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class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Définitions formelles</span> </button> <ul id="toc-Définitions_formelles-sublist" class="vector-toc-list"> <li id="toc-Définition_fonctionnelle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Définition_fonctionnelle"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Définition fonctionnelle</span> </div> </a> <ul id="toc-Définition_fonctionnelle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Définition_relationnelle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Définition_relationnelle"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Définition relationnelle</span> </div> </a> <ul id="toc-Définition_relationnelle-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Exemple_concret" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exemple_concret"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Exemple concret</span> </div> </a> <ul id="toc-Exemple_concret-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exemples_et_contre-exemples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exemples_et_contre-exemples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Exemples et contre-exemples</span> </div> </a> <ul id="toc-Exemples_et_contre-exemples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propriétés" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propriétés"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Propriétés</span> </div> </a> <ul id="toc-Propriétés-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes et références</span> </div> </a> <button aria-controls="toc-Notes_et_références-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 Notes et références</span> </button> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</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-2"> <a class="vector-toc-link" href="#Références"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Références</span> </div> </a> <ul id="toc-Références-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Article_connexe" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Article_connexe"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Article connexe</span> </div> </a> <ul id="toc-Article_connexe-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">Bijection</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%82%D8%A7%D8%A8%D9%84_(%D8%AF%D8%A7%D9%84%D8%A9)" title="تقابل (دالة) – arabe" lang="ar" hreflang="ar" data-title="تقابل (دالة)" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%91%D1%96%D0%B5%D0%BA%D1%86%D1%8B%D1%8F" title="Біекцыя – biélorusse" lang="be" hreflang="be" data-title="Біекцыя" data-language-autonym="Беларуская" data-language-local-name="biélorusse" 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%91%D0%B8%D0%B5%D0%BA%D1%86%D0%B8%D1%8F" title="Биекция – bulgare" lang="bg" hreflang="bg" data-title="Биекция" data-language-autonym="Български" data-language-local-name="bulgare" 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/Bijekcija" title="Bijekcija – bosniaque" lang="bs" hreflang="bs" data-title="Bijekcija" data-language-autonym="Bosanski" data-language-local-name="bosniaque" 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/Funci%C3%B3_bijectiva" title="Funció bijectiva – catalan" lang="ca" hreflang="ca" data-title="Funció bijectiva" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Biiezzioni" title="Biiezzioni – corse" lang="co" hreflang="co" data-title="Biiezzioni" data-language-autonym="Corsu" data-language-local-name="corse" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bijekce" title="Bijekce – tchèque" lang="cs" hreflang="cs" data-title="Bijekce" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bijektiv" title="Bijektiv – danois" lang="da" hreflang="da" data-title="Bijektiv" 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/Bijektive_Funktion" title="Bijektive Funktion – allemand" lang="de" hreflang="de" data-title="Bijektive Funktion" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Bijection" title="Bijection – anglais" lang="en" hreflang="en" data-title="Bijection" 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/Dissur%C4%B5eto" title="Dissurĵeto – espéranto" lang="eo" hreflang="eo" data-title="Dissurĵeto" 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/Funci%C3%B3n_biyectiva" title="Función biyectiva – espagnol" lang="es" hreflang="es" data-title="Función biyectiva" 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/Bijektiivne_funktsioon" title="Bijektiivne funktsioon – estonien" lang="et" hreflang="et" data-title="Bijektiivne funktsioon" 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/Bijekzio" title="Bijekzio – basque" lang="eu" hreflang="eu" data-title="Bijekzio" 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%AA%D9%86%D8%A7%D8%B8%D8%B1_%D8%AF%D9%88%D8%B3%D9%88%DB%8C%D9%87" title="تناظر دوسویه – persan" lang="fa" hreflang="fa" data-title="تناظر دوسویه" 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/Bijektio" title="Bijektio – finnois" lang="fi" hreflang="fi" data-title="Bijektio" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_bixectiva" title="Función bixectiva – galicien" lang="gl" hreflang="gl" data-title="Función bixectiva" 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%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%97%D7%93-%D7%97%D7%93-%D7%A2%D7%A8%D7%9B%D7%99%D7%AA_%D7%95%D7%A2%D7%9C" title="פונקציה חד-חד-ערכית ועל – hébreu" lang="he" hreflang="he" data-title="פונקציה חד-חד-ערכית ועל" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%85%E0%A4%82%E0%A4%A4%E0%A4%A5%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%AA%E0%A4%A3" title="द्विअंतथक्षेपण – hindi" lang="hi" hreflang="hi" data-title="द्विअंतथक्षेपण" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Bijekcija" title="Bijekcija – croate" lang="hr" hreflang="hr" data-title="Bijekcija" data-language-autonym="Hrvatski" data-language-local-name="croate" 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/Bijekci%C3%B3" title="Bijekció – hongrois" lang="hu" hreflang="hu" data-title="Bijekció" 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/%D5%93%D5%B8%D5%AD%D5%B4%D5%AB%D5%A1%D6%80%D5%AA%D5%A5%D6%84_%D5%B0%D5%A1%D5%B4%D5%A1%D5%BA%D5%A1%D5%BF%D5%A1%D5%BD%D5%AD%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Փոխմիարժեք համապատասխանություն – arménien" lang="hy" hreflang="hy" data-title="Փոխմիարժեք համապատասխանություն" data-language-autonym="Հայերեն" data-language-local-name="arménien" 