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cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Distributivité en arithmétique</span> </button> <ul id="toc-Distributivité_en_arithmétique-sublist" class="vector-toc-list"> <li id="toc-Distributivité_en_calcul_élémentaire" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distributivité_en_calcul_élémentaire"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Distributivité en calcul élémentaire</span> </div> </a> <ul id="toc-Distributivité_en_calcul_élémentaire-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distributivité_à_droite_et_à_gauche" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distributivité_à_droite_et_à_gauche"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Distributivité à droite et à gauche</span> </div> </a> <ul id="toc-Distributivité_à_droite_et_à_gauche-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entiers_de_Gauss" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Entiers_de_Gauss"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Entiers de Gauss</span> </div> </a> <ul id="toc-Entiers_de_Gauss-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Distributivité_en_algèbre_générale" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Distributivité_en_algèbre_générale"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Distributivité en algèbre générale</span> </div> </a> <button aria-controls="toc-Distributivité_en_algèbre_générale-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>Afficher / masquer la sous-section Distributivité en algèbre générale</span> </button> <ul id="toc-Distributivité_en_algèbre_générale-sublist" class="vector-toc-list"> <li id="toc-Anneaux_et_corps_commutatifs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Anneaux_et_corps_commutatifs"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Anneaux et corps commutatifs</span> </div> </a> <ul id="toc-Anneaux_et_corps_commutatifs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Anneaux_ℤ/nℤ" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Anneaux_ℤ/nℤ"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Anneaux ℤ/nℤ</span> </div> </a> <ul id="toc-Anneaux_ℤ/nℤ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quaternions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quaternions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Quaternions</span> </div> </a> <ul id="toc-Quaternions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Identités_remarquables_dans_les_anneaux_non_commutatifs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Identités_remarquables_dans_les_anneaux_non_commutatifs"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Identités remarquables dans les anneaux non commutatifs</span> </div> </a> <ul id="toc-Identités_remarquables_dans_les_anneaux_non_commutatifs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Espaces_vectoriels" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Espaces_vectoriels"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Espaces vectoriels</span> </div> </a> <ul id="toc-Espaces_vectoriels-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ensemble_des_parties_d'un_ensemble" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ensemble_des_parties_d'un_ensemble"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Ensemble des parties d'un ensemble</span> </div> </a> <ul id="toc-Ensemble_des_parties_d'un_ensemble-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Treillis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Treillis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Treillis</span> </div> </a> <ul id="toc-Treillis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Articles connexes</span> </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Références"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Références</span> </div> </a> <ul id="toc-Références-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="Sommaire" 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="Basculer la table des matières" > <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">Basculer la table des matières</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">Distributivité</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="Aller à un article dans une autre langue. Disponible en 55 langues." > <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-55" 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">55 langues</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%AA%D9%88%D8%B2%D9%8A%D8%B9%D9%8A%D8%A9" title="توزيعية – arabe" lang="ar" hreflang="ar" data-title="توزيعية" data-language-autonym="العربية" data-language-local-name="arabe" 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/Distributivid%C3%A1" title="Distributividá – asturien" lang="ast" hreflang="ast" data-title="Distributividá" data-language-autonym="Asturianu" data-language-local-name="asturien" 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%94%D0%B8%D1%81%D1%82%D1%80%D0%B8%D0%B1%D1%83%D1%82%D0%B8%D0%B2%D0%BB%D1%8B%D2%A1" title="Дистрибутивлыҡ – bachkir" lang="ba" hreflang="ba" data-title="Дистрибутивлыҡ" data-language-autonym="Башҡортса" data-language-local-name="bachkir" 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%94%D1%8B%D1%81%D1%82%D1%80%D1%8B%D0%B1%D1%83%D1%82%D1%8B%D1%9E%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Дыстрыбутыўнасць – biélorusse" lang="be" hreflang="be" data-title="Дыстрыбутыўнасць" data-language-autonym="Беларуская" data-language-local-name="biélorusse" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Propietat_distributiva" title="Propietat distributiva – catalan" lang="ca" hreflang="ca" data-title="Propietat distributiva" data-language-autonym="Català" data-language-local-name="catalan" 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/%D8%AF%D8%A7%D8%A8%DB%95%D8%B4%D8%A8%D9%88%D9%88%D9%86" title="دابەشبوون – sorani" lang="ckb" hreflang="ckb" data-title="دابەشبوون" data-language-autonym="کوردی" data-language-local-name="sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Distributivita" title="Distributivita – tchèque" lang="cs" hreflang="cs" data-title="Distributivita" data-language-autonym="Čeština" data-language-local-name="tchèque" 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%94%D0%B8%D1%81%D1%82%D1%80%D0%B8%D0%B1%D1%83%D1%82%D0%B8%D0%B2%D0%BB%C4%83%D1%85" title="Дистрибутивлăх – tchouvache" lang="cv" hreflang="cv" data-title="Дистрибутивлăх" data-language-autonym="Чӑвашла" data-language-local-name="tchouvache" 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/Distributivitet" title="Distributivitet – danois" lang="da" hreflang="da" data-title="Distributivitet" data-language-autonym="Dansk" data-language-local-name="danois" 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/Distributivgesetz" title="Distributivgesetz – allemand" lang="de" hreflang="de" data-title="Distributivgesetz" data-language-autonym="Deutsch" data-language-local-name="allemand" 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%95%CF%80%CE%B9%CE%BC%CE%B5%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="Επιμεριστική ιδιότητα – grec" lang="el" hreflang="el" data-title="Επιμεριστική ιδιότητα" data-language-autonym="Ελληνικά" data-language-local-name="grec" 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/Distributive_property" title="Distributive property – anglais" lang="en" hreflang="en" data-title="Distributive property" data-language-autonym="English" data-language-local-name="anglais" 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/Distribueco" title="Distribueco – espéranto" lang="eo" hreflang="eo" data-title="Distribueco" data-language-autonym="Esperanto" data-language-local-name="espéranto" 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/Distributividad" title="Distributividad – espagnol" lang="es" hreflang="es" data-title="Distributividad" data-language-autonym="Español" data-language-local-name="espagnol" 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/Distributiivsus" title="Distributiivsus – estonien" lang="et" hreflang="et" data-title="Distributiivsus" data-language-autonym="Eesti" data-language-local-name="estonien" 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/Banakortasun" title="Banakortasun – basque" lang="eu" hreflang="eu" data-title="Banakortasun" data-language-autonym="Euskara" data-language-local-name="basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AE%D8%A7%D8%B5%DB%8C%D8%AA_%D8%AA%D9%88%D8%B2%DB%8C%D8%B9%E2%80%8C%D9%BE%D8%B0%DB%8C%D8%B1%DB%8C" title="خاصیت توزیع‌پذیری – persan" lang="fa" hreflang="fa" data-title="خاصیت توزیع‌پذیری" data-language-autonym="فارسی" data-language-local-name="persan" 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/Osittelulaki" title="Osittelulaki – finnois" lang="fi" hreflang="fi" data-title="Osittelulaki" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Distributiifgesets" title="Distributiifgesets – frison septentrional" lang="frr" hreflang="frr" data-title="Distributiifgesets" data-language-autonym="Nordfriisk" data-language-local-name="frison septentrional" 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_dh%C3%A1ileach" title="Oibríocht dháileach – irlandais" lang="ga" hreflang="ga" data-title="Oibríocht dháileach" data-language-autonym="Gaeilge" data-language-local-name="irlandais" 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/Distributividade" title="Distributividade – galicien" lang="gl" hreflang="gl" data-title="Distributividade" data-language-autonym="Galego" data-language-local-name="galicien" 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%97%D7%95%D7%A7_%D7%94%D7%A4%D7%99%D7%9C%D7%95%D7%92" title="חוק הפילוג – hébreu" lang="he" hreflang="he" data-title="חוק הפילוג" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Disztributivit%C3%A1s" title="Disztributivitás – hongrois" lang="hu" hreflang="hu" data-title="Disztributivitás" data-language-autonym="Magyar" data-language-local-name="hongrois" 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%B2%D5%A1%D5%B7%D5%AD%D5%A1%D5%AF%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Բաշխականություն – arménien" lang="hy" hreflang="hy" data-title="Բաշխականություն" data-language-autonym="Հայերեն" data-language-local-name="arménien" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sifat_distributif" title="Sifat distributif – indonésien" lang="id" hreflang="id" data-title="Sifat distributif" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" 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/Dreifiregla" title="Dreifiregla – islandais" lang="is" hreflang="is" data-title="Dreifiregla" data-language-autonym="Íslenska" data-language-local-name="islandais" 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/Distributivit%C3%A0" title="Distributività – italien" lang="it" hreflang="it" data-title="Distributività" data-language-autonym="Italiano" data-language-local-name="italien" 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/%E5%88%86%E9%85%8D%E6%B3%95%E5%89%87" title="分配法則 – japonais" lang="ja" hreflang="ja" data-title="分配法則" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%84%EB%B0%B0%EB%B2%95%EC%B9%99" title="분배법칙 – coréen" lang="ko" hreflang="ko" data-title="분배법칙" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D1%82%D1%80%D0%B8%D0%B1%D1%83%D1%86%D0%B8%D1%8F%D0%BB%D1%83%D1%83%D0%BB%D1%83%D0%BA" title="Дистрибуциялуулук – kirghize" lang="ky" hreflang="ky" data-title="Дистрибуциялуулук" data-language-autonym="Кыргызча" data-language-local-name="kirghize" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Distributio_(mathematica)" title="Distributio (mathematica) – latin" lang="la" hreflang="la" data-title="Distributio (mathematica)" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D1%82%D1%80%D0%B8%D0%B1%D1%83%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82" title="Дистрибутивност – macédonien" lang="mk" hreflang="mk" data-title="Дистрибутивност" data-language-autonym="Македонски" data-language-local-name="macédonien" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kalis_agihan" title="Kalis agihan – malais" lang="ms" hreflang="ms" data-title="Kalis agihan" data-language-autonym="Bahasa Melayu" data-language-local-name="malais" 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/Distributiviteit" title="Distributiviteit – néerlandais" lang="nl" hreflang="nl" data-title="Distributiviteit" data-language-autonym="Nederlands" data-language-local-name="néerlandais" 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/Distributivitet" title="Distributivitet – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Distributivitet" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien 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/Distributiv_lov" title="Distributiv lov – norvégien bokmål" lang="nb" hreflang="nb" data-title="Distributiv lov" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozdzielno%C5%9B%C4%87" title="Rozdzielność – polonais" lang="pl" hreflang="pl" data-title="Rozdzielność" data-language-autonym="Polski" data-language-local-name="polonais" 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/Distributividade" title="Distributividade – portugais" lang="pt" hreflang="pt" data-title="Distributividade" data-language-autonym="Português" data-language-local-name="portugais" 