Associatività - Wikipedia

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vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non_associatività"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Non associatività</span> </div> </a> <ul id="toc-Non_associatività-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_esempi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_esempi"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Altri esempi</span> </div> </a> <ul id="toc-Altri_esempi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" 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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Associatività</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="Vai a una voce in un'altra lingua. Disponibile in 67 lingue" > <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 lingue</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 - arabo" lang="ar" hreflang="ar" data-title="عملية تجميعية" data-language-autonym="العربية" data-language-local-name="arabo" 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 - asturiano" lang="ast" hreflang="ast" data-title="Asociatividá" data-language-autonym="Asturianu" data-language-local-name="asturiano" 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) - baschiro" lang="ba" hreflang="ba" data-title="Ассоциативлыҡ (математика)" data-language-autonym="Башҡортса" data-language-local-name="baschiro" 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 - bielorusso" lang="be" hreflang="be" data-title="Асацыятыўная аперацыя" data-language-autonym="Беларуская" data-language-local-name="bielorusso" 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 - bulgaro" lang="bg" hreflang="bg" data-title="Асоциативност" data-language-autonym="Български" data-language-local-name="bulgaro" 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 - bosniaco" lang="bs" hreflang="bs" data-title="Asocijativnost" data-language-autonym="Bosanski" data-language-local-name="bosniaco" 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 - catalano" lang="ca" hreflang="ca" data-title="Propietat associativa" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DB%8C%DB%95%DA%A9%D8%AA%D8%B1%D8%A8%DB%95%D8%B3%D8%AA%D9%86" title="یەکتربەستن - curdo centrale" lang="ckb" hreflang="ckb" data-title="یەکتربەستن" data-language-autonym="کوردی" data-language-local-name="curdo centrale" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Asociativita" title="Asociativita - ceco" lang="cs" hreflang="cs" data-title="Asociativita" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D1%81%D1%81%D0%B0%D1%86%D0%B8%D0%B0%D1%82%D0%B8%D0%B2%D0%BB%C4%83%D1%85_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ассациативлăх (математика) - ciuvascio" lang="cv" hreflang="cv" data-title="Ассациативлăх (математика)" data-language-autonym="Чӑвашла" data-language-local-name="ciuvascio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Associativitet" title="Associativitet - danese" lang="da" hreflang="da" data-title="Associativitet" data-language-autonym="Dansk" data-language-local-name="danese" 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 - tedesco" lang="de" hreflang="de" data-title="Assoziativgesetz" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CF%81%CE%BF%CF%83%CE%B5%CF%84%CE%B1%CE%B9%CF%81%CE%B9%CF%83%CF%84%CE%B9%CE%BA%CE%AE_%CE%B9%CE%B4%CE%B9%CF%8C%CF%84%CE%B7%CF%84%CE%B1" title="Προσεταιριστική ιδιότητα - greco" lang="el" hreflang="el" data-title="Προσεταιριστική ιδιότητα" data-language-autonym="Ελληνικά" data-language-local-name="greco" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Associative_property" title="Associative property - inglese" lang="en" hreflang="en" data-title="Associative property" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Asocieco" title="Asocieco - esperanto" lang="eo" hreflang="eo" data-title="Asocieco" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Asociatividad_(%C3%A1lgebra)" title="Asociatividad (álgebra) - spagnolo" lang="es" hreflang="es" data-title="Asociatividad (álgebra)" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Assotsiatiivsus" title="Assotsiatiivsus - estone" lang="et" hreflang="et" data-title="Assotsiatiivsus" data-language-autonym="Eesti" data-language-local-name="estone" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Elkarkortasun" title="Elkarkortasun - basco" lang="eu" hreflang="eu" data-title="Elkarkortasun" data-language-autonym="Euskara" data-language-local-name="basco" 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 - persiano" lang="fa" hreflang="fa" data-title="خاصیت شرکت‌پذیری" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Liit%C3%A4nn%C3%A4isyys" title="Liitännäisyys - finlandese" lang="fi" hreflang="fi" data-title="Liitännäisyys" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Associativit%C3%A9" title="Associativité - francese" lang="fr" hreflang="fr" data-title="Associativité" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Asotsiatiifgesets" title="Asotsiatiifgesets - frisone settentrionale" lang="frr" hreflang="frr" data-title="Asotsiatiifgesets" data-language-autonym="Nordfriisk" data-language-local-name="frisone settentrionale" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Oibr%C3%ADocht_chomhthiomsaitheach" title="Oibríocht chomhthiomsaitheach - irlandese" lang="ga" hreflang="ga" data-title="Oibríocht chomhthiomsaitheach" data-language-autonym="Gaeilge" data-language-local-name="irlandese" 