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class="vector-toc-numb">2.2</span> <span>Seats and students of a classroom</span> </div> </a> <ul id="toc-Seats_and_students_of_a_classroom-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-More_mathematical_examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#More_mathematical_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>More mathematical examples</span> </div> </a> <ul id="toc-More_mathematical_examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverses" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Inverses"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Inverses</span> </div> </a> <ul id="toc-Inverses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Composition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Composition"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Composition</span> </div> </a> <ul id="toc-Composition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cardinality" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cardinality"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Cardinality</span> </div> </a> <ul id="toc-Cardinality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Category_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Category_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Category theory</span> </div> </a> <ul id="toc-Category_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalization_to_partial_functions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalization_to_partial_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Generalization to partial functions</span> </div> </a> <ul id="toc-Generalization_to_partial_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gallery" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gallery"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Gallery</span> </div> </a> <ul id="toc-Gallery-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">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="Go to an article in another language. Available in 55 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-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 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%82%D8%A7%D8%A8%D9%84_(%D8%AF%D8%A7%D9%84%D8%A9)" title="تقابل (دالة) – Arabic" lang="ar" hreflang="ar" data-title="تقابل (دالة)" data-language-autonym="العربية" data-language-local-name="Arabic" 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 – Belarusian" lang="be" hreflang="be" data-title="Біекцыя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D0%B8%D0%B5%D0%BA%D1%86%D0%B8%D1%8F" title="Биекция – Bulgarian" lang="bg" hreflang="bg" data-title="Биекция" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Bijekcija" title="Bijekcija – Bosnian" lang="bs" hreflang="bs" data-title="Bijekcija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/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-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bijekce" title="Bijekce – Czech" lang="cs" hreflang="cs" data-title="Bijekce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Biiezzioni" title="Biiezzioni – Corsican" lang="co" hreflang="co" data-title="Biiezzioni" data-language-autonym="Corsu" data-language-local-name="Corsican" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bijektiv" title="Bijektiv – Danish" lang="da" hreflang="da" data-title="Bijektiv" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bijektive_Funktion" title="Bijektive Funktion – German" lang="de" hreflang="de" data-title="Bijektive Funktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Bijektiivne_funktsioon" title="Bijektiivne funktsioon – Estonian" lang="et" hreflang="et" data-title="Bijektiivne funktsioon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</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 – Spanish" lang="es" hreflang="es" data-title="Función biyectiva" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Dissur%C4%B5eto" title="Dissurĵeto – Esperanto" lang="eo" hreflang="eo" data-title="Dissurĵeto" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/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 – Persian" lang="fa" hreflang="fa" data-title="تناظر دوسویه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Bijection" title="Bijection – French" lang="fr" hreflang="fr" data-title="Bijection" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</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 – Galician" lang="gl" hreflang="gl" data-title="Función bixectiva" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%84%EB%8B%A8%EC%82%AC_%ED%95%A8%EC%88%98" title="전단사 함수 – Korean" lang="ko" hreflang="ko" data-title="전단사 함수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%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 – Armenian" lang="hy" hreflang="hy" data-title="Փոխմիարժեք համապատասխանություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</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 – Croatian" lang="hr" hreflang="hr" data-title="Bijekcija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bijeksi" title="Bijeksi – Indonesian" lang="id" hreflang="id" data-title="Bijeksi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Gagnt%C3%A6k_v%C3%B6rpun" title="Gagntæk vörpun – Icelandic" lang="is" hreflang="is" data-title="Gagntæk vörpun" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Corrispondenza_biunivoca" title="Corrispondenza biunivoca – Italian" lang="it" hreflang="it" data-title="Corrispondenza biunivoca" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%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 – Hebrew" lang="he" hreflang="he" data-title="פונקציה חד-חד-ערכית ועל" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%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-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-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Bijekcija" title="Bijekcija – Lithuanian" lang="lt" hreflang="lt" data-title="Bijekcija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</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-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bijekci%C3%B3" title="Bijekció – Hungarian" lang="hu" hreflang="hu" data-title="Bijekció" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%B8%D1%98%D0%B5%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Бијекција – Macedonian" lang="mk" hreflang="mk" data-title="Бијекција" data-language-autonym="Македонски" data-language-local-name="Macedonian" 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 – Dutch" lang="nl" hreflang="nl" data-title="Bijectie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%85%A8%E5%8D%98%E5%B0%84" title="全単射 – Japanese" lang="ja" hreflang="ja" data-title="全単射" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bijeksjon" title="Bijeksjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Bijeksjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Bijeksjon" title="Bijeksjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Bijeksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/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 – Polish" lang="pl" hreflang="pl" data-title="Funkcja wzajemnie jednoznaczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_bijectiva" title="Função bijectiva – Portuguese" lang="pt" hreflang="pt" data-title="Função bijectiva" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Coresponden%C8%9B%C4%83_biunivoc%C4%83" title="Corespondență biunivocă – Romanian" lang="ro" hreflang="ro" data-title="Corespondență biunivocă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B8%D0%B5%D0%BA%D1%86%D0%B8%D1%8F" title="Биекция – Russian" lang="ru" hreflang="ru" data-title="Биекция" data-language-autonym="Русский" data-language-local-name="Russian" 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 – Scots" lang="sco" hreflang="sco" data-title="Bijection" data-language-autonym="Scots" data-language-local-name="Scots" 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 – Slovak" lang="sk" hreflang="sk" data-title="Bijektívne zobrazenie" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Bijektivna_preslikava" title="Bijektivna preslikava – Slovenian" lang="sl" hreflang="sl" data-title="Bijektivna preslikava" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" 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 – Serbian" lang="sr" hreflang="sr" data-title="Бијекција" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bijektio" title="Bijektio – Finnish" lang="fi" hreflang="fi" data-title="Bijektio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bijektiv_funktion" title="Bijektiv funktion – Swedish" lang="sv" hreflang="sv" data-title="Bijektiv funktion" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%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 – Tamil" lang="ta" hreflang="ta" data-title="இருவழிக்கோப்பு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%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 – Thai" lang="th" hreflang="th" data-title="ฟังก์ชันหนึ่งต่อหนึ่งทั่วถึง" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Birebir_%C3%B6rten_fonksiyon" title="Birebir örten fonksiyon – Turkish" lang="tr" hreflang="tr" data-title="Birebir örten fonksiyon" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li 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</nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <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">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</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">From Wikipedia, the free encyclopedia</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="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">One-to-one correspondence</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bijection.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/220px-Bijection.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/330px-Bijection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/440px-Bijection.svg.png 2x" data-file-width="200" data-file-height="200" /></a><figcaption>A bijective function, <i>f</i>: <i>X</i> → <i>Y</i>, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, <i>f</i>(1) = D.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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.sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="letter-spacing:0.0125em; background-color:#FFCC99"><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a></th></tr><tr><td class="sidebar-image"><span class="texhtml texhtml-big" style="font-size:250%;"><i>x</i> ↦ <i>f</i> (<i>x</i>)</span></td></tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px"> <a href="/wiki/History_of_the_function_concept" title="History of the function concept">History of the function concept</a></th></tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px"> Types by <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> and <a href="/wiki/Codomain" title="Codomain">codomain</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Boolean-valued_function" title="Boolean-valued function"><span class="texhtml"><span title="arbitrary set"><var>X</var></span> → <span title="Codomain of Booleans">𝔹</span></span></a></li> <li><a href="/wiki/Ordered_pair" title="Ordered pair"> <span class="texhtml"><span title="Domain of Booleans">𝔹</span> → <span title="arbitrary set"><var>X</var></span></span></a></li> <li><a href="/wiki/Boolean_function" title="Boolean function"> <span class="texhtml"><span title="several Boolean variables">𝔹<sup><var>n</var></sup></span> → <span title="Codomain of natural numbers"><var>X</var></span></span></a></li> <li><a href="/wiki/Integer-valued_function" title="Integer-valued function"> <span class="texhtml"><span title="arbitrary set"><var>X</var></span> → <span title="integers">ℤ</span></span></a></li> <li><a href="/wiki/Sequence" title="Sequence"> <span class="texhtml"><span title="integers">ℤ</span> → <span title="arbitrary set"><var>X</var></span></span></a></li> <li><a href="/wiki/Real-valued_function" title="Real-valued function"> <span class="texhtml"><span title="arbitrary set"><var>X</var></span> → <span title="real numbers">ℝ</span></span></a></li> <li><a href="/wiki/Function_of_a_real_variable" title="Function of a real variable"> <span class="texhtml"><span title="real numbers">ℝ</span> → <span title="arbitrary set"><var>X</var></span></span></a></li> <li><a href="/wiki/Function_of_several_real_variables" title="Function of several real variables"> <span class="texhtml"><span title="real coordinate (or Euclidean) space">ℝ<sup><var>n</var></sup></span> → <span title="arbitrary set"><var>X</var></span></span></a></li> <li><a href="/wiki/Complex-valued_function" class="mw-redirect" title="Complex-valued function"> <span class="texhtml"><span title="arbitrary set"><var>X</var></span> → <span title="complex numbers">ℂ</span></span></a></li> <li><a href="/wiki/Function_of_a_complex_variable" class="mw-redirect" title="Function of a complex variable"> <span class="texhtml"><span title="complex numbers">ℂ</span> → <span title="arbitrary set"><var>X</var></span></span></a></li> <li><a href="/wiki/Function_of_several_complex_variables" title="Function of several complex variables"> <span class="texhtml"><span title="complex coordinate space">ℂ<sup><var>n</var></sup></span> → <span title="arbitrary set"><var>X</var></span></span></a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px"> <a href="/wiki/List_of_types_of_functions" title="List of types of functions">Classes/properties</a> </th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Constant_function" title="Constant function">Constant</a></li> <li><a href="/wiki/Identity_function" title="Identity function">Identity</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear</a></li> <li><a href="/wiki/Polynomial" title="Polynomial">Polynomial</a></li> <li><a href="/wiki/Rational_function" title="Rational function">Rational</a></li> <li><a href="/wiki/Algebraic_function" title="Algebraic function">Algebraic</a></li> <li><a href="/wiki/Analytic_function" title="Analytic function">Analytic</a></li> <li><a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuous</a></li> <li><a href="/wiki/Measurable_function" title="Measurable function">Measurable</a></li> <li><a href="/wiki/Injective_function" title="Injective function">Injective</a></li> <li><a href="/wiki/Surjective_function" title="Surjective function">Surjective</a></li> <li><a class="mw-selflink selflink">Bijective</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px"> Constructions</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li> <li><a href="/wiki/Function_composition" title="Function composition">Composition</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">λ</a></li> <li><a href="/wiki/Inverse_function" title="Inverse function">Inverse</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px"> Generalizations </th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a>)</li> <li><a href="/wiki/Set-valued_function" title="Set-valued function">Set-valued</a></li> <li><a href="/wiki/Multivalued_function" title="Multivalued function">Multivalued</a></li> <li><a href="/wiki/Partial_function" title="Partial function">Partial</a></li> <li><a href="/wiki/Implicit_function" title="Implicit function">Implicit</a></li> <li><a href="/wiki/Function_space" title="Function space">Space</a></li> <li><a href="/wiki/Higher-order_function" title="Higher-order function">Higher-order</a></li> <li><a href="/wiki/Morphism" title="Morphism">Morphism</a></li> <li><a href="/wiki/Functor" title="Functor">Functor</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px"> <a href="/wiki/List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></th></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Functions" title="Template:Functions"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Functions" title="Template talk:Functions"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functions" title="Special:EditPage/Template:Functions"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>A <b>bijection</b>, <b>bijective function</b>, or <b>one-to-one correspondence</b> between two mathematical <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> such that each element of the second set (the <a href="/wiki/Codomain" title="Codomain">codomain</a>) is the image of exactly one element of the first set (the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a>). Equivalently, a bijection is a <a href="/wiki/Binary_relation" title="Binary relation">relation</a> between two sets such that each element of either set is paired with exactly one element of the other set. </p><p>A function is bijective <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it is <a href="/wiki/Inverse_function" title="Inverse function">invertible</a>; that is, a function <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\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"></span> is bijective if and only if there is a function <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:Y\to X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:Y\to X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57be2f492983d382259f5e84d03152a29940605d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.067ex; height:2.509ex;" alt="{\displaystyle g:Y\to X,}"></span> the <i>inverse</i> of <span class="texhtml mvar" style="font-style:italic;">f</span>, such that each of the two ways for <a href="/wiki/Function_composition" title="Function composition">composing</a> the two functions produces an <a href="/wiki/Identity_function" title="Identity function">identity function</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(f(x))=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(f(x))=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d12c5b0b3b9b020bfc9f5e330c074b809cc54ebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.771ex; height:2.843ex;" alt="{\displaystyle g(f(x))=x}"></span> for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g(y))=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g(y))=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e774231d71a255aa573dfbb55f22e4d9137bbeb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.423ex; height:2.843ex;" alt="{\displaystyle f(g(y))=y}"></span> for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}"></span> </p><p>For example, the <i>multiplication by two</i> defines a bijection from the <a href="/wiki/Integer" title="Integer">integers</a> to the <a href="/wiki/Even_number" class="mw-redirect" title="Even number">even numbers</a>, which has the <i>division by two</i> as its inverse function. </p><p>A function is bijective if and only if it is both <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a> (or <i>one-to-one</i>)—meaning that each element in the codomain is mapped from at most one element of the domain—and <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> (or <i>onto</i>)—meaning that each element of the codomain is mapped from at least one element of the domain. The term <i>one-to-one correspondence</i> must not be confused with <i><a href="/wiki/One-to-one_function" class="mw-redirect" title="One-to-one function">one-to-one function</a></i>, which means injective but not necessarily surjective. </p><p>The elementary operation of <a href="/wiki/Counting" title="Counting">counting</a> establishes a bijection from some <a href="/wiki/Finite_set" title="Finite set">finite set</a> to the first <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> <span class="texhtml">(1, 2, 3, ...)</span>, up to the number of elements in the counted set. It results that two finite sets have the same number of elements if and only if there exists a bijection between them. More generally, two sets are said to have the same <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a> if there exists a bijection between them. </p><p>A bijective function from a set to itself is also called a <a href="/wiki/Permutation" title="Permutation">permutation</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and the set of all permutations of a set forms its <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a>. </p><p>Some bijections with further properties have received specific names, which include <a href="/wiki/Automorphism" title="Automorphism">automorphisms</a>, <a href="/wiki/Isomorphism" title="Isomorphism">isomorphisms</a>, <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphisms</a>, <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a>, <a href="/wiki/Permutation_group" title="Permutation group">permutation groups</a>, and most <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformations</a>. <a href="/wiki/Galois_correspondence" class="mw-redirect" title="Galois correspondence">Galois correspondences</a> are bijections between sets of <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> of apparently very different nature. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> pairing elements of set <i>X</i> with elements of set <i>Y</i> to be a bijection, four properties must hold: </p> <ol><li>each element of <i>X</i> must be paired with at least one element of <i>Y</i>,</li> <li>no element of <i>X</i> may be paired with more than one element of <i>Y</i>,</li> <li>each element of <i>Y</i> must be paired with at least one element of <i>X</i>, and</li> <li>no element of <i>Y</i> may be paired with more than one element of <i>X</i>.</li></ol> <p>Satisfying properties (1) and (2) means that a pairing is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> with <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> <i>X</i>. It is more common to see properties (1) and (2) written as a single statement: Every element of <i>X</i> is paired with exactly one element of <i>Y</i>. Functions which satisfy property (3) are said to be "<a href="/wiki/Onto" class="mw-redirect" title="Onto">onto</a> <i>Y</i> " and are called <a href="/wiki/Surjective_function" title="Surjective function">surjections</a> (or <i>surjective functions</i>). Functions which satisfy property (4) are said to be "<a href="/wiki/One-to-one_function" class="mw-redirect" title="One-to-one function">one-to-one functions</a>" and are called <a href="/wiki/Injective_function" title="Injective function">injections</a> (or <i>injective functions</i>).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto".<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Batting_line-up_of_a_baseball_or_cricket_team">Batting line-up of a baseball or cricket team</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=3" title="Edit section: Batting line-up of a baseball or cricket team"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the <a href="/wiki/Batting_order_(baseball)" title="Batting order (baseball)">batting line-up</a> of a baseball or <a href="/wiki/Cricket" title="Cricket">cricket</a> team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The set <i>X</i> will be the players on the team (of size nine in the case of baseball) and the set <i>Y</i> will be the positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in the same position in the list. </p> <div class="mw-heading mw-heading3"><h3 id="Seats_and_students_of_a_classroom">Seats and students of a classroom</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=4" title="Edit section: Seats and students of a classroom"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a classroom there are a certain number of seats. A group of students enter the room and the instructor asks them to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. What the instructor observed in order to reach this conclusion was that: </p> <ol><li>Every student was in a seat (there was no one standing),</li> <li>No student was in more than one seat,</li> <li>Every seat had someone sitting there (there were no empty seats), and</li> <li>No seat had more than one student in it.