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id="toc-Surjections_as_right_invertible_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surjections_as_epimorphisms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surjections_as_epimorphisms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Surjections as epimorphisms</span> </div> </a> <ul id="toc-Surjections_as_epimorphisms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surjections_as_binary_relations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surjections_as_binary_relations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Surjections as binary relations</span> </div> </a> <ul id="toc-Surjections_as_binary_relations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cardinality_of_the_domain_of_a_surjection" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cardinality_of_the_domain_of_a_surjection"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Cardinality of the domain of a surjection</span> </div> </a> <ul id="toc-Cardinality_of_the_domain_of_a_surjection-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Composition_and_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Composition_and_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Composition and decomposition</span> </div> </a> <ul id="toc-Composition_and_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Induced_surjection_and_induced_bijection" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Induced_surjection_and_induced_bijection"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Induced surjection and induced bijection</span> </div> </a> <ul id="toc-Induced_surjection_and_induced_bijection-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-The_set_of_surjections" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_set_of_surjections"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>The set of surjections</span> </div> </a> <ul id="toc-The_set_of_surjections-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">5</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">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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">Surjective function</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 54 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-54" 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">54 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%AF%D8%A7%D9%84%D8%A9_%D8%BA%D8%A7%D9%85%D8%B1%D8%A9" title="دالة غامرة – Arabic" lang="ar" hreflang="ar" data-title="دالة غامرة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%AC%E0%A6%BF%E0%A6%95_%E0%A6%AB%E0%A6%BE%E0%A6%82%E0%A6%B6%E0%A6%A8" title="সার্বিক ফাংশন – Bangla" lang="bn" hreflang="bn" data-title="সার্বিক ফাংশন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%8E%D1%80%E2%80%99%D0%B5%D0%BA%D1%86%D1%8B%D1%8F" title="Сюр’екцыя – Belarusian" lang="be" hreflang="be" data-title="Сюр’екцыя" 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%A1%D1%8E%D1%80%D0%B5%D0%BA%D1%86%D0%B8%D1%8F" title="Сюрекция – Bulgarian" lang="bg" hreflang="bg" data-title="Сюрекция" 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/Surjektivna_funkcija" title="Surjektivna funkcija – Bosnian" lang="bs" hreflang="bs" data-title="Surjektivna funkcija" 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_exhaustiva" title="Funció exhaustiva – Catalan" lang="ca" hreflang="ca" data-title="Funció exhaustiva" 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/Surjekce" title="Surjekce – Czech" lang="cs" hreflang="cs" data-title="Surjekce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Surjektiv" title="Surjektiv – Danish" lang="da" hreflang="da" data-title="Surjektiv" 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/Surjektive_Funktion" title="Surjektive Funktion – German" lang="de" hreflang="de" data-title="Surjektive Funktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%80%CE%AF_(%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7)" title="Επί (συνάρτηση) – Greek" lang="el" hreflang="el" data-title="Επί (συνάρτηση)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_sobreyectiva" title="Función sobreyectiva – Spanish" lang="es" hreflang="es" data-title="Función sobreyectiva" 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/Sur%C4%B5eto" title="Surĵeto – Esperanto" lang="eo" hreflang="eo" data-title="Surĵeto" 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/Funtzio_supraiektibo" title="Funtzio supraiektibo – Basque" lang="eu" hreflang="eu" data-title="Funtzio supraiektibo" 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%D8%A7%D8%A8%D8%B9_%D9%BE%D9%88%D8%B4%D8%A7" title="تابع پوشا – Persian" lang="fa" hreflang="fa" data-title="تابع پوشا" 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/Surjection" title="Surjection – French" lang="fr" hreflang="fr" data-title="Surjection" 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_sobrexectiva" title="Función sobrexectiva – Galician" lang="gl" hreflang="gl" data-title="Función sobrexectiva" 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%EC%82%AC_%ED%95%A8%EC%88%98" title="전사 함수 – Korean" lang="ko" hreflang="ko" data-title="전사 함수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Surjektivna_funkcija" title="Surjektivna funkcija – Croatian" lang="hr" hreflang="hr" data-title="Surjektivna funkcija" 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/Surjektio" title="Surjektio – Ido" lang="io" hreflang="io" data-title="Surjektio" 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/Fungsi_surjektif" title="Fungsi surjektif – Indonesian" lang="id" hreflang="id" data-title="Fungsi surjektif" 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/Surjection" title="Surjection – Interlingua" lang="ia" hreflang="ia" data-title="Surjection" 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/%C3%81t%C3%A6k_v%C3%B6rpun" title="Átæk vörpun – Icelandic" lang="is" hreflang="is" data-title="Átæk 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/Funzione_suriettiva" title="Funzione