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) - galiziano" lang="gl" hreflang="gl" data-title="Asociatividade (álxebra)" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_%D7%90%D7%A1%D7%95%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%91%D7%99%D7%AA" title="פעולה אסוציאטיבית - ebraico" lang="he" hreflang="he" data-title="פעולה אסוציאטיבית" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Asocijativnost" title="Asocijativnost - croato" lang="hr" hreflang="hr" data-title="Asocijativnost" data-language-autonym="Hrvatski" data-language-local-name="croato" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Asszociativit%C3%A1s" title="Asszociativitás - ungherese" lang="hu" hreflang="hu" data-title="Asszociativitás" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B6%D5%B8%D6%82%D5%A3%D5%B8%D6%80%D5%A4%D5%A1%D5%AF%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Զուգորդականություն - armeno" lang="hy" hreflang="hy" data-title="Զուգորդականություն" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Associativitate" title="Associativitate - interlingua" lang="ia" hreflang="ia" data-title="Associativitate" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sifat_asosiatif" title="Sifat asosiatif - indonesiano" lang="id" hreflang="id" data-title="Sifat asosiatif" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tengiregla" title="Tengiregla - islandese" lang="is" hreflang="is" data-title="Tengiregla" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target"><span>Íslenska</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 - giapponese" lang="ja" hreflang="ja" data-title="結合法則" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%81%D1%81%D0%BE%D1%86%D0%B8%D0%B0%D1%82%D0%B8%D0%B2%D1%82%D1%96%D0%BA_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Ассоциативтік операция - kazako" lang="kk" hreflang="kk" data-title="Ассоциативтік операция" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B2%B0%ED%95%A9%EB%B2%95%EC%B9%99" title="결합법칙 - coreano" lang="ko" hreflang="ko" data-title="결합법칙" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Associativitas_(mathematica)" title="Associativitas (mathematica) - latino" lang="la" hreflang="la" data-title="Associativitas (mathematica)" data-language-autonym="Latina" data-language-local-name="latino" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Asociatyvumas" title="Asociatyvumas - lituano" lang="lt" hreflang="lt" data-title="Asociatyvumas" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Asociativit%C4%81te" title="Asociativitāte - lettone" lang="lv" hreflang="lv" data-title="Asociativitāte" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%90%D1%81%D0%BE%D1%86%D0%B8%D1%98%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82" title="Асоцијативност - macedone" lang="mk" hreflang="mk" data-title="Асоцијативност" data-language-autonym="Македонски" data-language-local-name="macedone" 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 - malese" lang="ms" hreflang="ms" data-title="Kalis sekutuan" data-language-autonym="Bahasa Melayu" data-language-local-name="malese" 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) - olandese" lang="nl" hreflang="nl" data-title="Associativiteit (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Assosiativitet" title="Assosiativitet - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Assosiativitet" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Assosiativ_lov" title="Assosiativ lov - norvegese bokmål" lang="nb" hreflang="nb" data-title="Assosiativ lov" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Associativitat" title="Associativitat - occitano" lang="oc" hreflang="oc" data-title="Associativitat" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/%C5%81%C4%85czno%C5%9B%C4%87_(matematyka)" title="Łączność (matematyka) - polacco" lang="pl" hreflang="pl" data-title="Łączność (matematyka)" data-language-autonym="Polski" data-language-local-name="polacco" 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 - portoghese" lang="pt" hreflang="pt" data-title="Associatividade" data-language-autonym="Português" data-language-local-name="portoghese" 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 - rumeno" lang="ro" hreflang="ro" data-title="Asociativitate" data-language-autonym="Română" data-language-local-name="rumeno" 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) - russo" lang="ru" hreflang="ru" data-title="Ассоциативность (математика)" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Asocijativnost" title="Asocijativnost - serbo-croato" lang="sh" hreflang="sh" data-title="Asocijativnost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croato" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Associativity" title="Associativity - Simple English" lang="en-simple" hreflang="en-simple" data-title="Associativity" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Asociat%C3%ADvnos%C5%A5" title="Asociatívnosť - slovacco" lang="sk" hreflang="sk" data-title="Asociatívnosť" data-language-autonym="Slovenčina" data-language-local-name="slovacco" 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 - sloveno" lang="sl" hreflang="sl" data-title="Asociativnost" data-language-autonym="Slovenščina" data-language-local-name="sloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Vetia_e_shoq%C3%ABrimit" title="Vetia e shoqërimit - albanese" lang="sq" hreflang="sq" data-title="Vetia e