more generally, given four sets <span class="texhtml mvar" style="font-style:italic;">M</span>, <span class="texhtml mvar" style="font-style:italic;">N</span>, <span class="texhtml mvar" style="font-style:italic;">P</span> and <span class="texhtml mvar" style="font-style:italic;">Q</span>, with <span class="texhtml"><var style="padding-right: 1px;">h</var> : <var style="padding-right: 1px;">M</var> → <var style="padding-right: 1px;">N</var></span>, <span class="texhtml"><var style="padding-right: 1px;">g</var> : <var style="padding-right: 1px;">N</var> → <var style="padding-right: 1px;">P</var></span>, and <span class="texhtml"><var style="padding-right: 1px;">f</var> : <var style="padding-right: 1px;">P</var> → <var style="padding-right: 1px;">Q</var></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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-display 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> as before. In short, composition of maps is always associative.</li><li>In <a href="/wiki/Category_theory" title="Category theory">category theory</a>, composition of morphisms is associative by definition. Associativity of functors and natural transformations follows from associativity of morphisms.</li><li>Consider a set with three elements, <span class="texhtml mvar" style="font-style:italic;">A</span>, <span class="texhtml mvar" style="font-style:italic;">B</span>, and <span class="texhtml mvar" style="font-style:italic;">C</span>. The following operation: <table class="wikitable" style="text-align:center"> <tbody><tr> <th>×</th> <th><span class="texhtml mvar" style="font-style:italic;">A</span></th> <th><span class="texhtml mvar" style="font-style:italic;">B</span></th> <th><span class="texhtml mvar" style="font-style:italic;">C</span> </th></tr> <tr> <th><span class="texhtml mvar" style="font-style:italic;">A</span> </th> <td><span class="texhtml mvar" style="font-style:italic;">A</span></td> <td><span class="texhtml mvar" style="font-style:italic;">A</span></td> <td><span class="texhtml mvar" style="font-style:italic;">A</span> </td></tr> <tr> <th><span class="texhtml mvar" style="font-style:italic;">B</span> </th> <td><span class="texhtml mvar" style="font-style:italic;">A</span></td> <td><span class="texhtml mvar" style="font-style:italic;">B</span></td> <td><span class="texhtml mvar" style="font-style:italic;">C</span> </td></tr> <tr> <th><span class="texhtml mvar" style="font-style:italic;">C</span> </th> <td><span class="texhtml mvar" style="font-style:italic;">A</span></td> <td><span class="texhtml mvar" style="font-style:italic;">A</span></td> <td><span class="texhtml mvar" style="font-style:italic;">A</span> </td></tr></tbody></table> is associative. Thus, for example, <span class="texhtml"><var style="padding-right: 1px;">A</var>(<var style="padding-right: 1px;">B</var><var style="padding-right: 1px;">C</var>) = (<var style="padding-right: 1px;">A</var><var style="padding-right: 1px;">B</var>)<var style="padding-right: 1px;">C</var> = <var style="padding-right: 1px;">A</var></span>. This operation is not commutative.</li><li>Because <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> represent <a href="/wiki/Linear_map" title="Linear map">linear functions</a>, and <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> represents function composition, one can immediately conclude that matrix multiplication is associative.<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><li>For <a href="/wiki/Real_number" title="Real number">real numbers</a> (and for any <a href="/wiki/Totally_ordered_set" class="mw-redirect" title="Totally ordered set">totally ordered set</a>), the minimum and maximum operation is associative: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(a,\max(b,c))=\max(\max(a,b),c)\quad {\text{ and }}\quad \min(a,\min(b,c))=\min(\min(a,b),c).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em" /> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(a,\max(b,c))=\max(\max(a,b),c)\quad {\text{ and }}\quad \min(a,\min(b,c))=\min(\min(a,b),c).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5cac21f54f4368b692d5578cd7fe51f6599847" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:85.272ex; height:2.843ex;" alt="{\displaystyle \max(a,\max(b,c))=\max(\max(a,b),c)\quad {\text{ and }}\quad \min(a,\min(b,c))=\min(\min(a,b),c).