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title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-複合函數與反函數" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#複合函數與反函數"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>複合函數與反函數</span> </div> </a> <ul id="toc-複合函數與反函數-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-雙射與勢" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#雙射與勢"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>雙射與勢</span> </div> </a> <ul id="toc-雙射與勢-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-例子與反例" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#例子與反例"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>例子與反例</span> </div> </a> <ul id="toc-例子與反例-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-性質" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#性質"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>性質</span> </div> </a> <ul id="toc-性質-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-雙射與範疇論" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#雙射與範疇論"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>雙射與範疇論</span> </div> </a> <ul id="toc-雙射與範疇論-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-另見" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#另見"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>另見</span> </div> </a> <ul id="toc-另見-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參考文獻" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參考文獻"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>參考文獻</span> </div> </a> <ul id="toc-參考文獻-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部連結" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部連結"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>外部連結</span> </div> </a> <ul id="toc-外部連結-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="目录" 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="开关目录" > <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">开关目录</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">双射</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="前往另一种语言写成的文章。55种语言可用" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-55" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">55种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%82%D8%A7%D8%A8%D9%84_(%D8%AF%D8%A7%D9%84%D8%A9)" title="تقابل (دالة) – 阿拉伯语" lang="ar" hreflang="ar" data-title="تقابل (دالة)" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%91%D1%96%D0%B5%D0%BA%D1%86%D1%8B%D1%8F" title="Біекцыя – 白俄罗斯语" lang="be" hreflang="be" data-title="Біекцыя" data-language-autonym="Беларуская" data-language-local-name="白俄罗斯语" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D0%B8%D0%B5%D0%BA%D1%86%D0%B8%D1%8F" title="Биекция – 保加利亚语" lang="bg" hreflang="bg" data-title="Биекция" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Bijekcija" title="Bijekcija – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Bijekcija" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_bijectiva" title="Funció bijectiva – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Funció bijectiva" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Biiezzioni" title="Biiezzioni – 科西嘉语" lang="co" hreflang="co" data-title="Biiezzioni" data-language-autonym="Corsu" data-language-local-name="科西嘉语" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bijekce" title="Bijekce – 捷克语" lang="cs" hreflang="cs" data-title="Bijekce" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bijektiv" title="Bijektiv – 丹麦语" lang="da" hreflang="da" data-title="Bijektiv" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bijektive_Funktion" title="Bijektive Funktion – 德语" lang="de" hreflang="de" data-title="Bijektive Funktion" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Bijection" title="Bijection – 英语" lang="en" hreflang="en" data-title="Bijection" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Dissur%C4%B5eto" title="Dissurĵeto – 世界语" lang="eo" hreflang="eo" data-title="Dissurĵeto" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_biyectiva" title="Función biyectiva – 西班牙语" lang="es" hreflang="es" data-title="Función biyectiva" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Bijektiivne_funktsioon" title="Bijektiivne funktsioon – 爱沙尼亚语" lang="et" hreflang="et" data-title="Bijektiivne funktsioon" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bijekzio" title="Bijekzio – 巴斯克语" lang="eu" hreflang="eu" data-title="Bijekzio" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%86%D8%A7%D8%B8%D8%B1_%D8%AF%D9%88%D8%B3%D9%88%DB%8C%D9%87" title="تناظر دوسویه – 波斯语" lang="fa" hreflang="fa" data-title="تناظر دوسویه" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bijektio" title="Bijektio – 芬兰语" lang="fi" hreflang="fi" data-title="Bijektio" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Bijection" title="Bijection – 法语" lang="fr" hreflang="fr" data-title="Bijection" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_bixectiva" title="Función bixectiva – 加利西亚语" lang="gl" hreflang="gl" data-title="Función bixectiva" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%97%D7%93-%D7%97%D7%93-%D7%A2%D7%A8%D7%9B%D7%99%D7%AA_%D7%95%D7%A2%D7%9C" title="פונקציה חד-חד-ערכית ועל – 希伯来语" lang="he" hreflang="he" data-title="פונקציה חד-חד-ערכית ועל" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%85%E0%A4%82%E0%A4%A4%E0%A4%A5%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%AA%E0%A4%A3" title="द्विअंतथक्षेपण – 印地语" lang="hi" hreflang="hi" data-title="द्विअंतथक्षेपण" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Bijekcija" title="Bijekcija – 克罗地亚语" lang="hr" hreflang="hr" data-title="Bijekcija" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bijekci%C3%B3" title="Bijekció – 匈牙利语" lang="hu" hreflang="hu" data-title="Bijekció" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%93%D5%B8%D5%AD%D5%B4%D5%AB%D5%A1%D6%80%D5%AA%D5%A5%D6%84_%D5%B0%D5%A1%D5%B4%D5%A1%D5%BA%D5%A1%D5%BF%D5%A1%D5%BD%D5%AD%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Փոխմիարժեք համապատասխանություն – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Փոխմիարժեք համապատասխանություն" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Bijection" title="Bijection – 国际语" lang="ia" hreflang="ia" data-title="Bijection" data-language-autonym="Interlingua" data-language-local-name="国际语" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bijeksi" title="Bijeksi – 印度尼西亚语" lang="id" hreflang="id" data-title="Bijeksi" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Bijektio" title="Bijektio – 伊多语" lang="io" hreflang="io" data-title="Bijektio" data-language-autonym="Ido" data-language-local-name="伊多语" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Gagnt%C3%A6k_v%C3%B6rpun" title="Gagntæk vörpun – 冰岛语" lang="is" hreflang="is" data-title="Gagntæk vörpun" data-language-autonym="Íslenska" data-language-local-name="冰岛语" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Corrispondenza_biunivoca" title="Corrispondenza biunivoca – 意大利语" lang="it" hreflang="it" data-title="Corrispondenza biunivoca" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%85%A8%E5%8D%98%E5%B0%84" title="全単射 – 日语" lang="ja" hreflang="ja" data-title="全単射" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D3%A8%D0%B7%D0%B0%D1%80%D0%B0_%D0%B1%D1%96%D1%80%D0%BC%D3%99%D0%BD%D0%B4%D1%96_%D1%81%D3%99%D0%B9%D0%BA%D0%B5%D1%81%D1%82%D1%96%D0%BA" title="Өзара бірмәнді сәйкестік – 哈萨克语" lang="kk" hreflang="kk" data-title="Өзара бірмәнді сәйкестік" data-language-autonym="Қазақша" data-language-local-name="哈萨克语" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%84%EB%8B%A8%EC%82%AC_%ED%95%A8%EC%88%98" title="전단사 함수 – 韩语" lang="ko" hreflang="ko" data-title="전단사 함수" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Functio_biiectiva" title="Functio biiectiva – 拉丁语" lang="la" hreflang="la" data-title="Functio biiectiva" data-language-autonym="Latina" data-language-local-name="拉丁语" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Bigezzion" title="Bigezzion – 倫巴底文" lang="lmo" hreflang="lmo" data-title="Bigezzion" data-language-autonym="Lombard" data-language-local-name="倫巴底文" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Bijekcija" title="Bijekcija – 立陶宛语" lang="lt" hreflang="lt" data-title="Bijekcija" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%B8%D1%98%D0%B5%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Бијекција – 马其顿语" lang="mk" hreflang="mk" data-title="Бијекција" data-language-autonym="Македонски" data-language-local-name="马其顿语" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bijectie" title="Bijectie – 荷兰语" lang="nl" hreflang="nl" data-title="Bijectie" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Bijeksjon" title="Bijeksjon – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Bijeksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bijeksjon" title="Bijeksjon – 书面挪威语" lang="nb" hreflang="nb" data-title="Bijeksjon" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Bijeccion" title="Bijeccion – 