suriettiva – Italian" lang="it" hreflang="it" data-title="Funzione suriettiva" 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%A2%D7%9C" title="פונקציה על – Hebrew" lang="he" hreflang="he" data-title="פונקציה על" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Functio_superiectiva" title="Functio superiectiva – Latin" lang="la" hreflang="la" data-title="Functio superiectiva" 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/Siurjekcija" title="Siurjekcija – Lithuanian" lang="lt" hreflang="lt" data-title="Siurjekcija" 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/Fonzion_suriettiva" title="Fonzion suriettiva – Lombard" lang="lmo" hreflang="lmo" data-title="Fonzion suriettiva" 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/Sz%C3%BCrjekci%C3%B3" title="Szürjekció – Hungarian" lang="hu" hreflang="hu" data-title="Szürjekció" 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%A1%D1%83%D1%80%D1%98%D0%B5%D0%BA%D1%82%D0%B8%D0%B2%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Сурјективна функција – Macedonian" lang="mk" hreflang="mk" data-title="Сурјективна функција" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A1%D1%8E%D1%80%D1%8A%D0%B5%D0%BA%D1%82%D0%B8%D0%B2_%D1%84%D1%83%D0%BD%D0%BA%D1%86" title="Сюръектив функц – Mongolian" lang="mn" hreflang="mn" data-title="Сюръектив функц" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Surjectie" title="Surjectie – Dutch" lang="nl" hreflang="nl" data-title="Surjectie" 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%B0%84" title="全射 – Japanese" lang="ja" hreflang="ja" data-title="全射" 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/Surjektiv_funksjon" title="Surjektiv funksjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Surjektiv funksjon" 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/Surjeksjon" title="Surjeksjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Surjeksjon" 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/Subrejeccion" title="Subrejeccion – Occitan" lang="oc" hreflang="oc" data-title="Subrejeccion" 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/Surjekcja" title="Surjekcja – Polish" lang="pl" hreflang="pl" data-title="Surjekcja" 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_sobrejectiva" title="Função sobrejectiva – Portuguese" lang="pt" hreflang="pt" data-title="Função sobrejectiva" 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/Func%C8%9Bie_surjectiv%C4%83" title="Funcție surjectivă – Romanian" lang="ro" hreflang="ro" data-title="Funcție surjectivă" 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%A1%D1%8E%D1%80%D1%8A%D0%B5%D0%BA%D1%86%D0%B8%D1%8F" title="Сюръекция – Russian" lang="ru" hreflang="ru" data-title="Сюръекция" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Surjective_function" title="Surjective function – Simple English" lang="en-simple" hreflang="en-simple" data-title="Surjective 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/Surjekt%C3%ADvne_zobrazenie" title="Surjektívne zobrazenie – Slovak" lang="sk" hreflang="sk" data-title="Surjektívne 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/Surjektivna_preslikava" title="Surjektivna preslikava – Slovenian" lang="sl" hreflang="sl" data-title="Surjektivna preslikava" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Surjekcyjo" title="Surjekcyjo – Silesian" lang="szl" hreflang="szl" data-title="Surjekcyjo" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D8%A7%D9%86%DA%A9%D8%B4%D9%86%DB%8C_%DA%AF%D8%B4%D8%AA%DA%AF%D8%B1" title="فانکشنی گشتگر – Central Kurdish" lang="ckb" hreflang="ckb" data-title="فانکشنی گشتگر" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" 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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 class="mw-selflink selflink">Surjective</a></li> <li><a href="/wiki/Bijection" title="Bijection">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>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>surjective function</b> (also known as <b>surjection</b>, or <b>onto function</b> <span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="/ɒ/: 'o' in 'body'">ɒ</span><span title="'n' in 'nigh'">n</span><span title="/./: syllable break">.</span><span title="'t' in 'tie'">t</span><span title="/uː/: 'oo' in 'goose'">uː</span></span>/</a></span></span>) is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="texhtml"><i>f</i></span> such that, for every element <span class="texhtml"><i>y</i></span> of the function's <a href="/wiki/Codomain" title="Codomain">codomain</a>, there exists <em>at least</em> one element <span class="texhtml"><i>x</i></span> in the function's <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> such that <span class="texhtml"><i>f</i>(<i>x</i>) = <i>y</i></span>. In other words, for a function <span class="texhtml"><i>f</i> : <i>X</i> → <i>Y</i></span>, the codomain <span class="texhtml"><i>Y</i></span> is the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of the function's domain <span class="texhtml"><i>X</i></span>.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It is not required that <span class="texhtml"><i>x</i></span> be <a href="/wiki/Unique_(mathematics)" class="mw-redirect" title="Unique (mathematics)">unique</a>; the function <span class="texhtml"><i>f</i></span> may map one or more elements of <span class="texhtml"><i>X</i></span> to the same element of <span class="texhtml"><i>Y</i></span>. </p><p>The term <i>surjective</i> and the related terms <i><a href="/wiki/Injective_function" title="Injective function">injective</a></i> and <i><a href="/wiki/Bijective_function" class="mw-redirect" title="Bijective function">bijective</a></i> were introduced by <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> a group of mainly <a href="/wiki/France" title="France">French</a> 20th-century <a href="/wiki/Mathematician" title="Mathematician">mathematicians</a> who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word <i><a