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class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%A7%80%EC%8B%9C+%ED%95%A8%EC%88%98" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o" class=""><span>로그인</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="더 많은 옵션" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="개인 도구" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%A7%80%EC%8B%9C+%ED%95%A8%EC%88%98" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%A7%80%EC%8B%9C+%ED%95%A8%EC%88%98" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</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"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</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/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</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="사이트"> <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="목차" 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">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</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">처음 위치</div> </a> </li> <li id="toc-정의" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>정의</span> </div> </a> <ul id="toc-정의-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-표기와_용어의_비평" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#표기와_용어의_비평"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>표기와 용어의 비평</span> </div> </a> <ul id="toc-표기와_용어의_비평-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-기본_속성" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#기본_속성"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>기본 속성</span> </div> </a> <ul id="toc-기본_속성-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-평균,_분산_그리고_공분산" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#평균,_분산_그리고_공분산"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>평균, 분산 그리고 공분산</span> </div> </a> <ul id="toc-평균,_분산_그리고_공분산-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-재귀_이론에서_특성함수,_괴델과_클레이니의_표현_함수" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#재귀_이론에서_특성함수,_괴델과_클레이니의_표현_함수"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>재귀 이론에서 특성함수, 괴델과 클레이니의 표현 함수</span> </div> </a> <ul id="toc-재귀_이론에서_특성함수,_괴델과_클레이니의_표현_함수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-퍼지_집합_이론에서_특성함수" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#퍼지_집합_이론에서_특성함수"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>퍼지 집합 이론에서 특성함수</span> </div> </a> <ul id="toc-퍼지_집합_이론에서_특성함수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-지시_함수의_미분" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#지시_함수의_미분"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>지시 함수의 미분</span> </div> </a> <ul id="toc-지시_함수의_미분-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-같이_보기" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#같이_보기"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-각주" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#각주"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>각주</span> </div> </a> <ul id="toc-각주-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-참고_문헌" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#참고_문헌"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>참고 문헌</span> </div> </a> <ul id="toc-참고_문헌-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="목차" 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="목차 토글" > <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">목차 토글</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">지시 함수</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="다른 언어로 문서를 방문합니다. 34개 언어로 읽을 수 있습니다" > <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-34" 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">34개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%BD%D0%B4%D0%B8%D0%BA%D0%B0%D1%82%D0%BE%D1%80%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Индикаторна функция – 불가리아어" lang="bg" hreflang="bg" data-title="Индикаторна функция" data-language-autonym="Български" data-language-local-name="불가리아어" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_caracter%C3%ADstica_(matem%C3%A0tiques)" title="Funció característica (matemàtiques) – 카탈로니아어" lang="ca" hreflang="ca" data-title="Funció característica (matemàtiques)" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Charakteristick%C3%A1_funkce" title="Charakteristická funkce – 체코어" lang="cs" hreflang="cs" data-title="Charakteristická funkce" data-language-autonym="Čeština" data-language-local-name="체코어" 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/Indikatorfunktion" title="Indikatorfunktion – 덴마크어" lang="da" hreflang="da" data-title="Indikatorfunktion" data-language-autonym="Dansk" data-language-local-name="덴마크어" 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/Indikatorfunktion" title="Indikatorfunktion – 독일어" lang="de" hreflang="de" data-title="Indikatorfunktion" data-language-autonym="Deutsch" data-language-local-name="독일어" 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/Indicator_function" title="Indicator function – 영어" lang="en" hreflang="en" data-title="Indicator function" data-language-autonym="English" data-language-local-name="영어" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_indicatriz" title="Función indicatriz – 스페인어" lang="es" hreflang="es" data-title="Función indicatriz" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Funtzio_adierazle" title="Funtzio adierazle – 바스크어" lang="eu" hreflang="eu" data-title="Funtzio adierazle" data-language-autonym="Euskara" data-language-local-name="바스크어" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D9%85%D8%B4%D8%AE%D8%B5%D9%87" title="تابع مشخصه – 페르시아어" lang="fa" hreflang="fa" data-title="تابع مشخصه" data-language-autonym="فارسی" data-language-local-name="페르시아어" 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/Indikaattorifunktio" title="Indikaattorifunktio – 핀란드어" lang="fi" hreflang="fi" data-title="Indikaattorifunktio" data-language-autonym="Suomi" data-language-local-name="핀란드어" 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/Fonction_caract%C3%A9ristique_(th%C3%A9orie_des_ensembles)" title="Fonction caractéristique (théorie des ensembles) – 프랑스어" lang="fr" hreflang="fr" data-title="Fonction caractéristique (théorie des ensembles)" data-language-autonym="Français" data-language-local-name="프랑스어" 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_indicadora" title="Función indicadora – 갈리시아어" lang="gl" hreflang="gl" data-title="Función indicadora" data-language-autonym="Galego" data-language-local-name="갈리시아어" 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%9E%D7%A6%D7%99%D7%99%D7%A0%D7%AA" title="פונקציה מציינת – 히브리어" lang="he" hreflang="he" data-title="פונקציה מציינת" data-language-autonym="עברית" data-language-local-name="히브리어" 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%B8%E0%A5%82%E0%A4%9A%E0%A4%95_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="सूचक फलन – 힌디어" lang="hi" hreflang="hi" data-title="सूचक फलन" data-language-autonym="हिन्दी" data-language-local-name="힌디어" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Karakterisztikus_f%C3%BCggv%C3%A9ny" title="Karakterisztikus függvény – 헝가리어" lang="hu" hreflang="hu" data-title="Karakterisztikus függvény" data-language-autonym="Magyar" data-language-local-name="헝가리어" 