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<span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Popular_culture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Popular_culture"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Popular culture</span> </div> </a> <ul id="toc-Popular_culture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Overview" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Overview"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Overview</span> </div> </a> <ul id="toc-Overview-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extensions_to_higher_numbers_of_sets" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extensions_to_higher_numbers_of_sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Extensions to higher numbers of sets</span> </div> </a> <button aria-controls="toc-Extensions_to_higher_numbers_of_sets-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Extensions to higher numbers of sets subsection</span> </button> <ul id="toc-Extensions_to_higher_numbers_of_sets-sublist" class="vector-toc-list"> <li id="toc-Edwards–Venn_diagrams" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Edwards–Venn_diagrams"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Edwards–Venn diagrams</span> </div> </a> <ul id="toc-Edwards–Venn_diagrams-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_diagrams" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_diagrams"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Other diagrams</span> </div> </a> <ul id="toc-Other_diagrams-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_concepts" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Related concepts</span> </div> </a> <ul id="toc-Related_concepts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Venn diagram</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 55 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-55" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">55 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AE%D8%B7%D8%B7_%D9%81%D9%86" title="مخطط فن – Arabic" lang="ar" hreflang="ar" data-title="مخطط فن" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Venn_diaqram%C4%B1" title="Venn diaqramı – Azerbaijani" lang="az" hreflang="az" data-title="Venn diaqramı" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AD%E0%A7%87%E0%A6%A8_%E0%A6%B0%E0%A7%87%E0%A6%96%E0%A6%BE%E0%A6%9A%E0%A6%BF%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="ভেন রেখাচিত্র – Bangla" lang="bn" hreflang="bn" data-title="ভেন রেখাচিত্র" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D0%B0%D0%B3%D1%80%D0%B0%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%92%D0%B5%D0%BD" title="Диаграма на Вен – Bulgarian" lang="bg" hreflang="bg" data-title="Диаграма на Вен" data-language-autonym="Български" data-language-local-name="Bulgarian" 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/Diagrama_de_Venn" title="Diagrama de Venn – Catalan" lang="ca" hreflang="ca" data-title="Diagrama de Venn" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Venn%C5%AFv_diagram" title="Vennův diagram – Czech" lang="cs" hreflang="cs" data-title="Vennův diagram" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Diagram_Venn" title="Diagram Venn – Welsh" lang="cy" hreflang="cy" data-title="Diagram Venn" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Venn-diagram" title="Venn-diagram – Danish" lang="da" hreflang="da" data-title="Venn-diagram" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-se mw-list-item"><a href="https://se.wikipedia.org/wiki/Venn-diagr%C3%A1mma" title="Venn-diagrámma – Northern Sami" lang="se" hreflang="se" data-title="Venn-diagrámma" data-language-autonym="Davvisámegiella" data-language-local-name="Northern Sami" class="interlanguage-link-target"><span>Davvisámegiella</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Mengendiagramm" title="Mengendiagramm – German" lang="de" hreflang="de" data-title="Mengendiagramm" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Venni_diagramm" title="Venni diagramm – Estonian" lang="et" hreflang="et" data-title="Venni diagramm" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%AC%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%B1_%CE%92%CE%B5%CE%BD" title="Διάγραμμα Βεν – Greek" lang="el" hreflang="el" data-title="Διάγραμμα Βεν" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://es.wikipedia.org/wiki/Diagrama_de_Venn" title="Diagrama de Venn – Spanish" lang="es" hreflang="es" data-title="Diagrama de Venn" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Venn-a_diagramo" title="Venn-a diagramo – Esperanto" lang="eo" hreflang="eo" data-title="Venn-a diagramo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Venn_diagrama" title="Venn diagrama – Basque" lang="eu" hreflang="eu" data-title="Venn diagrama" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D9%85%D9%88%D8%AF%D8%A7%D8%B1_%D9%88%D9%86" title="نمودار ون – Persian" lang="fa" hreflang="fa" data-title="نمودار ون" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Diagramme_de_Venn" title="Diagramme de Venn – French" lang="fr" hreflang="fr" data-title="Diagramme de Venn" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/L%C3%A9ar%C3%A1id_Venn" title="Léaráid Venn – Irish" lang="ga" hreflang="ga" data-title="Léaráid Venn" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Diagrama_de_Venn" title="Diagrama de Venn – Galician" lang="gl" hreflang="gl" data-title="Diagrama de Venn" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A4_%EB%8B%A4%EC%9D%B4%EC%96%B4%EA%B7%B8%EB%9E%A8" title="벤 다이어그램 – Korean" lang="ko" hreflang="ko" data-title="벤 다이어그램" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8E%D5%A5%D5%B6%D5%B6%D5%AB_%D5%A4%D5%AB%D5%A1%D5%A3%D6%80%D5%A1%D5%B4" title="Վեննի դիագրամ – Armenian" lang="hy" hreflang="hy" data-title="Վեննի դիագրամ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A5%87%E0%A4%A8_%E0%A4%86%E0%A4%B0%E0%A5%87%E0%A4%96" title="वेन आरेख – Hindi" lang="hi" hreflang="hi" data-title="वेन आरेख" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Diagramo_di_Venn" title="Diagramo di Venn – Ido" lang="io" hreflang="io" data-title="Diagramo di Venn" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Diagram_Venn" title="Diagram Venn – Indonesian" lang="id" hreflang="id" data-title="Diagram Venn" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Venn-mynd" title="Venn-mynd – Icelandic" lang="is" hreflang="is" data-title="Venn-mynd" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Diagramma_di_Venn" title="Diagramma di Venn – Italian" lang="it" hreflang="it" data-title="Diagramma di Venn" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%99%D7%90%D7%92%D7%A8%D7%9E%D7%AA_%D7%95%D7%9F" title="דיאגרמת ון – Hebrew" lang="he" hreflang="he" data-title="דיאגרמת ון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Venna_diagramma" title="Venna diagramma – Latvian" lang="lv" hreflang="lv" data-title="Venna diagramma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Veno_diagrama" title="Veno diagrama – Lithuanian" lang="lt" hreflang="lt" data-title="Veno diagrama" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Venn-diagram" title="Venn-diagram – Hungarian" lang="hu" hreflang="hu" data-title="Venn-diagram" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Gambar_rajah_Venn" title="Gambar rajah Venn – Malay" lang="ms" hreflang="ms" data-title="Gambar rajah Venn" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" 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/Venndiagram" title="Venndiagram – Dutch" lang="nl" hreflang="nl" data-title="Venndiagram" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%99%E3%83%B3%E5%9B%B3" title="ベン図 – Japanese" lang="ja" hreflang="ja" data-title="ベン図" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Venn-diagram" title="Venn-diagram – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Venn-diagram" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Venn-diagram" title="Venn-diagram – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Venn-diagram" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Eyler_Venn_diagrammasi" title="Eyler Venn diagrammasi – Uzbek" lang="uz" hreflang="uz" data-title="Eyler Venn diagrammasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Diagrama_d%27Euler-Venn" title="Diagrama d'Euler-Venn – Piedmontese" lang="pms" hreflang="pms" data-title="Diagrama d'Euler-Venn" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Diagram_Venna" title="Diagram Venna – Polish" lang="pl" hreflang="pl" data-title="Diagram Venna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Diagrama_de_Venn" title="Diagrama de Venn – Portuguese" lang="pt" hreflang="pt" data-title="Diagrama de Venn" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Diagram%C4%83_Venn" title="Diagramă Venn – Romanian" lang="ro" hreflang="ro" data-title="Diagramă Venn" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D0%B0%D0%B3%D1%80%D0%B0%D0%BC%D0%BC%D0%B0_%D0%92%D0%B5%D0%BD%D0%BD%D0%B0" title="Диаграмма Венна – Russian" lang="ru" hreflang="ru" data-title="Диаграмма Венна" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Venn_diagram" title="Venn diagram – Simple English" lang="en-simple" hreflang="en-simple" data-title="Venn