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/Distributivitate" title="Distributivitate – roumain" lang="ro" hreflang="ro" data-title="Distributivitate" data-language-autonym="Română" data-language-local-name="roumain" 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%94%D0%B8%D1%81%D1%82%D1%80%D0%B8%D0%B1%D1%83%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Дистрибутивность – russe" lang="ru" hreflang="ru" data-title="Дистрибутивность" data-language-autonym="Русский" data-language-local-name="russe" 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/Distributivnost" title="Distributivnost – serbo-croate" lang="sh" hreflang="sh" data-title="Distributivnost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croate" 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/Distributive_property" title="Distributive property – Simple English" lang="en-simple" hreflang="en-simple" data-title="Distributive property" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Distributivnost" title="Distributivnost – slovène" lang="sl" hreflang="sl" data-title="Distributivnost" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D1%82%D1%80%D0%B8%D0%B1%D1%83%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82" title="Дистрибутивност – serbe" lang="sr" hreflang="sr" data-title="Дистрибутивност" data-language-autonym="Српски / srpski" data-language-local-name="serbe" 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/Distributivitet" title="Distributivitet – suédois" lang="sv" hreflang="sv" data-title="Distributivitet" data-language-autonym="Svenska" data-language-local-name="suédois" 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%AA%E0%AE%99%E0%AF%8D%E0%AE%95%E0%AF%80%E0%AE%9F%E0%AF%8D%E0%AE%9F%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="பங்கீட்டுப் பண்பு – tamoul" lang="ta" hreflang="ta" data-title="பங்கீட்டுப் பண்பு" data-language-autonym="தமிழ்" data-language-local-name="tamoul" 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%81%E0%B8%88%E0%B8%81%E0%B9%81%E0%B8%88%E0%B8%87" title="สมบัติการแจกแจง – thaï" lang="th" hreflang="th" data-title="สมบัติการแจกแจง" data-language-autonym="ไทย" data-language-local-name="thaï" 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/Da%C4%9F%C4%B1lma_%C3%B6zelli%C4%9Fi" title="Dağılma özelliği – turc" lang="tr" hreflang="tr" data-title="Dağılma özelliği" data-language-autonym="Türkçe" data-language-local-name="turc" 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%94%D0%B8%D1%81%D1%82%D1%80%D0%B8%D0%B1%D1%83%D1%82%D0%B8%D0%B2%D0%BB%D1%8B%D0%BA" title="Дистрибутивлык – tatar" lang="tt" hreflang="tt" data-title="Дистрибутивлык" data-language-autonym="Татарча / tatarça" data-language-local-name="tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D1%82%D1%80%D0%B8%D0%B1%D1%83%D1%82%D0%B8%D0%B2%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Дистрибутивність – ukrainien" lang="uk" hreflang="uk" data-title="Дистрибутивність" data-language-autonym="Українська" data-language-local-name="ukrainien" 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/Distributivlik" title="Distributivlik – ouzbek" lang="uz" hreflang="uz" data-title="Distributivlik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ouzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Thu%E1%BB%99c_t%C3%ADnh_ph%C3%A2n_ph%E1%BB%91i" title="Thuộc tính phân phối – vietnamien" lang="vi" hreflang="vi" data-title="Thuộc tính phân phối" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%93%D7%99%D7%A1%D7%98%D7%A8%D7%99%D7%91%D7%95%D7%98%D7%99%D7%95%D7%95" title="דיסטריבוטיוו – yiddish" lang="yi" hreflang="yi" data-title="דיסטריבוטיוו" 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/%E5%88%86%E9%85%8D%E5%BE%8B" title="分配律 – chinois" lang="zh" hreflang="zh" data-title="分配律" data-language-autonym="中文" data-language-local-name="chinois" 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/%E5%88%86%E9%85%8D%E5%BE%8B" title="分配律 – cantonais" lang="yue" hreflang="yue" data-title="分配律" data-language-autonym="粵語" data-language-local-name="cantonais" 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/Q187959#sitelinks-wikipedia" title="Modifier les liens interlangues" class="wbc-editpage">Modifier les liens</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="Espaces de noms"> <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/Distributivit%C3%A9" title="Voir le contenu de la page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon 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href="/wiki/Sp%C3%A9cial:Pages_li%C3%A9es/Distributivit%C3%A9" title="Liste des pages liées qui pointent sur celle-ci [j]" accesskey="j"><span>Pages liées</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Suivi_des_liens/Distributivit%C3%A9" rel="nofollow" title="Liste des modifications récentes des pages appelées par celle-ci [k]" accesskey="k"><span>Suivi des pages liées</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Aide:Importer_un_fichier" title="Téléverser des fichiers [u]" accesskey="u"><span>Téléverser un fichier</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_sp%C3%A9ciales" title="Liste de toutes les pages spéciales [q]" accesskey="q"><span>Pages spéciales</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Distributivit%C3%A9&oldid=218921680" title="Adresse permanente de cette version de cette page"><span>Lien 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class="noprint">Un article de Wikipédia, l'encyclopédie libre.</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="fr" dir="ltr"><p>En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, plus précisément en <a href="/wiki/Arithm%C3%A9tique" title="Arithmétique">arithmétique</a> et en <a href="/wiki/Alg%C3%A8bre_g%C3%A9n%C3%A9rale" title="Algèbre générale">algèbre générale</a>, la <b>distributivité</b> d'une <a href="/wiki/Op%C3%A9ration_binaire" title="Opération binaire">opération</a> par rapport à une autre est une généralisation de la propriété élémentaire : « <i>le produit d'une somme est égal à la somme des produits</i> ». </p><p>Par exemple, dans l'<a href="/wiki/Expression_(math%C3%A9matiques)" title="Expression (mathématiques)">expression</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 2\times (5+3)=(2\times 5)+(2\times 3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>×</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>×</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times (5+3)=(2\times 5)+(2\times 3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cdc38adcb8c7646ecb8fac04245c1b816c7304d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.866ex; height:2.843ex;" alt="{\displaystyle 2\times (5+3)=(2\times 5)+(2\times 