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/Bijection" title="Bijection – interlingua" lang="ia" hreflang="ia" data-title="Bijection" 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/Bijeksi" title="Bijeksi – indonésien" lang="id" hreflang="id" data-title="Bijeksi" 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-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Bijektio" title="Bijektio – ido" lang="io" hreflang="io" data-title="Bijektio" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Gagnt%C3%A6k_v%C3%B6rpun" title="Gagntæk vörpun – islandais" lang="is" hreflang="is" data-title="Gagntæk vörpun" 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/Corrispondenza_biunivoca" title="Corrispondenza biunivoca – italien" lang="it" hreflang="it" data-title="Corrispondenza biunivoca" 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%85%A8%E5%8D%98%E5%B0%84" title="全単射 – japonais" lang="ja" hreflang="ja" data-title="全単射" 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-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D3%A8%D0%B7%D0%B0%D1%80%D0%B0_%D0%B1%D1%96%D1%80%D0%BC%D3%99%D0%BD%D0%B4%D1%96_%D1%81%D3%99%D0%B9%D0%BA%D0%B5%D1%81%D1%82%D1%96%D0%BA" title="Өзара бірмәнді сәйкестік – kazakh" lang="kk" hreflang="kk" data-title="Өзара бірмәнді сәйкестік" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%84%EB%8B%A8%EC%82%AC_%ED%95%A8%EC%88%98" title="전단사 함수 – coréen" lang="ko" hreflang="ko" data-title="전단사 함수" data-language-autonym="한국어" data-language-local-name="coréen" 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/Functio_biiectiva" title="Functio biiectiva – latin" lang="la" hreflang="la" data-title="Functio biiectiva" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Bigezzion" title="Bigezzion – lombard" lang="lmo" hreflang="lmo" data-title="Bigezzion" data-language-autonym="Lombard" data-language-local-name="lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Bijekcija" title="Bijekcija – lituanien" lang="lt" hreflang="lt" data-title="Bijekcija" data-language-autonym="Lietuvių" data-language-local-name="lituanien" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%B8%D1%98%D0%B5%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Бијекција – macédonien" lang="mk" hreflang="mk" data-title="Бијекција" data-language-autonym="Македонски" data-language-local-name="macédonien" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bijectie" title="Bijectie – néerlandais" lang="nl" hreflang="nl" data-title="Bijectie" 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/Bijeksjon" title="Bijeksjon – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Bijeksjon" 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/Bijeksjon" title="Bijeksjon – norvégien bokmål" lang="nb" hreflang="nb" data-title="Bijeksjon" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Bijeccion" title="Bijeccion – occitan" lang="oc" hreflang="oc" data-title="Bijeccion" data-language-autonym="Occitan" data-language-local-name="occitan" 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/Funkcja_wzajemnie_jednoznaczna" title="Funkcja wzajemnie jednoznaczna – polonais" lang="pl" hreflang="pl" data-title="Funkcja wzajemnie jednoznaczna" 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/Fun%C3%A7%C3%A3o_bijectiva" title="Função bijectiva – portugais" lang="pt" hreflang="pt" data-title="Função bijectiva" 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/Coresponden%C8%9B%C4%83_biunivoc%C4%83" title="Corespondență biunivocă – roumain" lang="ro" hreflang="ro" data-title="Corespondență biunivocă" 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%91%D0%B8%D0%B5%D0%BA%D1%86%D0%B8%D1%8F" title="Биекция – russe" lang="ru" hreflang="ru" data-title="Биекция" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Bijection" title="Bijection – écossais" lang="sco" hreflang="sco" data-title="Bijection" data-language-autonym="Scots" data-language-local-name="écossais" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Bijective_function" title="Bijective function – Simple English" lang="en-simple" hreflang="en-simple" data-title="Bijective function" 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/Bijekt%C3%ADvne_zobrazenie" title="Bijektívne zobrazenie – slovaque" lang="sk" hreflang="sk" data-title="Bijektívne zobrazenie" data-language-autonym="Slovenčina" data-language-local-name="slovaque" 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/Bijektivna_preslikava" title="Bijektivna preslikava – slovène" lang="sl" hreflang="sl" data-title="Bijektivna preslikava" 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%91%D0%B8%D1%98%D0%B5%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Бијекција – serbe" lang="sr" hreflang="sr" data-title="Бијекција" 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/Bijektiv_funktion" title="Bijektiv funktion – suédois" lang="sv" hreflang="sv" data-title="Bijektiv funktion" 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%87%E0%AE%B0%E0%AF%81%E0%AE%B5%E0%AE%B4%E0%AE%BF%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81" title="இருவழிக்கோப்பு – tamoul" lang="ta" hreflang="ta" data-title="இருவழிக்கோப்பு" 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%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99%E0%B8%AB%E0%B8%99%E0%B8%B6%E0%B9%88%E0%B8%87%E0%B8%95%E0%B9%88%E0%B8%AD%E0%B8%AB%E0%B8%99%E0%B8%B6%E0%B9%88%E0%B8%87%E0%B8%97%E0%B8%B1%E0%B9%88%E0%B8%A7%E0%B8%96%E0%B8%B6%E0%B8%87" title="ฟังก์ชันหนึ่งต่อหนึ่งทั่วถึง – thaï" lang="th" hreflang="th" data-title="ฟังก์ชันหนึ่งต่อหนึ่งทั่วถึง" 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/Birebir_%C3%B6rten_fonksiyon" title="Birebir örten fonksiyon – turc" lang="tr" hreflang="tr" data-title="Birebir örten fonksiyon" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D1%96%D1%94%D0%BA%D1%86%D1%96%D1%8F" title="Бієкція – ukrainien" lang="uk" hreflang="uk" data-title="Бієкція" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Song_%C3%A1nh" title="Song ánh – vietnamien" lang="vi" hreflang="vi" data-title="Song ánh" 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-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%8C%E5%B0%84" title="双射 – chinois" lang="zh" hreflang="zh" data-title="双射" 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%B0%8D%E5%B0%84%E5%87%BD%E6%95%B8" title="對射函數 – cantonais" lang="yue" hreflang="yue" data-title="對射函數" 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/Q180907#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/Bijection" title="Voir