diagram" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Vennov_diagram" title="Vennov diagram – Slovak" lang="sk" hreflang="sk" data-title="Vennov diagram" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Vennov_diagram" title="Vennov diagram – Slovenian" lang="sl" hreflang="sl" data-title="Vennov diagram" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Jaantusyada_Venn" title="Jaantusyada Venn – Somali" lang="so" hreflang="so" data-title="Jaantusyada Venn" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Venn-diagrammi" title="Venn-diagrammi – Finnish" lang="fi" hreflang="fi" data-title="Venn-diagrammi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Venndiagram" title="Venndiagram – Swedish" lang="sv" hreflang="sv" data-title="Venndiagram" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%86%E0%AE%A9%E0%AF%8D_%E0%AE%AA%E0%AE%9F%E0%AE%AE%E0%AF%8D" title="வென் படம் – Tamil" lang="ta" hreflang="ta" data-title="வென் படம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%9C%E0%B8%99%E0%B8%A0%E0%B8%B2%E0%B8%9E%E0%B9%80%E0%B8%A7%E0%B8%99%E0%B8%99%E0%B9%8C" title="แผนภาพเวนน์ – Thai" lang="th" hreflang="th" data-title="แผนภาพเวนน์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Venn_%C5%9Femas%C4%B1" title="Venn şeması – Turkish" lang="tr" hreflang="tr" data-title="Venn şeması" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D1%96%D0%B0%D0%B3%D1%80%D0%B0%D0%BC%D0%B0_%D0%92%D0%B5%D0%BD%D0%BD%D0%B0" title="Діаграма Венна – Ukrainian" lang="uk" hreflang="uk" data-title="Діаграма Венна" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%C6%A1_%C4%91%E1%BB%93_Venn" title="Sơ đồ Venn – Vietnamese" lang="vi" hreflang="vi" data-title="Sơ đồ Venn" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Diagrama_Venn" title="Diagrama Venn – Waray" lang="war" hreflang="war" data-title="Diagrama Venn" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%BA%AB%E6%B0%8F%E5%9C%96" title="溫氏圖 – Cantonese" lang="yue" hreflang="yue" data-title="溫氏圖" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%96%87%E6%B0%8F%E5%9B%BE" title="文氏图 – Chinese" lang="zh" hreflang="zh" data-title="文氏图" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Diagram that shows all possible logical relations between a collection of sets</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Venn_diagram_gr_la_ru.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Venn_diagram_gr_la_ru.svg/220px-Venn_diagram_gr_la_ru.svg.png" decoding="async" width="220" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Venn_diagram_gr_la_ru.svg/330px-Venn_diagram_gr_la_ru.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Venn_diagram_gr_la_ru.svg/440px-Venn_diagram_gr_la_ru.svg.png 2x" data-file-width="1001" data-file-height="965" /></a><figcaption>Venn diagram showing the uppercase <a href="/wiki/Glyph" title="Glyph">glyphs</a> shared by the <a href="/wiki/Greek_alphabet" title="Greek alphabet">Greek</a> (upper left), <a href="/wiki/Latin_alphabets" class="mw-redirect" title="Latin alphabets">Latin</a> (upper right), and <a href="/wiki/Russian_alphabet" title="Russian alphabet">Russian Cyrillic</a> (bottom) alphabets</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on <a href="/wiki/Statistics" title="Statistics">statistics</a></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Standard_deviation_diagram_micro.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/250px-Standard_deviation_diagram_micro.svg.png" decoding="async" width="250" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/375px-Standard_deviation_diagram_micro.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/500px-Standard_deviation_diagram_micro.svg.png 2x" data-file-width="400" data-file-height="200" /></a></span></td></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability" title="Probability">Probability</a> <ul><li><a href="/wiki/Probability_axioms" title="Probability axioms">Axioms</a></li></ul></li> <li><a href="/wiki/Determinism" title="Determinism">Determinism</a> <ul><li><a href="/wiki/Deterministic_system" title="Deterministic system">System</a></li></ul></li> <li><a href="/wiki/Indeterminism" title="Indeterminism">Indeterminism</a></li> <li><a href="/wiki/Randomness" title="Randomness">Randomness</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability_space" title="Probability space">Probability space</a></li> <li><a href="/wiki/Sample_space" title="Sample space">Sample space</a></li> <li><a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">Event</a> <ul><li><a href="/wiki/Collectively_exhaustive_events" title="Collectively exhaustive events">Collectively exhaustive events</a></li> <li><a href="/wiki/Elementary_event" title="Elementary event">Elementary event</a></li> <li><a href="/wiki/Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></li> <li><a href="/wiki/Outcome_(probability)" title="Outcome (probability)">Outcome</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li></ul></li> <li><a href="/wiki/Experiment_(probability_theory)" title="Experiment (probability theory)">Experiment</a> <ul><li><a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a></li></ul></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a> <ul><li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal distribution</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul></li> <li><a href="/wiki/Probability_measure" title="Probability measure">Probability measure</a></li> <li><a href="/wiki/Random_variable" title="Random variable">Random variable</a> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Continuous_or_discrete_variable" title="Continuous or discrete variable">Continuous or discrete</a></li> <li><a href="/wiki/Expected_value" title="Expected value">Expected value</a></li> <li><a href="/wiki/Variance" title="Variance">Variance</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Realization_(probability)" title="Realization (probability)">Observed value</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a></li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic process</a></li></ul></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Complementary_event" title="Complementary event">Complementary event</a></li> <li><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li> <li><a href="/wiki/Marginal_distribution" title="Marginal distribution">Marginal probability</a></li> <li><a href="/wiki/Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li> <li><a href="/wiki/Conditional_independence" title="Conditional independence">Conditional independence</a></li> <li><a href="/wiki/Law_of_total_probability" title="Law of total probability">Law of total probability</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a></li> <li><a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a></li> <li><a href="/wiki/Boole%27s_inequality" title="Boole's inequality">Boole's inequality</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a class="mw-selflink selflink">Venn diagram</a></li> <li><a href="/wiki/Tree_diagram_(probability_theory)" title="Tree diagram (probability theory)">Tree diagram</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_fundamentals" title="Template:Probability fundamentals"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_fundamentals" title="Template talk:Probability fundamentals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_fundamentals" title="Special:EditPage/Template:Probability fundamentals"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>A <b>Venn diagram</b> is a widely used <a href="/wiki/Diagram" title="Diagram">diagram</a> style that shows the logical relation between <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a>, popularized by <a href="/wiki/John_Venn" title="John Venn">John Venn</a> (1834–1923) in the 1880s. The diagrams are used to teach elementary <a href="/wiki/Set_theory" title="Set theory">set theory</a>, and to illustrate simple set relationships in <a href="/wiki/Probability" title="Probability">probability</a>, <a href="/wiki/Logic" title="Logic">logic</a>, <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Linguistics" title="Linguistics">linguistics</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a>. A Venn diagram uses simple closed curves drawn on a plane to represent sets. Very often, these curves are circles or ellipses. </p><p>Similar ideas had been proposed before Venn such as by <a href="/wiki/Christian_Weise" title="Christian Weise">Christian Weise</a> in 1712 (<i>Nucleus Logicoe Wiesianoe</i>) and <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> (<i><a href="/wiki/Letters_to_a_German_Princess" title="Letters to a German Princess">Letters to a German Princess</a></i>) in 1768. The idea was popularised by Venn in <i>Symbolic Logic</i>, Chapter V "Diagrammatic Representation", published in 1881. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Details">Details</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=1" title="Edit section: Details"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Primary"></span><span class="anchor" id="Simple"></span><span class="anchor" id="Cylindrical"></span><span class="anchor" id="Metric"></span><span class="anchor" id="2"></span><span class="anchor" id="3"></span>A Venn diagram, also called a <i>set diagram</i> or <i>logic diagram</i>, shows <i>all</i> possible logical relations between a finite collection of different sets. These diagrams depict <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">elements</a> as points in the plane, and sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. The points inside a curve labelled <i>S</i> represent elements of the set <i>S</i>, while points outside the boundary represent elements not in the set <i>S</i>. This lends itself to intuitive visualizations; for example, the set of all elements that are members of both sets <i>S</i> and <i>T</i>, denoted <i>S</i> ∩ <i>T</i> and read "the intersection of <i>S</i> and <i>T</i>", is represented visually by the area of overlap of the regions <i>S</i> and <i>T</i>.<sup id="cite_ref-Peil_2020_1-0" class="reference"><a href="#cite_note-Peil_2020-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>In Venn diagrams, the curves are overlapped in every possible way, showing all possible relations between the sets. They are thus a special case of <a href="/wiki/Euler_diagram" title="Euler diagram">Euler diagrams</a>, which do not necessarily show all relations. Venn diagrams were conceived around 1880 by John Venn. They are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics, and computer science. </p><p>A Venn diagram in which the area of each shape is proportional to the number of elements it contains is called an <b>area-proportional</b> (or <b>scaled</b>) <b>Venn diagram</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=2" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Venn_diagram_of_legs_and_flying.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Venn_diagram_of_legs_and_flying.svg/220px-Venn_diagram_of_legs_and_flying.svg.png" decoding="async" width="220" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Venn_diagram_of_legs_and_flying.svg/330px-Venn_diagram_of_legs_and_flying.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Venn_diagram_of_legs_and_flying.svg/440px-Venn_diagram_of_legs_and_flying.svg.png 2x" data-file-width="305" data-file-height="205" /></a><figcaption>Sets of creatures with two legs, and creatures that fly</figcaption></figure> <p>This example involves two sets of creatures, represented here as colored circles. The orange circle represents all types of creatures that have two legs. The blue circle represents creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that have two legs <i>and</i> can fly—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap. This overlapping region would only contain those elements (in this example, creatures) that are members of both the orange set (two-legged creatures) and the blue set (flying creatures). </p><p>Humans and penguins are bipedal, and so are in the orange circle, but since they cannot fly, they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes can fly, but have six, not two, legs, so the point for mosquitoes is in the part of the blue circle that does not overlap with the orange one. Creatures that are neither two-legged nor able to fly (for example, whales and spiders) would all be represented by points outside both circles. </p><p>The combined region of the two sets is called their <i><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></i>, denoted by <span class="nowrap">A ∪ B</span>, where A is the orange circle and B the blue. The union in this case contains all living creatures that either are two-legged or can fly (or both). The region included in both A and B, where the two sets overlap, is called the <i><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></i> of A and B, denoted by <span class="nowrap">A ∩ B</span>. </p> <div class="mw-heading mw-heading2"><h2 id="History"><span class="anchor" id="Symmetric"></span><span class="anchor" id="n"></span>History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=3" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Venn-stainedglass-gonville-caius.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Venn-stainedglass-gonville-caius.jpg/150px-Venn-stainedglass-gonville-caius.jpg" decoding="async" width="150" height="291" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Venn-stainedglass-gonville-caius.jpg/225px-Venn-stainedglass-gonville-caius.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Venn-stainedglass-gonville-caius.jpg/300px-Venn-stainedglass-gonville-caius.jpg 2x" data-file-width="380" data-file-height="736" /></a><figcaption><a href="/wiki/Stained-glass" class="mw-redirect" title="Stained-glass">Stained-glass</a> window with Venn diagram in <a href="/wiki/Gonville_and_Caius_College,_Cambridge" title="Gonville and Caius College, Cambridge">Gonville and Caius College, Cambridge</a></figcaption></figure> <p>Venn diagrams were introduced in 1880 by <a href="/wiki/John_Venn" title="John Venn">John Venn</a> in a paper entitled "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings"<sup id="cite_ref-Venn_2014_2-0" class="reference"><a href="#cite_note-Venn_2014-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> in the <i>Philosophical Magazine and Journal of Science</i>,<sup id="cite_ref-PM_3-0" class="reference"><a href="#cite_note-PM-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> about the different ways to represent <a href="/wiki/Proposition" title="Proposition">propositions</a> by diagrams.<sup id="cite_ref-Venn1880_1_4-0" class="reference"><a href="#cite_note-Venn1880_1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Venn1880_2_5-0" class="reference"><a href="#cite_note-Venn1880_2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Sandifer2003_6-0" class="reference"><a href="#cite_note-Sandifer2003-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The use of these types of diagrams in <a href="/wiki/Formal_logic" class="mw-redirect" title="Formal logic">formal logic</a>, according to <a href="/wiki/Frank_Ruskey" title="Frank Ruskey">Frank Ruskey</a> and Mark Weston, predates Venn but are "rightly associated" with him as he "comprehensively surveyed and formalized their usage, and was the first to generalize them".<sup id="cite_ref-Ruskey2005_7-0" class="reference"><a href="#cite_note-Ruskey2005-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Diagrams of overlapping circles representing unions and intersections were introduced by Catalan philosopher <a href="/wiki/Ramon_Llull" title="Ramon Llull">Ramon Llull</a> (c. 1232–1315/1316) in the 13th century, who used them to illustrate combinations of basic principles.<sup id="cite_ref-Baron_1969_8-0" class="reference"><a href="#cite_note-Baron_1969-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> (1646–1716) produced similar diagrams in the 17th century (though much of this work was unpublished), as did Johann Christian Lange in a work from 1712 describing <a href="/wiki/Christian_Weise" title="Christian Weise">Christian Weise</a>'s contributions to logic.<sup id="cite_ref-Leibniz_1690_9-0" class="reference"><a href="#cite_note-Leibniz_1690-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Baron_1969_8-1" class="reference"><a href="#cite_note-Baron_1969-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Euler_diagram" title="Euler diagram">Euler diagrams</a>, which are similar to Venn diagrams but don't necessarily contain all possible unions and intersections, were first made prominent by mathematician <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> in the 18th century.<sup id="cite_ref-NB_1_10-0" class="reference"><a href="#cite_note-NB_1-10"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Venn1881_11-0" class="reference"><a href="#cite_note-Venn1881-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gailand_1967_12-0" class="reference"><a href="#cite_note-Gailand_1967-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Venn did not use the term "Venn diagram" and referred to the concept as "Eulerian Circles".<sup id="cite_ref-Sandifer2003_6-1" class="reference"><a href="#cite_note-Sandifer2003-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> He became acquainted with Euler diagrams in 1862 and wrote that Venn diagrams did not occur to him "till much later", while attempting to adapt Euler diagrams to <a href="/wiki/Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a>.<sup id="cite_ref-Maths_Today_13-0" class="reference"><a href="#cite_note-Maths_Today-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> In the opening sentence of his 1880 article Venn wrote that Euler diagrams were the only diagrammatic representation of logic to gain "any general acceptance".<sup id="cite_ref-Venn1880_1_4-1" class="reference"><a href="#cite_note-Venn1880_1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Venn1880_2_5-1" class="reference"><a href="#cite_note-Venn1880_2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Venn viewed his diagrams as a pedagogical tool, analogous to verification of physical concepts through experiment. As an example of their applications, he noted that a three-set diagram could show the <a href="/wiki/Syllogism" title="Syllogism">syllogism</a>: 'All <i>A</i> is some <i>B</i>. No <i>B</i> is any <i>C</i>. Hence, no <i>A</i> is any <i>C</i>.'