}"></span></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="Propositional_logic">Propositional logic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associative_property&action=edit&section=4" title="Edit section: Propositional logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output 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dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Rule_of_inference" title="Rule of inference">Transformation rules</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;;background:#ddf;font-size:110%; border-bottom:1px #fefefe solid;"> <a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;"> <a href="/wiki/Rule_of_inference" title="Rule of inference">Rules of inference</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <ul><li><a href="/wiki/Conditional_proof" title="Conditional proof"><span>Implication introduction</span></a> / <a href="/wiki/Modus_ponens" title="Modus ponens"><span title="A→B,   A   ⊢   B">elimination (<i>modus ponens</i>)</span></a></li> <li><a href="/wiki/Biconditional_introduction" title="Biconditional introduction"><span title="A→B,   B→A   ⊢   A↔B">Biconditional introduction</span></a> / <a href="/wiki/Biconditional_elimination" title="Biconditional elimination"><span title="A↔B   ⊢   A→B">elimination</span></a></li> <li><a href="/wiki/Conjunction_introduction" title="Conjunction introduction"><span title="A,   B   ⊢   A∧B">Conjunction introduction</span></a> / <a href="/wiki/Conjunction_elimination" title="Conjunction elimination"><span title="A∧B   ⊢   A">elimination</span></a></li> <li><a href="/wiki/Disjunction_introduction" title="Disjunction introduction"><span title="A   ⊢   A∨B">Disjunction introduction</span></a> / <a href="/wiki/Disjunction_elimination" title="Disjunction elimination"><span title="A∨B,   A→C,   B→C   ⊢   C">elimination</span></a></li> <li><a href="/wiki/Disjunctive_syllogism" title="Disjunctive syllogism"><span title="A∨B,   ¬A   ⊢   B">Disjunctive</span></a> / <a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism"><span title="A→B,   B→C   ⊢   A→C">hypothetical syllogism</span></a></li> <li><a href="/wiki/Constructive_dilemma" title="Constructive dilemma"><span title="A→P,   B→Q,   A∨B   ⊢   P∨Q">Constructive</span></a> / <a href="/wiki/Destructive_dilemma" title="Destructive dilemma"><span title="A→P,   B→Q,   ¬P∨¬Q   ⊢   ¬A∨¬B">destructive dilemma</span></a></li> <li><a href="/wiki/Absorption_(logic)" title="Absorption (logic)"><span title="A→B   ⊢   A→A∧B">Absorption</span></a> / <a href="/wiki/Modus_tollens" title="Modus tollens"><span title="A→B,   ¬B   ⊢   ¬A"><i>modus tollens</i></span></a> / <a href="/wiki/Modus_ponendo_tollens" title="Modus ponendo tollens"><span title="¬(A∧B),   A   ⊢   ¬B"><i>modus ponendo tollens</i></span></a></li> <li><a href="/wiki/Negation_introduction" title="Negation introduction">Negation introduction</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;"> <a href="/wiki/Rule_of_replacement" title="Rule of replacement">Rules of replacement</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <div class="hlist"> <ul><li><a class="mw-selflink-fragment" href="#Propositional_logic"><span title="A∨(B∨C)   =   (A∨B)∨C">Associativity</span></a></li> <li><a href="/wiki/Commutative_property#Propositional_logic" title="Commutative property"><span title="A∨B   =   B∨A">Commutativity</span></a></li> <li><a href="/wiki/Distributive_property#Propositional_logic" title="Distributive property"><span title="A∧(B∨C)   =   (A∧B)∨(A∧C)">Distributivity</span></a></li> <li><a href="/wiki/Double_negation" title="Double negation"><span title="¬¬A   =   A">Double negation</span></a></li> <li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/Transposition_(logic)" class="mw-redirect" title="Transposition (logic)">Transposition</a></li> <li><a href="/wiki/Material_implication_(rule_of_inference)" title="Material implication (rule of inference)"><span title="A→B   ⊢   ¬A∨B">Material implication</span></a></li> <li><a href="/wiki/Exportation_(logic)" title="Exportation (logic)"><span title="(A∧B)→C   ⊢   A→(B→C)">Exportation</span></a></li> <li><a href="/wiki/Tautology_(rule_of_inference)" title="Tautology (rule of inference)"><span title="A∨A   =   A">Tautology</span></a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;;background:#ddf;font-size:110%;"> <a href="/wiki/First-order_logic" title="First-order logic">Predicate logic</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;"> <a href="/wiki/Rule_of_inference" title="Rule of inference">Rules of inference</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <ul><li><a href="/wiki/Universal_generalization" title="Universal generalization">Universal generalization</a> / <a href="/wiki/Universal_instantiation" title="Universal instantiation">instantiation</a></li> <li><a href="/wiki/Existential_generalization" title="Existential generalization">Existential generalization</a> / <a href="/wiki/Existential_instantiation" title="Existential instantiation">instantiation</a></li></ul></td> </tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Rule_of_replacement">Rule of replacement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associative_property&action=edit&section=5" title="Edit section: Rule of replacement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In standard truth-functional propositional logic, <i>association</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> or <i>associativity</i><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> are two <a href="/wiki/Validity_(logic)" title="Validity (logic)">valid</a> <a href="/wiki/Rule_of_replacement" title="Rule of replacement">rules of replacement</a>. The rules allow one to move parentheses in <a href="/wiki/Well-formed_formula" title="Well-formed formula">logical expressions</a> in <a href="/wiki/Formal_proof" title="Formal