</li></ol> <p>The instructor was able to conclude that there were just as many seats as there were students, without having to count either set. </p> <div class="mw-heading mw-heading2"><h2 id="More_mathematical_examples">More mathematical examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=5" title="Edit section: More mathematical examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:A_bijection_from_the_natural_numbers_to_the_integers.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/A_bijection_from_the_natural_numbers_to_the_integers.png/220px-A_bijection_from_the_natural_numbers_to_the_integers.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/A_bijection_from_the_natural_numbers_to_the_integers.png/330px-A_bijection_from_the_natural_numbers_to_the_integers.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/A_bijection_from_the_natural_numbers_to_the_integers.png/440px-A_bijection_from_the_natural_numbers_to_the_integers.png 2x" data-file-width="566" data-file-height="565" /></a><figcaption>A bijection from the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> to the <a href="/wiki/Integer" title="Integer">integers</a>, which maps 2<i>n</i> to −<i>n</i> and 2<i>n</i> − 1 to <i>n</i>, for <i>n</i> ≥ 0.</figcaption></figure> <ul><li>For any set <i>X</i>, the <a href="/wiki/Identity_function" title="Identity function">identity function</a> <b>1</b><sub><i>X</i></sub>: <i>X</i> → <i>X</i>, <b>1</b><sub><i>X</i></sub>(<i>x</i>) = <i>x</i> is bijective.</li> <li>The function <i>f</i>: <b>R</b> → <b>R</b>, <i>f</i>(<i>x</i>) = 2<i>x</i> + 1 is bijective, since for each <i>y</i> there is a unique <i>x</i> = (<i>y</i> − 1)/2 such that <i>f</i>(<i>x</i>) = <i>y</i>. More generally, any <a href="/wiki/Linear_function" title="Linear function">linear function</a> over the reals, <i>f</i>: <b>R</b> → <b>R</b>, <i>f</i>(<i>x</i>) = <i>ax</i> + <i>b</i> (where <i>a</i> is non-zero) is a bijection. Each real number <i>y</i> is obtained from (or paired with) the real number <i>x</i> = (<i>y</i> − <i>b</i>)/<i>a</i>.</li> <li>The function <i>f</i>: <b>R</b> → (−π/2, π/2), given by <i>f</i>(<i>x</i>) = arctan(<i>x</i>) is bijective, since each real number <i>x</i> is paired with exactly one angle <i>y</i> in the interval (−π/2, π/2) so that tan(<i>y</i>) = <i>x</i> (that is, <i>y</i> = arctan(<i>x</i>)). If the <a href="/wiki/Codomain" title="Codomain">codomain</a> (−π/2, π/2) was made larger to include an integer multiple of π/2, then this function would no longer be onto (surjective), since there is no real number which could be paired with the multiple of π/2 by this arctan function.</li> <li>The <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, <i>g</i>: <b>R</b> → <b>R</b>, <i>g</i>(<i>x</i>) = e<sup><i>x</i></sup>, is not bijective: for instance, there is no <i>x</i> in <b>R</b> such that <i>g</i>(<i>x</i>) = −1, showing that <i>g</i> is not onto (surjective). However, if the codomain is restricted to the positive real numbers <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} ^{+}\equiv \left(0,\infty \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>≡</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{+}\equiv \left(0,\infty \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca500b0e7d5e95360de2b1cccf05179fc977bbba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.617ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{+}\equiv \left(0,\infty \right)}"></span>, then <i>g</i> would be bijective; its inverse (see below) is the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> function ln.</li> <li>The function <i>h</i>: <b>R</b> → <b>R</b><sup>+</sup>, <i>h</i>(<i>x</i>) = <i>x</i><sup>2</sup> is not bijective: for instance, <i>h</i>(−1) = <i>h</i>(1) = 1, showing that <i>h</i> is not one-to-one (injective). However, if the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> is restricted to <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} _{0}^{+}\equiv \left[0,\infty \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>≡</mo> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{0}^{+}\equiv \left[0,\infty \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d91971a3b102da44c5007d7601345e236242856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.359ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} _{0}^{+}\equiv \left[0,\infty \right)}"></span>, then <i>h</i> would be bijective; its inverse is the positive square root function.</li> <li>By <a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a>, given any two sets <i>X</i> and <i>Y</i>, and two injective functions <i>f</i>: <i>X → Y</i> and <i>g</i>: <i>Y → X</i>, there exists a bijective function <i>h</i>: <i>X → Y</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Inverses">Inverses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=6" title="Edit section: Inverses"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A bijection <i>f</i> with domain <i>X</i> (indicated by <i>f</i>: <i>X → Y</i> in <a href="/wiki/Function_(mathematics)#Notation" title="Function (mathematics)">functional notation</a>) also defines a <a href="/wiki/Converse_relation" title="Converse relation">converse relation</a> starting in <i>Y</i> and going to <i>X</i> (by turning the arrows around). The process of "turning the arrows around" for an arbitrary function does not, <i>in general</i>, yield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain <i>Y</i>. Moreover, properties (1) and (2) then say that this inverse <i>function</i> is a surjection and an injection, that is, the <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a> exists and is also a bijection. Functions that have inverse functions are said to be <a href="/wiki/Invertible_function" class="mw-redirect" title="Invertible function">invertible</a>. A function is invertible if and only if it is a bijection. </p><p>Stated in concise mathematical notation, a function <i>f</i>: <i>X → Y</i> is bijective if and only if it satisfies the condition </p> <dl><dd>for every <i>y</i> in <i>Y</i> there is a unique <i>x</i> in <i>X</i> with <i>y</i> = <i>f</i>(<i>x</i>).</dd></dl> <p>Continuing with the baseball batting line-up example, the function that is being defined takes as input the name of one of the players and outputs the position of that player in the batting order. Since this function is a bijection, it has an inverse function which takes as input a position in the batting order and outputs the player who will be batting in that position. </p> <div class="mw-heading mw-heading2"><h2 id="Composition">Composition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=7" title="Edit section: Composition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Bijective_composition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Bijective_composition.svg/300px-Bijective_composition.svg.png" decoding="async" width="300" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Bijective_composition.svg/450px-Bijective_composition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Bijective_composition.svg/600px-Bijective_composition.svg.png 2x" data-file-width="300" data-file-height="200" /></a><figcaption>A bijection composed of an injection (X → Y) and a surjection (Y → Z).