<sup id="cite_ref-Maths_Today_13-1" class="reference"><a href="#cite_note-Maths_Today-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Charles_L._Dodgson" class="mw-redirect" title="Charles L. Dodgson">Charles L. Dodgson</a> (Lewis Carroll) includes "Venn's Method of Diagrams" as well as "Euler's Method of Diagrams" in an "Appendix, Addressed to Teachers" of his book <i>Symbolic Logic</i> (4th edition published in 1896). The term "Venn diagram" was later used by <a href="/wiki/Clarence_Irving_Lewis" class="mw-redirect" title="Clarence Irving Lewis">Clarence Irving Lewis</a> in 1918, in his book <i>A Survey of Symbolic Logic</i>.<sup id="cite_ref-Ruskey2005_7-1" class="reference"><a href="#cite_note-Ruskey2005-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lewis1918_14-0" class="reference"><a href="#cite_note-Lewis1918-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>In the 20th century, Venn diagrams were further developed. <a href="/wiki/David_Wilson_Henderson" class="mw-redirect" title="David Wilson Henderson">David Wilson Henderson</a> showed, in 1963, that the existence of an <i>n</i>-Venn diagram with <i>n</i>-fold <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> implied that <i>n</i> was a <a href="/wiki/Prime_number" title="Prime number">prime number</a>.<sup id="cite_ref-Henderson_1963_15-0" class="reference"><a href="#cite_note-Henderson_1963-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> He also showed that such symmetric Venn diagrams exist when <i>n</i> is five or seven. In 2002, Peter Hamburger found symmetric Venn diagrams for <i>n</i> = 11 and in 2003, Griggs, Killian, and Savage showed that symmetric Venn diagrams exist for all other primes. These combined results show that rotationally symmetric Venn diagrams exist, if and only if <i>n</i> is a prime number.<sup id="cite_ref-Ruskey_2006_16-0" class="reference"><a href="#cite_note-Ruskey_2006-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Venn diagrams and Euler diagrams were incorporated as part of instruction in set theory, as part of the <a href="/wiki/New_math" class="mw-redirect" title="New math">new math</a> movement in the 1960s. Since then, they have also been adopted in the curriculum of other fields such as reading.<sup id="cite_ref-Strategies_17-0" class="reference"><a href="#cite_note-Strategies-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Popular_culture">Popular culture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=4" title="Edit section: Popular culture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Venn diagrams have been commonly used in <a href="/wiki/Meme" title="Meme">memes</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> At least one politician has been mocked for misusing Venn diagrams.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=5" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Set_(mathematics)#Basic_operations" title="Set (mathematics)">Set (mathematics) § Basic operations</a></div> <style data-mw-deduplicate="TemplateStyles:r1248256098">@media all and (max-width:720px){.mw-parser-output .mod-gallery{width:100%!important}}.mw-parser-output .mod-gallery{display:table}.mw-parser-output .mod-gallery-default{background:transparent;margin-top:4px}.mw-parser-output .mod-gallery-center{margin-left:auto;margin-right:auto}.mw-parser-output .mod-gallery-left{float:left}.mw-parser-output .mod-gallery-right{float:right}.mw-parser-output .mod-gallery-none{float:none}.mw-parser-output .mod-gallery-collapsible{width:100%}.mw-parser-output .mod-gallery .title,.mw-parser-output .mod-gallery .main,.mw-parser-output .mod-gallery .footer{display:table-row}.mw-parser-output .mod-gallery .title>div{display:table-cell;padding:0 4px 4px;text-align:center;font-weight:bold}.mw-parser-output .mod-gallery .main>div{display:table-cell}.mw-parser-output .mod-gallery .gallery{line-height:1.35em}.mw-parser-output .mod-gallery .footer>div{display:table-cell;padding:4px;text-align:right;font-size:85%;line-height:1em}.mw-parser-output .mod-gallery .title>div *,.mw-parser-output .mod-gallery .footer>div *{overflow:visible}.mw-parser-output .mod-gallery .gallerybox img{background:none!important}.mw-parser-output .mod-gallery .bordered-images .thumb img{border:solid var(--background-color-neutral,#eaecf0)1px}.mw-parser-output .mod-gallery .whitebg .thumb{background:var(--background-color-base,#fff)!important}</style><div class="mod-gallery mod-gallery-default"><div class="main"><div><ul class="gallery mw-gallery-traditional nochecker bordered-images whitebg"> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Venn0001.svg" class="mw-file-description" title="Intersection of two sets '"`UNIQ--postMath-00000001-QINU`"'"><img alt="Intersection of two sets '"`UNIQ--postMath-00000001-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/180px-Venn0001.svg.png" decoding="async" width="180" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/270px-Venn0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/360px-Venn0001.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span></div> <div class="gallerytext"><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">Intersection</a> of two sets <span class="mwe-math-element"><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\cap B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A\cap B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea88d0d996a3bf31887919aecd945907f5715d0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.67ex; height:2.176ex;" alt="{\displaystyle ~A\cap B}"></span></div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Venn0111.svg" class="mw-file-description" title="Union of two sets '"`UNIQ--postMath-00000002-QINU`"'"><img alt="Union of two sets '"`UNIQ--postMath-00000002-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/180px-Venn0111.svg.png" decoding="async" width="180" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/270px-Venn0111.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/360px-Venn0111.svg.png 2x" data-file-width="380" data-file-height="280" /></a></span></div> <div class="gallerytext"><a href="/wiki/Union_(set_theory)" title="Union (set theory)">Union</a> of two sets <span class="mwe-math-element"><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\cup B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mo>∪</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A\cup B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad5f965785922f898935afd5fc0ad021d0971998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.67ex; height:2.176ex;" alt="{\displaystyle ~A\cup B}"></span></div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Venn0110.svg" class="mw-file-description" title="Symmetric difference of two sets '"`UNIQ--postMath-00000003-QINU`"'"><img alt="Symmetric difference of two sets '"`UNIQ--postMath-00000003-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/180px-Venn0110.svg.png" decoding="async" width="180" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/270px-Venn0110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/360px-Venn0110.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span></div> <div class="gallerytext"><a href="/wiki/Symmetric_difference" title="Symmetric difference">Symmetric difference</a> of two sets <span class="mwe-math-element"><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~\triangle ~B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mtext> </mtext> <mi mathvariant="normal">△</mi> <mtext> </mtext> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A~\triangle ~B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a58e82568ffbdc53d8141047a5c45c44ce3fc2a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.734ex; height:2.176ex;" alt="{\displaystyle A~\triangle ~B}"></span></div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Venn0010.svg" class="mw-file-description" title="Relative complement of A (left) in B (right) '"`UNIQ--postMath-00000004-QINU`"'"><img alt="Relative complement of A (left) in B (right) '"`UNIQ--postMath-00000004-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Venn0010.svg/180px-Venn0010.svg.png" decoding="async" width="180" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Venn0010.svg/270px-Venn0010.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Venn0010.svg/360px-Venn0010.svg.png 2x" data-file-width="380" data-file-height="280" /></a></span></div> <div class="gallerytext"><a href="/wiki/Complement_(set_theory)#Relative_complement" title="Complement (set theory)">Relative complement</a> of <i>A</i> (left) in <i>B</i> (right) <span class="mwe-math-element"><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}\cap B~=~B\setminus A}"> <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> <mo>∩</mo> <mi>B</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>B</mi> <mo class="MJX-variant">∖</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{c}\cap B~=~B\setminus A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/388738345c6e52341a9b4531f123cd2053231089" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.995ex; height:2.843ex;" alt="{\displaystyle A^{c}\cap B~=~B\setminus A}"></span></div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"><a href="/wiki/File:Venn1010.svg" class="mw-file-description" title="Absolute complement of A in U '"`UNIQ--postMath-00000005-QINU`"'"><img alt="Absolute complement of A in U '"`UNIQ--postMath-00000005-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Venn1010.svg/180px-Venn1010.svg.png" decoding="async" width="180" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Venn1010.svg/270px-Venn1010.