3)}"></span>, le facteur 2 est distribué<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup> à chacun des deux termes de la somme 5 + 3. L'égalité est alors bien vérifiée : à gauche <span class="texhtml">2 × 8 = 16</span>, à droite <span class="texhtml">10 + 6 = 16</span>. </p><p>Cette propriété est vraie pour tout <a href="/wiki/N-uplet" class="mw-redirect" title="N-uplet">triplet</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 (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</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/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span> d'<a href="/wiki/Entier_naturel" title="Entier naturel">entiers naturels</a>, d'<a href="/wiki/Entier_relatif" title="Entier relatif">entiers relatifs</a>, de <a href="/wiki/Nombre_rationnel" title="Nombre rationnel">nombres rationnels</a>, de <a href="/wiki/Nombre_r%C3%A9el" title="Nombre réel">nombres réels</a> ou de <a href="/wiki/Nombre_complexe" title="Nombre complexe">nombres complexes</a> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\times (y+z)=(x\times y)+(x\times z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\times (y+z)=(x\times y)+(x\times z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e43732917cdcb7e8082d91b7605cc5d1dcca46d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.205ex; height:2.843ex;" alt="{\displaystyle x\times (y+z)=(x\times y)+(x\times z)}"></span></dd></dl> <p>On parle alors de <b>distributivité de la <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> par rapport à l'<a href="/wiki/Addition" title="Addition">addition</a></b><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>. </p><p>En <a href="/wiki/Alg%C3%A8bre_g%C3%A9n%C3%A9rale" title="Algèbre générale">algèbre générale</a>, la distributivité est généralisée à d'autres opérations que l'addition et la multiplication. Une <a href="/wiki/Loi_de_composition_interne" title="Loi de composition interne">loi de composition interne</a> ∘ est distributive par rapport à une autre loi interne ∗ dans un ensemble <i>E</i> si pour tout triplet <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"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</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/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span> d'éléments de <i>E</i>, on a les propriétés suivantes<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup> : </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\circ (y\ast z)=(x\circ y)\ast (x\circ z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∗</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∘</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∘</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\circ (y\ast z)=(x\circ y)\ast (x\circ z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c92480053b8a727c0c1cae0bcba22932b84c116" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.976ex; height:2.843ex;" alt="{\displaystyle x\circ (y\ast z)=(x\circ y)\ast (x\circ z)}"></span>   (<b>distributivité à gauche</b>)</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\ast y)\circ z=(x\circ z)\ast (y\circ z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∘</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>∘</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x\ast y)\circ z=(x\circ z)\ast (y\circ z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1a527c86e0a8e2c9ca7f14f226c3c12c1e67c2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.735ex; height:2.843ex;" alt="{\displaystyle (x\ast y)\circ z=(x\circ z)\ast (y\circ z)}"></span>   (<b>distributivité à droite</b>)</dd></dl> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fichier:FactorComun.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/FactorComun.svg/220px-FactorComun.svg.png" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/FactorComun.svg/330px-FactorComun.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/FactorComun.svg/440px-FactorComun.svg.png 2x" data-file-width="504" data-file-height="381" /></a><figcaption>Une illustration géométrique de la distributivité</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Distributivité_en_arithmétique"><span id="Distributivit.C3.A9_en_arithm.C3.A9tique"></span>Distributivité en arithmétique</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=1" title="Modifier la section : Distributivité en arithmétique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=1" title="Modifier le code source de la section : Distributivité en arithmétique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En arithmétique, les deux opérations considérées lorsqu'on parle de distributivité sont l'addition et la multiplication. La multiplication est distributive par rapport à l'addition : </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\times (y+z)=(x\times y)+(x\times z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\times (y+z)=(x\times y)+(x\times z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e43732917cdcb7e8082d91b7605cc5d1dcca46d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.205ex; height:2.843ex;" alt="{\displaystyle x\times (y+z)=(x\times y)+(x\times z)}"></span></dd></dl> <p>mais l'addition n'est pas distributive par rapport à la multiplication : sauf cas spéciaux (comme <span class="texhtml"><i>x</i> = 0</span>), en général, </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\times z)\neq (x+y)\times (x+z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>×</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+(y\times z)\neq (x+y)\times (x+z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3311b744608349af20d67b758b841915f4bb561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.205ex; height:2.843ex;" alt="{\displaystyle x+(y\times z)\neq (x+y)\times (x+z)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Distributivité_en_calcul_élémentaire"><span id="Distributivit.C3.A9_en_calcul_.C3.A9l.C3.A9mentaire"></span>Distributivité en calcul élémentaire</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=2" title="Modifier la section : Distributivité en calcul élémentaire" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=2" title="Modifier le code source de la section : Distributivité en calcul élémentaire"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si les facteurs d'un produit sont des sommes, on peut effectuer les produits terme à terme puis effectuer la somme. Cette propriété est souvent utilisée, en <a href="/wiki/Calcul_mental" title="Calcul mental">calcul mental</a> ou en <a href="/wiki/Informatique" title="Informatique">informatique</a>, pour <a href="/wiki/Algorithme_de_multiplication" class="mw-redirect" title="Algorithme de multiplication">calculer un produit d'entiers de façon efficace</a>. </p> <dl><dt>Exemple 1</dt> <dd>235 × 99 = 235 × (100 – 1) = 23 500 – 235 = 23 265</dd></dl> <p>De même, la multiplication par les <a href="/wiki/Nombre_uniforme" title="Nombre uniforme">nombres uniformes</a> 9, 99, 999, etc. se ramène à une <a href="/wiki/Soustraction" title="Soustraction">soustraction</a> en utilisant la distributivité. </p> <dl><dt>Exemple 2</dt> <dd>458 × 592 = (400 + 50 + 8) × (500 + 90 + 2) = 200 000 + 36 000 + 800 + 25 000 + 4 500 + 100 + 4000 + 720 + 16 = 271 136.