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%8E%D5%A5%D5%AF%D5%BF%D5%B8%D6%80%D5%AB_%D5%AB%D5%B6%D5%A4%D5%AB%D5%AF%D5%A1%D5%BF%D5%B8%D6%80" title="Վեկտորի ինդիկատոր – 아르메니아어" lang="hy" hreflang="hy" data-title="Վեկտորի ինդիկատոր" data-language-autonym="Հայերեն" data-language-local-name="아르메니아어" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_indikator" title="Fungsi indikator – 인도네시아어" lang="id" hreflang="id" data-title="Fungsi indikator" data-language-autonym="Bahasa Indonesia" data-language-local-name="인도네시아어" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Kennifall_(mengjafr%C3%A6%C3%B0i)" title="Kennifall (mengjafræði) – 아이슬란드어" lang="is" hreflang="is" data-title="Kennifall (mengjafræði)" data-language-autonym="Íslenska" data-language-local-name="아이슬란드어" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_indicatrice" title="Funzione indicatrice – 이탈리아어" lang="it" hreflang="it" data-title="Funzione indicatrice" data-language-autonym="Italiano" data-language-local-name="이탈리아어" 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/%E6%8C%87%E7%A4%BA%E9%96%A2%E6%95%B0" title="指示関数 – 일본어" lang="ja" hreflang="ja" data-title="指示関数" data-language-autonym="日本語" data-language-local-name="일본어" 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/%D0%A1%D0%B8%D0%BF%D0%B0%D1%82%D1%82%D0%B0%D1%83%D1%8B%D1%88_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Сипаттауыш функция – 카자흐어" lang="kk" hreflang="kk" data-title="Сипаттауыш функция" data-language-autonym="Қазақша" data-language-local-name="카자흐어" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Fonzion_carateristega" title="Fonzion carateristega – Lombard" lang="lmo" hreflang="lmo" data-title="Fonzion carateristega" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Fungsi_penunjuk" title="Fungsi penunjuk – 말레이어" lang="ms" hreflang="ms" data-title="Fungsi penunjuk" data-language-autonym="Bahasa Melayu" data-language-local-name="말레이어" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Indicatorfunctie" title="Indicatorfunctie – 네덜란드어" lang="nl" hreflang="nl" data-title="Indicatorfunctie" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja_charakterystyczna_zbioru" title="Funkcja charakterystyczna zbioru – 폴란드어" lang="pl" hreflang="pl" data-title="Funkcja charakterystyczna zbioru" data-language-autonym="Polski" data-language-local-name="폴란드어" 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_indicadora" title="Função indicadora – 포르투갈어" lang="pt" hreflang="pt" data-title="Função indicadora" data-language-autonym="Português" data-language-local-name="포르투갈어" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%BD%D0%B4%D0%B8%D0%BA%D0%B0%D1%82%D0%BE%D1%80_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Индикатор (математика) – 러시아어" lang="ru" hreflang="ru" data-title="Индикатор (математика)" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Funksioni_tregues" title="Funksioni tregues – 알바니아어" lang="sq" hreflang="sq" data-title="Funksioni tregues" data-language-autonym="Shqip" data-language-local-name="알바니아어" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a 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src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Indicator_function_illustration.png/220px-Indicator_function_illustration.png" decoding="async" width="220" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Indicator_function_illustration.png/330px-Indicator_function_illustration.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Indicator_function_illustration.png/440px-Indicator_function_illustration.png 2x" data-file-width="813" data-file-height="516" /></a><figcaption>2차원 집합의 지시 함수의 그래프.</figcaption></figure> <p><a href="/wiki/%EC%88%98%ED%95%99" title="수학">수학</a>에서 <b>지시 함수</b>(指示函數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">indicator function</span>), <b>정의 함수</b>(定義函數), 또는 <b>특성 함수</b>(特性函數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">characteristic function</span>)는 특정 집합에 특정 값이 속하는지를 표시하는 <a href="/wiki/%ED%95%A8%EC%88%98" title="함수">함수</a>로, 특정 값이 집합에 속한다면 1, 속하지 않는다면 0의 값을 가진다. 기호 1이나 I로 표기되며 아래첨자로 나타내는 집합을 표시한다. 때로는 굵은글씨 또는 <a href="/wiki/%EC%B9%A0%ED%8C%90_%EB%B3%BC%EB%93%9C%EC%B2%B4" title="칠판 볼드체">칠판 볼드체</a>로 쓰인다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="정의"><span id=".EC.A0.95.EC.9D.98"></span>정의</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%80%EC%8B%9C_%ED%95%A8%EC%88%98&action=edit&section=1" title="부분 편집: 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EC%A7%91%ED%95%A9" title="집합">집합</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>의 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%A7%91%ED%95%A9" class="mw-redirect" title="부분 집합">부분 집합</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 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>에 대한 <b>지시 함수</b> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4af98405fdac1f966221d16a150f13fd114d8eda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.113ex; height:2.843ex;" alt="{\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\}}"></span></dd></dl> <p>는 다음과 같은 함수이다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{A}(x):={\begin{cases}1&x\in A\\0&x\notin A\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>x</mi> <mo>∈</mo> <mi>A</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>x</mi> <mo>∉</mo> <mi>A</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A}(x):={\begin{cases}1&x\in A\\0&x\notin A\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bcffaba683a4e4322d0ed098e05657f7f582edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.579ex; height:6.176ex;" alt="{\displaystyle \mathbf {1} _{A}(x):={\begin{cases}1&x\in A\\0&x\notin A\end{cases}}}"></span></dd></dl> <p>또한 <a href="/wiki/%EC%95%84%EC%9D%B4%EB%B2%84%EC%8A%A8_%EA%B4%84%ED%98%B8" title="아이버슨 괄호">아이버슨 괄호</a> 표기법으로 지시 함수<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 \mathbf {1} _{A}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f932fcbec6f7a11b3bee1f02714ed10c0235dcdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.94ex; height:2.843ex;" alt="{\displaystyle \mathbf {1} _{A}(x)}"></span>를 <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\in A]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x\in A]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/485b189d561529e8b87162430c7dc1dc6859e5d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle [x\in A]}"></span>으로 나타낼 수 있다. </p><p>지시 함수는 <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 \mathbf {1} _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4a6cdf0babfdaa63fa048de3d9b8d0a4c06522" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.801ex; height:2.509ex;" alt="{\displaystyle \mathbf {1} _{A}}"></span> 외에도 <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 \mathbf {I} _{A}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {I} _{A}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/098976f651a178d4753a7e8a23037b2f34936237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.618ex; height:2.843ex;" alt="{\displaystyle \mathbf {I} _{A}(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 \chi _{A}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{A}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89b9161990fcda15db1ab01bce8b2832f8517857" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.059ex; height:2.843ex;" alt="{\displaystyle \chi _{A}(x)}"></span>, <i>K<sub>A</sub></i> 또는 간단하게 <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>로 표기할 수 있다.