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Venn1010.svg/360px-Venn1010.svg.png 2x" data-file-width="380" data-file-height="280" /></a></span></div> <div class="gallerytext"><a href="/wiki/Complement_(set_theory)#Absolute_complement" title="Complement (set theory)">Absolute complement</a> of A in U <span class="mwe-math-element"><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}~=~U\setminus A}"> <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> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>U</mi> <mo class="MJX-variant">∖</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{c}~=~U\setminus A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01b9c239ba186b145183487f61c948a3d1d6fe6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.667ex; height:2.843ex;" alt="{\displaystyle A^{c}~=~U\setminus A}"></span></div> </li> </ul></div></div></div> <p>A Venn diagram is constructed with a collection of simple closed curves drawn in a plane. According to Lewis,<sup id="cite_ref-Lewis1918_14-1" class="reference"><a href="#cite_note-Lewis1918-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> the "principle of these diagrams is that classes [or <i>sets</i>] be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram. That is, the diagram initially leaves room for any possible relation of the classes, and the actual or given relation, can then be specified by indicating that some particular region is null or is not-null".<sup id="cite_ref-Lewis1918_14-2" class="reference"><a href="#cite_note-Lewis1918-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 157">: 157 </span></sup> </p><p>Venn diagrams normally comprise overlapping <a href="/wiki/Circle" title="Circle">circles</a>. The interior of the circle symbolically represents the elements of the set, while the exterior represents elements that are not members of the set. For instance, in a two-set Venn diagram, one circle may represent the group of all <a href="/wiki/Wood" title="Wood">wooden</a> objects, while the other circle may represent the set of all tables. The overlapping region, or <i>intersection</i>, would then represent the set of all wooden tables. Shapes other than circles can be employed as shown below by Venn's own higher set diagrams. Venn diagrams do not generally contain information on the relative or absolute sizes (<a href="/wiki/Cardinality" title="Cardinality">cardinality</a>) of sets. That is, they are <a href="/wiki/Schematic" title="Schematic">schematic</a> diagrams generally not drawn to scale. </p><p>Venn diagrams are similar to Euler diagrams. However, a Venn diagram for <i>n</i> component sets must contain all 2<sup><i>n</i></sup> hypothetically possible zones, that correspond to some combination of inclusion or exclusion in each of the component sets.<sup id="cite_ref-Weisstein_2020_20-0" class="reference"><a href="#cite_note-Weisstein_2020-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Euler diagrams contain only the actually possible zones in a given context. In Venn diagrams, a shaded zone may represent an empty zone, whereas in an Euler diagram, the corresponding zone is missing from the diagram. For example, if one set represents <i>dairy products</i> and another <i>cheeses</i>, the Venn diagram contains a zone for cheeses that are not dairy products. Assuming that in the context <i>cheese</i> means some type of dairy product, the Euler diagram has the cheese zone entirely contained within the dairy-product zone—there is no zone for (non-existent) non-dairy cheese. This means that as the number of contours increases, Euler diagrams are typically less visually complex than the equivalent Venn diagram, particularly if the number of non-empty intersections is small.<sup id="cite_ref-Kent_2004_21-0" class="reference"><a href="#cite_note-Kent_2004-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The difference between Euler and Venn diagrams can be seen in the following example. Take the three sets: </p> <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 A=\{1,\,2,\,5\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>5</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\{1,\,2,\,5\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734fb65746344a257af9a238488d5f29487b20d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.496ex; height:2.843ex;" alt="{\displaystyle A=\{1,\,2,\,5\}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{1,\,6\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>6</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{1,\,6\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/579e4e8195e70be156aa8a4878510095582fc069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.933ex; height:2.843ex;" alt="{\displaystyle B=\{1,\,6\}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\{4,\,7\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>4</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>7</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\{4,\,7\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08a1a9c06eaa7bde1c206c7e1e02b3e7462c8fbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.936ex; height:2.843ex;" alt="{\displaystyle C=\{4,\,7\}}"></span></li></ul> <p>The Euler and the Venn diagram of those sets are: </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 335px"> <div class="thumb" style="width: 330px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:3-set_Euler_diagram.svg" class="mw-file-description" title="Euler diagram"><img alt="Euler diagram" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/3-set_Euler_diagram.svg/177px-3-set_Euler_diagram.svg.png" decoding="async" width="177" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/3-set_Euler_diagram.svg/266px-3-set_Euler_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/3-set_Euler_diagram.svg/354px-3-set_Euler_diagram.svg.png 2x" data-file-width="512" data-file-height="347" /></a></span></div> <div class="gallerytext">Euler diagram</div> </li> <li class="gallerybox" style="width: 335px"> <div class="thumb" style="width: 330px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:3-set_Venn_diagram.svg" class="mw-file-description" title="Venn diagram"><img alt="Venn diagram" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/3-set_Venn_diagram.svg/117px-3-set_Venn_diagram.svg.png" decoding="async" width="117" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/3-set_Venn_diagram.svg/176px-3-set_Venn_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/3-set_Venn_diagram.svg/234px-3-set_Venn_diagram.svg.png 2x" data-file-width="512" data-file-height="525" /></a></span></div> <div class="gallerytext">Venn diagram</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Extensions_to_higher_numbers_of_sets"><span class="anchor" id="Elegant"></span><span class="anchor" id="4"></span><span class="anchor" id="5"></span><span class="anchor" id="6"></span><span class="anchor" id="7"></span><span class="anchor" id="8"></span><span class="anchor" id="9"></span><span class="anchor" id="10"></span><span class="anchor" id="11"></span>Extensions to higher numbers of sets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=6" title="Edit section: Extensions to higher numbers of sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Venn diagrams typically represent two or three sets, but there are forms that allow for higher numbers. Shown below, four intersecting spheres form the highest order Venn diagram that has the symmetry of a <a href="/wiki/Simplex" title="Simplex">simplex</a> and can be visually represented. The 16 intersections correspond to the vertices of a <a href="/wiki/Tesseract" title="Tesseract">tesseract</a> (or the cells of a <a href="/wiki/16-cell" title="16-cell">16-cell</a>, respectively). </p> <table class="wikitable" style="text-align:center; width: 100%;"> <tbody><tr> <td style="vertical-align:top;"><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_00,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/4_spheres%2C_cell_00%2C_solid.png/180px-4_spheres%2C_cell_00%2C_solid.png" decoding="async" width="180" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/4_spheres%2C_cell_00%2C_solid.png/270px-4_spheres%2C_cell_00%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/4_spheres%2C_cell_00%2C_solid.png/360px-4_spheres%2C_cell_00%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span> </td> <td style="vertical-align:top;"><span typeof="mw:File"><a href="/wiki/File:4_spheres,_weight_1,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/4_spheres%2C_weight_1%2C_solid.png/180px-4_spheres%2C_weight_1%2C_solid.png" decoding="async" width="180" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/4_spheres%2C_weight_1%2C_solid.png/270px-4_spheres%2C_weight_1%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/4_spheres%2C_weight_1%2C_solid.png/360px-4_spheres%2C_weight_1%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><br /> <p><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_01,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/4_spheres%2C_cell_01%2C_solid.png/45px-4_spheres%2C_cell_01%2C_solid.png" decoding="async" width="45" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/4_spheres%2C_cell_01%2C_solid.png/68px-4_spheres%2C_cell_01%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/4_spheres%2C_cell_01%2C_solid.png/90px-4_spheres%2C_cell_01%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_02,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/4_spheres%2C_cell_02%2C_solid.png/45px-4_spheres%2C_cell_02%2C_solid.png" decoding="async" width="45" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/4_spheres%2C_cell_02%2C_solid.png/68px-4_spheres%2C_cell_02%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/4_spheres%2C_cell_02%2C_solid.png/90px-4_spheres%2C_cell_02%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_04,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/4_spheres%2C_cell_04%2C_solid.png/45px-4_spheres%2C_cell_04%2C_solid.png" decoding="async" width="45" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/4_spheres%2C_cell_04%2C_solid.png/68px-4_spheres%2C_cell_04%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/4_spheres%2C_cell_04%2C_solid.png/90px-4_spheres%2C_cell_04%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_08,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/4_spheres%2C_cell_08%2C_solid.png/45px-4_spheres%2C_cell_08%2C_solid.png" decoding="async" width="45" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/4_spheres%2C_cell_08%2C_solid.png/68px-4_spheres%2C_cell_08%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/4_spheres%2C_cell_08%2C_solid.png/90px-4_spheres%2C_cell_08%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span> </p> </td> <td style="vertical-align:top;"><span typeof="mw:File"><a href="/wiki/File:4_spheres,_weight_2,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/4_spheres%2C_weight_2%2C_solid.png/180px-4_spheres%2C_weight_2%2C_solid.png" decoding="async" width="180" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/4_spheres%2C_weight_2%2C_solid.png/270px-4_spheres%2C_weight_2%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/4_spheres%2C_weight_2%2C_solid.png/360px-4_spheres%2C_weight_2%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><br /> <p><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_03,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/4_spheres%2C_cell_03%2C_solid.png/30px-4_spheres%2C_cell_03%2C_solid.png" decoding="async" width="30" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/4_spheres%2C_cell_03%2C_solid.png/45px-4_spheres%2C_cell_03%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/4_spheres%2C_cell_03%2C_solid.png/60px-4_spheres%2C_cell_03%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_05,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/4_spheres%2C_cell_05%2C_solid.png/30px-4_spheres%2C_cell_05%2C_solid.png" decoding="async" width="30" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/4_spheres%2C_cell_05%2C_solid.png/45px-4_spheres%2C_cell_05%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/4_spheres%2C_cell_05%2C_solid.png/60px-4_spheres%2C_cell_05%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_06,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/4_spheres%2C_cell_06%2C_solid.png/30px-4_spheres%2C_cell_06%2C_solid.png" decoding="async" width="30" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/4_spheres%2C_cell_06%2C_solid.png/45px-4_spheres%2C_cell_06%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/4_spheres%2C_cell_06%2C_solid.png/60px-4_spheres%2C_cell_06%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_09,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/4_spheres%2C_cell_09%2C_solid.png/30px-4_spheres%2C_cell_09%2C_solid.png" decoding="async" width="30" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/4_spheres%2C_cell_09%2C_solid.png/45px-4_spheres%2C_cell_09%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ac/4_spheres%2C_cell_09%2C_solid.png/60px-4_spheres%2C_cell_09%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_10,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/4_spheres%2C_cell_10%2C_solid.png/30px-4_spheres%2C_cell_10%2C_solid.png" decoding="async" width="30" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/4_spheres%2C_cell_10%2C_solid.png/45px-4_spheres%2C_cell_10%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/4_spheres%2C_cell_10%2C_solid.png/60px-4_spheres%2C_cell_10%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_12,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/4_spheres%2C_cell_12%2C_solid.png/30px-4_spheres%2C_cell_12%2C_solid.png" decoding="async" width="30" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/4_spheres%2C_cell_12%2C_solid.png/45px-4_spheres%2C_cell_12%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/4_spheres%2C_cell_12%2C_solid.png/60px-4_spheres%2C_cell_12%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span> </p> </td> <td style="vertical-align:top;"><span typeof="mw:File"><a href="/wiki/File:4_spheres,_weight_3,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/4_spheres%2C_weight_3%2C_solid.png/180px-4_spheres%2C_weight_3%2C_solid.png" decoding="async" width="180" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/4_spheres%2C_weight_3%2C_solid.png/270px-4_spheres%2C_weight_3%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/4_spheres%2C_weight_3%2C_solid.png/360px-4_spheres%2C_weight_3%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><br /> <p><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_07,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/4_spheres%2C_cell_07%2C_solid.png/45px-4_spheres%2C_cell_07%2C_solid.png" decoding="async" width="45" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/4_spheres%2C_cell_07%2C_solid.png/68px-4_spheres%2C_cell_07%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/4_spheres%2C_cell_07%2C_solid.png/90px-4_spheres%2C_cell_07%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_11,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/4_spheres%2C_cell_11%2C_solid.png/45px-4_spheres%2C_cell_11%2C_solid.png" decoding="async" width="45" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/4_spheres%2C_cell_11%2C_solid.png/68px-4_spheres%2C_cell_11%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/4_spheres%2C_cell_11%2C_solid.png/90px-4_spheres%2C_cell_11%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_13,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/4_spheres%2C_cell_13%2C_solid.png/45px-4_spheres%2C_cell_13%2C_solid.png" decoding="async" width="45" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/4_spheres%2C_cell_13%2C_solid.png/68px-4_spheres%2C_cell_13%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/4_spheres%2C_cell_13%2C_solid.png/90px-4_spheres%2C_cell_13%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_14,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/4_spheres%2C_cell_14%2C_solid.png/45px-4_spheres%2C_cell_14%2C_solid.png" decoding="async" width="45" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/4_spheres%2C_cell_14%2C_solid.png/68px-4_spheres%2C_cell_14%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/4_spheres%2C_cell_14%2C_solid.png/90px-4_spheres%2C_cell_14%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span> </p> </td> <td style="vertical-align:top;"><span typeof="mw:File"><a href="/wiki/File:4_spheres,_cell_15,_solid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4_spheres%2C_cell_15%2C_solid.png/180px-4_spheres%2C_cell_15%2C_solid.png" decoding="async" width="180" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4_spheres%2C_cell_15%2C_solid.png/270px-4_spheres%2C_cell_15%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4_spheres%2C_cell_15%2C_solid.png/360px-4_spheres%2C_cell_15%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></span> </td></tr></tbody></table> <p>For higher numbers of sets, some loss of symmetry in the diagrams is unavoidable. Venn was keen to find "symmetrical figures ... elegant in themselves,"<sup id="cite_ref-Venn1881_11-1" class="reference"><a href="#cite_note-Venn1881-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> that represented higher numbers of sets, and he devised an <i>elegant</i> four-set diagram using <a href="/wiki/Ellipse" title="Ellipse">ellipses</a> (see below). He also gave a construction for Venn diagrams for <i>any</i> number of sets, where each successive curve that delimits a set interleaves with previous curves, starting with the three-circle diagram. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Venn4.svg" class="mw-file-description" title="Venn's construction for four sets (use Gray code to compute, the digit 1 means in the set, and the digit 0 means not in the set)"><img alt="Venn's construction for four sets (use Gray code to compute, the digit 1 means in the set, and the digit 0 means not in the set)" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Venn4.svg/150px-Venn4.svg.png" decoding="async" width="150" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Venn4.svg/225px-Venn4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Venn4.svg/300px-Venn4.svg.png 2x" data-file-width="813" data-file-height="650" /></a></span></div> <div class="gallerytext">Venn's construction for four sets (use <a href="/wiki/Gray_code" title="Gray code">Gray code</a> to compute, the digit 1 means in the set, and the digit 0 means not in the set)</div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Venn5.svg" class="mw-file-description" title="Venn's construction for five sets"><img alt="Venn's construction for five sets" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Venn5.svg/150px-Venn5.svg.png" decoding="async" width="150" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Venn5.svg/225px-Venn5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Venn5.svg/300px-Venn5.svg.png 2x" data-file-width="813" data-file-height="650" /></a></span></div> <div class="gallerytext">Venn's construction for five sets</div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Venn6.svg" class="mw-file-description" title="Venn's construction for six sets"><img alt="Venn's construction for six sets" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Venn6.svg/150px-Venn6.svg.png" decoding="async" width="150" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Venn6.svg/225px-Venn6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Venn6.svg/300px-Venn6.svg.png 2x" data-file-width="813" data-file-height="650" /></a></span></div> <div class="gallerytext">Venn's construction for six sets</div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Venn%27s_four_ellipse_construction.svg" class="mw-file-description" title="Venn's four-set diagram using ellipses"><img alt="Venn's four-set diagram using ellipses" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Venn%27s_four_ellipse_construction.svg/145px-Venn%27s_four_ellipse_construction.svg.png" decoding="async" width="145" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Venn%27s_four_ellipse_construction.svg/217px-Venn%27s_four_ellipse_construction.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Venn%27s_four_ellipse_construction.svg/289px-Venn%27s_four_ellipse_construction.svg.png 2x" data-file-width="470" data-file-height="390" /></a></span></div> <div class="gallerytext">Venn's four-set diagram using ellipses</div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:CirclesN4xb.svg" class="mw-file-description" title="Non-example: This Euler diagram is not a Venn diagram for four sets as it has only 14 regions as opposed to 24 = 16 regions (including the white region); there is no region where only the yellow and blue, or only the red and green circles meet."><img alt="Non-example: This Euler diagram is not a Venn diagram for four sets as it has only 14 regions as opposed to 24 = 16 regions (including the white region); there is no region where only the yellow and blue, or only the red and green circles meet." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/CirclesN4xb.svg/129px-CirclesN4xb.svg.png" decoding="async" width="129" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/CirclesN4xb.svg/194px-CirclesN4xb.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/CirclesN4xb.svg/258px-CirclesN4xb.svg.png 2x" data-file-width="519" data-file-height="482" /></a></span></div> <div class="gallerytext"><b>Non-example:</b> This <a href="/wiki/Euler_diagram" title="Euler diagram">Euler diagram</a> is <em>not</em> a Venn diagram for four sets as it has only 14 regions as opposed to 2<sup>4</sup> = 16 regions (including the white region); there is no region where only the yellow and blue, or only the red and green circles meet.</div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Symmetrical_5-set_Venn_diagram.svg" class="mw-file-description" title="Five-set Venn diagram using congruent ellipses in a five-fold rotationally symmetrical arrangement devised by Branko Grünbaum. Labels have been simplified for greater readability; for example, A denotes A ∩ Bc ∩ Cc ∩ Dc ∩ Ec, while BCE denotes Ac ∩ B ∩ C ∩ Dc ∩ E."><img alt="Five-set Venn diagram using congruent ellipses in a five-fold rotationally symmetrical arrangement devised by Branko Grünbaum. Labels have been simplified for greater readability; for example, A denotes A ∩ Bc ∩ Cc ∩ Dc ∩ Ec, while BCE denotes Ac ∩ B ∩ C ∩ Dc ∩ E." src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Symmetrical_5-set_Venn_diagram.svg/120px-Symmetrical_5-set_Venn_diagram.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Symmetrical_5-set_Venn_diagram.svg/180px-Symmetrical_5-set_Venn_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Symmetrical_5-set_Venn_diagram.svg/240px-Symmetrical_5-set_Venn_diagram.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="gallerytext">Five-set Venn diagram using congruent ellipses in a five-fold <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotationally symmetrical</a> arrangement devised by <a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Branko Grünbaum</a>. Labels have been simplified for greater readability; for example, <b>A</b> denotes <span class="nowrap"><b>A</b> ∩ <b>B</b><sup>c</sup> ∩ <b>C</b><sup>c</sup> ∩ <b>D</b><sup>c</sup> ∩ <b>E</b><sup>c</sup></span>, while <b>BCE</b> denotes <span class="nowrap"><b>A</b><sup>c</sup> ∩ <b>B</b> ∩ <b>C</b> ∩ <b>D</b><sup>c</sup> ∩ <b>E</b></span>.</div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:6-set_Venn_diagram.svg" class="mw-file-description" title="Six-set Venn diagram made of only triangles (interactive version)"><img alt="Six-set Venn diagram made of only triangles (interactive version)" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/6-set_Venn_diagram.svg/120px-6-set_Venn_diagram.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/6-set_Venn_diagram.svg/180px-6-set_Venn_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/6-set_Venn_diagram.svg/240px-6-set_Venn_diagram.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="gallerytext">Six-set Venn diagram made of only triangles <a class="external text" href="https://upload.wikimedia.org/wikipedia/commons/5/56/6-set_Venn_diagram_SMIL.svg">(interactive version)</a></div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Edwards–Venn_diagrams"><span id="Edwards.E2.80.93Venn_diagrams"></span><span class="anchor" id="Edwards-Venn"></span><span class="anchor" id="Adelaide"></span><span class="anchor" id="Hamilton"></span><span class="anchor" id="Massey"></span><span class="anchor" id="Victoria"></span><span class="anchor" id="Palmerston_North"></span><span class="anchor" id="Manawatu"></span>Edwards–Venn diagrams</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=7" title="Edit section: Edwards–Venn diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 185px"> <div class="thumb" style="width: 180px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Venn-three.svg" class="mw-file-description" title="Three sets"><img alt="Three sets" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Venn-three.svg/150px-Venn-three.svg.png" decoding="async" width="150" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Venn-three.svg/225px-Venn-three.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Venn-three.svg/300px-Venn-three.svg.png 2x" data-file-width="548" data-file-height="370" /></a></span></div> <div class="gallerytext"> Three sets</div> </li> <li class="gallerybox" style="width: 185px"> <div class="thumb" style="width: 180px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Edwards-Venn-four.svg" class="mw-file-description" title="Four sets"><img alt="Four sets" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Edwards-Venn-four.svg/150px-Edwards-Venn-four.svg.png" decoding="async" width="150" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Edwards-Venn-four.svg/225px-Edwards-Venn-four.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Edwards-Venn-four.svg/300px-Edwards-Venn-four.svg.png 2x" data-file-width="548" data-file-height="370" /></a></span></div> <div class="gallerytext"> Four sets</div> </li> <li class="gallerybox" style="width: 185px"> <div class="thumb" style="width: 180px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Edwards-Venn-five.svg" class="mw-file-description" title="Five sets"><img alt="Five sets" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Edwards-Venn-five.svg/150px-Edwards-Venn-five.svg.png" decoding="async" width="150" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Edwards-Venn-five.svg/225px-Edwards-Venn-five.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Edwards-Venn-five.svg/300px-Edwards-Venn-five.svg.png 2x" data-file-width="548" data-file-height="370" /></a></span></div> <div class="gallerytext"> Five sets</div> </li> <li class="gallerybox" style="width: 185px"> <div class="thumb" style="width: 180px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Edwards-Venn-six.svg" class="mw-file-description" title="Six sets"><img alt="Six sets" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Edwards-Venn-six.svg/150px-Edwards-Venn-six.svg.png" decoding="async" width="150" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Edwards-Venn-six.svg/225px-Edwards-Venn-six.