href="https://en.wiktionary.org/wiki/sur#French" class="extiw" title="wikt:sur">sur</a></i> means <i>over</i> or <i>above</i>, and relates to the fact that the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of the domain of a surjective function completely covers the function's codomain. </p><p>Any function induces a surjection by <a href="/wiki/Restriction_of_a_function" class="mw-redirect" title="Restriction of a function">restricting</a> its codomain to the image of its domain. Every surjective function has a <a href="/wiki/Inverse_function#Left_and_right_inverses" title="Inverse function">right inverse</a> assuming the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>, and every function with a right inverse is necessarily a surjection. The <a href="/wiki/Function_composition" title="Function composition">composition</a> of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection. </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=Surjective_function&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information on notation: <a href="/wiki/Function_(mathematics)#Notation" title="Function (mathematics)">Function (mathematics) § Notation</a></div> <p>A <b>surjective function</b> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> whose <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> is equal to its <a href="/wiki/Codomain" title="Codomain">codomain</a>. Equivalently, 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> with <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <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 codomain <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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> is surjective if for every <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> </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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> there exists at least one <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> with <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)=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5080a8b0a963407ea74ffa50702563771518d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle f(x)=y}"></span>.<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Surjections are sometimes denoted by a two-headed rightwards arrow (<span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced"><a href="/wiki/Unicode" title="Unicode">U+</a>21A0</span> </span><span style="font-size:125%;line-height:1em">↠</span> <span style="font-variant: small-caps; text-transform: lowercase;">RIGHTWARDS TWO HEADED ARROW</span>),<sup id="cite_ref-Unicode_Arrows_5-0" class="reference"><a href="#cite_note-Unicode_Arrows-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> as 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 f\colon X\twoheadrightarrow 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\colon X\twoheadrightarrow Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/245f8c3c6bf7968f9dbdbeda676e959e08870d50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\twoheadrightarrow Y}"></span>. </p><p>Symbolically, </p> <dl><dd>If <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\colon X\rightarrow 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\colon X\rightarrow Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f986e95e93b70de25a0084daf075cb02c3ccae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\rightarrow Y}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is said to be surjective if</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall y\in Y,\,\exists x\in X,\;\;f(x)=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall y\in Y,\,\exists x\in X,\;\;f(x)=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1c33ca492c6a0392d37a336f8fdca8722522e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.922ex; height:2.843ex;" alt="{\displaystyle \forall y\in Y,\,\exists x\in X,\;\;f(x)=y}"></span>.<sup id="cite_ref-:1_2-1" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></dd></dl> <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=Surjective_function&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Codomain2.SVG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Codomain2.SVG/250px-Codomain2.SVG.png" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Codomain2.SVG/375px-Codomain2.SVG.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Codomain2.SVG/500px-Codomain2.SVG.png 2x" data-file-width="800" data-file-height="600" /></a><figcaption><b>A non-surjective function</b> from <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> <i>X</i> to <a href="/wiki/Codomain" title="Codomain">codomain</a> <i>Y</i>. The smaller yellow oval inside <i>Y</i> is the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> (also called <a href="/wiki/Range_of_a_function" title="Range of a function">range</a>) of <i>f</i>. This function is <b>not</b> surjective, because the image does not fill the whole codomain. In other words, <i>Y</i> is colored in a two-step process: First, for every <i>x</i> in <i>X</i>, the point <i>f</i>(<i>x</i>) is colored yellow; Second, all the rest of the points in <i>Y</i>, that are not yellow, are colored blue. The function <i>f</i> would be surjective only if there were no blue points.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For more examples, see <a href="#Gallery">§ Gallery</a>.</div> <ul><li>For any set <i>X</i>, the <a href="/wiki/Identity_function" title="Identity function">identity function</a> id<sub><i>X</i></sub> on <i>X</i> is surjective.</li> <li>The function <span class="texhtml"><i>f</i> : <b>Z</b> → {0, 1}</span> defined by <i>f</i>(<i>n</i>) = <i>n</i> <b><a href="/wiki/Modular_arithmetic" title="Modular arithmetic">mod</a></b> 2 (that is, <a href="/wiki/Even_number" class="mw-redirect" title="Even number">even</a> <a href="/wiki/Integer" title="Integer">integers</a> are mapped to 0 and <a href="/wiki/Odd_number" class="mw-redirect" title="Odd number">odd</a> integers to 1) is surjective.