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</div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Sitio"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Conteníu" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Conteníu</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">despintar</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Entamu</div> </a> </li> <li id="toc-Definición_formal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definición_formal"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definición formal</span> </div> </a> <button aria-controls="toc-Definición_formal-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Definición formal</span> </button> <ul id="toc-Definición_formal-sublist" class="vector-toc-list"> <li id="toc-Llei_asociativa_xeneralizada" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Llei_asociativa_xeneralizada"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Llei asociativa xeneralizada</span> </div> </a> <ul id="toc-Llei_asociativa_xeneralizada-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-En_lóxica_proposicional" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#En_lóxica_proposicional"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>En lóxica proposicional</span> </div> </a> <button aria-controls="toc-En_lóxica_proposicional-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección En lóxica proposicional</span> </button> <ul id="toc-En_lóxica_proposicional-sublist" class="vector-toc-list"> <li id="toc-Regla_de_reemplazu" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Regla_de_reemplazu"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Regla de reemplazu</span> </div> </a> <ul id="toc-Regla_de_reemplazu-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conectivas_de_funciones_de_verdá" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conectivas_de_funciones_de_verdá"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Conectivas de funciones de verdá</span> </div> </a> <ul id="toc-Conectivas_de_funciones_de_verdá-sublist" class="vector-toc-list"> <li id="toc-Asociatividá_de_la_dixunción:" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Asociatividá_de_la_dixunción:"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Asociatividá de la dixunción:</span> </div> </a> <ul id="toc-Asociatividá_de_la_dixunción:-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Asociatividá_de_la_conxunción:" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Asociatividá_de_la_conxunción:"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Asociatividá de la conxunción:</span> </div> </a> <ul id="toc-Asociatividá_de_la_conxunción:-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Asociatividá_de_la_equivalencia:" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Asociatividá_de_la_equivalencia:"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>Asociatividá de la equivalencia:</span> </div> </a> <ul id="toc-Asociatividá_de_la_equivalencia:-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Operación_non_asociativa" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operación_non_asociativa"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Operación non asociativa</span> </div> </a> <button aria-controls="toc-Operación_non_asociativa-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Operación non asociativa</span> </button> <ul id="toc-Operación_non_asociativa-sublist" class="vector-toc-list"> <li id="toc-Non_asociatividá_del_cálculu_de_coma_flotante" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non_asociatividá_del_cálculu_de_coma_flotante"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Non asociatividá del cálculu de coma flotante</span> </div> </a> <ul id="toc-Non_asociatividá_del_cálculu_de_coma_flotante-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ver_tamién" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_tamién"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ver tamién</span> </div> </a> <ul id="toc-Ver_tamién-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referencies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referencies"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Referencies</span> </div> </a> <ul id="toc-Referencies-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Conteníu" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Cambiar a la tabla de contenidos" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Cambiar a la tabla de contenidos</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Asociatividá</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir a un artículo en otro idioma. Disponible en 67 idiomas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-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 llingües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D9%85%D9%84%D9%8A%D8%A9_%D8%AA%D8%AC%D9%85%D9%8A%D8%B9%D9%8A%D8%A9" title="عملية تجميعية – árabe" lang="ar" hreflang="ar" data-title="عملية تجميعية" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</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 – bielorrusu" lang="be" hreflang="be" data-title="Асацыятыўная аперацыя" data-language-autonym="Беларуская" data-language-local-name="bielorrusu" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%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 – búlgaru" lang="bg" hreflang="bg" data-title="Асоциативност" data-language-autonym="Български" data-language-local-name="búlgaru" 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 – bosniu" lang="bs" hreflang="bs" data-title="Asocijativnost" data-language-autonym="Bosanski" data-language-local-name="bosniu" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Propietat_associativa" title="Propietat