奥克语" lang="oc" hreflang="oc" data-title="Bijeccion" data-language-autonym="Occitan" data-language-local-name="奥克语" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja_wzajemnie_jednoznaczna" title="Funkcja wzajemnie jednoznaczna – 波兰语" lang="pl" hreflang="pl" data-title="Funkcja wzajemnie jednoznaczna" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_bijectiva" title="Função bijectiva – 葡萄牙语" lang="pt" hreflang="pt" data-title="Função bijectiva" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Coresponden%C8%9B%C4%83_biunivoc%C4%83" title="Corespondență biunivocă – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Corespondență biunivocă" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B8%D0%B5%D0%BA%D1%86%D0%B8%D1%8F" title="Биекция – 俄语" lang="ru" hreflang="ru" data-title="Биекция" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Bijection" title="Bijection – 苏格兰语" lang="sco" hreflang="sco" data-title="Bijection" data-language-autonym="Scots" data-language-local-name="苏格兰语" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Bijective_function" title="Bijective function – Simple English" lang="en-simple" hreflang="en-simple" data-title="Bijective function" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Bijekt%C3%ADvne_zobrazenie" title="Bijektívne zobrazenie – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Bijektívne zobrazenie" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Bijektivna_preslikava" title="Bijektivna preslikava – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Bijektivna preslikava" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%B8%D1%98%D0%B5%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Бијекција – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Бијекција" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bijektiv_funktion" title="Bijektiv funktion – 瑞典语" lang="sv" hreflang="sv" data-title="Bijektiv funktion" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%B0%E0%AF%81%E0%AE%B5%E0%AE%B4%E0%AE%BF%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81" title="இருவழிக்கோப்பு – 泰米尔语" lang="ta" hreflang="ta" data-title="இருவழிக்கோப்பு" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99%E0%B8%AB%E0%B8%99%E0%B8%B6%E0%B9%88%E0%B8%87%E0%B8%95%E0%B9%88%E0%B8%AD%E0%B8%AB%E0%B8%99%E0%B8%B6%E0%B9%88%E0%B8%87%E0%B8%97%E0%B8%B1%E0%B9%88%E0%B8%A7%E0%B8%96%E0%B8%B6%E0%B8%87" title="ฟังก์ชันหนึ่งต่อหนึ่งทั่วถึง – 泰语" lang="th" hreflang="th" data-title="ฟังก์ชันหนึ่งต่อหนึ่งทั่วถึง" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Birebir_%C3%B6rten_fonksiyon" title="Birebir örten fonksiyon – 土耳其语" lang="tr" hreflang="tr" data-title="Birebir örten fonksiyon" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D1%96%D1%94%D0%BA%D1%86%D1%96%D1%8F" title="Бієкція – 乌克兰语" lang="uk" hreflang="uk" data-title="Бієкція" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Song_%C3%A1nh" title="Song ánh – 越南语" lang="vi" hreflang="vi" data-title="Song ánh" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link 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class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">工具</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">隐藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多选项" > <div class="vector-menu-heading"> 操作 </div> <div 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class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%87%BA%E6%9B%B4%E6%94%B9/%E5%8F%8C%E5%B0%84" rel="nofollow" title="页面链出所有页面的更改[k]" accesskey="k"><span>相关更改</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Project:%E4%B8%8A%E4%BC%A0" title="上传图像或多媒体文件[u]" accesskey="u"><span>上传文件</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:%E7%89%B9%E6%AE%8A%E9%A1%B5%E9%9D%A2" title="全部特殊页面的列表[q]" accesskey="q"><span>特殊页面</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E5%8F%8C%E5%B0%84&oldid=80417614" title="此页面该修订版本的固定链接"><span>固定链接</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E5%8F%8C%E5%B0%84&action=info" title="关于此页面的更多信息"><span>页面信息</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:%E5%BC%95%E7%94%A8%E6%AD%A4%E9%A1%B5%E9%9D%A2&page=%E5%8F%8C%E5%B0%84&id=80417614&wpFormIdentifier=titleform" title="有关如何引用此页面的信息"><span>引用此页</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:URL%E7%BC%A9%E7%9F%AD%E7%A8%8B%E5%BA%8F&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E5%258F%258C%25E5%25B0%2584"><span>获取短链接</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E5%258F%258C%25E5%25B0%2584"><span>下载二维码</span></a></li> </ul> </div> </div> <div id="p-electronpdfservice-sidebar-portlet-heading" class="vector-menu mw-portlet mw-portlet-electronpdfservice-sidebar-portlet-heading" > <div class="vector-menu-heading"> 打印/导出 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="electron-print_pdf" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=%E5%8F%8C%E5%B0%84&action=show-download-screen"><span>下载为PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="javascript:print();" rel="alternate" title="本页面的可打印版本[p]" accesskey="p"><span>打印页面</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> 在其他项目中 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Bijectivity" hreflang="en"><span>维基共享资源</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q180907" title="链接到连接的数据仓库项目[g]" accesskey="g"><span>维基数据项目</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-noteTA-307af9e9" class="mw-indicator"><div class="mw-parser-output"><span class="skin-invert" typeof="mw:File"><span title="本页使用了标题或全文手工转换"><img alt="本页使用了标题或全文手工转换" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/35px-Zh_conversion_icon_m.svg.png" decoding="async" width="35" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/53px-Zh_conversion_icon_m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/70px-Zh_conversion_icon_m.svg.png 2x" data-file-width="32" data-file-height="20" /></span></span></div></div> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div 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.ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-os .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}</style><table class="box-No_footnotes plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/File:Text_document_with_red_question_mark.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">此條目已<a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E4%BE%86%E6%BA%90" class="mw-redirect" title="Wikipedia:列明來源">列出參考文獻</a>，但<b>因為沒有<a href="/wiki/Help:%E8%84%9A%E6%B3%A8" title="Help:脚注">文內引註</a>而使來源仍然不明</b>。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2022年4月22日</span>)</i></small><span class="hide-when-compact"><br /><small>请加上合适的文內引註来<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E5%8F%8C%E5%B0%84&action=edit">改善这篇条目</a>。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <div id="noteTA-307af9e9" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Math"></div></div></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Bijection.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/200px-Bijection.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/300px-Bijection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/400px-Bijection.svg.png 2x" data-file-width="200" data-file-height="200" /></a><figcaption>一个双射函数</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:" :"}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist-pipe dd::after,.mw-parser-output .hlist-pipe li::after{content:" | ";font-weight:normal}.mw-parser-output .hlist-hyphen dd::after,.mw-parser-output .hlist-hyphen li::after{content:" - ";font-weight:normal}.mw-parser-output .hlist-comma dd::after,.mw-parser-output .hlist-comma li::after{content:"、";font-weight:normal}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist 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.sidebar-below{background:inherit!important;color:inherit!important;border-color:#54595d!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar a:not(.new):not(.mw-selflink):link{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><style data-mw-deduplicate="TemplateStyles:r58896141">.mw-parser-output .serif{font-family:Times,serif}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><style data-mw-deduplicate="TemplateStyles:r82655520">.