shoqërimit" data-language-autonym="Shqip" data-language-local-name="albanese" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D1%81%D0%BE%D1%86%D0%B8%D1%98%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82" title="Асоцијативност - serbo" lang="sr" hreflang="sr" data-title="Асоцијативност" data-language-autonym="Српски / srpski" data-language-local-name="serbo" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Associativitet" title="Associativitet - svedese" lang="sv" hreflang="sv" data-title="Associativitet" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AF%87%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AE%A3%E0%AF%8D%E0%AE%AA%E0%AF%81" title="சேர்ப்புப் பண்பு - tamil" lang="ta" hreflang="ta" data-title="சேர்ப்புப் பண்பு" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%A1%E0%B8%9A%E0%B8%B1%E0%B8%95%E0%B8%B4%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%9B%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%99%E0%B8%AB%E0%B8%A1%E0%B8%B9%E0%B9%88" title="สมบัติการเปลี่ยนหมู่ - thailandese" lang="th" hreflang="th" data-title="สมบัติการเปลี่ยนหมู่" data-language-autonym="ไทย" data-language-local-name="thailandese" 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) - turco" lang="tr" hreflang="tr" data-title="Birleşme özelliği (ikili işlemler)" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%90%D1%81%D1%81%D0%BE%D1%86%D0%B8%D0%B0%D1%82%D0%B8%D0%B2%D0%BB%D1%8B%D0%BA" title="Ассоциативлык - tataro" lang="tt" hreflang="tt" data-title="Ассоциативлык" data-language-autonym="Татарча / tatarça" data-language-local-name="tataro" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D1%81%D0%BE%D1%86%D1%96%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Асоціативність - ucraino" lang="uk" hreflang="uk" data-title="Асоціативність" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Assotsiativlik" title="Assotsiativlik - uzbeco" lang="uz" hreflang="uz" data-title="Assotsiativlik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeco" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Propiet%C3%A0_asociativa" title="Propietà asociativa - veneto" lang="vec" hreflang="vec" data-title="Propietà asociativa" data-language-autonym="Vèneto" data-language-local-name="veneto" 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 - vietnamita" lang="vi" hreflang="vi" data-title="Tính kết hợp" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" 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" 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.hatnote.nota-disambigua{clear:both;margin-top:0;padding:.05em .5em}</style> <style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint nota-disambigua"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/18px-Nota_disambigua.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/27px-Nota_disambigua.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/36px-Nota_disambigua.svg.png 2x" data-file-width="200" data-file-height="200" /></span></span> <span class="hatnote-text"><a href="/wiki/Aiuto:Disambiguazione" title="Aiuto:Disambiguazione">Disambiguazione</a> – Se stai cercando l'associatività nell'architettura a memoria cache per le CPU, vedi <b><a href="/wiki/CPU_cache" title="CPU cache">CPU cache</a></b>.</span></div> </div> <style data-mw-deduplicate="TemplateStyles:r133964453">.mw-parser-output .avviso .mbox-text-div>div,.mw-parser-output .avviso .mbox-text-full-div>div{font-size:90%}.mw-parser-output .avviso .mbox-image{flex-basis:52px;flex-grow:0;flex-shrink:0}.mw-parser-output .avviso .mbox-text-full-div .hide-when-compact{display:block}</style><div style="" class="ambox metadata plainlinks avviso avviso-contenuto"> <div class="avviso-immagine mbox-image noprint"><span typeof="mw:File"><a href="/wiki/File:Question_book-4.svg" class="mw-file-description" title="Niente fonti!"><img alt="Niente fonti!" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/45px-Question_book-4.svg.png" decoding="async" width="45" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/68px-Question_book-4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/90px-Question_book-4.svg.png 2x" data-file-width="262" data-file-height="204" /></a></span></div> <div class="avviso-testo mbox-text"> <div class="mbox-text-div"><b>Questa voce o sezione  sull'argomento matematica <a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">non cita le fonti necessarie</a> o quelle presenti sono insufficienti</b>. <div class="hide-when-compact"> <div class="noprint"><hr />Puoi <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Associativit%C3%A0&action=edit">migliorare questa voce</a> aggiungendo citazioni da <a href="/wiki/Wikipedia:Fonti_attendibili" title="Wikipedia:Fonti attendibili">fonti attendibili</a> secondo le <a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">linee guida sull'uso delle fonti</a>. Segui i suggerimenti del <a href="/wiki/Progetto:Matematica" title="Progetto:Matematica">progetto di riferimento</a>.