</figcaption></figure> <p>The <a href="/wiki/Function_composition" title="Function composition">composition</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\,\circ \,f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mspace width="thinmathspace" /> <mo>∘</mo> <mspace width="thinmathspace" /> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\,\circ \,f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee184e9e8a33749be6531c4effac43a704b79e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.364ex; height:2.509ex;" alt="{\displaystyle g\,\circ \,f}"></span> of two bijections <i>f</i>: <i>X → Y</i> and <i>g</i>: <i>Y → Z</i> is a bijection, whose inverse is given by <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mspace width="thinmathspace" /> <mo>∘</mo> <mspace width="thinmathspace" /> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\,\circ \,f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee184e9e8a33749be6531c4effac43a704b79e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.364ex; height:2.509ex;" alt="{\displaystyle g\,\circ \,f}"></span> is <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)^{-1}\;=\;(f^{-1})\,\circ \,(g^{-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mo>∘</mo> <mspace width="thinmathspace" /> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>∘</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g\,\circ \,f)^{-1}\;=\;(f^{-1})\,\circ \,(g^{-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e413920bda230689cc331c12ed8edb608467164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.586ex; height:3.176ex;" alt="{\displaystyle (g\,\circ \,f)^{-1}\;=\;(f^{-1})\,\circ \,(g^{-1})}"></span>. </p><p>Conversely, if the composition <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mspace width="thinmathspace" /> <mo>∘</mo> <mspace width="thinmathspace" /> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\,\circ \,f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee184e9e8a33749be6531c4effac43a704b79e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.364ex; height:2.509ex;" alt="{\displaystyle g\,\circ \,f}"></span> of two functions is bijective, it only follows that <i>f</i> is <a href="/wiki/Injective_function" title="Injective function">injective</a> and <i>g</i> is <a href="/wiki/Surjective_function" title="Surjective function">surjective</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Cardinality">Cardinality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=8" title="Edit section: Cardinality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>X</i> and <i>Y</i> are <a href="/wiki/Finite_set" title="Finite set">finite sets</a>, then there exists a bijection between the two sets <i>X</i> and <i>Y</i> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <i>X</i> and <i>Y</i> have the same number of elements. Indeed, in <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a>, this is taken as the definition of "same number of elements" (<a href="/wiki/Equinumerosity" title="Equinumerosity">equinumerosity</a>), and generalising this definition to <a href="/wiki/Infinite_set" title="Infinite set">infinite sets</a> leads to the concept of <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a>, a way to distinguish the various sizes of infinite sets. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=9" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A function <i>f</i>: <b>R</b> → <b>R</b> is bijective if and only if its <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> meets every horizontal and vertical line exactly once.</li> <li>If <i>X</i> is a set, then the bijective functions from <i>X</i> to itself, together with the operation of functional composition (∘), form a <a href="/wiki/Group_(algebra)" class="mw-redirect" title="Group (algebra)">group</a>, the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> of <i>X</i>, which is denoted variously by S(<i>X</i>), <i>S<sub>X</sub></i>, or <i>X</i>! (<i>X</i> <a href="/wiki/Factorial" title="Factorial">factorial</a>).</li> <li>Bijections preserve <a href="/wiki/Cardinalities" class="mw-redirect" title="Cardinalities">cardinalities</a> of sets: for a subset <i>A</i> of the domain with cardinality |<i>A</i>| and subset <i>B</i> of the codomain with cardinality |<i>B</i>|, one has the following equalities: <dl><dd>|<i>f</i>(<i>A</i>)| = |<i>A</i>| and |<i>f</i><sup>−1</sup>(<i>B</i>)| = |<i>B</i>|.</dd></dl></li> <li>If <i>X</i> and <i>Y</i> are <a href="/wiki/Finite_set" title="Finite set">finite sets</a> with the same cardinality, and <i>f</i>: <i>X → Y</i>, then the following are equivalent: <ol><li><i>f</i> is a bijection.</li> <li><i>f</i> is a <a href="/wiki/Surjection" class="mw-redirect" title="Surjection">surjection</a>.</li> <li><i>f</i> is an <a href="/wiki/Injection_(mathematics)" class="mw-redirect" title="Injection (mathematics)">injection</a>.</li></ol></li> <li>For a finite set <i>S</i>, there is a bijection between the set of possible <a href="/wiki/Total_ordering" class="mw-redirect" title="Total ordering">total orderings</a> of the elements and the set of bijections from <i>S</i> to <i>S</i>. That is to say, the number of <a href="/wiki/Permutation" title="Permutation">permutations</a> of elements of <i>S</i> is the same as the number of total orderings of that set—namely, <i>n</i>!.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Category_theory">Category theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=10" title="Edit section: Category theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bijections are precisely the <a href="/wiki/Isomorphism" title="Isomorphism">isomorphisms</a> in the <a href="/wiki/Category_theory" title="Category theory">category</a> <i><a href="/wiki/Category_of_sets" title="Category of sets">Set</a></i> of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in the category <i><a href="/wiki/Category_of_groups" title="Category of groups">Grp</a></i> of <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, the morphisms must be <a href="/wiki/Homomorphism" title="Homomorphism">homomorphisms</a> since they must preserve the group structure, so the isomorphisms are <i>group isomorphisms</i> which are bijective homomorphisms. </p> <div class="mw-heading mw-heading2"><h2 id="Generalization_to_partial_functions">Generalization to partial functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=11" title="Edit section: Generalization to partial functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The notion of one-to-one correspondence generalizes to <a href="/wiki/Partial_functions" class="mw-redirect" title="Partial functions">partial functions</a>, where they are called <i>partial bijections</i>, although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a <a href="/wiki/Total_function" class="mw-redirect" title="Total function">total function</a>, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the <a href="/wiki/Symmetric_inverse_semigroup" title="Symmetric inverse semigroup">symmetric inverse semigroup</a>.<sup id="cite_ref-Hollings2014_4-0" class="reference"><a href="#cite_note-Hollings2014-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Another way of defining the same notion is to say that a partial bijection from <i>A</i> to <i>B</i> is any relation <i>R</i> (which turns out to be a partial function) with the property that <i>R</i> is the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph of</a> a bijection <i>f</i>:<i>A′</i>→<i>B′</i>, where <i>A′</i> is a <a href="/wiki/Subset" title="Subset">subset</a> of <i>A</i> and <i>B′</i> is a subset of <i>B</i>.<sup id="cite_ref-Borceux1994_5-0" class="reference"><a href="#cite_note-Borceux1994-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>When the partial bijection is on the same set, it is sometimes called a <i>one-to-one partial <a href="/wiki/Transformation_(function)" title="Transformation (function)">transformation</a></i>.<sup id="cite_ref-Grillet1995_6-0" class="reference"><a href="#cite_note-Grillet1995-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> An example is the <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformation</a> simply defined on the complex plane, rather than its completion to the extended complex plane.