proof">logical proofs</a>. The rules (using <a href="/wiki/Logical_connective#In_language" title="Logical connective">logical connectives</a> notation) are: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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-display 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> </p><p>and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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-display 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> </p><p>where "<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>" is a <a href="/wiki/Metalogic" title="Metalogic">metalogical</a> <a href="/wiki/Symbol_(formal)" title="Symbol (formal)">symbol</a> representing "can be replaced in a <a href="/wiki/Formal_proof" title="Formal proof">proof</a> with". </p> <div class="mw-heading mw-heading3"><h3 id="Truth_functional_connectives">Truth functional connectives</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associative_property&action=edit&section=6" title="Edit section: Truth functional connectives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Associativity</i> is a property of some <a href="/wiki/Logical_connective" title="Logical connective">logical connectives</a> of truth-functional <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>. The following <a href="/wiki/Logical_equivalence" title="Logical equivalence">logical equivalences</a> demonstrate that associativity is a property of particular connectives. The following (and their converses, since <span class="texhtml">↔</span> is commutative) are truth-functional <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautologies</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Stack Exchange is not a reliable source? (June 2022)">citation needed</span></a></i>]</sup> </p> <dl><dt>Associativity of disjunction</dt> <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> <dt>Associativity of conjunction</dt> <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> <dt>Associativity of equivalence</dt> <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></dl> <p><a href="/wiki/Logical_NOR" title="Logical NOR">Joint denial</a> is an example of a truth functional connective that is <i>not</i> associative. </p> <div class="mw-heading mw-heading2"><h2 id="Non-associative_operation">Non-associative operation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associative_property&action=edit&section=7" title="Edit section: Non-associative operation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A binary operation <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> on a set <i>S</i> that does not satisfy the associative law is called <b>non-associative</b>. Symbolically, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x*y)*z\neq x*(y*z)\qquad {\mbox{for some }}x,y,z\in S.}"> <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> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for some </mtext> </mstyle> </mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x*y)*z\neq x*(y*z)\qquad {\mbox{for some }}x,y,z\in S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fd446a4382f83e60d3229708be37fca0d2640e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.91ex; height:2.843ex;" alt="{\displaystyle (x*y)*z\neq x*(y*z)\qquad {\mbox{for some }}x,y,z\in S.}"></span> </p><p>For such an operation the order of evaluation <i>does</i> matter. For example: </p> <dl><dt><a href="/wiki/Subtraction" title="Subtraction">Subtraction</a></dt> <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" /> <mo>≠</mo> <mspace width="thinmathspace" /> <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> <dt><a href="/wiki/Division_(mathematics)" title="Division (mathematics)">Division</a></dt> <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 (4/2)/2\,\neq \,4/(2/2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mo>≠</mo> <mspace width="thinmathspace" /> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (4/2)/2\,\neq \,4/(2/2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce99bd10172c6b9da8f3a475168880aa004d2a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.116ex; height:2.843ex;" alt="{\displaystyle (4/2)/2\,\neq \,4/(2/2)}"></span></dd> <dt><a href="/wiki/Exponentiation" title="Exponentiation">Exponentiation</a></dt> <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" /> <mo>≠</mo> <mspace width="thinmathspace" /> <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> <dt><a href="/wiki/Vector_cross_product" class="mw-redirect" title="Vector cross product">Vector cross product</a></dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {i} \times (\mathbf {i} \times \mathbf {j} )&=\mathbf {i} \times \mathbf {k} =-\mathbf {j} \\(\mathbf {i} \times \mathbf {i} )\times \mathbf {j} &=\mathbf {0} \times \mathbf {j} =\mathbf {0} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {i} \times (\mathbf {i} \times \mathbf {j} )&=\mathbf {i} \times \mathbf {k} =-\mathbf {j} \\(\mathbf {i} \times \mathbf {i} )\times \mathbf {j} &=\mathbf {0} \times \mathbf {j} =\mathbf {0} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ea8d0b5788a070a6013f6cc5e1eebb914226e6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.357ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {i} \times (\mathbf {i} \times \mathbf {j} )&=\mathbf {i} \times \mathbf {k} =-\mathbf {j} \\(\mathbf {i} \times \mathbf {i} )\times \mathbf {j} &=\mathbf {0} \times \mathbf {j} =\mathbf {0} \end{aligned}}}"></span></dd></dl> <p>Also although addition is associative for finite sums, it is not associative inside infinite sums (<a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a>). For example, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+\dots =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+\dots =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f8f932b8e501383f363b4c13f90f7ee3cc523b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:76.723ex; height:2.843ex;" alt="{\displaystyle (1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+\dots =0}"></span> whereas <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+\dots =1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+\dots =1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b541e235fba8d4aac4f599e4bdbe1a82dc8f08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:81.373ex; height:2.843ex;" alt="{\displaystyle 1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+\dots =1.}"></span> </p><p>Some non-associative operations are fundamental in mathematics. They appear often as the multiplication in structures called <a href="/wiki/Non-associative_algebra" title="Non-associative algebra">non-associative algebras</a>, which have also an addition and a <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>. Examples are the <a href="/wiki/Octonion" title="Octonion">octonions</a> and <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a>. In Lie algebras, the multiplication satisfies <a href="/wiki/Jacobi_identity" title="Jacobi identity">Jacobi identity</a> instead of the associative law; this allows abstracting the algebraic nature of <a href="/wiki/Infinitesimal_transformation" title="Infinitesimal transformation">infinitesimal transformations</a>. </p><p>Other examples are <a href="/wiki/Quasigroup" title="Quasigroup">quasigroup</a>, <a href="/wiki/Quasifield" title="Quasifield">quasifield</a>, <a href="/wiki/Non-associative_ring" class="mw-redirect" title="Non-associative ring">non-associative ring</a>, and <a href="/wiki/Commutative_non-associative_magmas" class="mw-redirect" title="Commutative non-associative magmas">commutative non-associative magmas</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Nonassociativity_of_floating_point_calculation">Nonassociativity of floating point calculation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associative_property&action=edit&section=8" title="Edit section: Nonassociativity of floating point calculation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In mathematics, addition and multiplication of real numbers are associative. By contrast, in computer science, addition and multiplication of <a href="/wiki/Floating_point" class="mw-redirect" title="Floating point">floating point</a> numbers are <i>not</i> associative, as different rounding errors may be introduced when dissimilar-sized values are joined in a different order.<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>To illustrate this, consider a floating point representation with a 4-bit <a href="/wiki/Significand" title="Significand">significand</a>: </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent">(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><span style="color:red;">1</span></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></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">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><span style="color:red;">0</span></sup> + 1.000<sub>2</sub>×2<sup>4</sup> = 1.00<span style="color:red;">0</span><sub>2</sub>×2<sup>4</sup></div> <p>Even though most computers compute with 24 or 53 bits of significand,<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> this is still an important source of rounding error, and approaches such as the <a href="/wiki/Kahan_summation_algorithm" title="Kahan summation algorithm">Kahan summation algorithm</a> are ways to minimise the errors. It can be especially problematic in parallel computing.