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><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="letter-spacing:0.0125em; background-color:#FFCC99"><a href="/wiki/%E5%87%BD%E6%95%B8" class="mw-redirect" title="函數">各種函數</a></th></tr><tr><td class="sidebar-image"><span class="serif"><span class="texhtml texhtml-big" style="font-size:250%;margin-left:2px;margin-right:2px;"><i>x</i> ↦ <i>f</i> (<i>x</i>)</span></span></td></tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; text-decoration: overline; padding: 5px 0 3px"> 不同<a href="/wiki/%E5%AE%9A%E4%B9%89%E5%9F%9F" title="定义域">定義域</a>和<a href="/wiki/%E9%99%AA%E5%9F%9F" class="mw-redirect" title="陪域">陪域</a></th></tr><tr><td class="sidebar-content" style="text-align: left; padding-left: 1.5em;"> <div class="plainlist"> <ul><li><span class="nowrap"><span title="arbitrary set" style="color:gray; padding-right: 0.3em"><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></span> <span style="font-size:150%; letter-spacing:-0.125em; padding: 0 0.3em"><a href="/wiki/%E5%B8%83%E5%B0%94%E5%80%BC%E5%87%BD%E6%95%B0" title="布尔值函数">→</a></span> <span title="Codomain of booleans" style="padding: 0 0.3em"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f0c20e9335038ffebc3536dc301978226675a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {B} }"></span>,</span> <span title="Domain of booleans" style="padding: 0 0.3em"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f0c20e9335038ffebc3536dc301978226675a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {B} }"></span></span> <span style="font-size:150%; letter-spacing:-0.125em; padding: 0 0.3em"><a href="/wiki/%E6%9C%89%E5%BA%8F%E5%AF%B9" title="有序对">→</a></span> <span title="arbitrary set" style="color:gray; padding: 0 0.3em;"><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>,</span> <span title="several boolean variables" style="padding: 0 0.3em"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {B} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {B} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/774bb74188cbd4485eb30cf91a444a9949dad247" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {B} ^{n}}"></span></span> <span style="font-size:150%; letter-spacing:-0.125em; padding: 0 0.3em"><a href="/wiki/%E5%B8%83%E5%B0%94%E5%87%BD%E6%95%B0" title="布尔函数">→</a></span> <span title="Codomain of booleans" style="padding-left: 0.3em"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f0c20e9335038ffebc3536dc301978226675a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {B} }"></span></span></span></li> <li><span class="nowrap"><span title="arbitrary set" style="color:gray; padding-right: 0.3em"><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></span> <span style="font-size:150%; letter-spacing:-0.125em; padding: 0 0.3em"><span class="ilh-all" data-orig-title="整數值函數" data-lang-code="en" data-lang-name="英语" data-foreign-title="integer-valued function"><span class="ilh-page"><a href="/w/index.php?title=%E6%95%B4%E6%95%B8%E5%80%BC%E5%87%BD%E6%95%B8&action=edit&redlink=1" class="new" title="整數值函數（页面不存在）">→</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/integer-valued_function" class="extiw" title="en:integer-valued function"><span lang="en" dir="auto">integer-valued function</span></a></span>）</span></span></span> <span title="integers" style="padding: 0 0.3em"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>,</span> <span title="integers" style="padding: 0 0.3em"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/628778fcf14bd3629e9b9ebacffa172b0ad6ce41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.061ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ^{+}}"></span></span><span style="font-size:150%; letter-spacing:-0.125em; padding: 0 0.3em"><a href="/wiki/%E5%BA%8F%E5%88%97" title="序列">→</a></span> <span title="arbitrary set" style="color:gray; padding-left: 0.3em"><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></span></span></li> <li><span class="nowrap"><span title="arbitrary set" style="color:gray; padding-right: 0.3em"><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></span> <span style="font-size:150%; letter-spacing:-0.125em; padding: 0 0.3em"><span class="ilh-all" data-orig-title="實值函數" data-lang-code="en" data-lang-name="英语" data-foreign-title="real-valued function"><span class="ilh-page"><a href="/w/index.php?title=%E5%AF%A6%E5%80%BC%E5%87%BD%E6%95%B8&action=edit&redlink=1" class="new" title="實值函數（页面不存在）">→</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/real-valued_function" class="extiw" title="en:real-valued function"><span lang="en" dir="auto">real-valued function</span></a></span>）</span></span></span> <span title="real numbers" style="padding: 0 0.3em"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>,</span> <span title="real numbers" style="padding: 0 0.3em"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span></span> <span style="font-size:150%; letter-spacing:-0.125em; padding: 0 0.3em"><a href="/wiki/%E5%AE%9E%E5%8F%98%E5%87%BD%E6%95%B0" class="mw-redirect" title="实变函数">→</a></span> <span title="arbitrary set" style="color:gray; padding: 0 0.3em"><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>,</span> <span title="real coordinate (or Euclidean) space" style="padding: 0 0.3em"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span></span> <span style="font-size:150%; letter-spacing:-0.125em; padding: 0 0.3em"><span class="ilh-all" data-orig-title="多實變函數" data-lang-code="en" data-lang-name="英语" data-foreign-title="function of several real variables"><span class="ilh-page"><a href="/w/index.php?title=%E5%A4%9A%E5%AF%A6%E8%AE%8A%E5%87%BD%E6%95%B8&action=edit&redlink=1" class="new" title="多實變函數（页面不存在）">→</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/function_of_several_real_variables" class="extiw" title="en:function of several real variables"><span lang="en" dir="auto">function of several real variables</span></a></span>）</span></span></span> <span title="arbitrary set" style="color:gray; padding-left: 0.3em"><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></span></span></li> <li><span class="nowrap"><span title="arbitrary set" style="color:gray; padding-right:0.3em"><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></span> <span style="font-size:150%; letter-spacing:-0.125em; padding: 0 0.3em"><span class="ilh-all" data-orig-title="複值函數" data-lang-code="en" data-lang-name="英语" data-foreign-title="complex-valued function"><span class="ilh-page"><a href="/w/index.php?title=%E8%A4%87%E5%80%BC%E5%87%BD%E6%95%B8&action=edit&redlink=1" class="new" title="複值函數（页面不存在）">→</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/complex-valued_function" class="extiw" title="en:complex-valued function"><span lang="en" dir="auto">complex-valued function</span></a></span>）</span></span></span> <span title="complex numbers" style="padding: 0 0.3em"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>,</span> <span title="complex numbers" style="padding: 0 0.3em"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span></span> <span style="font-size:150%; letter-spacing:-0.125em; padding: 0 0.3em"><a href="/wiki/%E5%A4%8D%E5%8F%98%E5%87%BD%E6%95%B0" class="mw-redirect" title="复变函数">→</a></span> <span title="arbitrary set" style="color:gray; padding: 0 0.3em"><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>,</span> <span title="complex coordinate space" style="padding: 0 0.3em"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span></span> <span style="font-size:150%; letter-spacing:-0.125em; padding: 0 0.3em"><span class="ilh-all" data-orig-title="多複變函數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Function of several complex variables"><span class="ilh-page"><a href="/w/index.php?title=%E5%A4%9A%E8%A4%87%E8%AE%8A%E5%87%BD%E6%95%B8&action=edit&redlink=1" class="new" title="多複變函數（页面不存在）">→</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Function_of_several_complex_variables" class="extiw" title="en:Function of several complex variables"><span lang="en" dir="auto">Function of several complex variables</span></a></span>）</span></span></span> <span title="arbitrary set" style="color:gray; padding-left: 0.3em"><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></span></span></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; text-decoration: overline; padding: 5px 0 3px"> <a href="/wiki/%E5%87%BD%E6%95%B0%E7%A9%BA%E9%97%B4" title="函数空间">函數類/性質</a> </th></tr><tr><td class="sidebar-content"> <div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/%E5%B8%B8%E6%95%B8%E5%87%BD%E6%95%B8" title="常數函數">常值</a></li> <li><a href="/wiki/%E6%81%86%E7%AD%89%E5%87%BD%E6%95%B8" title="恆等函數">恆等</a></li> <li><a href="/wiki/%E7%B7%9A%E6%80%A7%E6%98%A0%E5%B0%84" class="mw-redirect" title="線性映射">線性</a></li> <li><a href="/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F" title="多項式">多項式</a></li> <li><a href="/wiki/%E6%9C%89%E7%90%86%E5%87%BD%E6%95%B8" title="有理函數">有理</a></li> <li><a href="/wiki/%E4%BB%A3%E6%95%B8%E5%87%BD%E6%95%B8" title="代數函數">代數</a></li> <li><a href="/wiki/%E8%A7%A3%E6%9E%90%E5%87%BD%E6%95%B0" title="解析函数">解析</a></li> <li><a href="/wiki/%E5%85%89%E6%BB%91%E5%87%BD%E6%95%B0" title="光滑函数">光滑</a></li> <li><a href="/wiki/%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0" title="连续函数">連續</a></li> <li><a href="/wiki/%E5%8F%AF%E6%B5%8B%E5%87%BD%E6%95%B0" title="可测函数">可測</a></li> <li><a href="/wiki/%E5%8D%95%E5%B0%84" title="单射">單射</a></li> <li><a href="/wiki/%E6%BB%A1%E5%B0%84" title="满射">满射</a></li> <li><a class="mw-selflink selflink">双射</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; text-decoration: overline; padding: 5px 0 3px"> 構造</th></tr><tr><td class="sidebar-content"> <div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/%E9%99%90%E5%88%B6_(%E6%95%B8%E5%AD%B8)" title="限制 (數學)">限制</a></li> <li><a