</li> <li>The function <span class="texhtml"><i>f</i> : <b>R</b> → <b>R</b></span> defined by <i>f</i>(<i>x</i>) = 2<i>x</i> + 1 is surjective (and even <a href="/wiki/Bijective_function" class="mw-redirect" title="Bijective function">bijective</a>), because for every <a href="/wiki/Real_number" title="Real number">real number</a> <i>y</i>, we have an <i>x</i> such that <i>f</i>(<i>x</i>) = <i>y</i>: such an appropriate <i>x</i> is (<i>y</i> − 1)/2.</li> <li>The function <span class="texhtml"><i>f</i> : <b>R</b> → <b>R</b></span> defined by <i>f</i>(<i>x</i>) = <i>x</i><sup>3</sup> − 3<i>x</i> is surjective, because the pre-image of any <a href="/wiki/Real_number" title="Real number">real number</a> <i>y</i> is the solution set of the cubic polynomial equation <i>x</i><sup>3</sup> − 3<i>x</i> − <i>y</i> = 0, and every cubic polynomial with real coefficients has at least one real root. However, this function is not <a href="/wiki/Injective_function" title="Injective function">injective</a> (and hence not <a href="/wiki/Bijective_function" class="mw-redirect" title="Bijective function">bijective</a>), since, for example, the pre-image of <i>y</i> = 2 is {<i>x</i> = −1, <i>x</i> = 2}. (In fact, the pre-image of this function for every <i>y</i>, −2 ≤ <i>y</i> ≤ 2 has more than one element.)</li> <li>The function <span class="texhtml"><i>g</i> : <b>R</b> → <b>R</b></span> defined by <span class="nowrap"><i>g</i>(<i>x</i>) = <i>x</i><sup>2</sup></span> is <i>not</i> surjective, since there is no real number <i>x</i> such that <span class="nowrap"><i>x</i><sup>2</sup> = −1</span>. However, the function <span class="texhtml"><i>g</i> : <b>R</b> → <b>R</b><sub>≥0</sub></span> defined by <span class="texhtml"><i>g</i>(<i>x</i>) = <i>x</i><sup>2</sup></span> (with the restricted codomain) <i>is</i> surjective, since for every <i>y</i> in the nonnegative real codomain <i>Y</i>, there is at least one <i>x</i> in the real domain <i>X</i> such that <span class="nowrap"><i>x</i><sup>2</sup> = <i>y</i></span>.</li> <li>The <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> function <span class="texhtml">ln : (0, +∞) → <b>R</b></span> is a surjective and even bijective (mapping from the set of positive real numbers to the set of all real numbers). Its inverse, the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, if defined with the set of real numbers as the domain and the codomain, is not surjective (as its range is the set of positive real numbers).</li> <li>The <a href="/wiki/Matrix_exponential" title="Matrix exponential">matrix exponential</a> is not surjective when seen as a map from the space of all <i>n</i>×<i>n</i> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> to itself. It is, however, usually defined as a map from the space of all <i>n</i>×<i>n</i> matrices to the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> of degree <i>n</i> (that is, the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of all <i>n</i>×<i>n</i> <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible matrices</a>). Under this definition, the matrix exponential is surjective for complex matrices, although still not surjective for real matrices.</li> <li>The <a href="/wiki/Projection_(set_theory)" title="Projection (set theory)">projection</a> from a <a href="/wiki/Cartesian_product" title="Cartesian product">cartesian product</a> <span class="texhtml"><i>A</i> × <i>B</i></span> to one of its factors is surjective, unless the other factor is empty.</li> <li>In a 3D video game, vectors are projected onto a 2D flat screen by means of a surjective function.</li></ul> <div style="clear:both;" class=""></div> <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=Surjective_function&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A function is <a href="/wiki/Bijective_function" class="mw-redirect" title="Bijective function">bijective</a> if and only if it is both surjective and <a href="/wiki/Injective_function" title="Injective function">injective</a>. </p><p>If (as is often done) a function is identified with its <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a>, then surjectivity is not a property of the function itself, but rather a property of the <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mapping</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This is, the function together with its codomain. Unlike injectivity, surjectivity cannot be read off of the graph of the function alone. </p> <div class="mw-heading mw-heading3"><h3 id="Surjections_as_right_invertible_functions">Surjections as right invertible functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surjective_function&action=edit&section=4" title="Edit section: Surjections as right invertible functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The function <span class="nowrap"><i>g</i> : <i>Y</i> → <i>X</i></span> is said to be a <a href="/wiki/Inverse_function#Left_and_right_inverses" title="Inverse function">right inverse</a> of the function <span class="nowrap"><i>f</i> : <i>X</i> → <i>Y</i></span> if <span class="nowrap"><i>f</i>(<i>g</i>(<i>y</i>)) = <i>y</i></span> for every <i>y</i> in <i>Y</i> (<i>g</i> can be undone by <i>f</i>). In other words, <i>g</i> is a right inverse of <i>f</i> if the <a href="/wiki/Function_composition" title="Function composition">composition</a> <span class="nowrap"><i>f</i> <small>o</small> <i>g</i></span> of <i>g</i> and <i>f</i> in that order is the <a href="/wiki/Identity_function" title="Identity function">identity function</a> on the domain <i>Y</i> of <i>g</i>. The function <i>g</i> need not be a complete <a href="/wiki/Inverse_function" title="Inverse function">inverse</a> of <i>f</i> because the composition in the other order, <span class="nowrap"><i>g</i> <small>o</small> <i>f</i></span>, may not be the identity function on the domain <i>X</i> of <i>f</i>. In other words, <i>f</i> can undo or "<i>reverse</i>" <i>g</i>, but cannot necessarily be reversed by it. </p><p>Every function with a right inverse is necessarily a surjection. The