</div> </div> </div> </div> </div> <p>In <a href="/wiki/Matematica" title="Matematica">matematica</a>, l'<b>associatività</b> (o <b>proprietà associativa</b>) è una proprietà che può avere un'<a href="/wiki/Operazione_binaria" title="Operazione binaria">operazione binaria</a>. Significa che l'ordine di valutazione è irrilevante se l'operazione appare più di una volta in un'espressione. Detta in altro modo, non sono richieste <a href="/wiki/Parentesi" title="Parentesi">parentesi</a> per un'operazione associativa. Si consideri ad esempio l'uguaglianza </p> <dl><dd>(5+2)+1 = 5+(2+1)</dd></dl> <p>Sommando 5 e 2 si ottiene 7, e sommando 1 si ottiene il risultato 8 per il membro a sinistra. Per valutare il membro a destra, si inizia a sommare 2 e 1 ottenendo 3, e quindi si somma 3 e 5 per ottenere 8 ancora. Quindi l'uguaglianza è verificata. Di fatto è verificata per <i>tutti</i> i <a href="/wiki/Numero_reale" title="Numero reale">numeri reali</a>, non solo per 5, 2, e 1. Diciamo che "l'addizione nell'insieme dei numeri reali è un'operazione associativa". </p><p>Le operazioni associative sono frequenti in matematica, e infatti molte <a href="/wiki/Struttura_algebrica" title="Struttura algebrica">strutture algebriche</a> richiedono esplicitamente che le loro operazioni binarie siano associative. Tuttavia, molte operazioni importanti non sono associative; un esempio comune è il <a href="/wiki/Prodotto_vettoriale" title="Prodotto vettoriale">prodotto vettoriale</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definizione">Definizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associativit%C3%A0&veaction=edit&section=1" title="Modifica la sezione Definizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Associativit%C3%A0&action=edit&section=1" title="Edit section's source code: Definizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formalmente, un'operazione binaria <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <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> su un <a href="/wiki/Insieme" title="Insieme">insieme</a> <i>S</i> è detta <b>associativa</b> se soddisfa la <b>legge associativa</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*y)*z=x*(y*z)\qquad {\mbox{per ogni }}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>per ogni </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=x*(y*z)\qquad {\mbox{per ogni }}x,y,z\in S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94f590a64389b6e2b8f7a1f9f5708ebdea12985f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.578ex; height:2.843ex;" alt="{\displaystyle (x*y)*z=x*(y*z)\qquad {\mbox{per ogni }}x,y,z\in S.}"></span></dd></dl> <p>L'ordine di valutazione non influisce sul valore di tale espressione, e si dimostra che lo stesso vale per le espressioni che contengono un numero arbitrario di operazioni <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>. Quindi, quando <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> è associativa, l'ordine di valutazione può essere lasciato non specificato senza causare ambiguità, omettendo le parentesi e scrivendo semplicemente: </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*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> </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/de5dc104a7addc815d049f8f9702d62f3336b7b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.61ex; height:2.009ex;" alt="{\displaystyle x*y*z.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Esempi">Esempi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associativit%C3%A0&veaction=edit&section=2" title="Modifica la sezione Esempi" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Associativit%C3%A0&action=edit&section=2" title="Edit section's source code: Esempi"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Seguono alcuni esempi di operazioni associative. </p> <ul><li>In <a href="/wiki/Aritmetica" title="Aritmetica">aritmetica</a>, l'<a href="/wiki/Addizione" title="Addizione">addizione</a> e la <a href="/wiki/Moltiplicazione" title="Moltiplicazione">moltiplicazione</a> di <a href="/wiki/Numero_reale" title="Numero reale">numeri reali</a> sono associative, cioè,</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{matrix}(x+y)+z=x+(y+z)=x+y+z\quad \\(x\,y)z=x(y\,z)=x\,y\,z\qquad \qquad \qquad \quad \ \ \,\end{matrix}}\right\}{\mbox{per ogni }}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>per ogni </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{per ogni }}x,y,z\in \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74eeb5f901caebaa5cd7a66d874de7b5eb77784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.251ex; 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{per ogni }}x,y,z\in \mathbb {R} .}"></span></dd></dl></dd></dl> <ul><li>L'addizione e la moltiplicazione dei <a href="/wiki/Numero_complesso" title="Numero complesso">numeri complessi</a> e dei <a href="/wiki/Quaternione" title="Quaternione">quaternioni</a> sono associative. La somma degli <a href="/wiki/Ottetto_(matematica)" title="Ottetto (matematica)">ottetti</a> è ancora associativa, ma la moltiplicazione degli ottetti non è associativa.</li> <li>Le funzioni <a href="/wiki/Massimo_comun_divisore" title="Massimo comun divisore">massimo comun divisore</a> e <a href="/wiki/Minimo_comune_multiplo" title="Minimo comune multiplo">minimo comune multiplo</a> agiscono associativamente:</li></ul> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{matrix}\operatorname {M.C.D.} (\operatorname {M.C.D.} (x,y),z)=\operatorname {M.C.D.} (x,\operatorname {M.C.D.} (y,z))=\operatorname {M.C.D.} (x,y,z)\ \quad \\\operatorname {m.c.m.} (\operatorname {m.c.m.} (x,y),z)=\operatorname {m.c.m.} (x,\operatorname {m.c.m.} (y,z))=\operatorname {m.c.m.} (x,y,z)\quad \end{matrix}}\right\}{\mbox{ per ogni }}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> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">M</mi> <mo>.</mo> <mi mathvariant="normal">C</mi> <mo>.</mo> <mi mathvariant="normal">D</mi> <mo>.</mo> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">M</mi> <mo>.</mo> <mi mathvariant="normal">C</mi> <mo>.</mo> <mi mathvariant="normal">D</mi> <mo>.