<sup id="cite_ref-Campbell2007_7-0" class="reference"><a href="#cite_note-Campbell2007-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Gallery">Gallery</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=12" title="Edit section: Gallery"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1248256098">@media all and (max-width:720px){.mw-parser-output .mod-gallery{width:100%!important}}.mw-parser-output .mod-gallery{display:table}.mw-parser-output .mod-gallery-default{background:transparent;margin-top:4px}.mw-parser-output .mod-gallery-center{margin-left:auto;margin-right:auto}.mw-parser-output .mod-gallery-left{float:left}.mw-parser-output .mod-gallery-right{float:right}.mw-parser-output .mod-gallery-none{float:none}.mw-parser-output .mod-gallery-collapsible{width:100%}.mw-parser-output .mod-gallery .title,.mw-parser-output .mod-gallery .main,.mw-parser-output .mod-gallery .footer{display:table-row}.mw-parser-output .mod-gallery .title>div{display:table-cell;padding:0 4px 4px;text-align:center;font-weight:bold}.mw-parser-output .mod-gallery .main>div{display:table-cell}.mw-parser-output .mod-gallery .gallery{line-height:1.35em}.mw-parser-output .mod-gallery .footer>div{display:table-cell;padding:4px;text-align:right;font-size:85%;line-height:1em}.mw-parser-output .mod-gallery .title>div *,.mw-parser-output .mod-gallery .footer>div *{overflow:visible}.mw-parser-output .mod-gallery .gallerybox img{background:none!important}.mw-parser-output .mod-gallery .bordered-images .thumb img{border:solid var(--background-color-neutral,#eaecf0)1px}.mw-parser-output .mod-gallery .whitebg .thumb{background:var(--background-color-base,#fff)!important}</style><div class="mod-gallery mod-gallery-default mod-gallery-center"><div class="main"><div><ul class="gallery mw-gallery-traditional nochecker bordered-images whitebg"> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Injection.svg" class="mw-file-description" title="An injective non-surjective function (injection, not a bijection)"><img alt="An injective non-surjective function (injection, not a bijection)" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Injection.svg/180px-Injection.svg.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Injection.svg/270px-Injection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Injection.svg/360px-Injection.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext">An injective non-surjective function (injection, not a bijection)</div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Bijection.svg" class="mw-file-description" title="An injective surjective function (bijection)"><img alt="An injective surjective function (bijection)" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/180px-Bijection.svg.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/270px-Bijection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/360px-Bijection.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext">An injective surjective function (<b>bijection</b>)</div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Surjection.svg" class="mw-file-description" title="A non-injective surjective function (surjection, not a bijection)"><img alt="A non-injective surjective function (surjection, not a bijection)" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Surjection.svg/180px-Surjection.svg.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Surjection.svg/270px-Surjection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Surjection.svg/360px-Surjection.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext">A non-injective surjective function (surjection, not a bijection)</div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Not-Injection-Surjection.svg" class="mw-file-description" title="A non-injective non-surjective function (also not a bijection)"><img alt="A non-injective non-surjective function (also not a bijection)" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Not-Injection-Surjection.svg/180px-Not-Injection-Surjection.svg.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Not-Injection-Surjection.svg/270px-Not-Injection-Surjection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Not-Injection-Surjection.svg/360px-Not-Injection-Surjection.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext">A non-injective non-surjective function (also not a bijection)</div> </li> </ul></div></div></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <ul><li><a href="/wiki/Ax%E2%80%93Grothendieck_theorem" title="Ax–Grothendieck theorem">Ax–Grothendieck theorem</a></li> <li><a href="/wiki/Bijection,_injection_and_surjection" title="Bijection, injection and surjection">Bijection, injection and surjection</a></li> <li><a href="/wiki/Bijective_numeration" title="Bijective numeration">Bijective numeration</a></li> <li><a href="/wiki/Bijective_proof" title="Bijective proof">Bijective proof</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Multivalued_function" title="Multivalued function">Multivalued function</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=14" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall1959">Hall 1959</a>, p. 3</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">There are names associated to properties (1) and (2) as well. A relation which satisfies property (1) is called a <i>total relation</i> and a relation satisfying (2) is a <i>single valued relation</i>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://brilliant.org/wiki/bijection-injection-and-surjection/">"Bijection, Injection, And Surjection | Brilliant Math & Science Wiki"</a>. <i>brilliant.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">7 December</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=brilliant.org&rft.atitle=Bijection%2C+Injection%2C+And+Surjection+%7C+Brilliant+Math+%26+Science+Wiki&rft_id=https%3A%2F%2Fbrilliant.org%2Fwiki%2Fbijection-injection-and-surjection%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></span> </li> <li id="cite_note-Hollings2014-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hollings2014_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristopher_Hollings2014" class="citation book cs1">Christopher Hollings (16 July 2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=O9wJBAAAQBAJ&pg=PA251"><i>Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups</i></a>. American Mathematical Society. p. 251. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-1493-1" title="Special:BookSources/978-1-4704-1493-1"><bdi>978-1-4704-1493-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+across+the+Iron+Curtain%3A+A+History+of+the+Algebraic+Theory+of+Semigroups&rft.pages=251&rft.pub=American+Mathematical+Society&rft.date=2014-07-16&rft.isbn=978-1-4704-1493-1&rft.au=Christopher+Hollings&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DO9wJBAAAQBAJ%26pg%3DPA251&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></span> </li> <li id="cite_note-Borceux1994-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Borceux1994_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrancis_Borceux1994" class="citation book cs1">Francis Borceux (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5i2v9q0m5XAC&pg=PA289"><i>Handbook of Categorical Algebra: Volume 2, Categories and Structures</i></a>. Cambridge University Press. p. 289. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-44179-7" title="Special:BookSources/978-0-521-44179-7"><bdi>978-0-521-44179-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Categorical+Algebra%3A+Volume+2%2C+Categories+and+Structures&rft.pages=289&rft.pub=Cambridge+University+Press&rft.date=1994&rft.isbn=978-0-521-44179-7&rft.au=Francis+Borceux&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5i2v9q0m5XAC%26pg%3DPA289&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></span> </li> <li id="cite_note-Grillet1995-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Grillet1995_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPierre_A._Grillet1995" class="citation book cs1">Pierre A. Grillet (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yM544W1N2UUC&pg=PA228"><i>Semigroups: An Introduction to the Structure Theory</i></a>. CRC Press. p. 228. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8247-9662-4" title="Special:BookSources/978-0-8247-9662-4"><bdi>978-0-8247-9662-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Semigroups%3A+An+Introduction+to+the+Structure+Theory&rft.pages=228&rft.pub=CRC+Press&rft.date=1995&rft.isbn=978-0-8247-9662-4&rft.au=Pierre+A.