proposition that every surjective function has a right inverse is equivalent to the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>. </p><p>If <span class="nowrap"><i>f</i> : <i>X</i> → <i>Y</i></span> is surjective and <i>B</i> is a <a href="/wiki/Subset" title="Subset">subset</a> of <i>Y</i>, then <span class="nowrap"><i>f</i>(<i>f</i><sup> −1</sup>(<i>B</i>)) = <i>B</i></span>. Thus, <i>B</i> can be recovered from its <a href="/wiki/Preimage" class="mw-redirect" title="Preimage">preimage</a> <span class="nowrap"><i>f</i><sup> −1</sup>(<i>B</i>)</span>. </p><p>For example, in the first illustration in the <a href="#Gallery">gallery</a>, there is some function <i>g</i> such that <i>g</i>(<i>C</i>) = 4. There is also some function <i>f</i> such that <i>f</i>(4) = <i>C</i>. It doesn't matter that <i>g</i> is not unique (it would also work if <i>g</i>(<i>C</i>) equals 3); it only matters that <i>f</i> "reverses" <i>g</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Surjections_as_epimorphisms">Surjections as epimorphisms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surjective_function&action=edit&section=5" title="Edit section: Surjections as epimorphisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A function <span class="nowrap"><i>f</i> : <i>X</i> → <i>Y</i></span> is surjective if and only if it is <a href="/wiki/Right-cancellative" class="mw-redirect" title="Right-cancellative">right-cancellative</a>:<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> given any functions <span class="nowrap"><i>g</i>,<i>h</i> : <i>Y</i> → <i>Z</i></span>, whenever <span class="nowrap"><i>g</i> <small>o</small> <i>f</i> = <i>h</i> <small>o</small> <i>f</i></span>, then <span class="nowrap"><i>g</i> = <i>h</i></span>. This property is formulated in terms of functions and their <a href="/wiki/Function_composition" title="Function composition">composition</a> and can be generalized to the more general notion of the <a href="/wiki/Morphism" title="Morphism">morphisms</a> of a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> and their composition. Right-cancellative morphisms are called <a href="/wiki/Epimorphism" title="Epimorphism">epimorphisms</a>. Specifically, surjective functions are precisely the epimorphisms in the <a href="/wiki/Category_of_sets" title="Category of sets">category of sets</a>. The prefix <i>epi</i> is derived from the Greek preposition <i>ἐπί</i> meaning <i>over</i>, <i>above</i>, <i>on</i>. </p><p>Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse <i>g</i> of a morphism <i>f</i> is called a <a href="/wiki/Section_(category_theory)" title="Section (category theory)">section</a> of <i>f</i>. A morphism with a right inverse is called a <a href="/wiki/Split_epimorphism" class="mw-redirect" title="Split epimorphism">split epimorphism</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Surjections_as_binary_relations">Surjections as binary relations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surjective_function&action=edit&section=6" title="Edit section: Surjections as binary relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any function with domain <i>X</i> and codomain <i>Y</i> can be seen as a <a href="/wiki/Left-total_relation" class="mw-redirect" title="Left-total relation">left-total</a> and <a href="/wiki/Right-unique_relation" class="mw-redirect" title="Right-unique relation">right-unique</a> binary relation between <i>X</i> and <i>Y</i> by identifying it with its <a href="/wiki/Function_graph" class="mw-redirect" title="Function graph">function graph</a>. A surjective function with domain <i>X</i> and codomain <i>Y</i> is then a binary relation between <i>X</i> and <i>Y</i> that is right-unique and both left-total and <a href="/wiki/Right-total_relation" class="mw-redirect" title="Right-total relation">right-total</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Cardinality_of_the_domain_of_a_surjection">Cardinality of the domain of a surjection</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surjective_function&action=edit&section=7" title="Edit section: Cardinality of the domain of a surjection"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If <span class="nowrap"><i>f</i> : <i>X</i> → <i>Y</i></span> is a surjective function, then <i>X</i> has at least as many elements as <i>Y</i>, in the sense of <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a>. (The proof appeals to the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> to show that a function <span class="nowrap"><i>g</i> : <i>Y</i> → <i>X</i></span> satisfying <i>f</i>(<i>g</i>(<i>y</i>)) = <i>y</i> for all <i>y</i> in <i>Y</i> exists. <i>g</i> is easily seen to be injective, thus the <a href="/wiki/Cardinal_number#Formal_definition" title="Cardinal number">formal definition</a> of |<i>Y</i>| ≤ |<i>X</i>| is satisfied.) </p><p>Specifically, if both <i>X</i> and <i>Y</i> are <a href="/wiki/Finite_set" title="Finite set">finite</a> with the same number of elements, then <span class="nowrap"><i>f</i> : <i>X</i> → <i>Y</i></span> is surjective if and only if <i>f</i> is <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a>. </p><p>Given two sets <i>X</i> and <i>Y</i>, the notation <span class="nowrap"><i>X</i> ≤<sup>*</sup> <i>Y</i></span> is used to say that either <i>X</i> is empty or that there is a surjection from <i>Y</i> onto <i>X</i>. Using the axiom of choice one can show that <span class="nowrap"><i>X</i> ≤<sup>*</sup> <i>Y</i></span> and <span class="nowrap"><i>Y</i> ≤<sup>*</sup> <i>X</i></span> together imply that <span class="nowrap">|<i>Y</i>| = |<i>X</i>|,</span> a variant of the <a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Composition_and_decomposition">Composition and decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surjective_function&action=edit&section=8" title="Edit section: Composition and decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Function_composition" title="Function composition">composition</a> of surjective functions is