le contenu de la page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussion:Bijection" rel="discussion" title="Discussion au sujet de cette page de contenu [t]" accesskey="t"><span>Discussion</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Modifier la variante de langue" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">français</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Affichages"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Bijection"><span>Lire</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bijection&veaction=edit" title="Modifier cette page [v]" accesskey="v"><span>Modifier</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bijection&action=edit" title="Modifier le wikicode de cette page [e]" accesskey="e"><span>Modifier le code</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bijection&action=history" title="Historique des versions de cette page [h]" accesskey="h"><span>Voir l’historique</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Outils de la page"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Outils" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Outils</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Outils</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">masquer</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Plus d’options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Bijection"><span>Lire</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bijection&veaction=edit" title="Modifier cette page [v]" accesskey="v"><span>Modifier</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bijection&action=edit" title="Modifier le wikicode de cette page [e]" accesskey="e"><span>Modifier le code</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bijection&action=history"><span>Voir l’historique</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Général </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_li%C3%A9es/Bijection" 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/Bijection" 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=Bijection&oldid=212827581" title="Adresse permanente de cette version de cette page"><span>Lien permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Bijection&action=info" title="Davantage d’informations sur cette page"><span>Informations sur la page</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:Citer&page=Bijection&id=212827581&wpFormIdentifier=titleform" title="Informations sur la manière de citer cette page"><span>Citer cette page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:UrlQ%C4%B1sald%C4%B1c%C4%B1s%C4%B1&url=https%3A%2F%2Ffr.wikipedia.org%2Fwiki%2FBijection"><span>Obtenir l'URL raccourcie</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:QrKodu&url=https%3A%2F%2Ffr.wikipedia.org%2Fwiki%2FBijection"><span>Télécharger le code QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimer / exporter </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:Livre&bookcmd=book_creator&referer=Bijection"><span>Créer un livre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:DownloadAsPdf&page=Bijection&action=show-download-screen"><span>Télécharger comme PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Bijection&printable=yes" title="Version imprimable de cette page [p]" accesskey="p"><span>Version imprimable</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Dans d’autres projets </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Bijectivity" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q180907" title="Lien vers l’élément dans le dépôt de données connecté [g]" accesskey="g"><span>Élément 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="Outils de la page"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apparence"> <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">Apparence</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</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">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>, une <b>bijection</b> ou <b>application bijective</b> (parfois appelée <b>correspondance biunivoque</b><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>) est une <a href="/wiki/Application_(math%C3%A9matiques)" title="Application (mathématiques)">application</a> qui est à la fois <i><a href="/wiki/Injection_(math%C3%A9matiques)" title="Injection (mathématiques)">injective</a></i> et <i><a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a></i>, autrement dit pour laquelle tout élément de son <a href="/wiki/Ensemble_d%27arriv%C3%A9e" title="Ensemble d'arrivée">ensemble d'arrivée</a> possède un et un seul <i><a href="/wiki/Ant%C3%A9c%C3%A9dent_(math%C3%A9matiques)" title="Antécédent (mathématiques)">antécédent</a></i><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>Note 1<span class="cite_crochet">]</span></a></sup>. </p><p>Une propriété des bijections est que s'il existe une bijection <i>f</i> d'un <a href="/wiki/Ensemble" title="Ensemble">ensemble</a> <i>E</i> dans un ensemble <i>F</i> alors il existe une <a href="/wiki/Bijection_r%C3%A9ciproque" title="Bijection réciproque">bijection réciproque</a> de <i>F</i> dans <i>E</i> qui à chaque élément de <i>F</i> associe son antécédent par <i>f</i>. Les deux ensembles sont dits en bijection, ou <a href="/wiki/%C3%89quipotence" title="Équipotence">équipotents</a>. </p><p><a href="/wiki/Georg_Cantor" title="Georg Cantor">Cantor</a> a le premier démontré que s'il existe une injection de <i>E</i> vers <i>F</i> et une injection de <i>F</i> vers <i>E</i> (non nécessairement surjectives), alors <i>E</i> et <i>F</i> sont équipotents (c'est le <a href="/wiki/Th%C3%A9or%C3%A8me_de_Cantor-Bernstein" title="Théorème de Cantor-Bernstein">théorème de Cantor-Bernstein</a>). </p><p>Si deux <a href="/wiki/Ensemble_fini" title="Ensemble fini">ensembles finis</a> sont équipotents alors ils ont le même nombre d'éléments. L'extension de cette équivalence aux <a href="/wiki/Ensemble_infini" title="Ensemble infini">ensembles <i>infinis</i></a> a mené à la notion de <a href="/wiki/Nombre_cardinal" title="Nombre cardinal">cardinal</a> d'un ensemble, et à distinguer différentes tailles d'ensembles infinis, qui sont des classes d'équipotence. Ainsi, on peut par exemple montrer que l'ensemble des <a href="/wiki/Entier_naturel" title="Entier naturel">entiers naturels</a> est de même taille que l'ensemble des <a href="/wiki/Nombre_rationnel" title="Nombre rationnel">rationnels</a>, mais de taille strictement inférieure à l'ensemble des <a href="/wiki/Nombre_r%C3%A9el" title="Nombre réel">réels</a>. En effet, 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 \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span> dans <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>,</span> il existe des injections mais pas de surjection. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Définitions_formelles"><span id="D.C3.A9finitions_formelles"></span>Définitions formelles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&veaction=edit&section=1" title="Modifier la section : Définitions formelles" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bijection&action=edit&section=1" title="Modifier le code source de la section : Définitions formelles"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Définition_fonctionnelle"><span id="D.C3.A9finition_fonctionnelle"></span>Définition fonctionnelle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&veaction=edit&section=2" title="Modifier la section : Définition fonctionnelle" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bijection&action=edit&section=2" title="Modifier le code source de la section : Définition fonctionnelle"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Une <a href="/wiki/Application_(math%C3%A9matiques)" title="Application (mathématiques)">application</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 f:E\to F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:E\to F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ed16e79049ec4d5edcc1333c82255cd50904d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.346ex; height:2.509ex;" alt="{\displaystyle f:E\to F}"></span> est bijective si tout élément de l'ensemble d'arrivée <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> a exactement un <a href="/wiki/Ant%C3%A9c%C3%A9dent_(math%C3%A9matiques)" title="Antécédent (mathématiques)">antécédent</a> (dans <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>) par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, ce qui s'écrit formellement : </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 y\in F,\ \exists !\ x\in E,\quad f(x)=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∃</mi> <mo>!</mo> <mtext> </mtext> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mspace width="1em" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall y\in F,\ \exists !\ x\in E,\quad f(x)=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135d0dff7e59cd2f7c343d5492cba52282df8dc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.138ex; height:2.843ex;" alt="{\displaystyle \forall y\in F,\ \exists !\ x\in E,\quad f(x)=y}"></span></dd></dl> <p>ou, ce qui est équivalent, s'il existe une application <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:F\to E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>F</mi> <mo stretchy="false">→</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:F\to E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9b38691a1a314d4e27332cfc7f79cfb59089e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.184ex; height:2.509ex;" alt="{\displaystyle g:F\to E}"></span> qui, <a href="/wiki/Composition_de_fonctions" title="Composition de fonctions">composée</a> à gauche ou à droite par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, donne l'<a href="/wiki/Application_identit%C3%A9" title="Application identité">application identité</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 g\circ f=\operatorname {id} _{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f=\operatorname {id} _{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d45e053348675a087b40c0ee7b8023a4603fb272" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.115ex; height:2.509ex;" alt="{\displaystyle g\circ f=\operatorname {id} _{E}}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g=\operatorname {id} _{F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>=</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g=\operatorname {id} _{F}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/371b4818e9b73ae6e746b646cb192e81d63a1658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.09ex; height:2.509ex;" alt="{\displaystyle f\circ g=\operatorname {id} _{F}}"></span>,</dd></dl> <p>c'est-à-dire: </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 x\in E,\ \forall y\in F,\quad f(x)=y\Longleftrightarrow g(y)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> <mspace width="1em" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mo stretchy="false">⟺</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in E,\ \forall y\in F,\quad f(x)=y\Longleftrightarrow g(y)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b39e846f09dd03d8293c20421238818c495a23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.026ex; height:2.843ex;" alt="{\displaystyle \forall x\in E,\ \forall y\in F,\quad f(x)=y\Longleftrightarrow g(y)=x}"></span>.</dd></dl> <p>Une telle application <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> est alors déterminée de manière unique par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>. On l'appelle la <i><a href="/wiki/Bijection_r%C3%A9ciproque" title="Bijection réciproque">bijection réciproque</a></i> 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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> et on la note <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}"></span>. C'est aussi une bijection, et sa réciproque est <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Définition_relationnelle"><span id="D.C3.A9finition_relationnelle"></span>Définition relationnelle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&veaction=edit&section=3" title="Modifier la section : Définition relationnelle" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bijection&action=edit&section=3" title="Modifier le code source de la section : Définition relationnelle"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Une bijection 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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> dans <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> est une <a href="/wiki/Relation_binaire#Définition_formelle" title="Relation binaire">relation binaire <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> dans <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span></a> qui est une application et dont la <a href="/wiki/Relation_binaire#Réciproque" title="Relation binaire">relation réciproque</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 