</mo> </mrow> <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> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">M</mi> <mo>.</mo> <mi mathvariant="normal">C</mi> <mo>.</mo> <mi mathvariant="normal">D</mi> <mo>.</mo> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">M</mi> <mo>.</mo> <mi mathvariant="normal">C</mi> <mo>.</mo> <mi mathvariant="normal">D</mi> <mo>.</mo> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">M</mi> <mo>.</mo> <mi mathvariant="normal">C</mi> <mo>.</mo> <mi mathvariant="normal">D</mi> <mo>.</mo> </mrow> <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> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">m</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> <mi mathvariant="normal">m</mi> <mo>.</mo> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">m</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> <mi mathvariant="normal">m</mi> <mo>.</mo> </mrow> <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> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">m</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> <mi mathvariant="normal">m</mi> <mo>.</mo> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">m</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> <mi mathvariant="normal">m</mi> <mo>.</mo> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">m</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> <mi mathvariant="normal">m</mi> <mo>.</mo> </mrow> <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> per ogni </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 {M.C.D.} (\operatorname {M.C.D.} (x,y),z)=\operatorname {M.C.D.} (x,\operatorname {M.C.D.} (y,z))=\operatorname {M.C.D.} (x,y,z)\ \quad \\\operatorname {m.c.m.} (\operatorname {m.c.m.} (x,y),z)=\operatorname {m.c.m.} (x,\operatorname {m.c.m.} (y,z))=\operatorname {m.c.m.} (x,y,z)\quad \end{matrix}}\right\}{\mbox{ per ogni }}x,y,z\in \mathbb {Z} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49c448dfbc57a74284ef1840eab5da3a3b7abb21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:99.371ex; height:6.176ex;" alt="{\displaystyle \left.{\begin{matrix}\operatorname {M.C.D.} (\operatorname {M.C.D.} (x,y),z)=\operatorname {M.C.D.} (x,\operatorname {M.C.D.} (y,z))=\operatorname {M.C.D.} (x,y,z)\ \quad \\\operatorname {m.c.m.} (\operatorname {m.c.m.} (x,y),z)=\operatorname {m.c.m.} (x,\operatorname {m.c.m.} (y,z))=\operatorname {m.c.m.} (x,y,z)\quad \end{matrix}}\right\}{\mbox{ per ogni }}x,y,z\in \mathbb {Z} .}"></span> </p> <ul><li>L'<a href="/wiki/Intersezione_(insiemistica)" title="Intersezione (insiemistica)">intersezione</a> e l'<a href="/wiki/Unione_(insiemistica)" title="Unione (insiemistica)">unione</a> di <a href="/wiki/Insieme" title="Insieme">insiemi</a>:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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{per tutti gli insiemi }}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>per tutti gli insiemi </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{per tutti gli insiemi }}A,B,C.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/630c0760e50085729c02546d2161addd0cf84a90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:74.11ex; 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{per tutti gli insiemi }}A,B,C.}"></span></dd></dl></dd></dl> <ul><li>Se <i>M</i> è un dato insieme e <i>S</i> indica l'insieme di tutte le funzioni da <i>M</i> a <i>M</i>, allora l'operazione di <a href="/wiki/Composizione_di_funzioni" title="Composizione di funzioni">composizione di funzioni</a> su <i>S</i> è associativa:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h\qquad {\mbox{per ogni }}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>per ogni </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{per ogni }}f,g,h\in S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a263fab79f3eeaea27fab969d48019448821827c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.28ex; height:2.843ex;" alt="{\displaystyle (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h\qquad {\mbox{per ogni }}f,g,h\in S.}"></span></dd></dl></dd></dl> <ul><li>Leggermente più in generale, dati quattro insiemi <i>M</i>, <i>N</i>, <i>P</i> e <i>Q</i>, con f: <i>M</i> a <i>N</i>, <i>g</i>: <i>N</i> a <i>P</i>, e <i>h</i>: <i>P</i> a <i>Q</i>, allora</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>h</mi> <mo>=</mo> <mi>f</mi> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac9ff2f14721c0ad9348126aad6a77a0a014b8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.185ex; height:2.843ex;" alt="{\displaystyle (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h}"></span></dd></dl></dd></dl> <dl><dd>come prima. In breve, la composizione di mappe è sempre associativa.</dd></dl> <ul><li>Una <a href="/wiki/Matrice" title="Matrice">matrice</a> rappresenta una <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazione lineare</a> fra <a href="/wiki/Spazi_vettoriali" class="mw-redirect" title="Spazi vettoriali">spazi vettoriali</a> rispetto a <a href="/wiki/Base_(algebra_lineare)" title="Base (algebra lineare)">basi</a> fissate, e il prodotto di matrici corrisponde alla composizione delle trasformazioni lineari corrispondenti. Dunque dall'associatività della composizione di funzioni segue l'associatività del prodotto di matrici.