associativa – catalán" lang="ca" hreflang="ca" data-title="Propietat associativa" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DB%8C%DB%95%DA%A9%D8%AA%D8%B1%D8%A8%DB%95%D8%B3%D8%AA%D9%86" title="یەکتربەستن – kurdu central" lang="ckb" hreflang="ckb" data-title="یەکتربەستن" data-language-autonym="کوردی" data-language-local-name="kurdu central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Asociativita" title="Asociativita – checu" lang="cs" hreflang="cs" data-title="Asociativita" data-language-autonym="Čeština" data-language-local-name="checu" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D1%81%D1%81%D0%B0%D1%86%D0%B8%D0%B0%D1%82%D0%B8%D0%B2%D0%BB%C4%83%D1%85_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ассациативлăх (математика) – chuvash" lang="cv" hreflang="cv" data-title="Ассациативлăх (математика)" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Associativitet" title="Associativitet – danés" lang="da" hreflang="da" data-title="Associativitet" data-language-autonym="Dansk" data-language-local-name="danés" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Assoziativgesetz" title="Assoziativgesetz – alemán" lang="de" hreflang="de" data-title="Assoziativgesetz" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%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 – griegu" lang="el" hreflang="el" data-title="Προσεταιριστική ιδιότητα" data-language-autonym="Ελληνικά" data-language-local-name="griegu" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Associative_property" title="Associative property – inglés" lang="en" hreflang="en" data-title="Associative property" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/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) – español" lang="es" hreflang="es" data-title="Asociatividad (álgebra)" data-language-autonym="Español" data-language-local-name="español" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Assotsiatiivsus" title="Assotsiatiivsus – estoniu" lang="et" hreflang="et" data-title="Assotsiatiivsus" data-language-autonym="Eesti" data-language-local-name="estoniu" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Elkarkortasun" title="Elkarkortasun – vascu" lang="eu" hreflang="eu" data-title="Elkarkortasun" data-language-autonym="Euskara" data-language-local-name="vascu" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%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 – persa" lang="fa" hreflang="fa" data-title="خاصیت شرکت‌پذیری" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Liit%C3%A4nn%C3%A4isyys" title="Liitännäisyys – finlandés" lang="fi" hreflang="fi" data-title="Liitännäisyys" data-language-autonym="Suomi" data-language-local-name="finlandés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Associativit%C3%A9" title="Associativité – francés" lang="fr" hreflang="fr" data-title="Associativité" data-language-autonym="Français" data-language-local-name="francés" 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 – frisón del norte" lang="frr" hreflang="frr" data-title="Asotsiatiifgesets" data-language-autonym="Nordfriisk" data-language-local-name="frisón del norte" 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 – irlandés" lang="ga" hreflang="ga" data-title="Oibríocht chomhthiomsaitheach" data-language-autonym="Gaeilge" data-language-local-name="irlandés" 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) – gallegu" lang="gl" hreflang="gl" data-title="Asociatividade (álxebra)" data-language-autonym="Galego" data-language-local-name="gallegu" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_%D7%90%D7%A1%D7%95%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%91%D7%99%D7%AA" title="פעולה אסוציאטיבית – hebréu" lang="he" hreflang="he" data-title="פעולה אסוציאטיבית" data-language-autonym="עברית" data-language-local-name="hebréu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Asocijativnost" title="Asocijativnost – croata" lang="hr" hreflang="hr" data-title="Asocijativnost" data-language-autonym="Hrvatski" data-language-local-name="croata" 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 – húngaru" lang="hu" hreflang="hu" data-title="Asszociativitás" data-language-autonym="Magyar" data-language-local-name="húngaru" 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 – armeniu" lang="hy" hreflang="hy" data-title="Զուգորդականություն" data-language-autonym="Հայերեն" data-language-local-name="armeniu" 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 – indonesiu" lang="id" hreflang="id" data-title="Sifat asosiatif" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiu" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tengiregla" title="Tengiregla – islandés" lang="is" hreflang="is" data-title="Tengiregla" data-language-autonym="Íslenska" data-language-local-name="islandés" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Associativit%C3%A0" title="Associatività – italianu" lang="it" hreflang="it" data-title="Associatività" data-language-autonym="Italiano" data-language-local-name="italianu" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B5%90%E5%90%88%E6%B3%95%E5%89%87" title="結合法則 – xaponés" lang="ja" hreflang="ja" data-title="結合法則" data-language-autonym="日本語" data-language-local-name="xaponés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-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 – kazaquistanín" lang="kk" hreflang="kk" data-title="Ассоциативтік операция" data-language-autonym="Қазақша" data-language-local-name="kazaquistanín" 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 – coreanu" lang="ko" hreflang="ko" data-title="결합법칙" data-language-autonym="한국어" data-language-local-name="coreanu" 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) – llatín" lang="la" hreflang="la" data-title="Associativitas (mathematica)" data-language-autonym="Latina" data-language-local-name="llatín" 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 – lituanu" lang="lt" hreflang="lt" data-title="Asociatyvumas" data-language-autonym="Lietuvių" data-language-local-name="lituanu" 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 – letón" lang="lv" hreflang="lv" data-title="Asociativitāte" data-language-autonym="Latviešu" data-language-local-name="letón" 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 – macedoniu" lang="mk" hreflang="mk" data-title="Асоцијативност" data-language-autonym="Македонски" data-language-local-name="macedoniu" 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 – malayu" lang="ms" hreflang="ms" data-title="Kalis sekutuan" data-language-autonym="Bahasa Melayu" data-language-local-name="malayu" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Associativiteit_(wiskunde)" title="Associativiteit (wiskunde) – neerlandés" lang="nl" hreflang="nl" data-title="Associativiteit (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Assosiativitet" title="Assosiativitet – noruegu Nynorsk" lang="nn" hreflang="nn" data-title="Assosiativitet" data-language-autonym="Norsk nynorsk" data-language-local-name="noruegu Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Assosiativ_lov" title="Assosiativ lov – noruegu Bokmål" lang="nb" hreflang="nb" data-title="Assosiativ lov" data-language-autonym="Norsk bokmål" data-language-local-name="noruegu Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Associativitat" title="Associativitat – occitanu" lang="oc" hreflang="oc" data-title="Associativitat" data-language-autonym="Occitan" data-language-local-name="occitanu" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/%C5%81%C4%85czno%C5%9B%C4%87_(matematyka)" title="Łączność (matematyka) – polacu" lang="pl" hreflang="pl" data-title="Łączność (matematyka)" data-language-autonym="Polski" data-language-local-name="polacu" 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 – portugués" lang="pt" hreflang="pt" data-title="Associatividade" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Asociativitate" title="Asociativitate – rumanu" lang="ro" hreflang="ro" data-title="Asociativitate" data-language-autonym="Română" data-language-local-name="rumanu" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%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) – rusu" lang="ru" hreflang="ru" data-title="Ассоциативность (математика)" data-language-autonym="Русский" data-language-local-name="rusu" 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-croata" lang="sh" hreflang="sh" data-title="Asocijativnost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/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 – eslovacu" lang="sk" hreflang="sk" data-title="Asociatívnosť" data-language-autonym="Slovenčina" data-language-local-name="eslovacu" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Asociativnost" title="Asociativnost – eslovenu" lang="sl" hreflang="sl" data-title="Asociativnost" data-language-autonym="Slovenščina" data-language-local-name="eslovenu" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Vetia_e_shoq%C3%ABrimit" title="Vetia e shoqërimit – albanu" lang="sq" hreflang="sq" data-title="Vetia e shoqërimit" data-language-autonym="Shqip" data-language-local-name="albanu" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%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 – serbiu" lang="sr" hreflang="sr" data-title="Асоцијативност" data-language-autonym="Српски / srpski" data-language-local-name="serbiu" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Associativitet" title="Associativitet – suecu" lang="sv" hreflang="sv" data-title="Associativitet" data-language-autonym="Svenska" data-language-local-name="suecu" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%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 – tailandés" lang="th" hreflang="th" data-title="สมบัติการเปลี่ยนหมู่" data-language-autonym="ไทย" data-language-local-name="tailandés" 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) – turcu" lang="tr" hreflang="tr" data-title="Birleşme özelliği (ikili işlemler)" data-language-autonym="Türkçe" data-language-local-name="turcu" 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 – tártaru" lang="tt" hreflang="tt" data-title="Ассоциативлык" data-language-autonym="Татарча / tatarça" data-language-local-name="tártaru" 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 – ucraín" lang="uk" hreflang="uk" data-title="Асоціативність" data-language-autonym="Українська" data-language-local-name="ucraín" 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 – uzbequistanín" lang="uz" hreflang="uz" data-title="Assotsiativlik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbequistanín" 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 – venecianu" lang="vec" hreflang="vec" data-title="Propietà asociativa" data-language-autonym="Vèneto" data-language-local-name="venecianu" 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 – vietnamín" lang="vi" hreflang="vi" data-title="Tính kết hợp" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamín" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BB%93%E5%90%88%E5%BE%8B" title="结合律 – chinu wu" lang="wuu" hreflang="wuu" data-title="结合律" data-language-autonym="吴语" data-language-local-name="chinu wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%A1%D7%90%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%95%D7%95%D7%99%D7%98%D7%A2%D7%98" title="אסאציאטיוויטעט – yiddish" lang="yi" hreflang="yi" data-title="אסאציאטיוויטעט" data-language-autonym="ייִדיש" data-language-local-name="yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BB%93%E5%90%88%E5%BE%8B" title="结合律 – chinu" lang="zh" hreflang="zh" data-title="结合律" data-language-autonym="中文" data-language-local-name="chinu" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B5%90%E5%90%88%E5%BE%8B" title="結合律 – cantonés" lang="yue" hreflang="yue" data-title="結合律" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q177251#sitelinks-wikipedia" title="Editar los enllaces d'interllingua" class="wbc-editpage">Editar los enllaces</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espacios de nome"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Asociativid%C3%A1" title="Ver la páxina de conteníu [c]" accesskey="c"><span>Páxina</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Alderique:Asociativid%C3%A1&action=edit&redlink=1" rel="discussion" class="new" title="Alderique tocante al conteníu de la páxina (la páxina nun esiste) [t]" accesskey="t"><span>Alderique</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input 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accesskey="g"><span>Elementu de Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Ferramientes de páxina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apariencia"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la barra llateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">despintar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De Wikipedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ast" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheru:Associativity_of_binary_operations_(without_question_marks).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Associativity_of_binary_operations_%28without_question_marks%29.svg/220px-Associativity_of_binary_operations_%28without_question_marks%29.svg.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Associativity_of_binary_operations_%28without_question_marks%29.svg/330px-Associativity_of_binary_operations_%28without_question_marks%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Associativity_of_binary_operations_%28without_question_marks%29.svg/440px-Associativity_of_binary_operations_%28without_question_marks%29.svg.png 2x" data-file-width="390" data-file-height="283" /></a><figcaption>Diagrama de la propiedá asociativa n'<a href="/wiki/Operaci%C3%B3n_binaria" title="Operación binaria">operaciones binaries</a>.</figcaption></figure> <p>La <b>asociatividá</b> (o <b>propiedá asociativa</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> ye una propiedá na <a href="/wiki/%C3%81lxebra" title="Álxebra">álxebra</a> que significa que reorganizar los parentesís nuna <a href="/wiki/Operaci%C3%B3n_binaria" title="Operación binaria">operación binaria</a> nun va camudar la resultancia. Na <a href="/wiki/L%C3%B3xica_proposicional" title="Lóxica proposicional">lóxica proposicional</a> ye una <a href="/w/index.php?title=Regla_de_reemplazu&action=edit&redlink=1" class="new" title="Regla de reemplazu (la páxina nun esiste)">regla de reemplazu</a> <a href="/w/index.php?title=Validez_(l%C3%B3xica)&action=edit&redlink=1" class="new" title="Validez (lóxica) (la páxina nun esiste)">válida</a> n'<a href="/w/index.php?title=F%C3%B3rmula_Bien_Formada&action=edit&redlink=1" class="new" title="Fórmula Bien Formada (la páxina nun esiste)">espresiones lóxiques</a> usaes en <a href="/w/index.php?title=Prueba_formal&action=edit&redlink=1" class="new" title="Prueba formal (la páxina nun esiste)">pruebes lóxiques</a>. </p><p>Esto ye, nuna espresión asociativa con dos o más escurrimientos siguíos d'un mesmu operador asociativu, l'orde en que s'executen les operaciones nun alteria la resultancia, siempres y cuando se caltenga intacta la secuencia de los operandos. </p><p>Nun se debe confundir la asociatividá cola <a href="/wiki/Conmutativid%C3%A1" title="Conmutatividá">conmutatividá</a>, que establez que sí se puede camudar l'orde de los operandos ensin afectar la resultancia final. </p><p>Les operaciones asociatives son abondosa en matemátiques; ello ye que en munches <a href="/w/index.php?title=Estructura_alxebr%C3%A1ica&action=edit&redlink=1" class="new" title="Estructura alxebráica (la páxina nun esiste)">estructures alxebráiques</a> explicitamente ríquese que les operciones binaries sían asociatives. </p><p>Sicasí, munches operaciones importantes son <i>non-asociatives</i>; por exemplu <a href="/wiki/Resta" title="Resta">restar</a>, la <a href="/wiki/Potenciaci%C3%B3n" title="Potenciación">potenciación</a> o'l <a href="/wiki/Productu_vectorial" title="Productu vectorial">productu vectorial</a>. </p><p>Un exemplu d'operación asociativa ye la <a href="/wiki/Adici%C3%B3n_(matem%C3%A1tica)" title="Adición (matemática)">suma</a> o la <a href="/wiki/Multiplicaci%C3%B3n" title="Multiplicación">multiplicación</a> en <a href="/wiki/N%C3%BAmberu_real" title="Númberu real">númberos reales</a>. Considerése les siguientes ecuaciones onde a pesar d'alteriase la disposición de los paréntesis, la resultancia nun se ve alteriáu. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2+3)+4=2+(3+4)=9\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>4</mn> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>9</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2+3)+4=2+(3+4)=9\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b33314f4fc13b0ee84b3386ca9d0755137b026ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.701ex; height:2.843ex;" alt="{\displaystyle (2+3)+4=2+(3+4)=9\,}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times (3\times 4)=(2\times 3)\times 4=24.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>×</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>×</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mn>4</mn> <mo>=</mo> <mn>24.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times (3\times 4)=(2\times 3)\times 4=24.