(<a href="/wiki/%EA%B7%B8%EB%A6%AC%EC%8A%A4_%EB%AC%B8%EC%9E%90" title="그리스 문자">그리스 문자</a> <a href="/wiki/%CE%A7" title="Χ">χ</a>는 '특성(characteristic)'이라는 말의 <a href="/wiki/%EA%B7%B8%EB%A6%AC%EC%8A%A4%EC%96%B4" title="그리스어">그리스어</a> 어원χαρακτήρ의 첫 글자이다) </p><p><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>에서 정의된 모든 지시함수는 <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>의 <a href="/wiki/%EB%A9%B1%EC%A7%91%ED%95%A9" title="멱집합">멱집합</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 {\mathcal {P}}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ed5b6b7f1ad70cba0f7b3cf4603bf627321b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.493ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(X)}"></span>으로 정의할 수 있으며, 결과적으로 두 집합은 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a83de58939a4ef8e4837ab6dd554da4278f557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.795ex; height:2.676ex;" alt="{\displaystyle 2^{X}}"></span>로 표기될 수 있다. 이것은 특별한 집합(<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=\{0,1\}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\{0,1\}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4a7370dad27d6810040774b3232f70fd3bf7c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.816ex; height:2.843ex;" alt="{\displaystyle Y=\{0,1\}=2}"></span>)의 표기법<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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b4c76e7ea7aad9de5488491e1c6c7363c0bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}"></span>이 모든 함수<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>의 집합을 표시하기 때문이다. </p> <div class="mw-heading mw-heading2"><h2 id="표기와_용어의_비평"><span id=".ED.91.9C.EA.B8.B0.EC.99.80_.EC.9A.A9.EC.96.B4.EC.9D.98_.EB.B9.84.ED.8F.89"></span>표기와 용어의 비평</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%80%EC%8B%9C_%ED%95%A8%EC%88%98&action=edit&section=2" title="부분 편집: 표기와 용어의 비평"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><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 1_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a15eaa9285cd4654e86a76f3318c6ab2aad95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle 1_{A}}"></span>는 <i>A</i>의 <a href="/wiki/%ED%95%AD%EB%93%B1%ED%95%A8%EC%88%98" class="mw-redirect" title="항등함수">항등함수</a>를 표기할 때도 쓰인다.</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 \chi _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49baf1caaa804f2d77bfc7570d102ee4a3cafa26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.009ex;" alt="{\displaystyle \chi _{A}}"></span>는 <a href="/w/index.php?title=%EB%B3%BC%EB%A1%9D_%ED%95%B4%EC%84%9D%ED%95%99&action=edit&redlink=1" class="new" title="볼록 해석학 (없는 문서)">볼록 해석학</a>에서 <a href="/w/index.php?title=%ED%8A%B9%EC%84%B1_%ED%95%A8%EC%88%98_(%EB%B3%BC%EB%A1%9D_%ED%95%B4%EC%84%9D%ED%95%99)&action=edit&redlink=1" class="new" title="특성 함수 (볼록 해석학) (없는 문서)">특성함수</a>를 표기할 때에도 쓰인다.</li></ul> <p><a href="/wiki/%ED%95%B4%EC%84%9D%ED%95%99" class="mw-disambig" title="해석학">해석학</a>에서 관련 개념은 <a href="/wiki/%EA%B0%80%EB%B3%80%EC%88%98" title="가변수">가변수</a>(dummy variables)이다.(수학에서 사용되는 <a href="/w/index.php?title=%EC%9E%90%EC%9C%A0%EB%B3%80%EC%88%98%EC%99%80_%EB%B0%94%EC%9A%B4%EB%93%9C_%EB%B3%80%EC%88%98&action=edit&redlink=1" class="new" title="자유변수와 바운드 변수 (없는 문서)">바운드 변수</a>라고도 하는"더미변수"(dummy varriables)와 헷갈리면 안 된다.) </p><p><a href="/wiki/%ED%8A%B9%EC%84%B1%ED%95%A8%EC%88%98_(%ED%99%95%EB%A5%A0%EB%A1%A0)" title="특성함수 (확률론)">특성함수</a>라는 표현은 <a href="/wiki/%ED%99%95%EB%A5%A0%EB%A1%A0" title="확률론">확률론</a>에서는 전혀 관계없는 의미를 가진다. 이런 이유로, 다른 대부분의 수학자들은 특성함수라는 표현을 집합의 원소인지를 나타내는 함수로 나타내는 반면에 <a href="/w/index.php?title=%ED%99%95%EB%A5%A0%EB%A1%A0%EC%9E%90&action=edit&redlink=1" class="new" title="확률론자 (없는 문서)">확률론자</a>들은 <b>지시 함수</b>라는 용어를 거의 이 함수를 가리키는데 사용한다. </p> <div class="mw-heading mw-heading2"><h2 id="기본_속성"><span id=".EA.B8.B0.EB.B3.B8_.EC.86.8D.EC.84.B1"></span>기본 속성</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%80%EC%8B%9C_%ED%95%A8%EC%88%98&action=edit&section=3" title="부분 편집: 기본 속성"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>어떤 집합인 <i>X</i>의 부분집합인 <i>A</i>의 <b>지시 함수</b> 또는 <b>특성함수</b>는 <i>X</i>의 원소를 <a href="/wiki/%EC%B9%98%EC%97%AD" title="치역">구간</a>{0,1}으로 <a href="/wiki/%ED%95%A8%EC%88%98" title="함수">대응</a>시킨다. </p><p>이 함수는 <i>A</i>가 공집합이 아닌 <i>X</i>의 <a href="/wiki/%EB%B6%80%EB%B6%84%EC%A7%91%ED%95%A9" title="부분집합">진부분집합</a> 일 경우에만 <a href="/wiki/%EC%A0%84%EC%82%AC_%ED%95%A8%EC%88%98" title="전사 함수">전사 함수</a>이다. 만약<i>A</i> ≡ <i>X</i>이면 지시함수 <b>1</b><sub><i>A</i></sub> = 1이다. 비슷하게 <i>A</i> ≡ Ø일때는 <b>1</b><sub><i>A</i></sub> = 0이다. </p><p>다음에서 점은 곱셈을 의미한다 1·1 = 1, 1·0 = 0 등등. "+"와 "-"는 덧셈과 뻴셈을 나타낸다. <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 \cap }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cap }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4e886e6f5a28a33e073fb108440c152ecfe2d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cap }"></span>"과 "<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 \cup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∪</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ff7d0293ad19b43524a133ae5129f3d71f2040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cup }"></span>"은 각각 교집합과 합집합을 나타낸다. </p><p>만약 <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>와 <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>가 <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>의 집합이라면 다음과 같다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{A\cap B}=\min\{\mathbf {1} _{A},\mathbf {1} _{B}\}=\mathbf {1} _{A}\cdot \mathbf {1} _{B},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A\cap B}=\min\{\mathbf {1} _{A},\mathbf {1} _{B}\}=\mathbf {1} _{A}\cdot \mathbf {1} _{B},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/610f3f40b4f3878626da8f9e21c09c29eed6d585" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.137ex; height:2.843ex;" alt="{\displaystyle \mathbf {1} _{A\cap B}=\min\{\mathbf {1} _{A},\mathbf {1} _{B}\}=\mathbf {1} _{A}\cdot \mathbf {1} _{B},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{A\cup B}=\max\{{\mathbf {1} _{A},\mathbf {1} _{B}}\}=\mathbf {1} _{A}+\mathbf {1} _{B}-\mathbf {1} _{A}\cdot \mathbf {1} _{B},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A\cup B}=\max\{{\mathbf {1} _{A},\mathbf {1} _{B}}\}=\mathbf {1} _{A}+\mathbf {1} _{B}-\mathbf {1} _{A}\cdot \mathbf {1} _{B},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/174c602a65bbb67627cc7824d34d31cd755eb1ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.887ex; height:2.843ex;" alt="{\displaystyle \mathbf {1} _{A\cup B}=\max\{{\mathbf {1} _{A},\mathbf {1} _{B}}\}=\mathbf {1} _{A}+\mathbf {1} _{B}-\mathbf {1} _{A}\cdot \mathbf {1} _{B},}"></span></dd></dl> <p>또한 <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>의 <a href="/wiki/%EC%97%AC%EC%A7%91%ED%95%A9" title="여집합">여집합</a>, 즉 <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^{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d03504da9a53f91a23a003bbaede82cc3afafcab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.224ex; height:2.676ex;" alt="{\displaystyle A^{C}}"></span>의 지시함수는 다음과 같다: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{A^{\complement }}=1-\mathbf {1} _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>∁</mi> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A^{\complement }}=1-\mathbf {1} _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724b249a01b8dc9fea42777835b4c2ea54eff626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.536ex; height:3.009ex;" alt="{\displaystyle \mathbf {1} _{A^{\complement }}=1-\mathbf {1} _{A}}"></span></dd></dl> <p>더 일반적으로, <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_{1},\dotsc ,A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},\dotsc ,A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4f7c36fd0979b4182d7a918670b7d518788bd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.937ex; height:2.509ex;" alt="{\displaystyle A_{1},\dotsc ,A_{n}}"></span>가 <i>X</i>의 부분집합들이라고 하면, 모든<i>x</i> ∈ <i>X</i>에 대해서 다음 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{k\in I}(1-\mathbf {1} _{A_{k}}(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{k\in I}(1-\mathbf {1} _{A_{k}}(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9a2ebdf576b2b24a29f2163391fdd5f5f07b400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.582ex; height:5.676ex;" alt="{\displaystyle \prod _{k\in I}(1-\mathbf {1} _{A_{k}}(x))}"></span></dd></dl> <p>는 명백히 0들과 1들의 곱이다.이 곱은 <i>x</i> ∈ <i>X</i>가 집합<i>A<sub>k</sub></i>에 포함되지 않을 경우에만 1이고, 다른경우에는 0이다. 이것은 다음과 같다: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{k\in I}(1-\mathbf {1} _{A_{k}})=\mathbf {1} _{X-\bigcup _{k}A_{k}}=1-\mathbf {1} _{\bigcup _{k}A_{k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>−</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{k\in I}(1-\mathbf {1} _{A_{k}})=\mathbf {1} _{X-\bigcup _{k}A_{k}}=1-\mathbf {1} _{\bigcup _{k}A_{k}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afe346b30ea8d23440abe3cc02676e2aa02b14e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.521ex; height:5.676ex;" alt="{\displaystyle \prod _{k\in I}(1-\mathbf {1} _{A_{k}})=\mathbf {1} _{X-\bigcup _{k}A_{k}}=1-\mathbf {1} _{\bigcup _{k}A_{k}}.}"></span></dd></dl> <p>이 식의 좌변을 확장하면 다음과 같다: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{\bigcup _{k}A_{k}}=1-\sum _{F\subseteq \{1,2,\dotsc ,n\}}(-1)^{|F|}\mathbf {1} _{\bigcap _{F}A_{k}}=\sum _{\emptyset \neq F\subseteq \{1,2,\dotsc ,n\}}(-1)^{|F|+1}\mathbf {1} _{\bigcap _{F}A_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∅</mi> <mo>≠</mo> <mi>F</mi> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{\bigcup _{k}A_{k}}=1-\sum _{F\subseteq \{1,2,\dotsc ,n\}}(-1)^{|F|}\mathbf {1} _{\bigcap _{F}A_{k}}=\sum _{\emptyset \neq F\subseteq \{1,2,\dotsc ,n\}}(-1)^{|F|+1}\mathbf {1} _{\bigcap _{F}A_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a8f812c3c1adee89e1acbad7442a91b539b195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:68.317ex; height:6.176ex;" alt="{\displaystyle \mathbf {1} _{\bigcup _{k}A_{k}}=1-\sum _{F\subseteq \{1,2,\dotsc ,n\}}(-1)^{|F|}\mathbf {1} _{\bigcap _{F}A_{k}}=\sum _{\emptyset \neq F\subseteq \{1,2,\dotsc ,n\}}(-1)^{|F|+1}\mathbf {1} _{\bigcap _{F}A_{k}}}"></span></dd></dl> <p>여기서 |<i>F</i>|는 <i>F</i>의 <a href="/wiki/%EC%A7%91%ED%95%A9%EC%9D%98_%ED%81%AC%EA%B8%B0" title="집합의 크기">크기</a>이다. 이것은 <a href="/wiki/%ED%8F%AC%ED%95%A8%EB%B0%B0%EC%A0%9C%EC%9D%98_%EC%9B%90%EB%A6%AC" title="포함배제의 원리">포함-배제 원리</a>의 한 부분이다. </p><p>앞에서 보았듯이, 지시 함수는 <a href="/wiki/%EC%A1%B0%ED%95%A9%EB%A1%A0" title="조합론">조합론</a>에서 유용한 표기 장치이다. 이 표기는 다른곳에서도 사용된다. 예를 들면 <a href="/wiki/%ED%99%95%EB%A5%A0%EB%A1%A0" title="확률론">확률론</a>이 있다: <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>가 확률 측정<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 {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1053af9e662ceaf56c4455f90e0f67273422eded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }"></span>의 <a href="/wiki/%ED%99%95%EB%A5%A0%EA%B3%B5%EA%B0%84" class="mw-redirect" title="확률공간">확률공간</a>이고, <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>가 측정가능한 집합 일 때 <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 \mathbf {1} _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4a6cdf0babfdaa63fa048de3d9b8d0a4c06522" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.801ex; height:2.509ex;" alt="{\displaystyle \mathbf {1} _{A}}"></span>는 인 <a href="/wiki/%ED%99%95%EB%A5%A0%EB%B3%80%EC%88%98" class="mw-redirect" title="확률변수">확률변수</a>가 되고, <a href="/wiki/%EA%B8%B0%EB%8C%93%EA%B0%92" title="기댓값">기댓값</a>이 <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>의 확률이 된다: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (\mathbf {1} _{A})=\int _{X}\mathbf {1} _{A}(x)\,d\mathbb {P} =\int _{A}d\mathbb {P} =\operatorname {P} (A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (\mathbf {1} _{A})=\int _{X}\mathbf {1} _{A}(x)\,d\mathbb {P} =\int _{A}d\mathbb {P} =\operatorname {P} (A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/279dd40a36dfb775d53ef3969fc572c5e15671b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.68ex; height:5.676ex;" alt="{\displaystyle \operatorname {E} (\mathbf {1} _{A})=\int _{X}\mathbf {1} _{A}(x)\,d\mathbb {P} =\int _{A}d\mathbb {P} =\operatorname {P} (A)}"></span>.</dd></dl> <p>이 항등식은 <a href="/wiki/%EB%A7%88%EB%A5%B4%EC%BD%94%ED%94%84_%EB%B6%80%EB%93%B1%EC%8B%9D" title="마르코프 부등식">마르코프 부등식</a>의 증명에 사용된다. </p><p><a href="/wiki/%EC%88%9C%EC%84%9C%EB%A1%A0" title="순서론">순서론</a>같이 많은 경우에서 지시함수의 역함수를 정의할 수 있다. 보통 초등 <a href="/wiki/%EC%A0%95%EC%88%98%EB%A1%A0" class="mw-redirect" title="정수론">정수론</a>에서 지시함수의 역함수, <a href="/wiki/%EB%AB%BC%EB%B9%84%EC%9A%B0%EC%8A%A4_%ED%95%A8%EC%88%98" title="뫼비우스 함수">뫼비우스 함수</a>의 일반화로써 <a href="/wiki/%EA%B7%BC%EC%A0%91_%EB%8C%80%EC%88%98" title="근접 대수">일반화된 뫼비우스 함수</a>라고 불린다.(고전 재귀 이론에서 역함수의 활용에 대해서는 아래 단락을 참조하자.) </p> <div class="mw-heading mw-heading2"><h2 id="평균,_분산_그리고_공분산"><span id=".ED.8F.89.EA.B7.A0.2C_.EB.B6.84.EC.82.B0_.EA.B7.B8.EB.A6.AC.EA.B3.A0_.EA.B3.B5.EB.B6.84.EC.82.B0"></span>평균, 분산 그리고 공분산</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%80%EC%8B%9C_%ED%95%A8%EC%88%98&action=edit&section=4" title="부분 편집: 평균, 분산 그리고 공분산"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%ED%99%95%EB%A5%A0_%EA%B3%B5%EA%B0%84" title="확률 공간">확률 공간</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 (\Omega ,{\mathcal {F}},\mathbb {P} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb8743f7565082ed1a9ee0490d9d71be82eafaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.902ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}"></span>과 <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\in {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cdf27fd56b3c06d1ddf9ad1efd7cee0a81cb2dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.51ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {F}}}"></span>가 주어지면, 지시함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{A}\colon \Omega \rightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A}\colon \Omega \rightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bf9c733f236a81cb5fb333d5736b042a1b7fdc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.806ex; height:2.509ex;" alt="{\displaystyle \mathbf {1} _{A}\colon \Omega \rightarrow \mathbb {R} }"></span>은</dd></dl> <p><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 \omega \in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42c12942fe5218b6931fc65b8246bf7eabe174ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.03ex; height:2.176ex;" alt="{\displaystyle \omega \in A}"></span>일 때,<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 \mathbf {1} _{A}(\omega )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A}(\omega )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa59290831b91729b70c1ee20ca0fe474a0d9f96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.317ex; height:2.843ex;" alt="{\displaystyle \mathbf {1} _{A}(\omega )=1}"></span> 이고, 그렇지 않으면 <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 \mathbf {1} _{A}(\omega )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A}(\omega )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52973449b4c5c98b06aba45914cf007ec94b304b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.317ex; height:2.843ex;" alt="{\displaystyle \mathbf {1} _{A}(\omega )=0}"></span>이다 </p> <dl><dt><a href="/wiki/%ED%8F%89%EA%B7%A0" title="평균">평균</a></dt> <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 \operatorname {E} (\mathbf {1} _{A}(\omega ))=\operatorname {P} (A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (\mathbf {1} _{A}(\omega ))=\operatorname {P} (A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a7a0d8ebb0e2b01a3a1a10d261df9aedf4c13ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.682ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} (\mathbf {1} _{A}(\omega ))=\operatorname {P} (A)}"></span></dd></dl> <dl><dt><a href="/wiki/%EB%B6%84%EC%82%B0" title="분산">분산</a></dt> <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 \operatorname {Var} (\mathbf {1} _{A}(\omega ))=\operatorname {P} (A)(1-\operatorname {P} (A))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (\mathbf {1} _{A}(\omega ))=\operatorname {P} (A)(1-\operatorname {P} (A))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5755cf68d5b01bdec34c2bfdcf52a9d2bc0a35f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.864ex; height:2.843ex;" alt="{\displaystyle \operatorname {Var} (\mathbf {1} _{A}(\omega ))=\operatorname {P} (A)(1-\operatorname {P} (A))}"></span></dd></dl> <dl><dt><a href="/wiki/%EA%B3%B5%EB%B6%84%EC%82%B0" title="공분산">공분산</a></dt> <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 \operatorname {Cov} (\mathbf {1} _{A}(\omega ),\mathbf {1} _{B}(\omega ))=\operatorname {P} (A\cap B)-\operatorname {P} (A)\operatorname {P} (B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} (\mathbf {1} _{A}(\omega ),\mathbf {1} _{B}(\omega ))=\operatorname {P} (A\cap B)-\operatorname {P} (A)\operatorname {P} (B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b75b8cfb72225b09b0f9d86cbcf15988e450aab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.138ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cov} (\mathbf {1} _{A}(\omega ),\mathbf {1} _{B}(\omega ))=\operatorname {P} (A\cap B)-\operatorname {P} (A)\operatorname {P} (B)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="재귀_이론에서_특성함수,_괴델과_클레이니의_표현_함수"><span id=".EC.9E.AC.EA.B7.80_.EC.9D.B4.EB.A1.A0.EC.97.90.EC.84.9C_.ED.8A.B9.EC.84.B1.ED.95.A8.EC.88.98.2C_.EA.B4.B4.EB.8D.B8.EA.B3.BC_.ED.81.B4.EB.A0.88.EC.9D.B4.EB.8B.88.EC.9D.98_.ED.91.9C.ED.98.84_.ED.95.A8.EC.88.98"></span>재귀 이론에서 특성함수, 괴델과 클레이니의 표현 함수</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%80%EC%8B%9C_%ED%95%A8%EC%88%98&action=edit&section=5" title="부분 편집: 재귀 이론에서 특성함수, 괴델과 클레이니의 표현 함수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EC%BF%A0%EB%A5%B4%ED%8A%B8_%EA%B4%B4%EB%8D%B8" title="쿠르트 괴델">쿠르트 괴델</a>은 1934년에 논문 "On Undecidable Propositions of Formal Mathematical Systems"에서 <i>표현 함수</i>를 설명했다.(이 논문은 <a href="/wiki/%EB%A7%88%ED%8B%B4_%EB%8D%B0%EC%9D%B4%EB%B9%84%EC%8A%A4" class="mw-disambig" title="마틴 데이비스">마틴 데이비스</a> (Martin Davis)의 <i>The Undecidable</i>의 pp. 41-74에 게재되어있다.) </p> <dl><dd>"각각의 클래스 또는 관계 R의 표현 함수φ는 R(x<sub>1</sub>, . . ., x<sub>n</sub>)이면 φ(x<sub>1</sub>, . . ., x<sub>n</sub>) = 0이고, ~R(x<sub>1</sub>, . . ., x<sub>n</sub>)일 때는 φ(x<sub>1</sub>, . . ., x<sub>n</sub>) = 1이다." (p. 42; the "~" indicates logical inversion i.e. "NOT")</dd></dl> <p><a href="/wiki/%EC%8A%A4%ED%8B%B0%EB%B8%90_%ED%81%B4%EB%A0%88%EC%9D%B4%EB%8B%88" title="스티븐 클레이니">스티븐 클레이니</a> (1952) (p.  227)는 <a href="/wiki/%EC%9B%90%EC%8B%9C_%EC%9E%AC%EA%B7%80_%ED%95%A8%EC%88%98" title="원시 재귀 함수">원시 재귀 함수</a>의 내용 중에서 같은 정의를 논리가 거짓이면 1이고 참이면 0이 나오는 논리 P의 함수 φ를 제공했다. </p><p>예를 들어, 어떤 하나의 함수가 0이면 표현 함수의 곱은 φ<sub>1</sub>*φ<sub>2</sub>* . . . *φ<sub>n</sub> = 0이기 때문에 논리연산 "또는"의 역할을 한다. 만약 φ<sub>1</sub> = 0 이거나 φ<sub>2</sub> = 0 이거나 . . . 이거나 φ<sub>n</sub> = 0 이라면 그 곱은 0이다. 독자들이 보았을 때 논리적인 반전이라고 생각하는 것, 즉 표현 함수가 함수 R이 "참"일 때, 또는 "만족"할 때 0이 되는 점은 클레이니의 다음의 연산에서 중요한 역할을 한다:논리 연산 OR, AND, 그리고 IMPLY(p. 228) 제한- (p. 228) 과 무제한- (p. 279ff) <a href="/w/index.php?title=%EB%AE%A4_%EC%97%B0%EC%82%B0%EC%9E%90&action=edit&redlink=1" class="new" title="뮤 연산자 (없는 문서)">뮤 연산자</a>들과 (Kleene (1952)) CASE 함수이다(p. 229). </p> <div class="mw-heading mw-heading2"><h2 id="퍼지_집합_이론에서_특성함수"><span id=".ED.8D.BC.EC.A7.80_.EC.A7.91.ED.95.A9_.EC.9D.B4.EB.A1.A0.EC.97.90.EC.84.9C_.ED.8A.B9.EC.84.B1.ED.95.A8.EC.88.98"></span>퍼지 집합 이론에서 특성함수</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%80%EC%8B%9C_%ED%95%A8%EC%88%98&action=edit&section=6" title="부분 편집: 퍼지 집합 이론에서 특성함수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>고전 수학에서 집합의 특성함수는 집합의 원소이면 1, 아니면 0을 낸다. <a href="/w/index.php?title=%ED%8D%BC%EC%A7%80_%EC%A7%91%ED%95%A9_%EC%9D%B4%EB%A1%A0&action=edit&redlink=1" class="new" title="퍼지 집합 이론 (없는 문서)">퍼지 집합 이론</a>에서 특성함수는 실수 구간 [0, 1]에서 반환하도록 일반화 되거나, 심지어 <a href="/w/index.php?title=%EB%B3%B4%ED%8E%B8%EC%A0%81_%EB%8C%80%EC%88%98%ED%95%99&action=edit&redlink=1" class="new" title="보편적 대수학 (없는 문서)">대수</a> 또는 [[구조 (논리학}|구조]](보통 적어도 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="부분 순서 집합">부분 순서 집합</a> 또는 <a href="/wiki/%EA%B2%A9%EC%9E%90_(%EC%88%9C%EC%84%9C%EB%A1%A0)" title="격자 (순서론)">격자</a>가 되어야 한다)에서 값을 반환하기도 한다. 이런 일반화된 특성함수는 대부분 <a href="/w/index.php?title=%EB%A9%A4%EB%B2%84%EC%8B%AD_%ED%95%A8%EC%88%98&action=edit&redlink=1" class="new" title="멤버십 함수 (없는 문서)">멤버십 함수</a>라고 하며, 해당 집합은 <a href="/wiki/%ED%8D%BC%EC%A7%80_%EC%A7%91%ED%95%A9" title="퍼지 집합">퍼지 집합</a>이라고 한다. 퍼지집합은 "키가 크다", "덥다" 등과 같이 많은 실생활에서 쓰이는 서술어에 나타나는 회원 <a href="/w/index.php?title=%EC%A7%84%EB%A6%AC%EC%9D%98_%EC%A0%95%EB%8F%84&action=edit&redlink=1" class="new" title="진리의 정도 (없는 문서)">등급</a>의 점진적인 변화를 모델링한다. </p> <div class="mw-heading mw-heading2"><h2 id="지시_함수의_미분"><span id=".EC.A7.80.EC.8B.9C_.ED.95.A8.EC.88.98.EC.9D.98_.EB.AF.B8.EB.B6.84"></span>지시 함수의 미분</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%80%EC%8B%9C_%ED%95%A8%EC%88%98&action=edit&section=7" title="부분 편집: 지시 함수의 미분"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>특정한 지시함수는 <a href="/wiki/%EB%8B%A8%EC%9C%84_%EA%B3%84%EB%8B%A8_%ED%95%A8%EC%88%98" title="단위 계단 함수">단위 계단 함수</a>이다. 단위 계단 함수는 일 차원 양수 구간 [0, ∞)의 지시함수이다. 헤비사이드 계단 함수 <i>H</i>(<i>x</i>)의 <a href="/wiki/%EB%B6%84%ED%8F%AC_(%ED%95%B4%EC%84%9D%ED%95%99)#미분" title="분포 (해석학)">분포 미분</a>은 <a href="/wiki/%EB%94%94%EB%9E%99_%EB%8D%B8%ED%83%80_%ED%95%A8%EC%88%98" title="디랙 델타 함수">디랙 델타 함수</a>와 같다: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)={\tfrac {dH(x)}{dx}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>d</mi> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)={\tfrac {dH(x)}{dx}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026372f914e09f133db324f9fb4c035e11796c86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.308ex; height:4.343ex;" alt="{\displaystyle \delta (x)={\tfrac {dH(x)}{dx}},}"></span></dd></dl> <p>이것은 다음의 성질을 따른다: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (x)dx=f(0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (x)dx=f(0).