href="/wiki/%E5%A4%8D%E5%90%88%E5%87%BD%E6%95%B0" title="复合函数">複合</a></li> <li><a href="/wiki/%CE%9B%E6%BC%94%E7%AE%97" title="Λ演算">λ</a></li> <li><a href="/wiki/%E5%8F%8D%E5%87%BD%E6%95%B8" title="反函數">逆</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; text-decoration: overline; padding: 5px 0 3px"> 推廣</th></tr><tr><td class="sidebar-content"> <div class="hlist" style="margin-left: 0em;"> <ul><li><span class="ilh-all" data-orig-title="偏函數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Partial function"><span class="ilh-page"><a href="/wiki/%E5%81%8F%E5%87%BD%E6%95%B0" class="mw-redirect" title="偏函数">偏</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Partial_function" class="extiw" title="en:Partial function"><span lang="en" dir="auto">Partial function</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%A4%9A%E5%80%BC%E5%87%BD%E6%95%B0" title="多值函数">多值</a></li> <li><a href="/wiki/%E9%9A%B1%E5%87%BD%E6%95%B8" class="mw-redirect" title="隱函數">隱</a></li></ul> </div></td> </tr><tr><td class="sidebar-navbar" style="line-height:1.6"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><style data-mw-deduplicate="TemplateStyles:r84244141">.mw-parser-output .navbar{display:inline;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:110%;margin:0 8em}.mw-parser-output .navbar-ct-mini{font-size:110%;margin:0 5em}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:%E5%90%84%E7%A8%AE%E5%87%BD%E6%95%B8" title="Template:各種函數"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E5%90%84%E7%A8%AE%E5%87%BD%E6%95%B8" title="Template talk:各種函數"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E5%90%84%E7%A8%AE%E5%87%BD%E6%95%B8" title="Special:编辑页面/Template:各種函數"><abbr title="编辑该模板">编</abbr></a></li></ul></div></td></tr></tbody></table> <p><a href="/wiki/%E6%95%B8%E5%AD%B8" class="mw-redirect" title="數學">數學</a>中，一個由<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</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><a href="/wiki/%E6%98%A0%E5%B0%84" title="映射">映射</a>至集合<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>的<a href="/wiki/%E5%87%BD%E6%95%B8" class="mw-redirect" title="函數">函數</a>，若對每一在<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>內的<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>，存在唯一一個在<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>內的<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>与其对应，且對每一在<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>內的<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>，存在唯一一個在<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>內的<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>与其对应，則此函數為<b>對射函數</b>。 </p><p>換句話說，如果其為兩集合間的<b>一一對應</b>，则<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>是雙射的。即，同時為<a href="/wiki/%E5%8D%95%E5%B0%84" title="单射">單射</a>和<a href="/wiki/%E6%BB%BF%E5%B0%84" class="mw-redirect" title="滿射">滿射</a>。 </p><p>例如，由<a href="/wiki/%E6%95%B4%E6%95%B8" class="mw-redirect" title="整數">整數</a>集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>至<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>的函數<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 \operatorname {succ} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>succ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {succ} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c47f9bab66ee2d9e9015a89cce54ef2a025e6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.274ex; height:1.676ex;" alt="{\displaystyle \operatorname {succ} }"></span>，其將每一個整數<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>連結至整數<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 \operatorname {succ} (x)=x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>succ</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {succ} (x)=x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a533bf92b956da08f787bcbc4eb867f6ae8656" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.844ex; height:2.843ex;" alt="{\displaystyle \operatorname {succ} (x)=x+1}"></span>，這是一個雙射函數；再看一個例子，函數<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 \operatorname {sumdif} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sumdif</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sumdif} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05a0a6235de6f455922052fdef24c5c0bc867e0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.153ex; width:6.949ex; height:2.176ex;" alt="{\displaystyle \operatorname {sumdif} }"></span>，其將每一對實數<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span>連結至<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 \operatorname {sumdif} (x,y)=(x+y,x-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sumdif</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sumdif} (x,y)=(x+y,x-y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb252ee525b8b7c6214b749bb2cc3f19b526c9f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.717ex; height:2.843ex;" alt="{\displaystyle \operatorname {sumdif} (x,y)=(x+y,x-y)}"></span>，這也是個雙射函數。 </p><p>一雙射函數亦簡稱為<b>雙射</b>（英語：<span lang="en">bijection</span>）或<b><a href="/wiki/%E7%BD%AE%E6%8F%9B" class="mw-redirect" title="置換">置換</a></b>。後者一般較常使用在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b549c2e838e7da9dfd27fe4e1eac711331bdef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle X=Y}"></span>時。以由<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>至<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>的所有雙射組成的集合標記為<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\leftrightarrow Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">↔</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\leftrightarrow Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b69b60af3edb830806b7a11982101e7facaa3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.367ex; height:2.176ex;" alt="{\displaystyle X\leftrightarrow Y}"></span>。 </p><p>雙射函數在許多數學領域扮演著很基本的角色，如在<a href="/wiki/%E5%90%8C%E6%A7%8B" class="mw-redirect" title="同構">同構</a>的定義（以及如<a href="/wiki/%E5%90%8C%E8%83%9A" title="同胚">同胚</a>和<a href="/w/index.php?title=%E5%BE%AE%E5%88%86%E5%90%8C%E6%A7%8B&action=edit&redlink=1" class="new" title="微分同構（页面不存在）">微分同構</a>等相關概念）、<a href="/wiki/%E7%BD%AE%E6%8F%9B%E7%BE%A4" class="mw-redirect" title="置換群">置換群</a>、<a href="/wiki/%E6%8A%95%E5%BD%B1%E6%98%A0%E5%B0%84" class="mw-redirect" title="投影映射">投影映射</a>及許多其他概念的基本上。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="複合函數與反函數"><span id=".E8.A4.87.E5.90.88.E5.87.BD.E6.95.B8.E8.88.87.E5.8F.8D.E5.87.BD.E6.95.B8"></span>複合函數與反函數</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8F%8C%E5%B0%84&action=edit&section=1" title="编辑章节：複合函數與反函數"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一函數<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>為雙射的若且唯若其<a href="/w/index.php?title=%E9%80%86%E9%97%9C%E4%BF%82&action=edit&redlink=1" class="new" title="逆關係（页面不存在）">逆關係</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}"></span>也是個函數。在這情況，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}"></span>也會是雙射函數。 </p><p>兩個雙射函數<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\leftrightarrow Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">↔</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\leftrightarrow Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/250a7f8d1f561ae2476b116ae48759dd1a76faff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\leftrightarrow Y}"></span>及<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:Y\leftrightarrow Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">↔</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:Y\leftrightarrow Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca791db452490402e1cd8a1d103867316b6eb689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.121ex; height:2.509ex;" alt="{\displaystyle g:Y\leftrightarrow Z}"></span>的<a href="/wiki/%E5%A4%8D%E5%90%88%E5%87%BD%E6%95%B0" title="复合函数">複合函數</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"></span>亦為雙射函數。其反函數為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g\circ f)^{-1}=(f^{-1})\circ (g^{-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∘</mo> <mo stretchy="false">(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g\circ f)^{-1}=(f^{-1})\circ (g^{-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4734a58f6efc8ae56e243330b4fb909e078cfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.747ex; height:3.176ex;" alt="{\displaystyle (g\circ f)^{-1}=(f^{-1})\circ (g^{-1})}"></span>。 </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Bijective_composition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Bijective_composition.svg/300px-Bijective_composition.svg.png" decoding="async" width="300" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Bijective_composition.svg/450px-Bijective_composition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Bijective_composition.svg/600px-Bijective_composition.svg.png 2x" data-file-width="300" data-file-height="200" /></a><figcaption>一个複合所得的双射，左侧为单射，右侧为满射。