R^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91122db9e35b74f39c755e207749062fffa55e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.097ex; height:2.676ex;" alt="{\displaystyle R^{-1}}"></span> est aussi une application. De façon plus détaillée, <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> doit posséder les quatre propriétés suivantes : </p> <ul><li><i>Fonctionnalité</i> : tout élément 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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> a au plus une image par <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>, c'est-à-dire</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in E,\ \forall y,y'\in F,\quad [(xRy{\text{ et }}xRy')\Rightarrow y=y']}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>∈</mo> <mi>F</mi> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>R</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> et </mtext> </mrow> <mi>x</mi> <mi>R</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mi>y</mi> <mo>=</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in E,\ \forall y,y'\in F,\quad [(xRy{\text{ et }}xRy')\Rightarrow y=y']}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/597d77a6d942bb44ecaba2f3344c7fc457682322" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.221ex; height:3.009ex;" alt="{\displaystyle \forall x\in E,\ \forall y,y'\in F,\quad [(xRy{\text{ et }}xRy')\Rightarrow y=y']}"></span> ;</dd></dl></dd></dl> <ul><li><i>Applicativité</i> : tout élément 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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> a au moins une image par <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>, c'est-à-dire</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in E,\ \exists y\in F,\quad xRy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∃</mi> <mi>y</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mi>R</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in E,\ \exists y\in F,\quad xRy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4d7873d6c8b6ff374d32f359790d7a2f0e79e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.488ex; height:2.509ex;" alt="{\displaystyle \forall x\in E,\ \exists y\in F,\quad xRy}"></span> ;</dd></dl></dd></dl> <ul><li><i>Injectivité</i> : tout élément 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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> a au plus un antécédent par <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>, c'est-à-dire</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,x'\in E,\ \forall y\in F,\quad [(xRy{\text{ et }}x'Ry)\Rightarrow x=x']}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>R</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> et </mtext> </mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>R</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,x'\in E,\ \forall y\in F,\quad [(xRy{\text{ et }}x'Ry)\Rightarrow x=x']}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b07abad238e4224379b2c3168c01d9b118c5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.728ex; height:3.009ex;" alt="{\displaystyle \forall x,x'\in E,\ \forall y\in F,\quad [(xRy{\text{ et }}x'Ry)\Rightarrow x=x']}"></span></dd></dl></dd></dl> <ul><li><i>Surjectivité</i> : tout élément 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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> a au moins un antécédent par <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>, c'est-à-dire</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall y\in F,\ \exists x\in E,\quad xRy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mi>R</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall y\in F,\ \exists x\in E,\quad xRy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92cfba910417adfeaf22b6ff4e15970976ab70d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.488ex; height:2.509ex;" alt="{\displaystyle \forall y\in F,\ \exists x\in E,\quad xRy}"></span>.</dd></dl></dd></dl> <p>L'injectivité 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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> équivaut à la fonctionnalité 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 R^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91122db9e35b74f39c755e207749062fffa55e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.097ex; height:2.676ex;" alt="{\displaystyle R^{-1}}"></span> et la surjectivité 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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> équivaut à l'applicativité 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 R^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91122db9e35b74f39c755e207749062fffa55e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.097ex; height:2.676ex;" alt="{\displaystyle R^{-1}}"></span>. </p><p>Il est usuel de représenter une <a href="/wiki/Relation_binaire#Relation_fonctionnelle" title="Relation binaire">relation binaire <i>fonctionnelle</i></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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> par une <i>fonction</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> en posant </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 x\in E,\ \forall y\in F,\quad f(x)=y\Longleftrightarrow xRy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> <mspace width="1em" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mo stretchy="false">⟺</mo> <mi>x</mi> <mi>R</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in E,\ \forall y\in F,\quad f(x)=y\Longleftrightarrow xRy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12cc95dd5bc8a3cca615a019e792a62a19180e8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.767ex; height:2.843ex;" alt="{\displaystyle \forall x\in E,\ \forall y\in F,\quad f(x)=y\Longleftrightarrow xRy}"></span>.