+Grillet&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyM544W1N2UUC%26pg%3DPA228&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></span> </li> <li id="cite_note-Campbell2007-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Campbell2007_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Meakin2007" class="citation book cs1">John Meakin (2007). "Groups and semigroups: connections and contrasts". In C.M. Campbell; M.R. Quick; E.F. Robertson; G.C. Smith (eds.). <i>Groups St Andrews 2005 Volume 2</i>. Cambridge University Press. p. 367. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-69470-4" title="Special:BookSources/978-0-521-69470-4"><bdi>978-0-521-69470-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Groups+and+semigroups%3A+connections+and+contrasts&rft.btitle=Groups+St+Andrews+2005+Volume+2&rft.pages=367&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-69470-4&rft.au=John+Meakin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span> <a rel="nofollow" class="external text" href="http://www.math.unl.edu/~jmeakin2/groups%20and%20semigroups.pdf">preprint</a> citing <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLawson1998" class="citation journal cs1">Lawson, M. V. (1998). <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjabr.1997.7242">"The Möbius Inverse Monoid"</a>. <i>Journal of Algebra</i>. <b>200</b> (2): 428–438. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjabr.1997.7242">10.1006/jabr.1997.7242</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Algebra&rft.atitle=The+M%C3%B6bius+Inverse+Monoid&rft.volume=200&rft.issue=2&rft.pages=428-438&rft.date=1998&rft_id=info%3Adoi%2F10.1006%2Fjabr.1997.7242&rft.aulast=Lawson&rft.aufirst=M.+V.&rft_id=https%3A%2F%2Fdoi.org%2F10.1006%252Fjabr.1997.7242&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory. Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic may be found in any of these: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall1959" class="citation book cs1"><a href="/wiki/Marshall_Hall_(mathematician)" title="Marshall Hall (mathematician)">Hall, Marshall Jr.</a> (1959). <i>The Theory of Groups</i>. MacMillan.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Groups&rft.pub=MacMillan&rft.date=1959&rft.aulast=Hall&rft.aufirst=Marshall+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolf1998" class="citation book cs1">Wolf (1998). <i>Proof, Logic and Conjecture: A Mathematician's Toolbox</i>. Freeman.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proof%2C+Logic+and+Conjecture%3A+A+Mathematician%27s+Toolbox&rft.pub=Freeman&rft.date=1998&rft.au=Wolf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSundstrom2003" class="citation book cs1">Sundstrom (2003). <i>Mathematical Reasoning: Writing and Proof</i>. Prentice-Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Reasoning%3A+Writing+and+Proof&rft.pub=Prentice-Hall&rft.date=2003&rft.au=Sundstrom&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmithEggenSt.Andre2006" class="citation book cs1">Smith; Eggen; St.Andre (2006). <i>A Transition to Advanced Mathematics (6th Ed.)</i>. Thomson (Brooks/Cole).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Transition+to+Advanced+Mathematics+%286th+Ed.%29&rft.pub=Thomson+%28Brooks%2FCole%29&rft.date=2006&rft.au=Smith&rft.au=Eggen&rft.au=St.Andre&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchumacher1996" class="citation book cs1">Schumacher (1996). <i>Chapter Zero: Fundamental Notions of Abstract Mathematics</i>. Addison-Wesley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chapter+Zero%3A+Fundamental+Notions+of+Abstract+Mathematics&rft.pub=Addison-Wesley&rft.date=1996&rft.au=Schumacher&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'Leary2003" class="citation book cs1">O'Leary (2003). <i>The Structure of Proof: With Logic and Set Theory</i>. Prentice-Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Structure+of+Proof%3A+With+Logic+and+Set+Theory&rft.pub=Prentice-Hall&rft.date=2003&rft.au=O%27Leary&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorash" class="citation book cs1">Morash. <i>Bridge to Abstract Mathematics</i>. Random House.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bridge+to+Abstract+Mathematics&rft.pub=Random+House&rft.au=Morash&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaddox2002" class="citation book cs1">Maddox (2002). <i>Mathematical Thinking and Writing</i>. Harcourt/ Academic Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Thinking+and+Writing&rft.pub=Harcourt%2F+Academic+Press&rft.date=2002&rft.au=Maddox&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLay2001" class="citation book cs1">Lay (2001). <i>Analysis with an introduction to proof</i>. Prentice Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+with+an+introduction+to+proof&rft.pub=Prentice+Hall&rft.date=2001&rft.au=Lay&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGilbertVanstone2005" class="citation book cs1">Gilbert; Vanstone (2005). <i>An Introduction to Mathematical Thinking</i>. Pearson Prentice-Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Mathematical+Thinking&rft.pub=Pearson+Prentice-Hall&rft.date=2005&rft.au=Gilbert&rft.au=Vanstone&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFletcherPatty" class="citation book cs1">Fletcher; Patty. <i>Foundations of Higher Mathematics</i>. PWS-Kent.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Higher+Mathematics&rft.pub=PWS-Kent&rft.au=Fletcher&rft.au=Patty&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIglewiczStoyle" class="citation book cs1">Iglewicz; Stoyle. <i>An Introduction to Mathematical Reasoning</i>. MacMillan.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Mathematical+Reasoning&rft.pub=MacMillan&rft.au=Iglewicz&rft.au=Stoyle&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDevlin2004" class="citation book cs1">Devlin, Keith (2004). <i>Sets, Functions, and Logic: An Introduction to Abstract Mathematics</i>. Chapman & Hall/ CRC Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sets%2C+Functions%2C+and+Logic%3A+An+Introduction+to+Abstract+Mathematics&rft.pub=Chapman+%26+Hall%2F+CRC+Press&rft.date=2004&rft.aulast=Devlin&rft.aufirst=Keith&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD'AngeloWest2000" class="citation book cs1">D'Angelo; West (2000). <i>Mathematical Thinking: Problem Solving and Proofs</i>. Prentice Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Thinking%3A+Problem+Solving+and+Proofs&rft.pub=Prentice+Hall&rft.date=2000&rft.au=D%27Angelo&rft.au=West&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCupillari1989" class="citation book cs1"><a href="/wiki/Antonella_Cupillari" title="Antonella Cupillari">Cupillari</a> (1989). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/nutsboltsofproof00anto"><i>The Nuts and Bolts of Proofs</i></a></span>. Wadsworth. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780534103200" title="Special:BookSources/9780534103200"><bdi>9780534103200</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Nuts+and+Bolts+of+Proofs&rft.pub=Wadsworth&rft.date=1989&rft.isbn=9780534103200&rft.au=Cupillari&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnutsboltsofproof00anto&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBond" class="citation book cs1">Bond. <i>Introduction to Abstract Mathematics</i>. Brooks/Cole.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Abstract+Mathematics&rft.pub=Brooks%2FCole&rft.au=Bond&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarnierFeldman2000" class="citation book cs1">Barnier; Feldman (2000). <i>Introduction to Advanced Mathematics</i>. Prentice Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Advanced+Mathematics&rft.pub=Prentice+Hall&rft.date=2000&rft.au=Barnier&rft.au=Feldman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAsh" class="citation book cs1">Ash. <i>A Primer of Abstract Mathematics</i>. MAA.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Primer+of+Abstract+Mathematics&rft.pub=MAA&rft.au=Ash&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bijection&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Bijectivity" class="extiw" title="commons:Category:Bijectivity">Bijectivity</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Bijection">"Bijection"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bijection&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DBijection&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Bijection"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Bijection.html">"Bijection"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Bijection&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FBijection.