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4322e3534b2318b8f51a29363c45fed9754c3019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.124ex; height:2.843ex;" alt="{\displaystyle 2\times (3\times 4)=(2\times 3)\times 4=24.}"></span> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definición_formal"><span id="Definici.C3.B3n_formal"></span>Definición formal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asociativid%C3%A1&veaction=edit&section=1" title="Editar seición: Definición formal" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Asociativid%C3%A1&action=edit&section=1" title="Editar el código fuente de la sección: Definición formal"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheru: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>Una operación ∗ en un <a href="/wiki/Conxuntu" title="Conxuntu">conxuntu</a> S ye asociativa cuando esti diagrama conmuta. Esto ye, cuando los dos caminos de S×S×S a S se <a href="/wiki/Funci%C3%B3n_compuesta" title="Función compuesta">componen</a> a la mesma funcción de S×S×S a S.</figcaption></figure> <p>Sía <b>A</b> un <a href="/wiki/Conxuntu" title="Conxuntu">conxuntu</a> nel cual definióse una <a href="/wiki/Operaci%C3%B3n_binaria" title="Operación binaria">operación binaria interna</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 \circledcirc }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circledcirc }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf70a53592b87a725eabcbb2dffc880e9aa9b66c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \circledcirc }"></span> tal que: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rccl}\circledcirc :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\circledcirc b\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>⊚</mo> <mo>:</mo> </mtd> <mtd> <mi>A</mi> <mo>×</mo> <mi>A</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>A</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">⟼</mo> </mtd> <mtd> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⊚</mo> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rccl}\circledcirc :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\circledcirc b\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/460b69c33c8d0a173b8d927f1c02f59409439fff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.125ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{rccl}\circledcirc :&A\times A&\longrightarrow &A\\&(a,b)&\longmapsto &c=a\circledcirc b\end{array}}}"></span> </p><p>Dizse que la operación <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circledcirc }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circledcirc }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf70a53592b87a725eabcbb2dffc880e9aa9b66c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \circledcirc }"></span> ye <b>asociativa</b> si: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a,b,c\in A\;:\quad a\circledcirc (b\circledcirc c)=(a\circledcirc b)\circledcirc c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mi>a</mi> <mo>⊚</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⊚</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⊚</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⊚</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a,b,c\in A\;:\quad a\circledcirc (b\circledcirc c)=(a\circledcirc b)\circledcirc c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebacc19f945992c2a4a61e2a144f4e6059136d5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.63ex; height:2.843ex;" alt="{\displaystyle \forall a,b,c\in A\;:\quad a\circledcirc (b\circledcirc c)=(a\circledcirc b)\circledcirc c}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Llei_asociativa_xeneralizada">Llei asociativa xeneralizada</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asociativid%C3%A1&veaction=edit&section=2" title="Editar seición: Llei asociativa xeneralizada" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Asociativid%C3%A1&action=edit&section=2" title="Editar el código fuente de la sección: Llei asociativa xeneralizada"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheru: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>N'ausencia de la propiedá asociativa, cinco factores a, b, c, d, y dan como resultáu una <a href="/w/index.php?title=Rede_de_Tamari&action=edit&redlink=1" class="new" title="Rede de Tamari (la páxina nun esiste)">rede de Tamari</a> d'orde cuatro, posiblemente productos distintos.</figcaption></figure> <p>Si una operación binaria ye asociativa, la aplcación repitida de la operación produz el mesmu resultáu ensin importar de cuaántas pareyes de paréntesis válides hai inxertaes na espresión. Esto conozse cómo <b>llei asociativa xeneralizada</b>.<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> Por exemplu, el productu de cuatro elementos puede escribise, ensin camudar l'orde de los factores, en cinco posibles opciones: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((ab)c)d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((ab)c)d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1483412ecf78ec5c5d17fb5d7bebfa2a2166e107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.069ex; height:2.843ex;" alt="{\displaystyle ((ab)c)d}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (ab)(cd)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ab)(cd)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3f01186cb5d5152b51841e993d8ae657b701fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.069ex; height:2.843ex;" alt="{\displaystyle (ab)(cd)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a(bc))d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a(bc))d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b76ed950d0e00a64c6526257b8e5f6ed797b8db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.069ex; height:2.843ex;" alt="{\displaystyle (a(bc))d}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a((bc)d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a((bc)d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/816d85fb33e74ec781a83bc54c883bcccfbe135b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.069ex; height:2.843ex;" alt="{\displaystyle a((bc)d)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(b(cd))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(b(cd))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a7009d4cbf69b3e7cb09bd386d39df221551c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.069ex; height:2.843ex;" alt="{\displaystyle a(b(cd))}"></span></dd></dl> <p>Si la operación del productu ye asociativa, la llei asociativa xeneralizada diz que toes estes fórmules van dar el mesmu resultáu. Poro, nun siendo que la fórmula con paréntesis omitíos yá tenga un significáu distintu (ver más embaxo), los paréntesis pueden considerase innecesarios y "el productu puede escribise ensin ambigüedaes como </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle abcd.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle abcd.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ae535cf143a8c140f062ba5630d55d525f621f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.097ex; height:2.176ex;" alt="{\displaystyle abcd.}"></span></dd></dl> <p>A medida que aumenta el número de elementos, el número de formas posibles de insertar paréntesis crece rápidamente, pero siguen siendo innecesarias para la desambiguación. </p><p>Un exemplu onde esto nun funciona ye'l <a href="/w/index.php?title=Bicondicional_l%C3%B3xicu&action=edit&redlink=1" class="new" title="Bicondicional lóxicu (la páxina nun esiste)">bicondicional lóxicu</a> ↔. Ye asociativu, por tanto A ↔ (B ↔C) ye equivalente a (A ↔ B) ↔ C, pero A ↔ B ↔ C más comúnmente significa (A ↔B y B ↔ C), que nun ye equivalente. </p> <div class="mw-heading mw-heading2"><h2 id="En_lóxica_proposicional"><span id="En_l.C3.B3xica_proposicional"></span>En lóxica proposicional</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asociativid%C3%A1&veaction=edit&section=3" title="Editar seición: En lóxica proposicional" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Asociativid%C3%A1&action=edit&section=3" title="Editar el código fuente de la sección: En lóxica proposicional"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Regla_de_reemplazu">Regla de reemplazu</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asociativid%C3%A1&veaction=edit&section=4" title="Editar seición: Regla de reemplazu" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Asociativid%C3%A1&action=edit&section=4" title="Editar el código fuente de la sección: Regla de reemplazu"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Na <a href="/wiki/L%C3%B3xica_proposicional" title="Lóxica proposicional">lóxica proposicional</a> estándar, l'<i>asociación</i>, o <i>asociatividá</i><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><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> son dos <a href="/w/index.php?title=Riegla_de_reemplazu&action=edit&redlink=1" class="new" title="Riegla de reemplazu (la páxina nun esiste)">regles de reemplazu</a> <a href="/w/index.php?title=Validez_(l%C3%B3xica)&action=edit&redlink=1" class="new" title="Validez (lóxica) (la páxina nun esiste)">válides</a>. Estes riegles dexen mover los paréntesis n'<a href="/w/index.php?title=F%C3%B3rmula_Bien_Formada&action=edit&redlink=1" class="new" title="Fórmula Bien Formada (la páxina nun esiste)">espresiones lóxiques</a> usaes en <a href="/w/index.php?title=Prueba_formal&action=edit&redlink=1" class="new" title="Prueba formal (la páxina nun esiste)">pruebes lóxiques</a>. Les riegles son: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (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> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (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> </p><p>onde "⇔" ye un símbolu metalógico que representa "pue ser reemplazáu nuna prueba por." </p> <div class="mw-heading mw-heading3"><h3 id="Conectivas_de_funciones_de_verdá"><span id="Conectivas_de_funciones_de_verd.C3.A1"></span>Conectivas de funciones de verdá</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asociativid%C3%A1&veaction=edit&section=5" title="Editar seición: Conectivas de funciones de verdá" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Asociativid%C3%A1&action=edit&section=5" title="Editar el código fuente de la sección: Conectivas de funciones de verdá"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Asociatividá ye una propiedá de delles <a href="/w/index.php?title=Conectiva_l%C3%B3xica&action=edit&redlink=1" class="new" title="Conectiva lóxica (la páxina nun esiste)">conectivas lóxiques</a> nes <a href="/w/index.php?title=Funci%C3%B3n_de_verd%C3%A1&action=edit&redlink=1" class="new" title="Función de verdá (la páxina nun esiste)">funciones de verdá</a> de la <a href="/wiki/L%C3%B3xica_proposicional" title="Lóxica proposicional">lóxica proposicional</a>. Les siguientes <a href="/w/index.php?title=Equivalencia_l%C3%B3xica&action=edit&redlink=1" class="new" title="Equivalencia lóxica (la páxina nun esiste)">equivalencies lóxiques</a> demuestren que la asociatividá ye una propiedá de conectivas lóxiques particulares. Son coles mesmes <a href="/w/index.php?title=Tautolox%C3%ADa&action=edit&redlink=1" class="new" title="Tautoloxía (la páxina nun esiste)">tautoloxíes</a> de funciones de verdá.