always surjective: If <i>f</i> and <i>g</i> are both surjective, and the codomain of <i>g</i> is equal to the domain of <i>f</i>, then <span class="nowrap"><i>f</i> <small>o</small> <i>g</i></span> is surjective. Conversely, if <span class="nowrap"><i>f</i> <small>o</small> <i>g</i></span> is surjective, then <i>f</i> is surjective (but <i>g</i>, the function applied first, need not be). These properties generalize from surjections in the <a href="/wiki/Category_of_sets" title="Category of sets">category of sets</a> to any <a href="/wiki/Epimorphism" title="Epimorphism">epimorphisms</a> in any <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>. </p><p>Any function can be decomposed into a surjection and an <a href="/wiki/Injective_function" title="Injective function">injection</a>: For any function <span class="nowrap"><i>h</i> : <i>X</i> → <i>Z</i></span> there exist a surjection <span class="nowrap"><i>f</i> : <i>X</i> → <i>Y</i></span> and an injection <span class="nowrap"><i>g</i> : <i>Y</i> → <i>Z</i></span> such that <span class="nowrap"><i>h</i> = <i>g</i> <small>o</small> <i>f</i></span>. To see this, define <i>Y</i> to be the set of <a href="/wiki/Preimage" class="mw-redirect" title="Preimage">preimages</a> <span class="nowrap"><i>h</i><sup>−1</sup>(<i>z</i>)</span> where <i>z</i> is in <span class="nowrap"><i>h</i>(<i>X</i>)</span>. These preimages are <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a> and <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> <i>X</i>. Then <i>f</i> carries each <i>x</i> to the element of <i>Y</i> which contains it, and <i>g</i> carries each element of <i>Y</i> to the point in <i>Z</i> to which <i>h</i> sends its points. Then <i>f</i> is surjective since it is a projection map, and <i>g</i> is injective by definition. </p> <div class="mw-heading mw-heading3"><h3 id="Induced_surjection_and_induced_bijection">Induced surjection and induced bijection</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surjective_function&action=edit&section=9" title="Edit section: Induced surjection and induced bijection"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any function induces a surjection by restricting its codomain to its range. Any surjective function induces a bijection defined on a <a href="/wiki/Quotient_set" class="mw-redirect" title="Quotient set">quotient</a> of its domain by collapsing all arguments mapping to a given fixed image. More precisely, every surjection <span class="nowrap"><i>f</i> : <i>A</i> → <i>B</i></span> can be factored as a projection followed by a bijection as follows. Let <i>A</i>/~ be the <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of <i>A</i> under the following <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>: <i>x</i> ~ <i>y</i> if and only if <i>f</i>(<i>x</i>) = <i>f</i>(<i>y</i>). Equivalently, <i>A</i>/~ is the set of all preimages under <i>f</i>. Let <i>P</i>(~) : <i>A</i> → <i>A</i>/~ be the <a href="/wiki/Projection_map" class="mw-redirect" title="Projection map">projection map</a> which sends each <i>x</i> in <i>A</i> to its equivalence class [<i>x</i>]<sub>~</sub>, and let <i>f</i><sub><i>P</i></sub> : <i>A</i>/~ → <i>B</i> be the well-defined function given by <i>f</i><sub><i>P</i></sub>([<i>x</i>]<sub>~</sub>) = <i>f</i>(<i>x</i>). Then <i>f</i> = <i>f</i><sub><i>P</i></sub> o <i>P</i>(~). </p> <div class="mw-heading mw-heading2"><h2 id="The_set_of_surjections">The set of surjections</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surjective_function&action=edit&section=10" title="Edit section: The set of surjections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given fixed finite sets <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span>, one can form the set of surjections <span class="texhtml"><i>A</i> ↠ <i>B</i></span>. The <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of this set is one of the twelve aspects of Rota's <a href="/wiki/Twelvefold_way" title="Twelvefold way">Twelvefold way</a>, and 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="{\textstyle |B|!{\begin{Bmatrix}|A|\\|B|\end{Bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |B|!{\begin{Bmatrix}|A|\\|B|\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c570b070912f5b4c8de4f467e02d173276ca21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11ex; height:6.176ex;" alt="{\textstyle |B|!{\begin{Bmatrix}|A|\\|B|\end{Bmatrix}}}"></span>, where <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="{\textstyle {\begin{Bmatrix}|A|\\|B|\end{Bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\begin{Bmatrix}|A|\\|B|\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d7e1ebeb5522c0f6519865f40aa1d354809c5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.295ex; height:6.176ex;" alt="{\textstyle {\begin{Bmatrix}|A|\\|B|\end{Bmatrix}}}"></span> denotes a <a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling number of the second kind</a>. </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=Surjective_function&action=edit&section=11" 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: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 <b>surjective</b> 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: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 <b>surjective</b> function (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: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:Not-Injection-Surjection.svg" class="mw-file-description" title="A non-injective non-surjective function (neither a bijection)"><img alt="A non-injective non-surjective function (neither 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 (neither a bijection)</div> </li> </ul></div></div></div> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Surjective_composition.svg" class="mw-file-description" title="Surjective composition: the first function need not be surjective."