</dd></dl> <p>Si l'on précise que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> est une <i>application</i>, on suppose que <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> est fonctionnelle <i>et applicative</i> (voir <a href="/wiki/Application_(math%C3%A9matiques)#Fonction_et_application" title="Application (mathématiques)">Application_(mathématiques)#Fonction_et_application</a> pour les différences entre <i>application</i> et <i>fonction</i>, qui peuvent varier selon les auteurs). </p><p>La symétrie entre fonctionnalité et injectivité d'une part, et entre applicativité et surjectivité d'autre part, donne que si <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> est une relation bijective alors <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 R^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91122db9e35b74f39c755e207749062fffa55e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.097ex; height:2.676ex;" alt="{\displaystyle R^{-1}}"></span> l'est aussi. </p> <div class="mw-heading mw-heading2"><h2 id="Exemple_concret">Exemple concret</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&veaction=edit&section=4" title="Modifier la section : Exemple concret" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bijection&action=edit&section=4" title="Modifier le code source de la section : Exemple concret"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Prenons le cas d'une station de vacances où un groupe de touristes doit être logé dans un hôtel. Chaque façon de répartir ces touristes dans les chambres de l'hôtel peut être représentée par une application de l'ensemble <i>X</i> des touristes vers l'ensemble <i>Y</i> des chambres (à chaque touriste est associée une chambre). </p> <ul><li>L'hôtelier souhaite que l'application soit <b>surjective</b>, c'est-à-dire que <i>chaque chambre soit occupée</i>. Cela n'est possible que s'il y a au moins autant de touristes que de chambres.</li> <li>Les touristes souhaitent que l'application soit <b>injective</b>, c'est-à-dire que <i>chacun d'entre eux ait une chambre individuelle</i>. Cela n'est possible que si le nombre de touristes ne dépasse pas le nombre de chambres.</li> <li>Ces souhaits sont incompatibles si le nombre de touristes est différent du nombre de chambres. Dans le cas contraire, il sera possible de répartir les touristes de telle sorte qu'il y en ait un seul par chambre, et que toutes les chambres soient occupées : on dira alors que l'application est à la fois injective et surjective ; elle est <b>bijective</b>.</li></ul> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fichier:Surjection_Injection_Bijection-fr.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Surjection_Injection_Bijection-fr.svg/700px-Surjection_Injection_Bijection-fr.svg.png" decoding="async" width="700" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Surjection_Injection_Bijection-fr.svg/1050px-Surjection_Injection_Bijection-fr.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Surjection_Injection_Bijection-fr.svg/1400px-Surjection_Injection_Bijection-fr.svg.png 2x" data-file-width="700" data-file-height="250" /></a></span> </p> <div class="mw-heading mw-heading2"><h2 id="Exemples_et_contre-exemples">Exemples et contre-exemples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&veaction=edit&section=5" title="Modifier la section : Exemples et contre-exemples" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bijection&action=edit&section=5" title="Modifier le code source de la section : Exemples et contre-exemples"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>La <a href="/wiki/Fonction_affine" title="Fonction affine">fonction affine</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 f:\mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3a10a3ad05781f5cf9c2d875a02227e21a8448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }"></span> définie par <span class="texhtml"><i>f</i>(<i>x</i>) = 2<i>x</i> + 1</span> est bijective, puisque pour tout réel <span class="texhtml"><i>y</i></span>, il existe exactement une solution réelle de l’équation <span class="texhtml"><i>y</i> = 2<i>x</i> + 1</span> d'inconnue <span class="texhtml"><i>x</i></span>, à savoir : <span class="texhtml"><i>x</i> = (<i>y</i> − 1)/2</span>.</li> <li>La <a href="/wiki/Fonction_carr%C3%A9" title="Fonction carré">fonction carré</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 g:\mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdfd1e16b7f932cdc2716a1b6bbe345089b250cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.023ex; height:2.509ex;" alt="{\displaystyle g:\mathbb {R} \to \mathbb {R} }"></span> définie par <span class="texhtml"><i>g</i>(<i>x</i>) = <i>x</i><sup>2</sup></span> n’est <i>pas</i> bijective, pour deux raisons. La première est que l'on a (par exemple) <span class="texhtml"><i>g</i>(1) = 1 = <i>g</i>(−1)</span>, et donc <span class="texhtml"><i>g</i></span> n’est pas injective ; la seconde est qu'il n'y a (par exemple) aucun réel <span class="texhtml"><i>x</i></span> tel que <span class="texhtml"><i>x</i><sup>2</sup> = −1</span>, et donc <span class="texhtml"><i>g</i></span> n’est pas surjective non plus. L'une ou l'autre de ces constatations est suffisante pour affirmer que <span class="texhtml"><i>g</i></span> n'est pas bijective.<br />En revanche, l'application <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{+}\to \mathbb {R} _{+},\,x\mapsto x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{+}\to \mathbb {R} _{+},\,x\mapsto x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/228f0756937cd3f30a040cea6d4cb8fb9a12b145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.741ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} _{+}\to \mathbb {R} _{+},\,x\mapsto x^{2}}"></span> <i>est</i> bijective. L'explication est que pour tout réel positif <span class="texhtml"><i>y</i></span>, il existe exactement une solution réelle positive de l’équation <span class="texhtml"><i>y = x</i><sup>2</sup></span>, qui est <span class="texhtml"><i>x</i> = <span class="racine texhtml">√<span style="border-top:1px solid; padding:0 0.1em;"><i>y</i></span></span></span>. La fonction <a href="/wiki/Racine_carr%C3%A9e" title="Racine carrée">racine carrée</a> est donc la bijection réciproque de la fonction carré sur ces ensembles.