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Non_associatività"><span id="Non_associativit.C3.A0"></span>Non associatività</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associativit%C3%A0&veaction=edit&section=3" title="Modifica la sezione Non associatività" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Associativit%C3%A0&action=edit&section=3" title="Edit section's source code: Non associatività"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un'operazione binaria <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <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> su un insieme <i>S</i> che non soddisfa la legge associativa è detta <b>non associativa</b>. In simboli, </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*y)*z\neq x*(y*z)\qquad {\mbox{per qualche }}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>per qualche </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{per qualche }}x,y,z\in S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ebbb5ca25e8240bdeeebded81cc398b10018e2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50ex; height:2.843ex;" alt="{\displaystyle (x*y)*z\neq x*(y*z)\qquad {\mbox{per qualche }}x,y,z\in S.}"></span></dd></dl> <p>Per tale operazione l'ordine di valutazione <i>è</i> importante. La <a href="/wiki/Sottrazione" title="Sottrazione">sottrazione</a>, la <a href="/wiki/Divisione_(matematica)" title="Divisione (matematica)">divisione</a> e l'<a href="/wiki/Esponenziazione" class="mw-redirect" title="Esponenziazione">esponenziazione</a> sono esempi ben noti di operazioni non associative: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}(5-3)-2\neq 5-(3-2)\quad \\(4/2)/2\neq 4/(2/2)\qquad \qquad \\2^{(1^{2})}\neq (2^{1})^{2}.\quad \qquad \qquad \end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>5</mn> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mo>≠</mo> <mn>5</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <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> <mo>≠</mo> <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> <mspace width="2em" /> <mspace width="2em" /> </mtd> </mtr> <mtr> <mtd> <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> <mo>≠</mo> <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> <mo>.</mo> <mspace width="1em" /> <mspace width="2em" /> <mspace width="2em" /> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}(5-3)-2\neq 5-(3-2)\quad \\(4/2)/2\neq 4/(2/2)\qquad \qquad \\2^{(1^{2})}\neq (2^{1})^{2}.\quad \qquad \qquad \end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c1818086d41ca717f42a94eadeb5b683d2cd16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.281ex; margin-bottom: -0.224ex; width:28.383ex; height:10.176ex;" alt="{\displaystyle {\begin{matrix}(5-3)-2\neq 5-(3-2)\quad \\(4/2)/2\neq 4/(2/2)\qquad \qquad \\2^{(1^{2})}\neq (2^{1})^{2}.\quad \qquad \qquad \end{matrix}}}"></span></dd></dl> <p>In generale, le parentesi devono essere usate per indicare l'ordine di valutazione, se un'operazione non associativa appare più di una volta in un'espressione. Tuttavia i <a href="/wiki/Matematici" class="mw-redirect" title="Matematici">matematici</a> si accordano su un particolare ordine di valutazione per molte operazioni non associative comuni. Questa è una convenzione, e non una verità matematica. </p><p>Un'operazione <b>associativa a sinistra</b> è un'operazione non associativa che viene valutata convenzionalmente da sinistra a destra, cioè, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{matrix}x*y*z=(x*y)*z\qquad \qquad \quad \,\\w*x*y*z=((w*x)*y)*z\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{per ogni }}w,x,y,z\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>z</mi> <mspace width="2em" /> <mspace width="2em" /> <mspace width="1em" /> <mspace width="thinmathspace" /> </mtd> </mtr> <mtr> <mtd> <mi>w</mi> <mo>∗</mo> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo>∗</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>z</mi> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>etc.</mtext> </mstyle> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <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>per ogni </mtext> </mstyle> </mrow> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{matrix}x*y*z=(x*y)*z\qquad \qquad \quad \,\\w*x*y*z=((w*x)*y)*z\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{per ogni }}w,x,y,z\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e96e9826512a51d4756c96cff2f3f95f476bde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:57.581ex; height:9.509ex;" alt="{\displaystyle \left.{\begin{matrix}x*y*z=(x*y)*z\qquad \qquad \quad \,\\w*x*y*z=((w*x)*y)*z\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{per ogni }}w,x,y,z\in S}"></span></dd></dl> <p>mentre un'operazione <b>associativa a destra</b> è valutata convenzionalmente da destra a sinistra: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{matrix}x*y*z=x*(y*z)\qquad \qquad \quad \,\\w*x*y*z=w*(x*(y*z))\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{per ogni }}w,x,y,z\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="2em" /> <mspace width="2em" /> <mspace width="1em" /> <mspace width="thinmathspace" /> </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> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>etc.</mtext> </mstyle> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <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>per ogni </mtext> </mstyle> </mrow> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{matrix}x*y*z=x*(y*z)\qquad \qquad \quad \,\\w*x*y*z=w*(x*(y*z))\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{per ogni }}w,x,y,z\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8babdea8cb618d3aa1b430fda610edc582dedc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:57.581ex; height:9.509ex;" alt="{\displaystyle \left.