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0234aefefc245a1be39666acb61fa90129684a7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.367ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (x)dx=f(0).}"></span></dd></dl> <p>단위 계단 함수의 미분은 양의 절반 선에 의해 주어진 영역의 '경계'에서 '내부 정상 도함수'로 볼 수 있다. 고차원에서는 단위 계단 함수는 일부 정의역 <i>D</i>의 지시 함수로 일반화되는 반면, 도함수는 내부 정상 도함수로 자연스럽게 일반화된다. <i>D</i>의 표면을 <i>S</i>로 표현하면, <a href="/w/index.php?title=%EB%9D%BC%ED%94%8C%EB%9D%BC%EC%8B%9C%EC%95%88_%ED%91%9C%EC%8B%9C%EA%B8%B0&action=edit&redlink=1" class="new" title="라플라시안 표시기 (없는 문서)">지시 함수의 내부 정상 도함수</a>가 '표면 델타 함수'δ<sub><i>S</i></sub>(<b>x</b>)를 발생 시킨다는 것을 알 수 있다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{S}(\mathbf {x} )=-\mathbf {n} _{x}\cdot \nabla _{x}\mathbf {1} _{\mathbf {x} \in D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>∈</mo> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{S}(\mathbf {x} )=-\mathbf {n} _{x}\cdot \nabla _{x}\mathbf {1} _{\mathbf {x} \in D}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d41aea1e537b5157f8bf23ebd5d30c2cbb5d581" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.92ex; height:2.843ex;" alt="{\displaystyle \delta _{S}(\mathbf {x} )=-\mathbf {n} _{x}\cdot \nabla _{x}\mathbf {1} _{\mathbf {x} \in D}}"></span></dd></dl> <p>여기서 <i>n</i>는 <i>S</i>의 바깥쪽 <a href="/w/index.php?title=%EB%B2%95%EC%84%A0&action=edit&redlink=1" class="new" title="법선 (없는 문서)">법선</a>이다. <i>표면 델타 함수</i>는 다음과 같은 특성을 가지고 있다:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\int _{\mathbf {R} ^{n}}f(\mathbf {x} )\,\mathbf {n} _{x}\cdot \nabla _{x}\mathbf {1} _{\mathbf {x} \in D}\;d^{n}\mathbf {x} =\oint _{S}\,f(\mathbf {\beta } )\;d^{n-1}\mathbf {\beta } .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>∈</mo> <mi>D</mi> </mrow> </msub> <mspace width="thickmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\int _{\mathbf {R} ^{n}}f(\mathbf {x} )\,\mathbf {n} _{x}\cdot \nabla _{x}\mathbf {1} _{\mathbf {x} \in D}\;d^{n}\mathbf {x} =\oint _{S}\,f(\mathbf {\beta } )\;d^{n-1}\mathbf {\beta } .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0af12beca2911281ee39fd7fe982c5ad053b6da8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.374ex; height:5.676ex;" alt="{\displaystyle -\int _{\mathbf {R} ^{n}}f(\mathbf {x} )\,\mathbf {n} _{x}\cdot \nabla _{x}\mathbf {1} _{\mathbf {x} \in D}\;d^{n}\mathbf {x} =\oint _{S}\,f(\mathbf {\beta } )\;d^{n-1}\mathbf {\beta } .}"></span></dd></dl> <p>함수 <i>f</i>를 1로 두면서 <a href="/w/index.php?title=%EB%9D%BC%ED%94%8C%EB%9D%BC%EC%8B%9C%EC%95%88_%ED%91%9C%EC%8B%9C%EA%B8%B0&action=edit&redlink=1" class="new" title="라플라시안 표시기 (없는 문서)">지시 함수의 내부 정상 도함수</a>는 <a href="/wiki/%ED%91%9C%EB%A9%B4%EC%A0%81" class="mw-redirect" title="표면적">표면적</a><i>S</i>의 수치적 수로 통합된다. </p> <div class="mw-heading mw-heading2"><h2 id="같이_보기"><span id=".EA.B0.99.EC.9D.B4_.EB.B3.B4.EA.B8.B0"></span>같이 보기</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%80%EC%8B%9C_%ED%95%A8%EC%88%98&action=edit&section=8" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r34752755">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="div-col"> <ul><li><a href="/w/index.php?title=%EB%94%94%EB%9E%99_%EC%B8%A1%EB%8F%84&action=edit&redlink=1" class="new" title="디랙 측도 (없는 문서)">디랙 측도</a>(en:Dirac measure)</li> <li><a href="/w/index.php?title=%EB%9D%BC%ED%94%8C%EB%9D%BC%EC%8B%9C%EC%95%88_%ED%91%9C%EC%8B%9C%EA%B8%B0&action=edit&redlink=1" class="new" title="라플라시안 표시기 (없는 문서)">라플라시안 표시기</a>(en:Laplacian of the indicator)</li> <li><a href="/wiki/%EB%94%94%EB%9E%99_%EB%8D%B8%ED%83%80_%ED%95%A8%EC%88%98" title="디랙 델타 함수">디랙 델타 함수</a>(en:Dirac delta)</li> <li><a href="/w/index.php?title=%ED%99%95%EC%9E%A5_(%EC%88%A0%EC%96%B4_%EB%85%BC%EB%A6%AC)&action=edit&redlink=1" class="new" title="확장 (술어 논리) (없는 문서)">확장 (술어 논리)</a>(en:Extension (predicate logic))</li> <li><a href="/w/index.php?title=%EC%9E%90%EC%9C%A0%EB%B3%80%EC%88%98%EC%99%80_%EC%9C%A0%EA%B3%84_%EB%B3%80%EC%88%98&action=edit&redlink=1" class="new" title="자유변수와 유계 변수 (없는 문서)">자유변수와 유계 변수</a>(en:Free variables and bound variables)</li> <li><a href="/wiki/%EB%8B%A8%EC%9C%84_%EA%B3%84%EB%8B%A8_%ED%95%A8%EC%88%98" title="단위 계단 함수">단위 계단 함수</a>(en:Heaviside step function)</li> <li><a href="/wiki/%EC%95%84%EC%9D%B4%EB%B2%84%EC%8A%A8_%EA%B4%84%ED%98%B8" title="아이버슨 괄호">아이버슨 괄호</a>(en:Iverson bracket)</li> <li><a href="/wiki/%ED%81%AC%EB%A1%9C%EB%84%A4%EC%BB%A4_%EB%8D%B8%ED%83%80" title="크로네커 델타">크로네커 델타</a>(en:Kronecker delta), 지시 함수와 <a href="/wiki/%EA%B0%99%EC%9D%8C" class="mw-redirect" title="같음">동등 관계</a>에 있다고 할 수 있는 함수이다.</li> <li><a href="/w/index.php?title=Macaulay_%EA%B4%84%ED%98%B8&action=edit&redlink=1" class="new" title="Macaulay 괄호 (없는 문서)">Macaulay 괄호</a>(en:Macaulay brackets)</li> <li><a href="/wiki/%EC%A4%91%EB%B3%B5_%EC%A7%91%ED%95%A9" class="mw-redirect" title="중복 집합">중복 집합</a>(en:Multiset)</li> <li><a href="/w/index.php?title=%EB%A9%A4%EB%B2%84%EC%8B%AD_%ED%95%A8%EC%88%98&action=edit&redlink=1" class="new" title="멤버십 함수 (없는 문서)">멤버십 함수</a>(en:Membership function (mathematics))</li> <li><a href="/wiki/%EB%8B%A8%EC%88%9C_%ED%95%A8%EC%88%98" title="단순 함수">단순 함수</a>(en:Simple function)</li> <li><a href="/wiki/%EA%B0%80%EB%B3%80%EC%88%98" title="가변수">가변수</a>(en:Dummy variable (statistics))</li> <li><a href="/wiki/%ED%86%B5%EA%B3%84%EC%A0%81_%EB%B6%84%EB%A5%98" title="통계적 분류">통계적 분류</a>(en:Statistical classification)</li> <li><a href="/wiki/%EC%86%90%EC%8B%A4_%ED%95%A8%EC%88%98" title="손실 함수">손실 함수</a>(en:Zero-one loss function)</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="각주"><span id=".EA.B0.81.EC.A3.BC"></span>각주</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%80%EC%8B%9C_%ED%95%A8%EC%88%98&action=edit&section=9" title="부분 편집: 각주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r35556958">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite id="CITEREFLange2012" class="citation">Lange, Rutger-Jan (2012), <a rel="nofollow" class="external text" href="http://link.springer.com/article/10.1007%2FJHEP11(2012)032">“Potential theory, path integrals and the Laplacian of the indicator”</a>, 《Journal of High Energy Physics》 (Springer) <b>2012</b> (11): 29–30, <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<a rel="nofollow" class="external text" href="//arxiv.org/abs/1302.0864">1302.0864</a>, <a href="/wiki/%EB%B9%84%EB%B8%8C%EC%BD%94%EB%93%9C" title="비브코드">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/2012JHEP...11..032L">2012JHEP...11..032L</a>, <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2FJHEP11%282012%29032">10.1007/JHEP11(2012)032</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+High+Energy+Physics&rft.atitle=Potential+theory%2C+path+integrals+and+the+Laplacian+of+the+indicator&rft.volume=2012&rft.issue=11&rft.pages=29-30&rft.date=2012&rft_id=info%3Aarxiv%2F1302.0864&rft_id=info%3Adoi%2F10.1007%2FJHEP11%282012%29032&rft_id=info%3Abibcode%2F2012JHEP...11..032L&rft.aulast=Lange&rft.aufirst=Rutger-Jan&rft_id=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%252FJHEP11%282012%29032&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A7%80%EC%8B%9C+%ED%95%A8%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="참고_문헌"><span id=".EC.B0.B8.EA.B3.A0_.EB.AC.B8.ED.97.8C"></span>참고 문헌</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%80%EC%8B%9C_%ED%95%A8%EC%88%98&action=edit&section=10" title="부분 편집: 참고 문헌"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Folland, G.B.; <i>Real Analysis: Modern Techniques and Their Applications</i>, 2nd ed, John Wiley & Sons, Inc., 1999.