</li> <li>De même, la fonction <a href="/wiki/Sinus_(math%C3%A9matiques)" title="Sinus (mathématiques)">sinus</a>, vue comme une application 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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> dans <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>, n'est ni injective, ni surjective, donc pas bijective ; <ul><li>sa corestriction <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 \sin :\mathbb {R} \to [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin :\mathbb {R} \to [-1,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da9c9bd73222a87d38204e3b32bdaa9c1f53d448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.546ex; height:2.843ex;" alt="{\displaystyle \sin :\mathbb {R} \to [-1,1]}"></span> est surjective mais pas injective (par exemple, <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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> et <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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> ont la même image) donc pas bijective ;</li> <li>sa restriction <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 \sin :[-{\pi /2},{\pi /2}]\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin :[-{\pi /2},{\pi /2}]\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4ee1a9318856fa53831b82e7d6c9ed710c0f2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.535ex; height:2.843ex;" alt="{\displaystyle \sin :[-{\pi /2},{\pi /2}]\to \mathbb {R} }"></span> est injective mais pas surjective (par exemple, <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> n'est l'image d'aucune valeur) donc pas bijective ;</li> <li>sa restriction-corestriction <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 \sin :[-{\pi /2},{\pi /2}]\to [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin :[-{\pi /2},{\pi /2}]\to [-1,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eae30493906c9f715c30e3cb8b6b293e29e3fbe3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.317ex; height:2.843ex;" alt="{\displaystyle \sin :[-{\pi /2},{\pi /2}]\to [-1,1]}"></span> est bijective (comme aussi une infinité d'autres de ses restrictions-corestrictions) ;</li> <li>sa bijection réciproque est alors <span class="texhtml"><a href="/wiki/Arc_sinus" title="Arc sinus">arcsin</a></span> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]\to [-{\pi /2},{\pi /2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1]\to [-{\pi /2},{\pi /2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a004cc1542da55cddcbd85ce79df866a2b2fe77f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.524ex; height:2.843ex;" alt="{\displaystyle [-1,1]\to [-{\pi /2},{\pi /2}]}"></span> ;</li> <li>cependant, la fonction arc sinus prenant les mêmes valeurs, mais vue comme une application 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 [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}"></span> dans <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>, est injective mais pas surjective (par exemple, <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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> n'est l'image d'aucune valeur) donc pas bijective.</li></ul></li> <li>La <a href="/wiki/Sigmo%C3%AFde_(math%C3%A9matiques)" title="Sigmoïde (mathématiques)">fonction sigmoïde</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 f:\mathbb {R} \to ]0,1[}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to ]0,1[}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999553802cb6200b672dc3021eb30e2f0ad1a08d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.515ex; height:2.843ex;" alt="{\displaystyle f:\mathbb {R} \to ]0,1[}"></span> définie par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{1+e^{-x}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{1+e^{-x}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faaa0c014ae28ac67db5c49b3f3e8b08415a3f2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.89ex; height:5.509ex;" alt="{\displaystyle f(x)={\frac {1}{1+e^{-x}}}}"></span> est bijective et est souvent utilisée en informatique, notamment dans les <a href="/wiki/R%C3%A9seau_de_neurones_artificiels" title="Réseau de neurones artificiels">réseaux de neurones</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Propriétés"><span id="Propri.C3.A9t.C3.A9s"></span>Propriétés</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&veaction=edit&section=6" title="Modifier la section : Propriétés" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bijection&action=edit&section=6" title="Modifier le code source de la section : Propriétés"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Les bijections sont les <a href="/wiki/Isomorphisme" title="Isomorphisme">isomorphismes</a> dans la <a href="/wiki/Cat%C3%A9gorie_des_ensembles" title="Catégorie des ensembles">catégorie des ensembles</a>.</li> <li>Soient <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:E\to F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:E\to F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ed16e79049ec4d5edcc1333c82255cd50904d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.346ex; height:2.509ex;" alt="{\displaystyle f:E\to F}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:F\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>:</mo> <mi>F</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h:F\to G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23b34137705bb37b633e8d876d8205f374287d8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.458ex; height:2.176ex;" alt="{\displaystyle h:F\to G}"></span>. <ul><li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> sont bijectives alors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\circ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b79b5100d10fab6e101d1a6de09debb03f0158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.812ex; height:2.509ex;" alt="{\displaystyle h\circ f}"></span> est bijective et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h\circ f)^{-1}=f^{-1}\circ h^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo>∘</mo> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∘</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h\circ f)^{-1}=f^{-1}\circ h^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abb766b9ca9d3397b9ec24c03fb5826a03433b98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.573ex; height:3.176ex;" alt="{\displaystyle (h\circ f)^{-1}=f^{-1}\circ h^{-1}}"></span>.</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\circ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b79b5100d10fab6e101d1a6de09debb03f0158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.812ex; height:2.509ex;" alt="{\displaystyle h\circ f}"></span> est bijective alors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> est injective et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> est surjective.</li></ul></li> <li>Pour tout ensemble <i>E</i>, les bijections de <i>E</i> sur lui-même s'appellent les <a href="/wiki/Permutation" title="Permutation">permutations</a> de <i>E</i>. Elles forment, avec l’opération ∘ de composition des applications, un <a href="/wiki/Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques)">groupe</a> appelé le <a href="/wiki/Groupe_sym%C3%A9trique" title="Groupe symétrique">groupe symétrique</a> de <i>E</i> et noté <i>S</i>(<i>E</i>) ou <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 {\mathfrak {S}}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {S}}(E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088c2f5dbcb92981f3ff0b0adfeb38990f033b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.512ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {S}}(E)}"></span>.