{\begin{matrix}x*y*z=x*(y*z)\qquad \qquad \quad \,\\w*x*y*z=w*(x*(y*z))\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{per ogni }}w,x,y,z\in S}"></span></dd></dl> <p>Esistono sia operazioni associative a sinistra che operazioni associative a destra; sotto sono dati alcuni esempi. </p> <div class="mw-heading mw-heading2"><h2 id="Altri_esempi">Altri esempi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associativit%C3%A0&veaction=edit&section=4" title="Modifica la sezione Altri esempi" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Associativit%C3%A0&action=edit&section=4" title="Edit section's source code: Altri esempi"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le operazioni associative a sinistra includono: </p> <ul><li>Sottrazione e divisione di numeri reali:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-y-z=(x-y)-z\qquad {\mbox{per ogni }}x,y,z\in \mathbb {R} ;}"> <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> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per ogni </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 x-y-z=(x-y)-z\qquad {\mbox{per ogni }}x,y,z\in \mathbb {R} ;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaafec83f9fb76b239ac88e705aa894352973ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.53ex; height:2.843ex;" alt="{\displaystyle x-y-z=(x-y)-z\qquad {\mbox{per ogni }}x,y,z\in \mathbb {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 x/y/z=(x/y)/z\qquad \qquad \quad {\mbox{per ogni }}x,y,z\in \mathbb {R} {\mbox{ con }}y\neq 0,z\neq 0.}"> <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> <mspace width="2em" /> <mspace width="2em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per ogni </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> con </mtext> </mstyle> </mrow> <mi>y</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mi>z</mi> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x/y/z=(x/y)/z\qquad \qquad \quad {\mbox{per ogni }}x,y,z\in \mathbb {R} {\mbox{ con }}y\neq 0,z\neq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19e70618997b266912322068eef5c1b0af0e313d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.234ex; height:2.843ex;" alt="{\displaystyle x/y/z=(x/y)/z\qquad \qquad \quad {\mbox{per ogni }}x,y,z\in \mathbb {R} {\mbox{ con }}y\neq 0,z\neq 0.}"></span></dd></dl></dd></dl> <p>Le operazioni associative a destra includono le seguenti: </p> <ul><li><a href="/wiki/Esponenziazione" class="mw-redirect" title="Esponenziazione">Esponenziazione</a> di numeri reali:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{y^{z}}=x^{(y^{z})}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{y^{z}}=x^{(y^{z})}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfb19d2e37612d1f6a44f86f48debc557532119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.368ex; height:2.843ex;" alt="{\displaystyle x^{y^{z}}=x^{(y^{z})}.}"></span></dd></dl></dd></dl> <dl><dd>La ragione per cui l'esponenziazione è associativa a destra è che un'esponenziazione associativa a sinistra ripetuta sarebbe meno pratica: ad esempio, la funzione <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 e^{x^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81500c5af225f59cf2585266a980dff95167c74f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.088ex; height:3.009ex;" alt="{\displaystyle e^{x^{2}}}"></span>senza parentesi verrebbe identificata con <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 e^{2x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{2x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40bfc27cb5a57737a31f4d19c04db672581a1b52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.078ex; height:2.676ex;" alt="{\displaystyle e^{2x}}"></span>. Le ripetizioni multiple possono (e, per chiarezza, vengono) riscritte con il simbolo di moltiplicazione:</dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{y})^{z}=x^{(yz)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </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/d9467bca5039122c11225f6edec4b5943fcbcb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.363ex; height:3.343ex;" alt="{\displaystyle (x^{y})^{z}=x^{(yz)}.}"></span></dd></dl></dd></dl> <ul><li>l'operatore di assegnazione in molti <a href="/wiki/Linguaggio_di_programmazione" title="Linguaggio di programmazione">linguaggi di programmazione</a> è associativo a destra, ad esempio nel caso del <a href="/wiki/Linguaggio_C" class="mw-redirect" title="Linguaggio C">linguaggio C</a>.</li></ul> <dl><dd><dl><dd><code>x = y = z;  </code><i>significa</i><code>  x = (y = z);  </code><i>e non</i><code>  (x = y) = z;</code></dd></dl></dd></dl> <dl><dd>In altre parole, l'istruzione assegna il valore di <code>z</code> sia a <code>y</code> che a <code>x</code>.</dd></dl> <p>Operazioni non associative per cui non è stato definito nessun ordine convenzionale di valutazione includono le seguenti: </p> <ul><li>Prendere la <a href="/wiki/Media_(statistica)" title="Media (statistica)">media</a> di numeri reali:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {(x+y)/2+z \over 2}\neq {x+(y+z)/2 \over 2}\neq {x+y+z \over 3}\qquad {\mbox{per qualche }}x,y,z\in \mathbb {R} .