<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><sup id="cite_ref-Goldberg_1991_10-0" class="reference"><a href="#cite_note-Goldberg_1991-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Notation_for_non-associative_operations">Notation for non-associative operations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associative_property&action=edit&section=9" title="Edit section: Notation for non-associative operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Operator_associativity" title="Operator associativity">Operator associativity</a></div> <p>In general, parentheses must be used to indicate the <a href="/wiki/Order_of_operations" title="Order of operations">order of evaluation</a> if a non-associative operation appears more than once in an expression (unless the notation specifies the order in another way, like <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 {\dfrac {2}{3/4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>2</mn> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {2}{3/4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f37f7089af4593a3be84e2e53b0191a2d69cc590" 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 {\dfrac {2}{3/4}}}"></span>). However, <a href="/wiki/Mathematician" title="Mathematician">mathematicians</a> agree on a particular order of evaluation for several common non-associative operations. This is simply a notational convention to avoid parentheses. </p><p>A <b>left-associative</b> operation is a non-associative operation that is conventionally evaluated from left to right, i.e., </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{array}{l}a*b*c=(a*b)*c\\a*b*c*d=((a*b)*c)*d\\a*b*c*d*e=(((a*b)*c)*d)*e\quad \\{\mbox{etc.}}\end{array}}\right\}{\mbox{for all }}a,b,c,d,e\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 columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo>∗</mo> <mi>c</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo>∗</mo> <mi>c</mi> <mo>∗</mo> <mi>d</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>d</mi> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo>∗</mo> <mi>c</mi> <mo>∗</mo> <mi>d</mi> <mo>∗</mo> <mi>e</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>e</mi> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>etc.</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for all </mtext> </mstyle> </mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{array}{l}a*b*c=(a*b)*c\\a*b*c*d=((a*b)*c)*d\\a*b*c*d*e=(((a*b)*c)*d)*e\quad \\{\mbox{etc.}}\end{array}}\right\}{\mbox{for all }}a,b,c,d,e\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/367f4133c7b3ca10534d63f860d0d15349ab75ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:63.091ex; height:12.843ex;" alt="{\displaystyle \left.{\begin{array}{l}a*b*c=(a*b)*c\\a*b*c*d=((a*b)*c)*d\\a*b*c*d*e=(((a*b)*c)*d)*e\quad \\{\mbox{etc.}}\end{array}}\right\}{\mbox{for all }}a,b,c,d,e\in S}"></span> </p><p>while a <b>right-associative</b> operation is conventionally evaluated from right to left: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{array}{l}x*y*z=x*(y*z)\\w*x*y*z=w*(x*(y*z))\quad \\v*w*x*y*z=v*(w*(x*(y*z)))\quad \\{\mbox{etc.}}\end{array}}\right\}{\mbox{for all }}z,y,x,w,v\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 columnalign="left" 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> </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" /> </mtd> </mtr> <mtr> <mtd> <mi>v</mi> <mo>∗</mo> <mi>w</mi> <mo>∗</mo> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo>=</mo> <mi>v</mi> <mo>∗</mo> <mo stretchy="false">(</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> <mo stretchy="false">)</mo> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>etc.</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for all </mtext> </mstyle> </mrow> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>w</mi> <mo>,</mo> <mi>v</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{array}{l}x*y*z=x*(y*z)\\w*x*y*z=w*(x*(y*z))\quad \\v*w*x*y*z=v*(w*(x*(y*z)))\quad \\{\mbox{etc.}}\end{array}}\right\}{\mbox{for all }}z,y,x,w,v\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9fa5227c1de94c8297d53d1ea2bdf1121db91d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:65.585ex; height:12.843ex;" alt="{\displaystyle \left.{\begin{array}{l}x*y*z=x*(y*z)\\w*x*y*z=w*(x*(y*z))\quad \\v*w*x*y*z=v*(w*(x*(y*z)))\quad \\{\mbox{etc.}}\end{array}}\right\}{\mbox{for all }}z,y,x,w,v\in S}"></span> </p><p>Both left-associative and right-associative operations occur. Left-associative operations include the following: </p> <dl><dt>Subtraction and division of real numbers<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><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-Bronstein_1987_15-0" class="reference"><a href="#cite_note-Bronstein_1987-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></dt> <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"> <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> </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/332b4de72030c6dcdd3471eb33a9596cffbf9c31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.416ex; height:2.843ex;" alt="{\displaystyle x-y-z=(x-y)-z}"></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 x/y/z=(x/y)/z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>z</mi> </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/b0745b1190349c3ea09b5c8269410feec9997198" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.704ex; height:2.843ex;" alt="{\displaystyle x/y/z=(x/y)/z}"></span></dd> <dt>Function application</dt> <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" /> <mi>x</mi> <mspace width="thinmathspace" /> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mspace width="thinmathspace" /> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <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> <p>This