><img alt="Surjective composition: the first function need not be surjective." src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Surjective_composition.svg/120px-Surjective_composition.svg.png" decoding="async" width="120" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Surjective_composition.svg/180px-Surjective_composition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Surjective_composition.svg/240px-Surjective_composition.svg.png 2x" data-file-width="300" data-file-height="200" /></a></span></div> <div class="gallerytext">Surjective composition: the first function need not be surjective.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Non-surjective_function2.svg" class="mw-file-description" title="Non-surjective functions in the Cartesian plane. Although some parts of the function are surjective, where elements y in Y do have a value x in X such that y = f(x), some parts are not. Left: There is y0 in Y, but there is no x0 in X such that y0 = f(x0). Right: There are y1, y2 and y3 in Y, but there are no x1, x2, and x3 in X such that y1 = f(x1), y2 = f(x2), and y3 = f(x3)."><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Non-surjective_function2.svg/120px-Non-surjective_function2.svg.png" decoding="async" width="120" height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Non-surjective_function2.svg/180px-Non-surjective_function2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Non-surjective_function2.svg/240px-Non-surjective_function2.svg.png 2x" data-file-width="1055" data-file-height="621" /></a></span></div> <div class="gallerytext"><b>Non-surjective functions</b> in the Cartesian plane. Although some parts of the function are surjective, where elements <i>y</i> in <i>Y</i> do have a value <i>x</i> in <i>X</i> such that <i>y</i> = <i>f</i>(<i>x</i>), some parts are not. <b>Left:</b> There is <i>y</i><sub>0</sub> in <i>Y</i>, but there is no <i>x</i><sub>0</sub> in <i>X</i> such that <i>y</i><sub>0</sub> = <i>f</i>(<i>x</i><sub>0</sub>). <b>Right:</b> There are <i>y</i><sub>1</sub>, <i>y</i><sub>2</sub> and <i>y</i><sub>3</sub> in <i>Y</i>, but there are no <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, and <i>x</i><sub>3</sub> in <i>X</i> such that <i>y</i><sub>1</sub> = <i>f</i>(<i>x</i><sub>1</sub>), <i>y</i><sub>2</sub> = <i>f</i>(<i>x</i><sub>2</sub>), and <i>y</i><sub>3</sub> = <i>f</i>(<i>x</i><sub>3</sub>).</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Surjective_function.svg" class="mw-file-description" title="Interpretation for surjective functions in the Cartesian plane, defined by the mapping f : X → Y, where y = f(x), X = domain of function, Y = range of function. Every element in the range is mapped onto from an element in the domain, by the rule f. There may be a number of domain elements which map to the same range element. That is, every y in Y is mapped from an element x in X, more than one x can map to the same y. Left: Only one domain is shown which makes f surjective. Right: two possible domains X1 and X2 are shown."><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Surjective_function.svg/120px-Surjective_function.svg.png" decoding="async" width="120" height="63" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Surjective_function.svg/180px-Surjective_function.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Surjective_function.svg/240px-Surjective_function.svg.png 2x" data-file-width="1000" data-file-height="522" /></a></span></div> <div class="gallerytext">Interpretation for <b>surjective functions</b> in the Cartesian plane, defined by the mapping <i>f</i> : <i>X</i> → <i>Y</i>, where <i>y</i> = <i>f</i>(<i>x</i>), <i>X</i> = domain of function, <i>Y</i> = range of function. Every element in the range is mapped onto from an element in the domain, by the rule <i>f</i>. There may be a number of domain elements which map to the same range element. That is, every <i>y</i> in <i>Y</i> is mapped from an element <i>x</i> in <i>X</i>, more than one <i>x</i> can map to the same <i>y</i>. <b>Left:</b> Only one domain is shown which makes <i>f</i> surjective. <b>Right:</b> two possible domains <i>X</i><sub>1</sub> and <i>X</i><sub>2</sub> are shown.</div> </li> </ul> <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=Surjective_function&action=edit&section=12" title="Edit section: See also"><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:Surjectivity" class="extiw" title="commons:Category:Surjectivity">Surjectivity</a></span>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/surjective" class="extiw" title="wiktionary:surjective">surjective</a></b></i>, <i><b><a href="https://en.wiktionary.org/wiki/surjection" class="extiw" title="wiktionary:surjection">surjection</a></b></i>, or <i><b><a href="https://en.wiktionary.org/wiki/onto" class="extiw" title="wiktionary:onto">onto</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><a href="/wiki/Bijection,_injection_and_surjection" title="Bijection, injection and surjection">Bijection, injection and surjection</a></li> <li><a href="/wiki/Cover_(algebra)" title="Cover (algebra)">Cover (algebra)</a></li> <li><a href="/wiki/Covering_map" class="mw-redirect" title="Covering map">Covering map</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li> <li><a href="/wiki/Index_set" title="Index set">Index set</a></li> <li><a href="/wiki/Section_(category_theory)" title="Section (category theory)">Section (category theory)</a></li></ul> <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=Surjective_function&action=edit&section=13" title="Edit section: References"><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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a></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://www.mathsisfun.com/sets/injective-surjective-bijective.html">"Injective, Surjective and Bijective"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Injective%2C+Surjective+and+Bijective&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fsets%2Finjective-surjective-bijective.