</li> <li>Le nombre de bijections entre deux ensembles finis de même <a href="/wiki/Cardinal_d%27un_ensemble_fini" class="mw-redirect" title="Cardinal d'un ensemble fini">cardinal</a> <i>n</i> est <a href="/wiki/Factorielle" title="Factorielle"><i>n</i>!</a>.</li> <li>Une application de ℝ dans ℝ est bijective si et seulement si son <a href="/wiki/Graphe_d%27une_fonction" title="Graphe d'une fonction">graphe</a> intersecte toute droite horizontale en exactement un point.</li> <li>Pour qu'une application d'un ensemble <i>fini</i> dans lui-même soit bijective, il suffit qu'elle soit injective <i>ou </i>surjective (elle est alors les deux). On peut le voir comme une application du <a href="/wiki/Principe_des_tiroirs" title="Principe des tiroirs">principe des tiroirs</a>. <dl><dd>NB : il peut exister une bijection entre deux ensembles infinis dont l'un est strictement inclus dans l'autre. On en trouve de <a href="/wiki/Ensemble_d%C3%A9nombrable#Quelques_exemples_d'ensembles_dénombrables" title="Ensemble dénombrable">nombreux exemples dans le cas dénombrable</a>.</dd></dl></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&veaction=edit&section=7" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bijection&action=edit&section=7" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&veaction=edit&section=8" 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=Bijection&action=edit&section=8" 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-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text">C'est-à-dire est <i><a href="/wiki/Image_(math%C3%A9matiques)" title="Image (mathématiques)">image</a></i> d'exactement un élément de son <a href="/wiki/Domaine_de_d%C3%A9finition" class="mw-redirect" title="Domaine de définition">domaine de définition</a>.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading3"><h3 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&veaction=edit&section=9" 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=Bijection&action=edit&section=9" title="Modifier le code source de la section : Références"><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">Dans <span class="ouvrage" id="Bourbaki"><span class="ouvrage" id="N._Bourbaki"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">N. <span class="nom_auteur">Bourbaki</span></a>, <cite class="italique"><a href="/wiki/%C3%89l%C3%A9ments_de_math%C3%A9matique" title="Éléments de mathématique">Éléments de mathématique</a> : Théorie des ensembles</cite> <small>[<a href="/wiki/R%C3%A9f%C3%A9rence:Th%C3%A9orie_des_ensembles_(Bourbaki)" title="Référence:Théorie des ensembles (Bourbaki)">détail des éditions</a>]</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%89l%C3%A9ments+de+math%C3%A9matique&rft.stitle=Th%C3%A9orie+des+ensembles&rft.aulast=Bourbaki&rft.aufirst=N.&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ABijection"></span></span></span> (édition de 1970 ou <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=VDGifaOQogcC&pg=SL252-PA17">de 2006</a>), ch. II, § 3, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 7, après la déf. 10, p. II. 17, on lit : <span class="citation">« Au lieu de dire que <i>f</i> est injective, on dit aussi que <i>f</i> est <i>biunivoque</i>. […] Si <i>f</i> [application de <i>A</i> dans <i>B</i>] est bijective, on dit aussi que <i>f met</i> A <i>et</i> B <i>en correspondance biunivoque</i>. »</span> Mais dans le « fascicule de résultats », à la fin du même volume, p. E.R.9, « biunivoque » n'est employé que dans le second sens.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Article_connexe">Article connexe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&veaction=edit&section=10" title="Modifier la section : Article connexe" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bijection&action=edit&section=10" title="Modifier le code source de la section : Article connexe"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Th%C3%A9or%C3%A8me_de_la_bijection" title="Théorème de la bijection">Théorème de la bijection</a> </p> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail des mathématiques"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/24px-Racine_carr%C3%A9e_bleue.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/36px-Racine_carr%C3%A9e_bleue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/48px-Racine_carr%C3%A9e_bleue.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail:Mathématiques">Portail des mathématiques</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Ce document provient de « <a dir="ltr" href="https://fr.wikipedia.org/w/index.php?title=Bijection&oldid=212827581">https://fr.wikipedia.org/w/index.php?title=Bijection&oldid=212827581</a> ».</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Cat%C3%A9gorie:Accueil" title="Catégorie:Accueil">Catégories</a> : <ul><li><a href="/wiki/Cat%C3%A9gorie:Th%C3%A9orie_des_ensembles" title="Catégorie:Théorie des ensembles">Théorie des ensembles</a></li><li><a href="/wiki/Cat%C3%A9gorie:Propri%C3%A9t%C3%A9_de_fonction" title="Catégorie:Propriété de fonction">Propriété de fonction</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Catégories cachées : <ul><li><a href="/wiki/Cat%C3%A9gorie:Portail:Math%C3%A9matiques/Articles_li%C3%A9s" title="Catégorie:Portail:Mathématiques/Articles liés">Portail:Mathématiques/Articles liés</a></li><li><a href="/wiki/Cat%C3%A9gorie:Portail:Sciences/Articles_li%C3%A9s" title="Catégorie:Portail:Sciences/Articles liés">Portail:Sciences/Articles liés</a></li><li><a href="/wiki/Cat%C3%A9gorie:Projet:Math%C3%A9matiques/Articles" title="Catégorie:Projet:Mathématiques/Articles">Projet:Mathématiques/Articles</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> La dernière modification de cette page a été faite le 26 février 2024 à 17:15.</li> <li id="footer-info-copyright"><span style="white-space: normal"><a href="/wiki/Wikip%C3%A9dia:Citation_et_r%C3%A9utilisation_du_contenu_de_Wikip%C3%A9dia" title="Wikipédia:Citation et réutilisation du contenu de Wikipédia">Droit d'auteur</a> : les textes sont disponibles sous <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.fr">licence Creative Commons attribution, partage dans les mêmes conditions</a> ; d’autres conditions peuvent s’appliquer. 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