notation can be motivated by the <a href="/wiki/Currying" title="Currying">currying</a> isomorphism, which enables partial application. </p><p>Right-associative operations include the following: </p> <dl><dt><a href="/wiki/Exponentiation" title="Exponentiation">Exponentiation</a> of real numbers in superscript notation</dt> <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><p>Exponentiation is commonly used with brackets or right-associatively because a repeated left-associative exponentiation operation is of little use. Repeated powers would mostly be rewritten with multiplication:</p></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 (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><p>Formatted correctly, the superscript inherently behaves as a set of parentheses; e.g. in the expression <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> the addition is performed <a href="/wiki/Order_of_operations" title="Order of operations">before</a> the exponentiation despite there being no explicit parentheses <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> wrapped around it. Thus given an expression such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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>, the full exponent <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> of the base <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> is evaluated first. However, in some contexts, especially in handwriting, the difference between <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> and <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> can be hard to see. In such a case, right-associativity is usually implied.</p></dd> <dt><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function definition</a></dt> <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 \mathbb {Z} \rightarrow \mathbb {Z} \rightarrow \mathbb {Z} =\mathbb {Z} \rightarrow (\mathbb {Z} \rightarrow \mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \rightarrow \mathbb {Z} \rightarrow \mathbb {Z} =\mathbb {Z} \rightarrow (\mathbb {Z} \rightarrow \mathbb {Z} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/328525848cc4d210ba95a8927ebe7dd9c3f1ad66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.666ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} \rightarrow \mathbb {Z} \rightarrow \mathbb {Z} =\mathbb {Z} \rightarrow (\mathbb {Z} \rightarrow \mathbb {Z} )}"></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 x\mapsto y\mapsto x-y=x\mapsto (y\mapsto x-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>y</mi> <mo stretchy="false">↦</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">↦</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto y\mapsto x-y=x\mapsto (y\mapsto x-y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/772c99f4ab4899b43ef9ae9facad1448fd7abd7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.986ex; height:2.843ex;" alt="{\displaystyle x\mapsto y\mapsto x-y=x\mapsto (y\mapsto x-y)}"></span><p>Using right-associative notation for these operations can be motivated by the <a href="/wiki/Curry%E2%80%93Howard_correspondence" title="Curry–Howard correspondence">Curry–Howard correspondence</a> and by the <a href="/wiki/Currying" title="Currying">currying</a> isomorphism.</p></dd></dl> <p>Non-associative operations for which no conventional evaluation order is defined include the following. </p> <dl><dt>Exponentiation of real numbers in infix notation<sup id="cite_ref-Codeplea_2016_16-0" class="reference"><a href="#cite_note-Codeplea_2016-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></dt> <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> <dt><a href="/wiki/Knuth%27s_up-arrow_notation" title="Knuth's up-arrow notation">Knuth's up-arrow operators</a></dt> <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> <dt>Taking the <a href="/wiki/Cross_product" title="Cross product">cross product</a> of three vectors</dt> <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 {\vec {a}}\times ({\vec {b}}\times {\vec {c}})\neq ({\vec {a}}\times {\vec {b}})\times {\vec {c}}\qquad {\mbox{ for some }}{\vec {a}},{\vec {b}},{\vec {c}}\in \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>≠</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for some </mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\times ({\vec {b}}\times {\vec {c}})\neq ({\vec {a}}\times {\vec {b}})\times {\vec {c}}\qquad {\mbox{ for some }}{\vec {a}},{\vec {b}},{\vec {c}}\in \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/017745f14a400226c68d119ac4425c871b433689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.58ex; height:3.343ex;" alt="{\displaystyle {\vec {a}}\times ({\vec {b}}\times {\vec {c}})\neq ({\vec {a}}\times {\vec {b}})\times {\vec {c}}\qquad {\mbox{ for some }}{\vec {a}},{\vec {b}},{\vec {c}}\in \mathbb {R} ^{3}}"></span></dd> <dt>Taking the pairwise <a href="/wiki/Average" title="Average">average</a> of real numbers</dt> <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" /> <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> <dt>Taking the <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">relative complement</a> of sets</dt> <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\backslash B)\backslash C\neq 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> <mo>≠</mo> <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\neq A\backslash (B\backslash C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c35c5844eb658cca6efcebc9d370454f494405f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.914ex; height:2.843ex;" alt="{\displaystyle (A\backslash B)\backslash C\neq A\backslash (B\backslash C)}"></span>.