}"> <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> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per qualche </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 {(x+y)/2+z \over 2}\neq {x+(y+z)/2 \over 2}\neq {x+y+z \over 3}\qquad {\mbox{per qualche }}x,y,z\in \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4a301d126d46610ce7f12f2609e80fb9a17c6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:72.272ex; height:5.676ex;" alt="{\displaystyle {(x+y)/2+z \over 2}\neq {x+(y+z)/2 \over 2}\neq {x+y+z \over 3}\qquad {\mbox{per qualche }}x,y,z\in \mathbb {R} .}"></span></dd></dl></dd></dl> <ul><li>Prendere il <a href="/wiki/Insieme_complemento" title="Insieme complemento">complemento relativo</a> di insiemi:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\backslash B)\backslash C\neq A\backslash (B\backslash C)\qquad {\mbox{per qualche insieme }}A,B,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> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>per qualche insieme </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 (A\backslash B)\backslash C\neq A\backslash (B\backslash C)\qquad {\mbox{per qualche insieme }}A,B,C.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90ab4be1790a40a1716dfc72be46411bced1f04a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.715ex; height:2.843ex;" alt="{\displaystyle (A\backslash B)\backslash C\neq A\backslash (B\backslash C)\qquad {\mbox{per qualche insieme }}A,B,C.}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associativit%C3%A0&veaction=edit&section=5" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Associativit%C3%A0&action=edit&section=5" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Distributivit%C3%A0" title="Distributività">Distributività</a></li> <li><a href="/wiki/Commutativit%C3%A0" title="Commutatività">Commutatività</a></li> <li><a href="/wiki/Semigruppo" title="Semigruppo">Semigruppo</a></li> <li><a href="/wiki/Associativit%C3%A0_della_potenza" title="Associatività della potenza">Associatività della potenza</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associativit%C3%A0&veaction=edit&section=6" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Associativit%C3%A0&action=edit&section=6" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><a href="https://it.wiktionary.org/wiki/associativit%C3%A0" class="extiw" title="wikt:associatività">Wikizionario</a></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://it.wiktionary.org/wiki/" title="Collabora a Wikizionario"><img alt="Collabora a Wikizionario" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/18px-Wiktionary_small.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/27px-Wiktionary_small.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/36px-Wiktionary_small.svg.png 2x" data-file-width="350" data-file-height="350" /></a></span> <a href="https://it.wiktionary.org/wiki/" class="extiw" title="wikt:">Wikizionario</a> contiene il lemma di dizionario «<b><a href="https://it.wiktionary.org/wiki/associativit%C3%A0" class="extiw" title="wikt:associatività">associatività</a></b>»</li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Associativit%C3%A0&veaction=edit&section=7" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Associativit%C3%A0&action=edit&section=7" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Associative.html"><span style="font-style:italic;">Associative</span></a>, su <span style="font-style:italic;"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span>, Wolfram Research.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q177251#P2812" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li></ul> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Crystal128-kmplot.svg" class="mw-file-description" title="Matematica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/25px-Crystal128-kmplot.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/38px-Crystal128-kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/50px-Crystal128-kmplot.svg.png 2x" data-file-width="245" data-file-height="244" /></a></span> <b><a href="/wiki/Portale:Matematica" title="Portale:Matematica">Portale Matematica</a></b>: accedi alle voci di Wikipedia che trattano di matematica</div></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Estratto da "<a dir="ltr" href="https://it.wikipedia.org/w/index.php?title=Associatività&oldid=138052662">https://it.wikipedia.org/w/index.php?title=Associatività&oldid=138052662</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categoria:Categorie" title="Categoria:Categorie">Categorie</a>: <ul><li><a href="/wiki/Categoria:Algebra_elementare" title="Categoria:Algebra elementare">Algebra elementare</a></li><li><a href="/wiki/Categoria:Strutture_algebriche" title="Categoria:Strutture algebriche">Strutture algebriche</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorie nascoste: <ul><li><a href="/wiki/Categoria:Senza_fonti_-_matematica" title="Categoria:Senza fonti - matematica">Senza fonti - matematica</a></li><li><a href="/wiki/Categoria:Senza_fonti_-_marzo_2012" title="Categoria:Senza fonti - marzo 2012">Senza fonti - marzo 2012</a></li><li><a href="/wiki/Categoria:P2812_letta_da_Wikidata" title="Categoria:P2812 letta da Wikidata">P2812 letta da Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Questa pagina è stata modificata per l'ultima volta il 25 feb 2024 alle 18:43.</li> <li id="footer-info-copyright">Il testo è disponibile secondo la <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.it">licenza Creative Commons Attribuzione-Condividi allo stesso modo</a>; possono applicarsi condizioni ulteriori. 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Associatività - Wikipedia