</figcaption></figure> <p>另一方面，若<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"></span>為雙射的，可知<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>是單射的且<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>是滿射的，但也僅限於此。 </p><p>一由<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>至<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>的關係<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>為雙射函數若且唯若存在另一由<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>至<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>的關係<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>，使得<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"></span>為<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>上的<a href="/wiki/%E6%81%86%E7%AD%89%E5%87%BD%E6%95%B8" title="恆等函數">恆等函數</a>，且<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}"></span>為<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>上的<a href="/wiki/%E6%81%86%E7%AD%89%E5%87%BD%E6%95%B8" title="恆等函數">恆等函數</a>。必然地，此兩個集合會有相同的<a href="/wiki/%E5%8A%BF_(%E6%95%B0%E5%AD%A6)" title="势 (数学)">勢</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="雙射與勢"><span id=".E9.9B.99.E5.B0.84.E8.88.87.E5.8B.A2"></span>雙射與勢</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8F%8C%E5%B0%84&action=edit&section=2" title="编辑章节：雙射與勢"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>若<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>和<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>為<a href="/wiki/%E6%9C%89%E9%99%90%E9%9B%86%E5%90%88" title="有限集合">有限集合</a>，則其存在一兩集合的雙射函數<a href="/wiki/%E8%8B%A5%E4%B8%94%E5%94%AF%E8%8B%A5" class="mw-redirect" title="若且唯若">若且唯若</a>兩個集合有相同的元素個數。確實，在<a href="/wiki/%E5%85%AC%E7%90%86%E9%9B%86%E5%90%88%E8%AB%96" class="mw-redirect" title="公理集合論">公理集合論</a>裡，這正是「相同元素個數」的<i>定義</i>，且廣義化至<a href="/wiki/%E7%84%A1%E7%AA%AE" class="mw-redirect" title="無窮">無限</a>集合，並導致了<a href="/wiki/%E5%9F%BA%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="基数 (数学)">基數</a>的概念，用以分辨<a href="/wiki/%E6%97%A0%E9%99%90%E9%9B%86%E5%90%88" title="无限集合">無限集合</a>的不同大小。 </p> <div class="mw-heading mw-heading2"><h2 id="例子與反例"><span id=".E4.BE.8B.E5.AD.90.E8.88.87.E5.8F.8D.E4.BE.8B"></span>例子與反例</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8F%8C%E5%B0%84&action=edit&section=3" title="编辑章节：例子與反例"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>對任一集合<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>，其<a href="/wiki/%E6%81%86%E7%AD%89%E5%87%BD%E6%95%B8" title="恆等函數">恆等函數</a>為雙射函數。</li> <li>函數<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e6e186aabef9e51814bbce62e625dc67e825f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }"></span>，其形式為<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)=2x+1}"> <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> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=2x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ca6b62bf1326a2e8672de9d2a8bfa95240fd76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.011ex; height:2.843ex;" alt="{\displaystyle f(x)=2x+1}"></span>，是雙射的，因為對任一<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>，存在一唯一<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(y-1)/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=(y-1)/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c384f4fa834723c844231d3d4d814a4436a7a98f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.721ex; height:2.843ex;" alt="{\displaystyle x=(y-1)/2}"></span>使得<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>。</li> <li><a href="/wiki/%E6%8C%87%E6%95%B8%E5%87%BD%E6%95%B8" class="mw-redirect" title="指數函數">指數函數</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf02b0de20abe9138ba32a40f4fe077f88ecd52d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.023ex; height:2.509ex;" alt="{\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }"></span>，其形式為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=e^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=e^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6141bc974ca74917c8ed2d48d781013d3c69c8f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.609ex; height:2.843ex;" alt="{\displaystyle g(x)=e^{x}}"></span>，不是雙射的：因為不存在一<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>內的<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>使得<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/816206666047292cf551c61f7a3c78e392656e73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.324ex; height:2.843ex;" alt="{\displaystyle g(x)=-1}"></span>，故<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>非為雙射。但若其<a href="/wiki/%E5%88%B0%E8%BE%BE%E5%9F%9F" title="到达域">陪域</a>改成正實數<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{+}=(0,+\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{+}=(0,+\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b417d9c06adc5e0bf3e83a4a4794a893ab408cb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.425ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{+}=(0,+\infty )}"></span>，則<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>便是雙射的了；其反函數為<a href="/wiki/%E8%87%AA%E7%84%B6%E5%B0%8D%E6%95%B8" title="自然對數">自然對數</a>函數<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 \ln }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0de5ba4f372ede555d00035e70c50ed0b9625d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \ln }"></span>。</li> <li>函數<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \rightarrow [0,+\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \rightarrow [0,+\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74a09a6dfe4418579f96387aa588050dbb1038b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.172ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \rightarrow [0,+\infty )}"></span>，其形式為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)=x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)=x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0f1d2ff490196a7920989fd59f70fdf9464519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.96ex; height:3.176ex;" alt="{\displaystyle h(x)=x^{2}}"></span>，不是雙射的：因為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(-1)=h(1)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(-1)=h(1)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a873d1a6e15c1783da97fc54bcd34bc3cc0501b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.789ex; height:2.843ex;" alt="{\displaystyle h(-1)=h(1)=1}"></span>，故<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>非為雙射。但如果把定義域也改成<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,+\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,+\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c0ec7f25cac88c59009a7fe528dc000ec7f58c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle [0,+\infty )}"></span>，則<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>便是雙射的了；其反函數為正平方根函數。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \to \mathbb {R} :x\mapsto (x-1)x(x+1)=x^{3}-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \to \mathbb {R} :x\mapsto (x-1)x(x+1)=x^{3}-x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f326bc190bd1ff6de1264e44a02002f68756aa08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.117ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} \to \mathbb {R} :x\mapsto (x-1)x(x+1)=x^{3}-x}"></span>不是雙射函數，因為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1,0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1,0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ecc0cf14129eadc29034c37c3514979e9a22f62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.167ex; height:2.509ex;" alt="{\displaystyle -1,0}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>都在其定義域裡且都映射至<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \to [-1,1]:x\mapsto \sin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \to [-1,1]:x\mapsto \sin(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb23cd8a0d8771ac197194b8610e7451013d752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.628ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \to [-1,1]:x\mapsto \sin(x)}"></span>不是雙射函數，因為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56c1a0cd8279cea58b0ccb583e75a0ee93975883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /3}"></span>和2<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56c1a0cd8279cea58b0ccb583e75a0ee93975883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /3}"></span>都在其定義域裡且都映射至<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 {\sqrt {3}}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3245e1141ec36a954dd702c886bba16d8c6cb057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt {3}}/2}"></span>。</li></ul> <div class="mw-heading mw-heading2"><h2 id="性質"><span id=".E6.80.A7.E8.B3.AA"></span>性質</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8F%8C%E5%B0%84&action=edit&section=4" title="编辑章节：性質"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>一由<a href="/wiki/%E5%AF%A6%E6%95%B8" class="mw-redirect" title="實數">實數</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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>至<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>的函數<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>是雙射的，若且唯若其<a href="/wiki/%E5%87%BD%E6%95%B0%E5%9B%BE%E5%83%8F" class="mw-redirect" title="函数图像">圖像</a>和任一水平線相交且只相交於一點。