<p>(Compare <a href="/wiki/Material_nonimplication" title="Material nonimplication">material nonimplication</a> in logic.)</p></dd></dl> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associative_property&action=edit&section=10" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> seems to have coined the term "associative property"<sup id="cite_ref-Hamilton_17-0" class="reference"><a href="#cite_note-Hamilton-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> around 1844, a time when he was contemplating the non-associative algebra of the <a href="/wiki/Octonions" class="mw-redirect" title="Octonions">octonions</a> he had learned about from <a href="/wiki/John_T._Graves" title="John T. Graves">John T. Graves</a>.<sup id="cite_ref-Baez_18-0" class="reference"><a href="#cite_note-Baez-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associative_property&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/associative_property" class="extiw" title="wiktionary:Special:Search/associative property">associative property</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><a href="/wiki/Light%27s_associativity_test" title="Light's associativity test">Light's associativity test</a></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping series</a>, the use of addition associativity for cancelling terms in an infinite <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a></li> <li>A <a href="/wiki/Semigroup" title="Semigroup">semigroup</a> is a set with an associative binary operation.</li> <li><a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">Commutativity</a> and <a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distributivity</a> are two other frequently discussed properties of binary operations.</li> <li><a href="/wiki/Power_associativity" title="Power associativity">Power associativity</a>, <a href="/wiki/Alternativity" title="Alternativity">alternativity</a>, <a href="/wiki/Flexible_algebra" title="Flexible algebra">flexibility</a> and <a href="/wiki/N-ary_associativity" title="N-ary associativity">N-ary associativity</a> are weak forms of associativity.</li> <li><a href="/wiki/Moufang_loop" title="Moufang loop">Moufang identities</a> also provide a weak form of associativity.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associative_property&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHungerford1974" class="citation book cs1">Hungerford, Thomas W. (1974). <i>Algebra</i> (1st ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. p. 24. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0387905181" title="Special:BookSources/978-0387905181"><bdi>978-0387905181</bdi></a>. <q>Definition 1.1 (i) a(bc) = (ab)c for all a, b, c in 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.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAssociative+property" class="Z3988"></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDurbin1992" class="citation book cs1">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> (3rd ed.). New York: Wiley. p. 78. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-51001-7" title="Special:BookSources/978-0-471-51001-7"><bdi>978-0-471-51001-7</bdi></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 width="thinmathspace" /> <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}\cdots 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}\cdots a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bacfc3e51ec6f5546c6bf77b2d7d1e695442d55" 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}\cdots 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%2Fen.wikipedia.org%3AAssociative+property" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://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">. Retrieved <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%2Fen.wikipedia.org%3AAssociative+property" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMooreParker2017" class="citation book cs1">Moore, Brooke Noel; Parker, Richard (2017). <i>Critical Thinking</i> (12th ed.). 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Education Place.</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"><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>. <a href="/wiki/Khan_Academy" title="Khan Academy">Khan Academy</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 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">Archived</a> 2022-07-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, section 9. 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