<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> </p> <div class="mw-heading mw-heading4"><h4 id="Asociatividá_de_la_dixunción:"><span id="Asociativid.C3.A1_de_la_dixunci.C3.B3n:"></span>Asociatividá de la dixunción:</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asociativid%C3%A1&veaction=edit&section=6" title="Editar seición: Asociatividá de la dixunción:" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Asociativid%C3%A1&action=edit&section=6" title="Editar el código fuente de la sección: Asociatividá de la dixunción:"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((P\lor Q)\lor R)\leftrightarrow (P\lor (Q\lor R))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∨</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((P\lor Q)\lor R)\leftrightarrow (P\lor (Q\lor R))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83022ecdfaf711fdaf67c742e94d33fdabad38bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.877ex; height:2.843ex;" alt="{\displaystyle ((P\lor Q)\lor R)\leftrightarrow (P\lor (Q\lor R))}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\lor (Q\lor R))\leftrightarrow ((P\lor Q)\lor R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∨</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\lor (Q\lor R))\leftrightarrow ((P\lor Q)\lor R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe6682d32332d4ff07effc195b334e497b308f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.877ex; height:2.843ex;" alt="{\displaystyle (P\lor (Q\lor R))\leftrightarrow ((P\lor Q)\lor R)}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Asociatividá_de_la_conxunción:"><span id="Asociativid.C3.A1_de_la_conxunci.C3.B3n:"></span>Asociatividá de la conxunción:</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asociativid%C3%A1&veaction=edit&section=7" title="Editar seición: Asociatividá de la conxunción:" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Asociativid%C3%A1&action=edit&section=7" title="Editar el código fuente de la sección: Asociatividá de la conxunción:"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((P\land Q)\land R)\leftrightarrow (P\land (Q\land R))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∧</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((P\land Q)\land R)\leftrightarrow (P\land (Q\land R))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/599ddb9e2cfe377e714700cc2045f57f0d546d05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.877ex; height:2.843ex;" alt="{\displaystyle ((P\land Q)\land R)\leftrightarrow (P\land (Q\land R))}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\land (Q\land R))\leftrightarrow ((P\land Q)\land R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∧</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\land (Q\land R))\leftrightarrow ((P\land Q)\land R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7042ca691d07414dd554984602225f064b1f66ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.877ex; height:2.843ex;" alt="{\displaystyle (P\land (Q\land R))\leftrightarrow ((P\land Q)\land R)}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Asociatividá_de_la_equivalencia:"><span id="Asociativid.C3.A1_de_la_equivalencia:"></span>Asociatividá de la equivalencia:</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asociativid%C3%A1&veaction=edit&section=8" title="Editar seición: Asociatividá de la equivalencia:" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Asociativid%C3%A1&action=edit&section=8" title="Editar el código fuente de la sección: Asociatividá de la equivalencia:"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((P\leftrightarrow Q)\leftrightarrow R)\leftrightarrow (P\leftrightarrow (Q\leftrightarrow R))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">↔</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">↔</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((P\leftrightarrow Q)\leftrightarrow R)\leftrightarrow (P\leftrightarrow (Q\leftrightarrow R))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1488e55751da92bcd3467470cfc8e8a7ce5819d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.003ex; height:2.843ex;" alt="{\displaystyle ((P\leftrightarrow Q)\leftrightarrow R)\leftrightarrow (P\leftrightarrow (Q\leftrightarrow R))}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\leftrightarrow (Q\leftrightarrow R))\leftrightarrow ((P\leftrightarrow Q)\leftrightarrow R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">↔</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">↔</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\leftrightarrow (Q\leftrightarrow R))\leftrightarrow ((P\leftrightarrow Q)\leftrightarrow R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6a49f0d06f9b7d211d1fd49c9d31b1a842cf696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.003ex; height:2.843ex;" alt="{\displaystyle (P\leftrightarrow (Q\leftrightarrow R))\leftrightarrow ((P\leftrightarrow Q)\leftrightarrow R)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Operación_non_asociativa"><span id="Operaci.C3.B3n_non_asociativa"></span>Operación non asociativa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asociativid%C3%A1&veaction=edit&section=9" title="Editar seición: Operación non asociativa" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Asociativid%C3%A1&action=edit&section=9" title="Editar el código fuente de la sección: Operación non asociativa"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una operación binaria ∗ nun conxuntu S que nun satisfai'l la llei asociativa denótase non asociativa. Simbólicamente, </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{para dalgun }}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>para dalgun </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{para dalgun }}x,y,z\in S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c017d9a275c4ed28b86e8d1ced4c875cb7878b25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.455ex; height:2.843ex;" alt="{\displaystyle (x*y)*z\neq x*(y*z)\qquad {\mbox{para dalgun }}x,y,z\in S.