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Distributivité_à_droite_et_à_gauche"><span id="Distributivit.C3.A9_.C3.A0_droite_et_.C3.A0_gauche"></span>Distributivité à droite et à gauche</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=3" title="Modifier la section : Distributivité à droite et à gauche" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=3" title="Modifier le code source de la section : Distributivité à droite et à gauche"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pour les <a href="/wiki/Entier_naturel" title="Entier naturel">entiers naturels</a>, les <a href="/wiki/Entier_relatif" title="Entier relatif">entiers relatifs</a>, les <a href="/wiki/Nombre_rationnel" title="Nombre rationnel">nombres rationnels</a>, les <a href="/wiki/Nombre_r%C3%A9el" title="Nombre réel">nombres réels</a> ou les <a href="/wiki/Nombre_complexe" title="Nombre complexe">nombres complexes</a>, l'addition et la multiplication sont des opérations <a href="/wiki/Loi_commutative" title="Loi commutative">commutatives</a>. On dit alors que la multiplication est distributive par rapport à l'addition, sans préciser <span class="nowrap">« à gauche</span> » ou <span class="nowrap">« à droite »</span>, car la distributivité à gauche implique la distributivité à droite (et réciproquement) du fait de la commutativité du produit. </p> <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Preuve</div><div class="NavContent" style="padding-bottom:0.4em"> <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\times (y+z)=(x\times y)+(x\times z)\Leftrightarrow (y+z)\times x=(x\times y)+(x\times z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\times (y+z)=(x\times y)+(x\times z)\Leftrightarrow (y+z)\times x=(x\times y)+(x\times z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8fe4c7687dd86c5fe3d802d898071c97eb05d68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.023ex; height:2.843ex;" alt="{\displaystyle x\times (y+z)=(x\times y)+(x\times z)\Leftrightarrow (y+z)\times x=(x\times y)+(x\times z)}"></span> (par commutativité de la multiplication dans le membre de gauche)</dd> <dd><span class="texhtml">⇔ (<i>y</i> + <i>z</i>) × <i>x</i> = (<i>y</i> × <i>x</i>) + (<i>x</i> × <i>z</i>)</span>   (par commutativité de la multiplication dans la <abbr class="abbr" title="Première">1<sup>re</sup></abbr> somme du membre de droite)</dd> <dd><span class="texhtml">⇔ (<i>y</i> + <i>z</i>) × <i>x</i> = (<i>y</i> × <i>x</i>) + (<i>z</i> × <i>x</i>)</span>   (par commutativité de la multiplication dans la <abbr class="abbr" title="Deuxième">2<sup>e</sup></abbr> somme du membre de droite)</dd></dl> </div><div class="clear" style="clear:both;"></div> </div> <p>Par contre, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {x+y}{z}}={\dfrac {x}{z}}+{\dfrac {y}{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mi>z</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>x</mi> <mi>z</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>y</mi> <mi>z</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {x+y}{z}}={\dfrac {x}{z}}+{\dfrac {y}{z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa78543b61e73be3f647947fe3ececb8c87621b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.258ex; height:5.176ex;" alt="{\displaystyle {\dfrac {x+y}{z}}={\dfrac {x}{z}}+{\dfrac {y}{z}}}"></span> mais <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {z}{x+y}}\neq {\dfrac {z}{x}}+{\dfrac {z}{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>z</mi> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>z</mi> <mi>x</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>z</mi> <mi>y</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {z}{x+y}}\neq {\dfrac {z}{x}}+{\dfrac {z}{y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ff4bc46b6834a8cc4c814c02838d58c85f3f53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.258ex; height:5.176ex;" alt="{\displaystyle {\dfrac {z}{x+y}}\neq {\dfrac {z}{x}}+{\dfrac {z}{y}}}"></span> et la division sera dite seulement <i>distributive à droite</i> par rapport à l'addition. </p> <div class="mw-heading mw-heading3"><h3 id="Entiers_de_Gauss">Entiers de Gauss</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=4" title="Modifier la section : Entiers de Gauss" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=4" title="Modifier le code source de la section : Entiers de Gauss"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Parmi les nombres complexes, un cas intéressant est celui des <a href="/wiki/Entier_de_Gauss" title="Entier de Gauss">entiers de Gauss</a>, qui s'écrivent sous la forme <span class="texhtml"><i>z</i> = <i>n</i> + <i>m</i>i</span> avec <i>n</i> et <i>m</i> entiers. On utilise la distributivité de la multiplication complexe pour montrer par exemple que (1 + i)<sup>2</sup> = 1 + 2i + i<sup>2</sup> = 2i, c'est-à-dire que 1 + i est une <a href="/wiki/Racine_carr%C3%A9e" title="Racine carrée">racine carrée</a> de 2i. Plus généralement, on montre que le produit de deux entiers de Gauss est un entier de Gauss. </p> <div class="mw-heading mw-heading2"><h2 id="Distributivité_en_algèbre_générale"><span id="Distributivit.C3.A9_en_alg.C3.A8bre_g.C3.A9n.C3.A9rale"></span>Distributivité en algèbre générale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=5" title="Modifier la section : Distributivité en algèbre générale" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=5" title="Modifier le code source de la section : Distributivité en algèbre générale"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En algèbre générale, on étudie les <a href="/wiki/Structure_alg%C3%A9brique" title="Structure algébrique">structures algébriques</a>, c'est-à-dire des ensembles munis de <a href="/wiki/Loi_de_composition" title="Loi de composition">lois de composition</a> ayant certaines propriétés. Dans ce cadre, la distributivité se généralise aux cas où : </p> <dl><dd><ul><li>les deux <a href="/wiki/Loi_de_composition_interne" title="Loi de composition interne">lois de composition interne</a> ne sont pas obligatoirement l'addition et la multiplication ;</li> <li>au moins une opération n'est pas commutative ;</li> <li>la première opération est une <a href="/wiki/Loi_de_composition_interne" title="Loi de composition interne">loi de composition interne</a> et la seconde opération est une <a href="/wiki/Loi_de_composition_externe" class="mw-redirect" title="Loi de composition externe">loi de composition externe</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup>. (Ce cas n'entre strictement parlant pas dans le cadre établi dans la préambule, où les trois éléments <i>x, y, z</i> sont supposés appartenir au même ensemble. Ici ce n'est pas le cas, et dans l'une parmi la distributivité à droite et celle à gauche, les deux * correspondent à des lois différents: voir le paragraphe "espaces vectoriels" ci-après.)