</li> <li>設<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>為一集合，則由<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>至其本身的雙射函數，加上其複合函數「<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 \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"></span>」的運算，會形成一個<a href="/wiki/%E7%BE%A4" title="群">群</a>，即為<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>的<a href="/wiki/%E5%AF%B9%E7%A7%B0%E7%BE%A4_(n%E6%AC%A1%E5%AF%B9%E7%A7%B0%E7%BE%A4)" title="对称群 (n次对称群)">對稱群</a>，其標記為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {S}}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {S}}(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e139938b7350f53d990871f82355d651f2ea584" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {S}}(X)}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {S}}_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {S}}_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20e578e741e7bcf81385426c3d9ef1ed670a9bdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.559ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {S}}_{X}}"></span>或<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> <mo>!</mo> </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/bbadb5c2252c9448fef9b280fae7a88ab82dd17e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X!}"></span>。</li> <li>取一定義域的子集<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>及一陪域的子集<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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>，則</li></ul> <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 |f(A)|=|A|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(A)|=|A|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0404ceeca4cdc53d14d914818c736f03a7049e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.26ex; height:2.843ex;" alt="{\displaystyle |f(A)|=|A|}"></span>且<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f^{-1}(B)|=|B|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f^{-1}(B)|=|B|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3b2e33fcab76d6fd68fcc4f7040906fdd3ac760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.676ex; height:3.176ex;" alt="{\displaystyle |f^{-1}(B)|=|B|}"></span>。</dd></dl> <ul><li>若<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>和<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>為具相同<a href="/wiki/%E5%8A%BF_(%E6%95%B0%E5%AD%A6)" title="势 (数学)">勢</a>的<a href="/wiki/%E6%9C%89%E9%99%90%E9%9B%86%E5%90%88" title="有限集合">有限集合</a>，且<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"></span>，則下列三種說法是等價的：</li></ul> <dl><dd><ol><li><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>為一雙射函數。</li> <li><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>為一滿射函數。</li> <li><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>為一單射函數。</li></ol></dd></dl> <ul><li>一个严格的单调函数是双射函数，但双射函数不一定是单调函数（例如<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=x^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x^{-3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8084158975ac0be71804cdfb0b782076c8ee0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.916ex; height:3.009ex;" alt="{\displaystyle y=x^{-3}}"></span>）。</li></ul> <div class="mw-heading mw-heading2"><h2 id="雙射與範疇論"><span id=".E9.9B.99.E5.B0.84.E8.88.87.E7.AF.84.E7.96.87.E8.AB.96"></span>雙射與範疇論</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8F%8C%E5%B0%84&action=edit&section=5" title="编辑章节：雙射與範疇論"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>形式上，雙射函數恰好是<a href="/wiki/%E9%9B%86%E5%90%88%E7%AF%84%E7%96%87" class="mw-redirect" title="集合範疇">集合範疇</a>內的<a href="/wiki/%E5%90%8C%E6%A7%8B" class="mw-redirect" title="同構">同構</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="另見"><span id=".E5.8F.A6.E8.A6.8B"></span>另見</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8F%8C%E5%B0%84&action=edit&section=6" title="编辑章节：另見"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E7%AD%89%E5%8A%BF" title="等势">等势</a></li> <li><a href="/wiki/%E5%8D%95%E5%B0%84" title="单射">單射</a></li> <li><a href="/wiki/%E5%90%8C%E6%A7%8B" class="mw-redirect" title="同構">同構</a></li> <li><a href="/wiki/%E7%BD%AE%E6%8F%9B" class="mw-redirect" title="置換">置換</a></li> <li><a href="/wiki/%E5%AF%B9%E7%A7%B0%E7%BE%A4_(n%E6%AC%A1%E5%AF%B9%E7%A7%B0%E7%BE%A4)" title="对称群 (n次对称群)">對稱群</a></li> <li><a href="/wiki/%E6%BB%A1%E5%B0%84" title="满射">满射</a></li> <li><a href="/w/index.php?title=%E9%9B%99%E5%B0%84%E8%A8%88%E6%95%B8%E6%B3%95&action=edit&redlink=1" class="new" title="雙射計數法（页面不存在）">雙射計數法</a></li> <li><a href="/wiki/%E6%B0%B4%E5%B9%B3%E7%B7%9A%E6%B8%AC%E8%A9%A6" title="水平線測試">水平线测试</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="參考文獻"><span id=".E5.8F.83.E8.80.83.E6.96.87.E7.8D.BB"></span>參考文獻</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8F%8C%E5%B0%84&action=edit&section=7" title="编辑章节：參考文獻"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r80540462">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .refbegin-100{font-size:100%}</style><div class="refbegin columns references-column-count references-column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2;"> <ul><li><cite class="citation book">Wolf. Proof, Logic and Conjecture: A Mathematician's Toolbox. Freeman. 1998.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Wolf&rft.btitle=Proof%2C+Logic+and+Conjecture%3A+A+Mathematician%27s+Toolbox&rft.date=1998&rft.genre=book&rft.pub=Freeman&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Sundstrom. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalreas0000sund">Mathematical Reasoning: Writing and Proof</a>. Prentice-Hall. 2003.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Sundstrom&rft.btitle=Mathematical+Reasoning%3A+Writing+and+Proof&rft.date=2003&rft.genre=book&rft.pub=Prentice-Hall&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalreas0000sund&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Smith; Eggen; St.Andre. A Transition to Advanced Mathematics (6th Ed.). Thomson (Brooks/Cole). 2006.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Eggen&rft.au=Smith&rft.au=St.Andre&rft.btitle=A+Transition+to+Advanced+Mathematics+%286th+Ed.%29&rft.date=2006&rft.genre=book&rft.pub=Thomson+%28Brooks%2FCole%29&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Schumacher. <a rel="nofollow" class="external text" href="https://archive.org/details/chapterzerofunda0000schu">Chapter Zero: Fundamental Notions of Abstract Mathematics</a>. Addison-Wesley. 1996.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Schumacher&rft.btitle=Chapter+Zero%3A+Fundamental+Notions+of+Abstract+Mathematics&rft.date=1996&rft.genre=book&rft.pub=Addison-Wesley&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fchapterzerofunda0000schu&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">O'Leary. The Structure of Proof: With Logic and Set Theory. Prentice-Hall. 2003.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=O%27Leary&rft.btitle=The+Structure+of+Proof%3A+With+Logic+and+Set+Theory&rft.date=2003&rft.genre=book&rft.pub=Prentice-Hall&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Morash. Bridge to Abstract Mathematics. Random House.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Morash&rft.btitle=Bridge+to+Abstract+Mathematics&rft.genre=book&rft.pub=Random+House&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Maddox. Mathematical Thinking and Writing. Harcourt/ Academic Press. 2002.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Maddox&rft.btitle=Mathematical+Thinking+and+Writing&rft.date=2002&rft.genre=book&rft.pub=Harcourt%2F+Academic+Press&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Lay. Analysis with an introduction to proof. Prentice Hall. 2001.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Lay&rft.btitle=Analysis+with+an+introduction+to+proof&rft.date=2001&rft.genre=book&rft.pub=Prentice+Hall&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Gilbert; Vanstone. An Introduction to Mathematical Thinking. Pearson Prentice-Hall. 2005.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Gilbert&rft.au=Vanstone&rft.btitle=An+Introduction+to+Mathematical+Thinking&rft.date=2005&rft.genre=book&rft.pub=Pearson+Prentice-Hall&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Fletcher; Patty. <a rel="nofollow" class="external text" href="https://archive.org/details/foundationsofhig0000flet">Foundations of Higher Mathematics</a>. PWS-Kent. 1992.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Fletcher&rft.au=Patty&rft.btitle=Foundations+of+Higher+Mathematics&rft.date=1992&rft.genre=book&rft.pub=PWS-Kent&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffoundationsofhig0000flet&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Iglewicz; Stoyle. An Introduction to Mathematical Reasoning. MacMillan.