}"></span></dd></dl> <p>Para tal una operación l'orde d'evaluación <i>sí</i> importa. </p> <div class="mw-heading mw-heading3"><h3 id="Non_asociatividá_del_cálculu_de_coma_flotante"><span id="Non_asociativid.C3.A1_del_c.C3.A1lculu_de_coma_flotante"></span>Non asociatividá del cálculu de coma flotante</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asociativid%C3%A1&veaction=edit&section=10" title="Editar seición: Non asociatividá del cálculu de coma flotante" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Asociativid%C3%A1&action=edit&section=10" title="Editar el código fuente de la sección: Non asociatividá del cálculu de coma flotante"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En matemátiques, la suma y multiplicación de númberos reales ye asociativa. Otra manera, na informática, la adición y multiplicación de <a href="/w/index.php?title=Coma_flotante&action=edit&redlink=1" class="new" title="Coma flotante (la páxina nun esiste)">númberos de coma flotante</a> nun ye asociativa, yá que s'introducen errores d'<a href="/w/index.php?title=Arredondiadura&action=edit&redlink=1" class="new" title="Arredondiadura (la páxina nun esiste)">arredondiadura</a> cuando los valores de distintu tamañu xunense ente sí.<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>Esi ye'l casu de esti exemplu con una <a href="/w/index.php?title=Mantisa&action=edit&redlink=1" class="new" title="Mantisa (la páxina nun esiste)">mantisa</a> de 4 bits. </p><p>(1.000<sub>2</sub>×2⁰ + 1.000<sub>2</sub>×2⁰) + 1.000<sub>2</sub>×2⁴ = 1.000<sub>2</sub>×2¹ + 1.000<sub>2</sub>×2⁴ = 1.00<span style="color:red;">1</span><sub>2</sub>×2⁴ </p><p>1.000<sub>2</sub>×2⁰ + (1.000<sub>2</sub>×2⁰ + 1.000<sub>2</sub>×2⁴) = 1.000<sub>2</sub>×2⁰ + 1.00<span style="color:red;">0</span><sub>2</sub>×2⁴ = 1.00<span style="color:red;">0</span><sub>2</sub>×2⁴ </p><p>Sin embargu, la mayoría d'ordenadores computen ente 24 y 53 bits de mantisa<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>, esto puede siguir siendo un orixe importante d'errores de arrendodio. </p> <div class="mw-heading mw-heading2"><h2 id="Ver_tamién"><span id="Ver_tami.C3.A9n"></span>Ver tamién</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asociativid%C3%A1&veaction=edit&section=11" title="Editar seición: Ver tamién" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Asociativid%C3%A1&action=edit&section=11" title="Editar el código fuente de la sección: Ver tamién"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Conmutativid%C3%A1" title="Conmutatividá">Conmutatividá</a></li> <li><a href="/wiki/Rellaci%C3%B3n_transitiva" title="Rellación transitiva">Rellación transitiva</a></li> <li><a href="/w/index.php?title=Rellaci%C3%B3n_reflexiva&action=edit&redlink=1" class="new" title="Rellación reflexiva (la páxina nun esiste)">Rellación reflexiva</a></li> <li><a href="/w/index.php?title=Rellaci%C3%B3n_sim%C3%A9trica&action=edit&redlink=1" class="new" title="Rellación simétrica (la páxina nun esiste)">Rellación simétrica</a></li> <li><a href="/w/index.php?title=Rellaci%C3%B3n_antisim%C3%A9trica&action=edit&redlink=1" class="new" title="Rellación antisimétrica (la páxina nun esiste)">Rellación antisimétrica</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencies">Referencies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Asociativid%C3%A1&veaction=edit&section=12" title="Editar seición: Referencies" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Asociativid%C3%A1&action=edit&section=12" title="Editar el código fuente de la sección: Referencies"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r3503771">@media only screen and (max-width:600px){.mw-parser-output .llistaref{column-count:1!important}}</style><div class="llistaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Hungerford, Thomas W. (1974). Algebra (1st ed.). Springer. p. 24. ISBN 978-0387905181. "Definition 1.1 (i) a(bc) = (ab)c for all a, b, c in G."</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Moore, Brooke Noel; Parker, Richard (2017). Critical Thinking (12th edition). New York: McGraw-Hill Education. p. 321. ISBN 9781259690877.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Moore, Brooke Noel; Parker, Richard (2017). Critical Thinking (12th edition). New York: McGraw-Hill Education. p. 321. ISBN 9781259690877.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2014). Introduction to Logic (14th edition). Essex: Pearson Education. p. 387. ISBN 9781292024820.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Hurley, Patrick J.; Watson, Lori (2016). A Concise Introduction to Logic (13th edition). Boston: Cengage Learning. p. 427. ISBN 9781305958098.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">"Symbolic Logic Proof of Associativity". Math.stackexchange.com. 22 March 2017.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">Knuth, Donald, The Art of Computer Programming, Volume 3, section 4.2.2</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">IEEE Computer Society (29 August 2008). IEEE Standard for Floating-Point Arithmetic. doi:10.1109/IEEESTD.2008.4610935. ISBN 978-0-7381-5753-5. IEEE Std 754-2008.</span> </li> </ol></div> <style data-mw-deduplicate="TemplateStyles:r2260362">.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid #c8ccd1;background-color:#eaecf0;color:#222222}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}</style><div class="mw-authority-control navigation-not-searchable"><div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r4075543">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline 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