</li> <li><a href="/w/index.php?title=Thomas_H._Cormen&action=edit&redlink=1" class="new" title="Thomas H. Cormen (없는 문서)">Thomas H. Cormen</a>, <a href="/w/index.php?title=Charles_E._Leiserson&action=edit&redlink=1" class="new" title="Charles E. Leiserson (없는 문서)">Charles E. Leiserson</a>, <a href="/w/index.php?title=Ronald_L._Rivest&action=edit&redlink=1" class="new" title="Ronald L. Rivest (없는 문서)">Ronald L. Rivest</a>, and <a href="/w/index.php?title=Clifford_Stein&action=edit&redlink=1" class="new" title="Clifford Stein (없는 문서)">Clifford Stein</a>. <i><a href="/wiki/Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a></i>, Second Edition. MIT Press and McGraw-Hill, 2001. <style data-mw-deduplicate="TemplateStyles:r38117996">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/0-262-03293-7" title="특수:책찾기/0-262-03293-7">0-262-03293-7</a>. Section 5.2: Indicator random variables, pp.94–99.</li> <li><a href="/w/index.php?title=Martin_Davis&action=edit&redlink=1" class="new" title="Martin Davis (없는 문서)">Martin Davis</a> ed. (1965), <i>The Undecidable</i>, Raven Press Books, Ltd., New York.</li> <li><a href="/w/index.php?title=Stephen_Kleene&action=edit&redlink=1" class="new" title="Stephen Kleene (없는 문서)">Stephen Kleene</a>, (1952), <i>Introduction to Metamathematics</i>, Wolters-Noordhoff Publishing and North Holland Publishing Company, Netherlands, Sixth Reprint with corrections 1971.</li> <li><a href="/w/index.php?title=George_Boolos&action=edit&redlink=1" class="new" title="George Boolos (없는 문서)">George Boolos</a>, <a href="/w/index.php?title=John_P._Burgess&action=edit&redlink=1" class="new" title="John P. Burgess (없는 문서)">John P. Burgess</a>, <a href="/w/index.php?title=Richard_C._Jeffrey&action=edit&redlink=1" class="new" title="Richard C. Jeffrey (없는 문서)">Richard C. Jeffrey</a> (2002), Cambridge University Press, Cambridge UK, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r38117996"><a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/0-521-00758-5" title="특수:책찾기/0-521-00758-5">0-521-00758-5</a>.</li> <li><a href="/w/index.php?title=Lotfi_A._Zadeh&action=edit&redlink=1" class="new" title="Lotfi A. Zadeh (없는 문서)">Lotfi A. Zadeh</a>, 1965, "Fuzzy sets". <i>Information and Control</i> <b>8</b>: 338–353. <a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20070622151801/http://www-bisc.cs.berkeley.edu/zadeh/papers/Fuzzy%20Sets-1965.pdf">[1]</a></li> <li><a href="/w/index.php?title=Joseph_Goguen&action=edit&redlink=1" class="new" title="Joseph Goguen (없는 문서)">Joseph Goguen</a>, 1967, "<i>L</i>-fuzzy sets". <i>Journal of Mathematical Analysis and Applications</i> <b>18</b>: 145–174</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r36480591">.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 dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li 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href="/wiki/%ED%8B%80%ED%86%A0%EB%A1%A0:%EC%A7%91%ED%95%A9%EB%A1%A0" title="틀토론:집합론"><abbr title="이 틀에 관해 토론하기" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">t</abbr></a></li><li class="nv-편집"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%AC%B8%EC%84%9C%ED%8E%B8%EC%A7%91/%ED%8B%80:%EC%A7%91%ED%95%A9%EB%A1%A0" title="특수:문서편집/틀:집합론"><abbr title="이 틀을 편집하기" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">e</abbr></a></li></ul></div><div id="집합론" style="font-size:114%;margin:0 4em"><a href="/wiki/%EC%A7%91%ED%95%A9%EB%A1%A0" title="집합론">집합론</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EC%A7%91%ED%95%A9" title="집합">집합</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%9B%90%EC%86%8C_(%EC%88%98%ED%95%99)" title="원소 (수학)">원소</a></li> <li><a href="/wiki/%EA%B3%B5%EC%A7%91%ED%95%A9" title="공집합">공집합</a></li> <li><a href="/wiki/%ED%95%9C%EC%9B%90%EC%86%8C_%EC%A7%91%ED%95%A9" title="한원소 집합">한원소 집합</a></li> <li><a href="/wiki/%EB%B6%80%EB%B6%84%EC%A7%91%ED%95%A9" title="부분집합">부분집합</a></li> <li><a href="/wiki/%EA%B5%90%EC%A7%91%ED%95%A9" title="교집합">교집합</a></li> <li><a href="/wiki/%ED%95%A9%EC%A7%91%ED%95%A9" title="합집합">합집합</a></li> <li><a href="/wiki/%EC%97%AC%EC%A7%91%ED%95%A9" title="여집합">여집합</a></li> <li><a href="/wiki/%EB%8C%80%EC%B9%AD%EC%B0%A8" title="대칭차">대칭차</a></li> <li><a href="/wiki/%EB%A9%B1%EC%A7%91%ED%95%A9" title="멱집합">멱집합</a></li> <li><a href="/wiki/%EA%B3%B1%EC%A7%91%ED%95%A9" title="곱집합">곱집합</a> <ul><li><a href="/wiki/%EC%88%9C%EC%84%9C%EC%8C%8D" title="순서쌍">순서쌍</a></li></ul></li> <li><a href="/wiki/%EC%9C%A0%ED%95%9C_%EC%A7%91%ED%95%A9" title="유한 집합">유한 집합</a></li> <li><a href="/wiki/%EB%AC%B4%ED%95%9C_%EC%A7%91%ED%95%A9" title="무한 집합">무한 집합</a> <ul><li><a href="/wiki/%EB%AC%B4%ED%95%9C_%EA%B3%B5%EB%A6%AC" title="무한 공리">무한 공리</a></li></ul></li> <li><a href="/wiki/%EA%B0%80%EC%82%B0_%EC%A7%91%ED%95%A9" title="가산 집합">가산 집합</a></li> <li><a href="/wiki/%EB%93%9C_%EB%AA%A8%EB%A5%B4%EA%B0%84%EC%9D%98_%EB%B2%95%EC%B9%99" title="드 모르간의 법칙">드 모르간의 법칙</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%ED%95%A8%EC%88%98" title="함수">함수</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%A0%95%EC%9D%98%EC%97%AD" title="정의역">정의역</a></li> <li><a href="/wiki/%EA%B3%B5%EC%97%AD" title="공역">공역</a></li> <li><a href="/wiki/%EC%B9%98%EC%97%AD" title="치역">치역</a></li> <li><a href="/wiki/%EC%83%81_(%EC%88%98%ED%95%99)" title="상 (수학)">상</a></li> <li><a href="/wiki/%EC%A0%84%EC%82%AC_%ED%95%A8%EC%88%98" title="전사 함수">전사 함수</a></li> <li><a href="/wiki/%EB%8B%A8%EC%82%AC_%ED%95%A8%EC%88%98" title="단사 함수">단사 함수</a></li> <li><a href="/wiki/%EC%A0%84%EB%8B%A8%EC%82%AC_%ED%95%A8%EC%88%98" title="전단사 함수">전단사 함수</a></li> <li><a href="/wiki/%EB%8B%A4%EA%B0%80_%ED%95%A8%EC%88%98" title="다가 함수">다가 함수</a></li> <li><a href="/wiki/%EC%97%AD%ED%95%A8%EC%88%98" title="역함수">역함수</a></li> <li><a href="/wiki/%ED%95%AD%EB%93%B1_%ED%95%A8%EC%88%98" title="항등 함수">항등 함수</a></li> <li><a href="/wiki/%EC%83%81%EC%88%98_%ED%95%A8%EC%88%98" title="상수 함수">상수 함수</a></li> <li><a class="mw-selflink selflink">지시 함수</a></li> <li><a href="/wiki/%ED%95%A8%EC%88%98%EC%9D%98_%ED%95%A9%EC%84%B1" title="함수의 합성">함수의 합성</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EA%B8%B0%EC%88%98_(%EC%88%98%ED%95%99)" title="기수 (수학)">기수</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%A7%91%ED%95%A9%EC%9D%98_%ED%81%AC%EA%B8%B0" title="집합의 크기">집합의 크기</a></li> <li><a href="/wiki/%EC%B9%B8%ED%86%A0%EC%96%B4%EC%9D%98_%EC%A0%95%EB%A6%AC" title="칸토어의 정리">칸토어의 정리</a> <ul><li><a href="/wiki/%EB%8C%80%EA%B0%81%EC%84%A0_%EB%85%BC%EB%B2%95" title="대각선 논법">대각선 논법</a></li></ul></li> <li><a href="/wiki/%EC%95%8C%EB%A0%88%ED%94%84_%EC%88%98" title="알레프 수">알레프 수</a></li> <li><a href="/wiki/%EB%B2%A0%ED%8A%B8_%EC%88%98" title="베트 수">베트 수</a></li> <li><a href="/wiki/%EA%B7%B9%ED%95%9C_%EA%B8%B0%EC%88%98" title="극한 기수">극한 기수</a></li> <li><a href="/wiki/%EA%B8%B0%EB%A9%9C_%ED%95%A8%EC%88%98" title="기멜 함수">기멜 함수</a></li> <li><a