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASurjective+function" class="Z3988"></span></span> </li> <li id="cite_note-:1-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><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">2019-12-07</span></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%3ASurjective+function" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiller" class="citation cs2">Miller, Jeff, "Injection, Surjection and Bijection", <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/i.html"><i>Earliest Uses of Some of the Words of Mathematics</i></a>, Tripod</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Injection%2C+Surjection+and+Bijection&rft.btitle=Earliest+Uses+of+Some+of+the+Words+of+Mathematics&rft.pub=Tripod&rft.aulast=Miller&rft.aufirst=Jeff&rft_id=http%3A%2F%2Fjeff560.tripod.com%2Fi.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASurjective+function" class="Z3988"></span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMashaal2006" class="citation book cs1">Mashaal, Maurice (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-CXn6y_1nJ8C&q=injection+surjection+bijection+bourbaki&pg=PA106"><i>Bourbaki</i></a>. American Mathematical Soc. p. 106. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3967-6" title="Special:BookSources/978-0-8218-3967-6"><bdi>978-0-8218-3967-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bourbaki&rft.pages=106&rft.pub=American+Mathematical+Soc.&rft.date=2006&rft.isbn=978-0-8218-3967-6&rft.aulast=Mashaal&rft.aufirst=Maurice&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-CXn6y_1nJ8C%26q%3Dinjection%2Bsurjection%2Bbijection%2Bbourbaki%26pg%3DPA106&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASurjective+function" class="Z3988"></span></span> </li> <li id="cite_note-Unicode_Arrows-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Unicode_Arrows_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2190.pdf">"Arrows – Unicode"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2013-05-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Arrows+%E2%80%93+Unicode&rft_id=https%3A%2F%2Fwww.unicode.org%2Fcharts%2FPDF%2FU2190.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASurjective+function" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFarlow" class="citation web cs1"><a href="/wiki/Stanley_Farlow" title="Stanley Farlow">Farlow, S. J.</a> <a rel="nofollow" class="external text" href="http://www.math.umaine.edu/~farlow/sec42.pdf">"Injections, Surjections, and Bijections"</a> <span class="cs1-format">(PDF)</span>. <i>math.umaine.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=math.umaine.edu&rft.atitle=Injections%2C+Surjections%2C+and+Bijections&rft.aulast=Farlow&rft.aufirst=S.+J.&rft_id=http%3A%2F%2Fwww.math.umaine.edu%2F~farlow%2Fsec42.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASurjective+function" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFT._M._Apostol1981" class="citation book cs1">T. M. Apostol (1981). <i>Mathematical Analysis</i>. Addison-Wesley. p. 35.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Analysis&rft.pages=35&rft.pub=Addison-Wesley&rft.date=1981&rft.au=T.+M.+Apostol&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASurjective+function" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldblatt2006" class="citation book cs1">Goldblatt, Robert (2006) [1984]. <a rel="nofollow" class="external text" href="http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3"><i>Topoi, the Categorial Analysis of Logic</i></a> (Revised ed.). <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45026-1" title="Special:BookSources/978-0-486-45026-1"><bdi>978-0-486-45026-1</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2009-11-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topoi%2C+the+Categorial+Analysis+of+Logic&rft.edition=Revised&rft.pub=Dover+Publications&rft.date=2006&rft.isbn=978-0-486-45026-1&rft.aulast=Goldblatt&rft.aufirst=Robert&rft_id=http%3A%2F%2Fhistorical.library.cornell.edu%2Fcgi-bin%2Fcul.math%2Fdocviewer%3Fdid%3DGold010%26id%3D3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASurjective+function" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surjective_function&action=edit&section=14" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="bourbaki" class="citation book cs1"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, N.</a> (2004) [1968]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7eclBQAAQBAJ&pg=PR1"><i>Theory of Sets</i></a>. <a href="/wiki/Elements_of_Mathematics" class="mw-redirect" title="Elements of Mathematics">Elements of Mathematics</a>. Vol. 1. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-59309-3">10.1007/978-3-642-59309-3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-22525-6" title="Special:BookSources/978-3-540-22525-6"><bdi>978-3-540-22525-6</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2004110815">2004110815</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Sets&rft.series=Elements+of+Mathematics&rft.pub=Springer&rft.date=2004&rft_id=info%3Alccn%2F2004110815&rft_id=info%3Adoi%2F10.1007%2F978-3-642-59309-3&rft.isbn=978-3-540-22525-6&rft.aulast=Bourbaki&rft.aufirst=N.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7eclBQAAQBAJ%26pg%3DPR1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASurjective+function" class="Z3988"></span></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 solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output 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.navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></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 class="mw-selflink selflink">Sur</a>/<a href="/wiki/Bijection" title="Bijection">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 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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> 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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 href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Concrete_category" title="Concrete 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