</li></ul></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Anneaux_et_corps_commutatifs">Anneaux et corps commutatifs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=6" title="Modifier la section : Anneaux et corps commutatifs" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=6" title="Modifier le code source de la section : Anneaux et corps commutatifs"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La distributivité de la seconde loi de composition interne sur la première loi de composition interne est une propriété fondamentale des <a href="/wiki/Anneau_(math%C3%A9matiques)" title="Anneau (mathématiques)">anneaux</a> (et donc des <a href="/wiki/Corps_(math%C3%A9matiques)" title="Corps (mathématiques)">corps</a>) : dans un anneau <i>A</i> muni de deux lois internes notées + et ×, la loi × doit être distributive (à droite et à gauche) par rapport à +. </p> <div class="mw-heading mw-heading3"><h3 id="Anneaux_ℤ/nℤ"><span id="Anneaux_.E2.84.A4.2Fn.E2.84.A4"></span>Anneaux ℤ/nℤ</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=7" title="Modifier la section : Anneaux ℤ/nℤ" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=7" title="Modifier le code source de la section : Anneaux ℤ/nℤ"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les <a href="/wiki/Anneau_%E2%84%A4/n%E2%84%A4" title="Anneau ℤ/nℤ">anneaux quotients de ℤ</a> héritent de l'addition et de la multiplication des entiers relatifs, et ces lois induites vérifient la distributivité de la multiplication par rapport à l'addition. </p> <div class="mw-heading mw-heading3"><h3 id="Quaternions">Quaternions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=8" title="Modifier la section : Quaternions" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=8" title="Modifier le code source de la section : Quaternions"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La distributivité de la multiplication sur la division reste valable pour les <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> de Hamilton, bien que la multiplication des quaternions ne soit pas <a href="/wiki/Commutativit%C3%A9" class="mw-redirect" title="Commutativité">commutative</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Identités_remarquables_dans_les_anneaux_non_commutatifs"><span id="Identit.C3.A9s_remarquables_dans_les_anneaux_non_commutatifs"></span>Identités remarquables dans les anneaux non commutatifs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=9" title="Modifier la section : Identités remarquables dans les anneaux non commutatifs" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=9" title="Modifier le code source de la section : Identités remarquables dans les anneaux non commutatifs"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Certaines <a href="/wiki/Identit%C3%A9_remarquable" title="Identité remarquable">identités remarquables</a> qui font intervenir la distributivité, par exemple <span class="texhtml">(<i>a</i>+<i>b</i>)<sup>2</sup> = <i>a</i><sup>2</sup> + 2<i>ab</i> + <i>b</i><sup>2</sup></span> et généralisations, utilisent également la commutativité et ne sont donc pas valides pour les anneaux non commutatifs tels que les anneaux de <a href="/wiki/Matrice_(math%C3%A9matiques)" title="Matrice (mathématiques)">matrices</a> ou les <a href="/wiki/Anneau_non_commutatif_de_polyn%C3%B4mes" title="Anneau non commutatif de polynômes">anneaux non commutatifs de polynômes</a>. Bien entendu, toute propriété résultant de la distributivité et qui ne nécessite pas la commutativité reste valable dans les anneaux non commutatifs. (Dans l'exemple en question, on aura <span class="texhtml">(<i>a</i>+<i>b</i>)<sup>2</sup> = <i>a</i><sup>2</sup> + <i>ab</i> + <i>ba</i> + <i>b</i><sup>2</sup></span> si <span class="texhtml"><i>ab</i> ≠ <i>ba</i></span> ; mais on a toujours <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 (1+x)^{n}=\sum _{k=0}^{n}C_{n}^{k}\,x^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+x)^{n}=\sum _{k=0}^{n}C_{n}^{k}\,x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c24135c61ca156e0c6dab6521320c138f30c4e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.893ex; height:7.009ex;" alt="{\displaystyle (1+x)^{n}=\sum _{k=0}^{n}C_{n}^{k}\,x^{k}}"></span> puisque tout <span class="texhtml mvar" style="font-style:italic;">x</span> commute avec 1 dans tout <a href="/wiki/Anneau_unitaire" title="Anneau unitaire">anneau unitaire</a>.) </p> <div class="mw-heading mw-heading3"><h3 id="Espaces_vectoriels">Espaces vectoriels</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=10" title="Modifier la section : Espaces vectoriels" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=10" title="Modifier le code source de la section : Espaces vectoriels"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans la définition d'un <a href="/wiki/Espace_vectoriel" title="Espace vectoriel">espace vectoriel</a>, la <a href="/wiki/Loi_de_composition" title="Loi de composition">multiplication externe</a> par des <a href="/wiki/Scalaire_(math%C3%A9matiques)" title="Scalaire (mathématiques)">scalaires</a> est distributive par rapport à l'addition des vecteurs. Ici, on a affaire à une loi de composition externe et non interne, mais la propriété de distributivité reste valable (aussi bien celle à gauche que celle à droite, qui elle ((λ+μ)•x = λ•x + μ•x) implique deux lois d'addition différentes: d'une part celle des scalaires, d'autre part celle des vecteurs). C'est donc une notion de distributivité plus générale qui n'est pas un cas particulier de celle définie dans la préambule de cet article, où tous les éléments appartiennent au même ensemble. </p> <div class="mw-heading mw-heading3"><h3 id="Ensemble_des_parties_d'un_ensemble"><span id="Ensemble_des_parties_d.27un_ensemble"></span>Ensemble des parties d'un ensemble</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=11" title="Modifier la section : Ensemble