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABijection" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/i.html">Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px 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.navbox{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"></div><div role="navigation" class="navbox" aria-labelledby="Set_theory" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Set_theory" title="Template:Set theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Set_theory" title="Template talk:Set theory"><abbr title="Discuss this template">t</abbr></a></li><li 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/150px-Venn_A_intersect_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/200px-Venn_A_intersect_B.svg.png 2x" data-file-width="350" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Axiom" title="Axiom">Axioms</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom_of_adjunction" title="Axiom of adjunction">Adjunction</a></li> <li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">Choice</a> <ul><li><a href="/wiki/Axiom_of_countable_choice" title="Axiom of countable choice">countable</a></li> <li><a href="/wiki/Axiom_of_dependent_choice" title="Axiom of dependent choice">dependent</a></li> <li><a href="/wiki/Axiom_of_global_choice" title="Axiom of global choice">global</a></li></ul></li> <li><a href="/wiki/Axiom_of_constructibility" title="Axiom of constructibility">Constructibility (V=L)</a></li> <li><a href="/wiki/Axiom_of_determinacy" title="Axiom of determinacy">Determinacy</a> <ul><li><a href="/wiki/Axiom_of_projective_determinacy" title="Axiom of projective determinacy">projective</a></li></ul></li> <li><a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">Extensionality</a></li> <li><a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">Infinity</a></li> <li><a href="/wiki/Axiom_of_limitation_of_size" title="Axiom of limitation of size">Limitation of size</a></li> <li><a href="/wiki/Axiom_of_pairing" title="Axiom of pairing">Pairing</a></li> <li><a href="/wiki/Axiom_of_power_set" title="Axiom of power set">Power set</a></li> <li><a href="/wiki/Axiom_of_regularity" title="Axiom of regularity">Regularity</a></li> <li><a href="/wiki/Axiom_of_union" title="Axiom of union">Union</a></li> <li><a href="/wiki/Martin%27s_axiom" title="Martin's axiom">Martin's axiom</a></li></ul> <ul><li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a> <ul><li><a href="/wiki/Axiom_schema_of_replacement" title="Axiom schema of replacement">replacement</a></li> <li><a href="/wiki/Axiom_schema_of_specification" title="Axiom schema of specification">specification</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)#Basic_operations" title="Set (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">Complement</a> (i.e. set difference)</li> <li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/Disjoint_union" title="Disjoint union">Disjoint union</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">Identities</a></li> <li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">Intersection</a></li> <li><a href="/wiki/Power_set" title="Power set">Power set</a></li> <li><a href="/wiki/Symmetric_difference" title="Symmetric difference">Symmetric difference</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">Union</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li>Concepts</li><li>Methods</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost" title="Almost">Almost</a></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal number</a> (<a href="/wiki/Large_cardinal" title="Large cardinal">large</a>)</li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li><a href="/wiki/Constructible_universe" title="Constructible universe">Constructible universe</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">Continuum hypothesis</a></li> <li><a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Diagonal argument</a></li> <li><a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a> <ul><li><a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a></li> <li><a href="/wiki/Tuple" title="Tuple">tuple</a></li></ul></li> <li><a href="/wiki/Family_of_sets" title="Family of sets">Family</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a class="mw-selflink selflink">One-to-one correspondence</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Set-builder_notation" title="Set-builder notation">Set-builder notation</a></li> <li><a href="/wiki/Transfinite_induction" title="Transfinite induction">Transfinite induction</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)" title="Set (mathematics)">Set</a> types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_set" title="Amorphous set">Amorphous</a></li> <li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a> (<a href="/wiki/Hereditarily_finite_set" title="Hereditarily finite set">hereditarily</a>)</li> <li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">Filter</a> <ul><li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">base</a></li> <li><a href="/wiki/Filter_(set_theory)#Filters_and_prefilters" title="Filter (set theory)">subbase</a></li> <li><a href="/wiki/Ultrafilter_on_a_set" title="Ultrafilter on a set">Ultrafilter</a></li></ul></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a> (<a href="/wiki/Dedekind-infinite_set" title="Dedekind-infinite set">Dedekind-infinite</a>)</li> <li><a href="/wiki/Computable_set" title="Computable set">Recursive</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Subset" title="Subset">Subset <b>·</b> Superset</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternative_set_theory" class="mw-redirect" title="Alternative set theory">Alternative</a></li> <li><a href="/wiki/Set_theory#Formalized_set_theory" title="Set theory">Axiomatic</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a></li></ul> <ul><li><a href="/wiki/Zermelo_set_theory" title="Zermelo set theory">Zermelo</a> <ul><li><a href="/wiki/General_set_theory" title="General set theory">General</a></li></ul></li> <li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> <ul><li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li></ul></li> <li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel </a> <ul><li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">von Neumann–Bernays–Gödel </a> <ul><li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li></ul></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes</a></li><li>Problems</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li> <li><a href="/wiki/Suslin%27s_problem" title="Suslin's problem">Suslin's problem</a></li> <li><a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Set_theorists" title="Category:Set theorists">Set theorists</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Paul_Bernays" title="Paul Bernays">Paul Bernays</a></li> <li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Paul_Cohen" title="Paul Cohen">Paul Cohen</a></li> <li><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li> <li><a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li> <li><a href="/wiki/Thomas_Jech" title="Thomas Jech">Thomas Jech</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a></li> <li><a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Quine</a></li> <li><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li> <li><a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a></li> <li><a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_logic" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a class="mw-selflink selflink">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a 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