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Iglewicz&rft.au=Stoyle&rft.btitle=An+Introduction+to+Mathematical+Reasoning&rft.genre=book&rft.pub=MacMillan&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Devlin, Keith. Sets, Functions, and Logic: An Introduction to Abstract Mathematics. Chapman & Hall/ CRC Press. 2004.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.aufirst=Keith&rft.aulast=Devlin&rft.btitle=Sets%2C+Functions%2C+and+Logic%3A+An+Introduction+to+Abstract+Mathematics&rft.date=2004&rft.genre=book&rft.pub=Chapman+%26+Hall%2F+CRC+Press&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">D'Angelo; West. <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_8800003757534">Mathematical Thinking: Problem Solving and Proofs</a>. Prentice Hall. 2000.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=D%27Angelo&rft.au=West&rft.btitle=Mathematical+Thinking%3A+Problem+Solving+and+Proofs&rft.date=2000&rft.genre=book&rft.pub=Prentice+Hall&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_8800003757534&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Cupillari. <a rel="nofollow" class="external text" href="https://archive.org/details/nutsboltsofproof00anto">The Nuts and Bolts of Proofs</a>. Wadsworth. 1989.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Cupillari&rft.btitle=The+Nuts+and+Bolts+of+Proofs&rft.date=1989&rft.genre=book&rft.pub=Wadsworth&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnutsboltsofproof00anto&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Bond. Introduction to Abstract Mathematics. Brooks/Cole.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Bond&rft.btitle=Introduction+to+Abstract+Mathematics&rft.genre=book&rft.pub=Brooks%2FCole&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Barnier; Feldman. Introduction to Advanced Mathematics. Prentice Hall. 2000.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Barnier&rft.au=Feldman&rft.btitle=Introduction+to+Advanced+Mathematics&rft.date=2000&rft.genre=book&rft.pub=Prentice+Hall&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Ash. <a rel="nofollow" class="external text" href="https://archive.org/details/primerofabstract0000ashr">A Primer of Abstract Mathematics</a>. MAA. 1998.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.au=Ash&rft.btitle=A+Primer+of+Abstract+Mathematics&rft.date=1998&rft.genre=book&rft.pub=MAA&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprimerofabstract0000ashr&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="外部連結"><span id=".E5.A4.96.E9.83.A8.E9.80.A3.E7.B5.90"></span>外部連結</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8F%8C%E5%B0%84&action=edit&section=8" title="编辑章节：外部連結"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r82655521">.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:#f9f9f9;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}}@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><div class="side-box side-box-right plainlinks sistersitebox" style="font-size:small;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r82655520"> <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/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/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/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"><a href="/wiki/%E7%BB%B4%E5%9F%BA%E5%85%B1%E4%BA%AB%E8%B5%84%E6%BA%90" title="维基共享资源">维基共享资源</a>中相关的多媒体资源：<b><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Bijectivity?uselang=zh">Bijectivity</a></b></div></div> </div> <ul><li><cite id="CITEREFHazewinkel2001" class="citation">Hazewinkel, Michiel (编), <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=p/b016230">Bijection</a>, <a href="/wiki/%E6%95%B0%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6" title="数学百科全书">数学百科全书</a>, <a href="/wiki/Springer_Science%2BBusiness_Media" class="mw-redirect" title="Springer Science+Business Media">Springer</a>, 2001, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-1-55608-010-4" title="Special:网络书源/978-1-55608-010-4"><span title="国际标准书号">ISBN</span> 978-1-55608-010-4</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.atitle=Bijection&rft.aufirst=Michiel&rft.aulast=Hazewinkel&rft.btitle=%E6%95%B0%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6&rft.date=2001&rft.genre=bookitem&rft.isbn=978-1-55608-010-4&rft.pub=Springer&rft_id=http%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3Dp%2Fb016230&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Bijection"><cite class="citation web"><a href="/wiki/%E5%9F%83%E9%87%8C%E5%85%8B%C2%B7%E9%9F%A6%E6%96%AF%E5%9D%A6%E5%9B%A0" title="埃里克·韦斯坦因">埃里克·韦斯坦因</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Bijection.html">Bijection</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8F%8C%E5%B0%84&rft.atitle=Bijection&rft.au=%E5%9F%83%E9%87%8C%E5%85%8B%C2%B7%E9%9F%A6%E6%96%AF%E5%9D%A6%E5%9B%A0&rft.genre=unknown&rft.jtitle=MathWorld&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FBijection.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span></li> <li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/i.html">Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.</a>（<a rel="nofollow" class="external text" href="//web.archive.org/web/20170817162925/http://jeff560.tripod.com/i.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><style data-mw-deduplicate="TemplateStyles:r84261037">.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 .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{text-align:center;padding-left:1em;padding-right:1em}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{border-color:#fdfdfd}.mw-parser-output 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.navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em;width:auto;padding-left:0.2em;position:absolute;left:1em}.mw-parser-output .navbox .mw-collapsible-toggle{margin-left:0.5em;position:absolute;right:1em}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><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"></div><div role="navigation" class="navbox" aria-labelledby="集合论" 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="collapsible-title navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84244141"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E9%9B%86%E5%90%88%E8%AE%BA" title="Template:集合论"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E9%9B%86%E5%90%88%E8%AE%BA" title="Template talk:集合论"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E9%9B%86%E5%90%88%E8%AE%BA" title="Special:编辑页面/Template:集合论"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="集合论" style="font-size:110%;margin:0 5em"><a href="/wiki/%E9%9B%86%E5%90%88%E8%AE%BA" title="集合论">集合论</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E5%85%AC%E7%90%86" title="公理">公理</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%80%89%E6%8B%A9%E5%85%AC%E7%90%86" title="选择公理">选择</a> <ul><li><a href="/wiki/%E5%8F%AF%E6%95%B0%E9%80%89%E6%8B%A9%E5%85%AC%E7%90%86" title="可数选择公理">可数</a></li> <li><a href="/wiki/%E4%BE%9D%E8%B3%B4%E9%81%B8%E6%93%87%E5%85%AC%E7%90%86" title="依賴選擇公理">依賴</a></li></ul></li> <li><a href="/wiki/%E5%A4%96%E5%BB%B6%E5%85%AC%E7%90%86" title="外延公理">外延</a></li> <li><a href="/wiki/%E6%97%A0%E7%A9%B7%E5%85%AC%E7%90%86" title="无穷公理">无穷</a></li> <li><a href="/wiki/%E9%85%8D%E5%AF%B9%E5%85%AC%E7%90%86" title="配对公理">配对</a></li> <li><a href="/wiki/%E5%B9%82%E9%9B%86%E5%85%AC%E7%90%86" title="幂集公理">幂集</a></li> <li><a href="/wiki/%E6%AD%A3%E5%88%99%E6%80%A7%E5%85%AC%E7%90%86" title="正则性公理">正则性</a></li> <li><a href="/wiki/%E5%B9%B6%E9%9B%86%E5%85%AC%E7%90%86" title="并集公理">并集</a></li> <li><a href="/wiki/%E9%A6%AC%E4%B8%81%E5%85%AC%E7%90%86" title="馬丁公理">马丁公理</a></li></ul> <ul><li><a href="/wiki/%E5%85%AC%E7%90%86%E6%A8%A1%E5%BC%8F" title="公理模式">公理模式</a> <ul><li><a href="/wiki/%E6%9B%BF%E4%BB%A3%E5%85%AC%E7%90%86" title="替代公理">替代</a></li> <li><a href="/wiki/%E5%88%86%E7%B1%BB%E5%85%AC%E7%90%86" title="分类公理">分类</a></li></ul></li></ul> </div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/%E7%BB%B4%E6%81%A9%E5%9B%BE" title="维恩图"><img alt="Venn diagram of set intersection" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/100px-Venn_A_intersect_B.svg.png" decoding="async" width="100" height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/150px-Venn_A_intersect_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/200px-Venn_A_intersect_B.svg.png 2x" data-file-width="350" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">运算</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%AC%9B%E5%8D%A1%E5%84%BF%E7%A7%AF" title="笛卡儿积">笛卡儿积</a></li> <li><a href="/wiki/%E5%BE%B7%E6%91%A9%E6%A0%B9%E5%AE%9A%E5%BE%8B" title="德摩根定律">德摩根定律</a></li> <li><a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a></li> <li><a href="/wiki/%E5%86%AA%E9%9B%86" title="冪集">冪集</a></li> <li><a href="/wiki/%E8%A1%A5%E9%9B%86" title="补集">补集</a></li> <li><a href="/wiki/%E5%AF%B9%E7%A7%B0%E5%B7%AE" title="对称差">对称差</a></li> <li><a