des parties d'un ensemble" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=11" title="Modifier le code source de la section : Ensemble des parties d'un ensemble"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Soit <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 {\mathcal {P}}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb9a5a97c7cb3db567ed6d470b51883427bb06a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.289ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(E)}"></span> l'<a href="/wiki/Ensemble_des_parties_d%27un_ensemble" title="Ensemble des parties d'un ensemble">ensemble des parties d'un ensemble</a> <i>E</i>. On munit <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 {\mathcal {P}}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb9a5a97c7cb3db567ed6d470b51883427bb06a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.289ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(E)}"></span> de deux lois de composition interne : la <a href="/wiki/Union_(math%C3%A9matiques)" title="Union (mathématiques)">réunion</a> ⋃ et <a href="/wiki/Intersection_(math%C3%A9matiques)" title="Intersection (mathématiques)">l'intersection</a> ⋂. Dans ce cas, les deux lois de composition interne sont distributives l'une par rapport à l'autre. Autrement dit, pour tout triplet (<i>A</i>, <i>B</i>, <i>C</i>) d'éléments de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb9a5a97c7cb3db567ed6d470b51883427bb06a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.289ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(E)}"></span> : </p> <dl><dd><span class="texhtml"><i>A</i> ∪ (<i>B</i> ∩ <i>C</i>) = (<i>A</i> ∪ <i>B</i>) ∩ (<i>A</i> ∪ <i>C</i>)</span></dd> <dd><span class="texhtml"><i>A</i> ∩ (<i>B</i> ∪ <i>C</i>) = (<i>A</i> ∩ <i>B</i>) ∪ (<i>A</i> ∩ <i>C</i>)</span></dd></dl> <p>La distributivité est également vérifiée si l'on considère la <a href="/wiki/Diff%C3%A9rence_sym%C3%A9trique" class="mw-redirect" title="Différence symétrique">différence symétrique</a> <span class="texhtml"><i>A</i> Δ <i>B</i> := (<i>A</i> ⋃ <i>B</i>) \ (<i>A</i> ⋂ <i>B</i>)</span> au lieu de la réunion. Contrairement à la réunion, cette opération confère la structure de <a href="/wiki/Groupe_ab%C3%A9lien" title="Groupe abélien">groupe abélien</a>, et avec l'intersection la structure d'<a href="/wiki/Anneau_de_Boole" title="Anneau de Boole">anneau de Boole</a> à <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 {\mathcal {P}}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb9a5a97c7cb3db567ed6d470b51883427bb06a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.289ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(E)}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Treillis">Treillis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=12" title="Modifier la section : Treillis" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=12" title="Modifier le code source de la section : Treillis"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un <a href="/wiki/Treillis_(ensemble_ordonn%C3%A9)" title="Treillis (ensemble ordonné)">treillis</a> est un <a href="/wiki/Ordre_partiel" class="mw-redirect" title="Ordre partiel">ensemble E partiellement ordonné</a> dans lequel toute <a href="/wiki/Paire" title="Paire">paire</a> {<i>x</i>, <i>y</i>} admet une <a href="/wiki/Borne_sup%C3%A9rieure" class="mw-redirect" title="Borne supérieure">borne supérieure</a> <i>x</i>⋁<i>y </i> et une <a href="/wiki/Borne_inf%C3%A9rieure" class="mw-redirect" title="Borne inférieure">borne inférieure</a> <i>x</i>⋀<i>y</i>. On dit que <i>E</i> est un <b>treillis distributif</b> si les deux lois de composition interne sont distributives l'une par rapport à l'autre. Dans ce cas, pour tout triplet (<i>x</i>, <i>y</i>, <i>z</i>) d'éléments de E, on a : </p> <dl><dd><span class="texhtml"><i>x</i> ⋁ (<i>y</i> ⋀ <i>z</i>) = (<i>x</i> ⋁ <i>y</i>) ⋀ (<i>x</i> ⋁ <i>z</i>)</span></dd> <dd><span class="texhtml"><i>x</i> ⋀ (<i>y</i> ⋁ <i>z</i>) = (<i>x</i> ⋀ <i>y</i>) ⋁ (<i>x</i> ⋀ <i>z</i>)</span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Articles_connexes">Articles connexes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=13" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=13" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r194021218">.mw-parser-output .autres-projets>.titre{text-align:center;margin:0.2em 0}.mw-parser-output .autres-projets>ul{margin:0;padding:0}.mw-parser-output .autres-projets>ul>li{list-style:none;margin:0.2em 0;text-indent:0;padding-left:24px;min-height:20px;text-align:left;display:block}.mw-parser-output .autres-projets>ul>li>a{font-style:italic}@media(max-width:720px){.mw-parser-output .autres-projets{float:none}}</style><div class="autres-projets boite-grise boite-a-droite noprint js-interprojets"> <p class="titre">Sur les autres projets Wikimedia :</p> <ul class="noarchive plainlinks"> <li class="wiktionary"><a href="https://fr.wiktionary.org/wiki/distributivit%C3%A9" class="extiw" title="wikt:distributivité">distributivité</a>, <span class="nowrap">sur le <span class="project">Wiktionnaire</span></span></li><li class="wikiversity"><a href="https://fr.wikiversity.org/wiki/Priorit%C3%A9s,_distributivit%C3%A9" class="extiw" title="v:Priorités, distributivité">Priorités, distributivité</a>, <span class="nowrap">sur <span class="project">Wikiversity</span></span></li> </ul> </div> <ul><li><a href="/wiki/Associativit%C3%A9" title="Associativité">Associativité</a></li> <li><a href="/wiki/Fran%C3%A7ois-Joseph_Servois" title="François-Joseph Servois">François-Joseph Servois</a> (le premier à employer l'adjectif « distributif » en mathématiques)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=14" title="Modifier la section : Notes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=14" title="Modifier le code source de la section : Notes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text">L'application de la distributivité à l'expression sous forme de produit s'appelle <a href="/wiki/D%C3%A9veloppement_(math%C3%A9matiques)" title="Développement (mathématiques)">développement</a>. L'application inverse de la propriété à une somme s'appelle factorisation ou mise en facteur commun.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text">Lang 1976, <abbr class="abbr" title="page">p.</abbr> 40</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text">Queysanne 1964, <abbr class="abbr" title="page">p.</abbr> 116.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink noprint"><a href="#cite_ref-4">↑</a> </span><span class="reference-text">Queysanne 1964, <abbr class="abbr" title="page">p.</abbr> 122.