href="/wiki/%EA%B0%80%EB%8A%A5_%EA%B3%B5%EC%A2%85%EB%8F%84" title="가능 공종도">가능 공종도</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EC%88%9C%EC%84%9C%EC%88%98" title="순서수">순서수</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%A0%95%EB%A0%AC_%EC%88%9C%EC%84%9C" class="mw-redirect" title="정렬 순서">정렬 순서</a></li> <li><a href="/wiki/%EC%B4%88%ED%95%9C%EA%B7%80%EB%82%A9%EB%B2%95" class="mw-redirect" title="초한귀납법">초한귀납법</a></li> <li><a href="/wiki/%EA%B3%B5%EC%A2%85%EB%8F%84" title="공종도">공종도</a> <ul><li><a href="/wiki/%EC%BE%A8%EB%8B%88%EA%B7%B8%EC%9D%98_%EC%A0%95%EB%A6%AC_(%EC%A7%91%ED%95%A9%EB%A1%A0)" title="쾨니그의 정리 (집합론)">쾨니그의 정리</a></li></ul></li> <li><a href="/wiki/%ED%95%98%EB%A5%B4%ED%86%A1%EC%8A%A4_%EC%88%98" title="하르톡스 수">하르톡스 수</a></li> <li><a href="/wiki/%EC%A0%95%EA%B7%9C_%ED%95%A8%EC%88%98" title="정규 함수">정규 함수</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">역설의 해소</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%86%8C%EB%B0%95%ED%95%9C_%EC%A7%91%ED%95%A9%EB%A1%A0" title="소박한 집합론">소박한 집합론</a></li> <li><a href="/wiki/%EB%9F%AC%EC%85%80%EC%9D%98_%EC%97%AD%EC%84%A4" title="러셀의 역설">러셀의 역설</a></li> <li><a href="/wiki/%EC%B9%B8%ED%86%A0%EC%96%B4_%EC%97%AD%EC%84%A4" title="칸토어 역설">칸토어 역설</a></li> <li><a href="/wiki/%EB%B6%80%EB%9E%84%EB%A6%AC%ED%8F%AC%EB%A5%B4%ED%8B%B0_%EC%97%AD%EC%84%A4" title="부랄리포르티 역설">부랄리포르티 역설</a></li> <li><a href="/wiki/%EB%AA%A8%EC%9E%84_(%EC%A7%91%ED%95%A9%EB%A1%A0)" title="모임 (집합론)">모임</a></li> <li><a href="/wiki/%EC%9C%A0%ED%98%95_%EC%9D%B4%EB%A1%A0" title="유형 이론">유형 이론</a> <ul><li>《<a href="/wiki/%EC%88%98%ED%95%99_%EC%9B%90%EB%A6%AC" title="수학 원리">수학 원리</a>》</li></ul></li> <li><a href="/wiki/%EC%B2%B4%EB%A5%B4%EB%A9%9C%EB%A1%9C-%ED%94%84%EB%A0%9D%EC%BC%88_%EC%A7%91%ED%95%A9%EB%A1%A0" title="체르멜로-프렝켈 집합론">체르멜로-프렝켈 집합론</a></li> <li><a href="/wiki/%EC%83%88_%EA%B8%B0%EC%B4%88" title="새 기초">새 기초</a></li> <li><a href="/wiki/%ED%8F%B0_%EB%85%B8%EC%9D%B4%EB%A7%8C-%EB%B2%A0%EB%A5%B4%EB%82%98%EC%9D%B4%EC%8A%A4-%EA%B4%B4%EB%8D%B8_%EC%A7%91%ED%95%A9%EB%A1%A0" title="폰 노이만-베르나이스-괴델 집합론">폰 노이만-베르나이스-괴델 집합론</a></li> <li><a href="/w/index.php?title=%EB%AA%A8%EC%8A%A4-%EC%BC%88%EB%A6%AC_%EC%A7%91%ED%95%A9%EB%A1%A0&action=edit&redlink=1" class="new" title="모스-켈리 집합론 (없는 문서)">모스-켈리 집합론</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" title="선택 공리">선택 공리</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EA%B0%80%EC%82%B0_%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" class="mw-redirect" title="가산 선택 공리">가산 선택 공리</a></li> <li><a href="/wiki/%EC%9D%98%EC%A1%B4%EC%A0%81_%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" class="mw-redirect" title="의존적 선택 공리">의존적 선택 공리</a></li> <li><a href="/wiki/%EC%B4%88%EB%A5%B8_%EB%B3%B4%EC%A1%B0%EC%A0%95%EB%A6%AC" title="초른 보조정리">초른 보조정리</a></li> <li><a href="/wiki/%EC%8A%88%ED%95%84%EB%9D%BC%EC%9D%B8_%ED%99%95%EC%9E%A5%EC%A0%95%EB%A6%AC" title="슈필라인 확장정리">슈필라인 확장정리</a></li> <li><a href="/wiki/%EB%B0%94%EB%82%98%ED%9D%90-%ED%83%80%EB%A5%B4%EC%8A%A4%ED%82%A4_%EC%97%AD%EC%84%A4" title="바나흐-타르스키 역설">바나흐-타르스키 역설</a></li> <li><a href="/wiki/%ED%8B%B0%ED%98%B8%EB%85%B8%ED%94%84_%EC%A0%95%EB%A6%AC" title="티호노프 정리">티호노프 정리</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">집합론의 <a href="/wiki/%EB%AA%A8%ED%98%95_%EC%9D%B4%EB%A1%A0" title="모형 이론">모형</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EA%B5%AC%EC%84%B1_%EA%B0%80%EB%8A%A5_%EC%A0%84%EC%B2%B4" title="구성 가능 전체">구성 가능 전체</a></li> <li><a href="/wiki/%EC%B6%94%EC%9D%B4%EC%A0%81_%EC%A7%91%ED%95%A9" title="추이적 집합">추이적 집합</a></li> <li><a href="/wiki/%EC%B6%94%EC%9D%B4%EC%A0%81_%EB%AA%A8%ED%98%95" title="추이적 모형">추이적 모형</a></li> <li><a href="/wiki/%EC%88%9C%EC%84%9C%EC%88%98_%EC%A0%95%EC%9D%98_%EA%B0%80%EB%8A%A5_%EC%A7%91%ED%95%A9" title="순서수 정의 가능 집합">순서수 정의 가능 집합</a></li> <li><a href="/wiki/%EA%B0%95%EC%A0%9C%EB%B2%95" title="강제법">강제법</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%ED%81%B0_%EA%B8%B0%EC%88%98" title="큰 기수">큰 기수</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EB%8F%84%EB%8B%AC_%EB%B6%88%EA%B0%80%EB%8A%A5%ED%95%9C_%EA%B8%B0%EC%88%98" title="도달 불가능한 기수">도달 불가능한 기수</a></li> <li><a href="/wiki/%EA%B0%80%EC%B8%A1_%EA%B8%B0%EC%88%98" title="가측 기수">가측 기수</a></li> <li><a href="/wiki/%EC%95%BD%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B8%B0%EC%88%98" title="약콤팩트 기수">약콤팩트 기수</a></li> <li><a href="/wiki/%EA%B0%95%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B8%B0%EC%88%98" title="강콤팩트 기수">강콤팩트 기수</a></li> <li><a href="/wiki/%EC%B4%88%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B8%B0%EC%88%98" title="초콤팩트 기수">초콤팩트 기수</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EC%B2%B4%EB%A5%B4%EB%A9%9C%EB%A1%9C-%ED%94%84%EB%A0%9D%EC%BC%88_%EC%A7%91%ED%95%A9%EB%A1%A0" title="체르멜로-프렝켈 집합론">ZF</a>와 독립된 명제</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%97%B0%EC%86%8D%EC%B2%B4_%EA%B0%80%EC%84%A4" title="연속체 가설">연속체 가설</a></li> <li><a href="/wiki/%ED%8A%B9%EC%9D%B4_%EA%B8%B0%EC%88%98_%EA%B0%80%EC%84%A4" title="특이 기수 가설">특이 기수 가설</a></li> <li><a href="/wiki/%EC%88%98%EC%8A%AC%EB%A6%B0_%EA%B0%80%EC%84%A4" class="mw-redirect" title="수슬린 가설">수슬린 가설</a></li> <li><a href="/wiki/%ED%99%94%EC%9D%B4%ED%8A%B8%ED%97%A4%EB%93%9C_%EB%AC%B8%EC%A0%9C" title="화이트헤드 문제">화이트헤드 문제</a></li> <li><a href="/wiki/%EA%B2%B0%EC%A0%95_%EC%A7%91%ED%95%A9" title="결정 집합">결정 공리</a></li> <li><a href="/wiki/%EB%A7%88%ED%8B%B4_%EA%B3%B5%EB%A6%AC" title="마틴 공리">마틴 공리</a></li> <li><a href="/wiki/%EB%8B%A4%EC%9D%B4%EC%95%84%EB%AA%AC%EB%93%9C_%EC%9B%90%EB%A6%AC" class="mw-redirect" title="다이아몬드 원리">다이아몬드 원리</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">원본 주소 "<a dir="ltr" href="https://ko.wikipedia.org/w/index.php?title=지시_함수&oldid=36854178">https://ko.wikipedia.org/w/index.php?title=지시_함수&oldid=36854178</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%B6%84%EB%A5%98" title="특수:분류">분류</a>: <ul><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%B8%A1%EB%8F%84%EB%A1%A0" title="분류:측도론">측도론</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%A0%81%EB%B6%84%ED%95%99" title="분류:적분학">적분학</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%8B%A4%ED%95%B4%EC%84%9D%ED%95%99" title="분류:실해석학">실해석학</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%A7%91%ED%95%A9%EB%A1%A0%EC%9D%98_%EA%B8%B0%EB%B3%B8_%EA%B0%9C%EB%85%90" title="분류:집합론의 기본 개념">집합론의 기본 개념</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EA%B3%84%EC%82%B0_%EA%B0%80%EB%8A%A5%EC%84%B1_%EC%9D%B4%EB%A1%A0" title="분류:계산 가능성 이론">계산 가능성 이론</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%ED%99%95%EB%A5%A0%EB%A1%A0" title="분류:확률론">확률론</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%ED%95%A8%EC%88%98%EC%9D%98_%EC%A2%85%EB%A5%98" title="분류:함수의 종류">함수의 종류</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">숨은 분류: <ul><li><a href="/wiki/%EB%B6%84%EB%A5%98:%ED%95%B4%EA%B2%B0%EB%90%98%EC%A7%80_%EC%95%8A%EC%9D%80_%EC%86%8D%EC%84%B1%EC%9D%B4_%EC%9E%88%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:해결되지 않은 속성이 있는 문서">해결되지 않은 속성이 있는 문서</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%9C%84%ED%82%A4%EB%8D%B0%EC%9D%B4%ED%84%B0_%EC%86%8D%EC%84%B1_P7859%EB%A5%BC_%EC%82%AC%EC%9A%A9%ED%95%98%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:위키데이터 속성 P7859를 사용하는 문서">위키데이터 속성 P7859를 사용하는 문서</a></li><li><a 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