href="/wiki/%E5%B9%B6%E9%9B%86" title="并集">并集</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li>概念</li><li>方法</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%8A%BF_(%E6%95%B0%E5%AD%A6)" title="势 (数学)">势</a></li> <li><a href="/wiki/%E5%9F%BA%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="基数 (数学)">基数</a>（<a href="/wiki/%E5%A4%A7%E5%9F%BA%E6%95%B0" title="大基数">大基数</a>）</li> <li><a href="/wiki/%E7%B1%BB_(%E6%95%B0%E5%AD%A6)" title="类 (数学)">类</a></li> <li><span class="ilh-all" data-orig-title="可构造全集" data-lang-code="en" data-lang-name="英语" data-foreign-title="Constructible universe"><span class="ilh-page"><a href="/w/index.php?title=%E5%8F%AF%E6%9E%84%E9%80%A0%E5%85%A8%E9%9B%86&action=edit&redlink=1" class="new" title="可构造全集（页面不存在）">可构造全集</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Constructible_universe" class="extiw" title="en:Constructible universe"><span lang="en" dir="auto">Constructible universe</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%BF%9E%E7%BB%AD%E7%BB%9F%E5%81%87%E8%AE%BE" title="连续统假设">连续统假设</a></li> <li><a href="/wiki/%E5%B0%8D%E8%A7%92%E8%AB%96%E8%AD%89%E6%B3%95" title="對角論證法">對角論證法</a></li> <li><a href="/wiki/%E5%85%83%E7%B4%A0_(%E6%95%B8%E5%AD%B8)" title="元素 (數學)">元素</a> <ul><li><a href="/wiki/%E6%9C%89%E5%BA%8F%E5%AF%B9" title="有序对">有序对</a></li> <li><a href="/wiki/%E5%85%83%E7%BB%84" title="元组">元组</a></li></ul></li> <li><a href="/wiki/%E9%9B%86%E5%90%88%E6%97%8F" title="集合族">集合族</a></li> <li><a href="/wiki/%E5%8A%9B%E8%BF%AB" title="力迫">力迫</a></li> <li><a class="mw-selflink selflink">一一对应</a></li> <li><a href="/wiki/%E5%BA%8F%E6%95%B0" title="序数">序数</a></li> <li><a href="/wiki/%E8%B6%85%E9%99%90%E5%BD%92%E7%BA%B3%E6%B3%95" title="超限归纳法">超限归纳法</a></li> <li><a href="/wiki/%E6%96%87%E6%B0%8F%E5%9B%BE" title="文氏图">文氏图</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>类型</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%8F%AF%E6%95%B8%E9%9B%86" title="可數集">可數集</a></li> <li><a href="/wiki/%E7%A9%BA%E9%9B%86" title="空集">空集</a></li> <li><a href="/wiki/%E6%9C%89%E9%99%90%E9%9B%86%E5%90%88" title="有限集合">有限集合</a>（<a href="/wiki/%E7%BB%A7%E6%89%BF%E6%9C%89%E9%99%90%E9%9B%86%E5%90%88" title="继承有限集合">继承有限集合</a>）</li> <li><a href="/wiki/%E6%A8%A1%E7%B3%8A%E9%9B%86" title="模糊集">模糊集</a></li> <li><a href="/wiki/%E6%97%A0%E9%99%90%E9%9B%86%E5%90%88" title="无限集合">无限集合</a></li> <li><a href="/wiki/%E9%80%92%E5%BD%92%E9%9B%86%E5%90%88" title="递归集合">递归集合</a></li> <li><a href="/wiki/%E5%AD%90%E9%9B%86" title="子集">子集</a></li> <li><a href="/wiki/%E4%BC%A0%E9%80%92%E9%9B%86%E5%90%88" title="传递集合">传递集合</a></li> <li><a href="/wiki/%E4%B8%8D%E5%8F%AF%E6%95%B8%E9%9B%86" title="不可數集">不可數集</a></li> <li><span class="ilh-all" data-orig-title="泛集" data-lang-code="en" data-lang-name="英语" data-foreign-title="Universal set"><span class="ilh-page"><a href="/w/index.php?title=%E6%B3%9B%E9%9B%86&action=edit&redlink=1" class="new" title="泛集（页面不存在）">泛集</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Universal_set" class="extiw" title="en:Universal set"><span lang="en" dir="auto">Universal set</span></a></span>）</span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">理论</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%8F%AF%E6%9B%BF%E4%BB%A3%E7%9A%84%E9%9B%86%E5%90%88%E8%AE%BA" title="可替代的集合论">可替代的集合论</a></li> <li><a href="/wiki/%E9%9B%86%E5%90%88%E8%AE%BA" title="集合论">集合论</a></li> <li><a href="/wiki/%E6%9C%B4%E7%B4%A0%E9%9B%86%E5%90%88%E8%AE%BA" title="朴素集合论">朴素集合论</a></li> <li><a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94%E5%AE%9A%E7%90%86" title="康托尔定理">康托尔定理</a></li></ul> <ul><li><a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛集合论">策梅洛</a> <ul><li><span class="ilh-all" data-orig-title="广义集合论" data-lang-code="en" data-lang-name="英语" data-foreign-title="General set theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%B9%BF%E4%B9%89%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="广义集合论（页面不存在）">广义</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/General_set_theory" class="extiw" title="en:General set theory"><span lang="en" dir="auto">General set theory</span></a></span>）</span></span></li></ul></li> <li>《<a href="/wiki/%E6%95%B0%E5%AD%A6%E5%8E%9F%E7%90%86" title="数学原理">数学原理</a>》 <ul><li><a href="/wiki/%E6%96%B0%E5%9F%BA%E7%A1%80%E9%9B%86%E5%90%88%E8%AE%BA" title="新基础集合论">新基础</a></li></ul></li> <li><a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B-%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛-弗兰克尔集合论">策梅洛-弗兰克</a> <ul><li><a href="/wiki/%E5%86%AF%E8%AF%BA%E4%BC%8A%E6%9B%BC-%E5%8D%9A%E5%86%85%E6%96%AF-%E5%93%A5%E5%BE%B7%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="冯诺伊曼-博内斯-哥德尔集合论">冯诺伊曼-博内斯-哥德尔</a> <ul><li><span class="ilh-all" data-orig-title="Morse–Kelley集合论" data-lang-code="en" data-lang-name="英语" data-foreign-title="Morse–Kelley set theory"><span class="ilh-page"><a href="/w/index.php?title=Morse%E2%80%93Kelley%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="Morse–Kelley集合论（页面不存在）">Morse–Kelley</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory" class="extiw" title="en:Morse–Kelley set theory"><span lang="en" dir="auto">Morse–Kelley set theory</span></a></span>）</span></span></li></ul></li> <li><span class="ilh-all" data-orig-title="克里普克–普拉特克集合论" data-lang-code="en" data-lang-name="英语" data-foreign-title="Kripke–Platek set theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%85%8B%E9%87%8C%E6%99%AE%E5%85%8B%E2%80%93%E6%99%AE%E6%8B%89%E7%89%B9%E5%85%8B%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="克里普克–普拉特克集合论（页面不存在）">克里普克–普拉特克</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory" class="extiw" title="en:Kripke–Platek set theory"><span lang="en" dir="auto">Kripke–Platek set theory</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="塔斯基–格罗滕迪克集合论" data-lang-code="en" data-lang-name="英语" data-foreign-title="Tarski–Grothendieck set theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%A1%94%E6%96%AF%E5%9F%BA%E2%80%93%E6%A0%BC%E7%BD%97%E6%BB%95%E8%BF%AA%E5%85%8B%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="塔斯基–格罗滕迪克集合论（页面不存在）">塔斯基–格罗滕迪克</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory" class="extiw" title="en:Tarski–Grothendieck set theory"><span lang="en" dir="auto">Tarski–Grothendieck set theory</span></a></span>）</span></span></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><span class="ilh-all" data-orig-title="集合论悖论" data-lang-code="en" data-lang-name="英语" data-foreign-title="Paradoxes of set theory"><span class="ilh-page"><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA%E6%82%96%E8%AE%BA&action=edit&redlink=1" class="new" title="集合论悖论（页面不存在）">悖论</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Paradoxes_of_set_theory" class="extiw" title="en:Paradoxes of set theory"><span lang="en" dir="auto">Paradoxes of set theory</span></a></span>）</span></span></li><li>问题</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%BD%97%E7%B4%A0%E6%82%96%E8%AE%BA" title="罗素悖论">罗素悖论</a></li> <li><a href="/wiki/%E8%98%87%E6%96%AF%E6%9E%97%E5%95%8F%E9%A1%8C" title="蘇斯林問題">蘇斯林問題</a></li> <li><a href="/wiki/ZFC%E7%B3%BB%E7%B5%B1%E7%84%A1%E6%B3%95%E7%A2%BA%E5%AE%9A%E7%9A%84%E5%91%BD%E9%A1%8C%E5%88%97%E8%A1%A8" title="ZFC系統無法確定的命題列表">ZFC系統無法確定的命題列表</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:%E9%9B%86%E5%90%88%E8%AB%96%E8%80%85" title="Category:集合論者">集合論者</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%BA%9A%E4%BC%AF%E6%8B%89%E7%BD%95%C2%B7%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94" title="亚伯拉罕·弗兰克尔">亚伯拉罕·弗兰克尔</a></li> <li><a href="/wiki/%E4%BC%AF%E7%89%B9%E5%85%B0%C2%B7%E7%BD%97%E7%B4%A0" title="伯特兰·罗素">伯特兰·罗素</a></li> <li><a href="/wiki/%E6%81%A9%E6%96%AF%E7%89%B9%C2%B7%E7%AD%96%E6%A2%85%E6%B4%9B" title="恩斯特·策梅洛">恩斯特·策梅洛</a></li> <li><a href="/wiki/%E6%A0%BC%E5%A5%A5%E5%B0%94%E6%A0%BC%C2%B7%E5%BA%B7%E6%89%98%E5%B0%94" title="格奥尔格·康托尔">格奥尔格·康托尔</a></li> <li><a href="/wiki/%E7%BA%A6%E7%BF%B0%C2%B7%E5%86%AF%C2%B7%E8%AF%BA%E4%BC%8A%E6%9B%BC" class="mw-redirect" title="约翰·冯·诺伊曼">约翰·冯·诺伊曼</a></li> <li><a href="/wiki/%E5%BA%93%E5%B0%94%E7%89%B9%C2%B7%E5%93%A5%E5%BE%B7%E5%B0%94" title="库尔特·哥德尔">库尔特·哥德尔</a></li> <li><a href="/wiki/%E7%9B%A7%E8%8F%B2%E7%89%B9%C2%B7%E6%BE%A4%E5%BE%B7" class="mw-redirect" title="盧菲特·澤德">盧菲特·澤德</a></li> <li><span class="ilh-all" data-orig-title="保罗·贝尔奈斯" data-lang-code="en" data-lang-name="英语" data-foreign-title="Paul Bernays"><span class="ilh-page"><a href="/w/index.php?title=%E4%BF%9D%E7%BD%97%C2%B7%E8%B4%9D%E5%B0%94%E5%A5%88%E6%96%AF&action=edit&redlink=1" class="new" title="保罗·贝尔奈斯（页面不存在）">保罗·贝尔奈斯</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Paul_Bernays" class="extiw" title="en:Paul Bernays"><span lang="en" dir="auto">Paul Bernays</span></a></span>）</span></span></li> <li><a href="/wiki/%E4%BF%9D%E7%BD%97%C2%B7%E5%AF%87%E6%81%A9" title="保罗·寇恩">保罗·寇恩</a></li> <li><a href="/wiki/%E7%90%86%E6%9F%A5%E5%BE%B7%C2%B7%E6%88%B4%E5%BE%B7%E9%87%91" title="理查德·戴德金">理查德·戴德金</a></li> <li><a href="/wiki/%E6%89%98%E9%A9%AC%E4%BB%80%C2%B7%E8%80%B6%E8%B5%AB" 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