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distributivit%C3%A9&veaction=edit&section=15" title="Modifier la section : Références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Distributivit%C3%A9&action=edit&section=15" title="Modifier le code source de la section : Références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="ouvrage" id="Lang1976"><span class="ouvrage" id="Serge_Lang1976"><a href="/wiki/Serge_Lang" title="Serge Lang">Serge <span class="nom_auteur">Lang</span></a>, <cite class="italique">Structures algébriques</cite>, InterEditions, <time>1976</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Structures+alg%C3%A9briques&rft.pub=InterEditions&rft.aulast=Lang&rft.aufirst=Serge&rft.date=1976&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ADistributivit%C3%A9"></span></span></span>.</li></ul> <ul><li><span class="ouvrage" id="Queysanne1964"><span class="ouvrage" id="Michel_Queysanne1964"><a href="/wiki/Michel_Queysanne" title="Michel Queysanne">Michel <span class="nom_auteur">Queysanne</span></a>, <cite class="italique">Algèbre : <abbr class="abbr" title="Premier">1<sup>er</sup></abbr> cycle scientifique - préparation aux grandes écoles</cite>, Armand Colin, <abbr class="abbr" title="collection">coll.</abbr> « U », <time>1964</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alg%C3%A8bre&rft.pub=Armand+Colin&rft.stitle=%3Cabbr+class%3D%22abbr%22+title%3D%22Premier%22%3E1%3Csup%3Eer%3C%2Fsup%3E%3C%2Fabbr%3E+cycle+scientifique+-+pr%C3%A9paration+aux+grandes+%C3%A9coles&rft.aulast=Queysanne&rft.aufirst=Michel&rft.date=1964&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ADistributivit%C3%A9"></span></span></span>.</li></ul> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Logique_math%C3%A9matique" title="Modèle:Palette Logique mathématique"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Logique_math%C3%A9matique&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%">Logique mathématique</div></th> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Calcul_des_propositions" title="Calcul des propositions">Calcul des propositions</a></th> <td class="navbox-list" style=""><table class="navbox-subgroup" style=""> <tbody><tr> <th class="navbox-group" style=""><a href="/wiki/R%C3%A8gle_d%27inf%C3%A9rence" title="Règle d'inférence">Règles d'inférence</a></th> <td class="navbox-list" style=";"><div class="liste-horizontale"> <ul><li><a href="/wiki/Modus_ponens" title="Modus ponens"><span title="A→B, A ⊢ B"><i>Modus ponens</i></span></a> / <a href="/wiki/Modus_tollens" title="Modus tollens"><span title="A→B, ¬B ⊢ ¬A"><i>Modus tollens</i></span></a></li> <li><a href="/wiki/%C3%89limination_de_la_conjonction" title="Élimination de la conjonction"><span title="A∧B ⊢ A">Élimination</span></a> / <a href="/wiki/R%C3%A8gle_d%27introduction_(logique)" title="Règle d'introduction (logique)"><span title="A, B ⊢ A∧B">introduction de la conjonction</span></a></li> <li><a href="/wiki/%C3%89limination_de_la_disjonction" title="Élimination de la disjonction"><span title="A∨B, A→C, B→C ⊢ C">Élimination</span></a> / <a href="/wiki/Introduction_de_la_disjonction" title="Introduction de la disjonction"><span title="A ⊢ A∨B">introduction de la disjonction</span></a></li> <li><a href="/wiki/Syllogisme_hypoth%C3%A9tique" title="Syllogisme hypothétique"><span title="A→B, B→C ⊢ A→C">Syllogisme hypothétique</span></a> / <a href="/wiki/Syllogisme_disjonctif" title="Syllogisme disjonctif"><span title="A∨B, ¬A ⊢ B">disjonctif</span></a></li> <li><a href="/wiki/Dilemme_constructif" title="Dilemme constructif"><span title="A→P, B→Q, A∨B ⊢ P∨Q">Dilemme constructif</span></a> / <a href="/wiki/Dilemme_destructif" title="Dilemme destructif"><span title="A→P, B→Q, ¬P∨¬Q ⊢ ¬A∨¬B">destructif</span></a></li> <li><a href="/w/index.php?title=Absorption_(logique)&action=edit&redlink=1" class="new" title="Absorption (logique) (page inexistante)"><span title="A→B ⊢ A→A∧B">Absorption</span></a></li> <li><i><a href="/wiki/Modus_ponendo_tollens" title="Modus ponendo tollens">Modus ponendo tollens</a></i></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/R%C3%A8gle_de_remplacement" title="Règle de remplacement">Règles de remplacement</a></th> <td class="navbox-list navbox-even" style=";"><div class="liste-horizontale"> <ul><li><a href="/wiki/Associativit%C3%A9" title="Associativité"><span title="A∨(B∨C) = (A∨B)∨C">Associativité</span></a></li> <li><a href="/wiki/Loi_commutative" title="Loi commutative">Commutativité</a></li> <li><a class="mw-selflink selflink"><span title="A∧(B∨C) = (A∧B)∨(A∧C)">Distributivité</span></a></li> <li><a href="/w/index.php?title=N%C3%A9gation_(logique)&action=edit&redlink=1" class="new" title="Négation (logique) (page inexistante)"><span title="¬¬A = A">Double négation</span></a></li> <li><a href="/wiki/Lois_de_De_Morgan" title="Lois de De Morgan">Lois de De Morgan</a></li> <li><a href="/wiki/Transposition_(logique)" title="Transposition (logique)">Transposition</a></li> <li><a href="/wiki/Implication_(logique)" title="Implication (logique)"><span title="A→B ⊢ ¬A∨B">Implication</span></a></li> <li><a href="/wiki/%C3%89quivalence_logique" title="Équivalence logique">Équivalence</a></li> <li><a href="/w/index.php?title=Exportation_(logique)&action=edit&redlink=1" class="new" title="Exportation (logique) (page inexistante)"><span title="(A∧B)→C ⊢ A→(B→C)">Exportation</span></a></li> <li><a href="/wiki/Tautologie_(logique)" title="Tautologie (logique)">Tautologie</a></li> <li><a href="/w/index.php?title=Introduction_de_la_n%C3%A9gation&action=edit&redlink=1" class="new" title="Introduction de la négation (page inexistante)">Introduction de la négation</a></li></ul> </div></td> </tr> </tbody></table></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Calcul_des_pr%C3%A9dicats" title="Calcul des prédicats">Calcul des prédicats</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/w/index.php?title=G%C3%A9n%C3%A9ralisation_universelle&action=edit&redlink=1" class="new" title="Généralisation universelle (page inexistante)">Généralisation</a> / <a href="/wiki/Instanciation_universelle" title="Instanciation universelle">instanciation universelle</a></li> <li><a href="/w/index.php?title=G%C3%A9n%C3%A9ralisation_existentielle&action=edit&redlink=1" class="new" title="Généralisation existentielle (page inexistante)">Généralisation</a> / <a href="/w/index.php?title=Instanciation_existentielle&action=edit&redlink=1" class="new" title="Instanciation existentielle (page inexistante)">instanciation existentielle</a></li></ul> </div></td> </tr> </tbody></table> </div> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span 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