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Edwards-Venn-six.svg/300px-Edwards-Venn-six.svg.png 2x" data-file-width="548" data-file-height="370" /></a></span></div> <div class="gallerytext"> Six sets</div> </li> </ul> <p><a href="/wiki/Anthony_William_Fairbank_Edwards" class="mw-redirect" title="Anthony William Fairbank Edwards">Anthony William Fairbank Edwards</a> constructed a series of Venn diagrams for higher numbers of sets by segmenting the surface of a sphere, which became known as Edwards–Venn diagrams.<sup id="cite_ref-Edwards_2004_22-0" class="reference"><a href="#cite_note-Edwards_2004-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> For example, three sets can be easily represented by taking three hemispheres of the sphere at right angles (<i>x</i> = 0, <i>y</i> = 0 and <i>z</i> = 0). A fourth set can be added to the representation, by taking a curve similar to the seam on a tennis ball, which winds up and down around the equator, and so on. The resulting sets can then be projected back to a plane, to give <i>cogwheel</i> diagrams with increasing numbers of teeth—as shown here. These diagrams were devised while designing a <a href="/wiki/Stained-glass" class="mw-redirect" title="Stained-glass">stained-glass</a> window in memory of Venn.<sup id="cite_ref-Edwards_2004_22-1" class="reference"><a href="#cite_note-Edwards_2004-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_diagrams">Other diagrams</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=8" title="Edit section: Other diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Edwards–Venn diagrams are <a href="/wiki/Topological_equivalence" class="mw-redirect" title="Topological equivalence">topologically equivalent</a> to diagrams devised by <a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Branko Grünbaum</a>, which were based around intersecting <a href="/wiki/Polygon" title="Polygon">polygons</a> with increasing numbers of sides. They are also two-dimensional representations of <a href="/wiki/Hypercube" title="Hypercube">hypercubes</a>. </p><p><a href="/wiki/Henry_John_Stephen_Smith" title="Henry John Stephen Smith">Henry John Stephen Smith</a> devised similar <i>n</i>-set diagrams using <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> curves<sup id="cite_ref-Edwards_2004_22-2" class="reference"><a href="#cite_note-Edwards_2004-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> with the series of equations <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}={\frac {\sin \left(2^{i}x\right)}{2^{i}}}{\text{ where }}0\leq i\leq n-1{\text{ and }}i\in \mathbb {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}={\frac {\sin \left(2^{i}x\right)}{2^{i}}}{\text{ where }}0\leq i\leq n-1{\text{ and }}i\in \mathbb {N} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc5d43e4d3ae72023ecdacbdded4c7b36ef5df9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.695ex; height:6.676ex;" alt="{\displaystyle y_{i}={\frac {\sin \left(2^{i}x\right)}{2^{i}}}{\text{ where }}0\leq i\leq n-1{\text{ and }}i\in \mathbb {N} .}"></span> </p><p><a href="/wiki/Charles_Lutwidge_Dodgson" class="mw-redirect" title="Charles Lutwidge Dodgson">Charles Lutwidge Dodgson</a> (also known as Lewis Carroll) devised a five-set diagram known as <a href="/wiki/Carroll%27s_square_(diagram)" class="mw-redirect" title="Carroll's square (diagram)">Carroll's square</a>. Joaquin and Boyles, on the other hand, proposed supplemental rules for the standard Venn diagram, in order to account for certain problem cases. For instance, regarding the issue of representing singular statements, they suggest to consider the Venn diagram circle as a representation of a set of things, and use <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> and set theory to treat categorical statements as statements about sets. Additionally, they propose to treat singular statements as statements about <a href="/wiki/Set_membership" class="mw-redirect" title="Set membership">set membership</a>. So, for example, to represent the statement "a is F" in this retooled Venn diagram, a small letter "a" may be placed inside the circle that represents the set F.<sup id="cite_ref-Joaquin_2017_23-0" class="reference"><a href="#cite_note-Joaquin_2017-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Related_concepts"><span class="anchor" id="Johnston"></span>Related concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=9" title="Edit section: Related concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Venn3tab.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn3tab.svg/220px-Venn3tab.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn3tab.svg/330px-Venn3tab.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn3tab.svg/440px-Venn3tab.svg.png 2x" data-file-width="296" data-file-height="296" /></a><figcaption>Venn diagram as a truth table</figcaption></figure> <p>Venn diagrams correspond to <a href="/wiki/Truth_table" title="Truth table">truth tables</a> for the propositions <span class="mwe-math-element"><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"> <mi>x</mi> <mo>∈</mo> <mi>A</mi> </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/27bcc9b2afb295d4234bc294860cd0c63bcad2ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle x\in A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aac01724708de4e1d41423bc64b35e9d94c9009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.934ex; height:2.176ex;" alt="{\displaystyle x\in B}"></span>, etc., in the sense that each region of Venn diagram corresponds to one row of the truth table.<sup id="cite_ref-Grimaldi_2004_24-0" class="reference"><a href="#cite_note-Grimaldi_2004-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Johnson_2001_25-0" class="reference"><a href="#cite_note-Johnson_2001-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> This type is also known as Johnston diagram. Another way of representing sets is with John F. Randolph's <a href="/wiki/R-diagram" class="mw-redirect" title="R-diagram">R-diagrams</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Existential_graph" title="Existential graph">Existential graph</a> (by <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a>)</li> <li><a href="/wiki/Logical_connectives" class="mw-redirect" title="Logical connectives">Logical connectives</a></li> <li><a href="/wiki/Information_diagram" title="Information diagram">Information diagram</a></li> <li><a href="/wiki/Marquand_diagram" class="mw-redirect" title="Marquand diagram">Marquand diagram</a> (and as further derivation <a href="/wiki/Veitch_chart" class="mw-redirect" title="Veitch chart">Veitch chart</a> and <a href="/wiki/Karnaugh_map" title="Karnaugh map">Karnaugh map</a>)</li> <li><a href="/wiki/Octahedron#Spherical_tiling" title="Octahedron">Spherical octahedron</a> – A stereographic projection of a regular octahedron makes a three-set Venn diagram, as three orthogonal great circles, each dividing space into two halves.</li> <li><a href="/wiki/Stanhope_Demonstrator" title="Stanhope Demonstrator">Stanhope Demonstrator</a></li> <li><a href="/wiki/Three_circles_model" title="Three circles model">Three circles model</a></li> <li><a href="/wiki/Triquetra" title="Triquetra">Triquetra</a></li> <li><a href="/wiki/Vesica_piscis" title="Vesica piscis">Vesica piscis</a></li> <li><a href="/wiki/UpSet_plot" title="UpSet plot">UpSet plot</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=11" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-NB_1-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-NB_1_10-0">^</a></b></span> <span class="reference-text">In Euler's <i>Lettres à une princesse d'Allemagne sur divers sujets de physique et de philosophie</i> [Letters to a German Princess on various physical and philosophical subjects] (Saint Petersburg, Russia: l'Academie Impériale des Sciences, 1768), volume 2, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gxsAAAAAQAAJ&pg=PA95">pages 95-126.</a> In Venn's article, however, he suggests that the diagrammatic idea predates Euler, and is attributable to <a href="/wiki/Christian_Weise" title="Christian Weise">Christian Weise</a> or Johann Christian Lange (in Lange's book <i>Nucleus Logicae Weisianae</i> (1712)).</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Peil_2020-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Peil_2020_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20200804163657/http://web.mnstate.edu/peil/MDEV102/U1/S3/Intersection4.htm">"Intersection of Sets"</a>. <i>web.mnstate.edu</i>. Archived from <a rel="nofollow" class="external text" href="http://web.mnstate.edu/peil/MDEV102/U1/S3/Intersection4.htm">the original</a> on 2020-08-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=web.mnstate.edu&rft.atitle=Intersection+of+Sets&rft_id=http%3A%2F%2Fweb.mnstate.edu%2Fpeil%2FMDEV102%2FU1%2FS3%2FIntersection4.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Venn_2014-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Venn_2014_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVenn" class="citation web cs1">Venn, John. <a rel="nofollow" class="external text" href="https://www.cis.upenn.edu/~bhusnur4/cit592_fall2014/venn%20diagrams.pdf">"On the Diagrammatic and Mechanical Representation of Propositions and Reasonings"</a> <span class="cs1-format">(PDF)</span>. <i>Penn Engineering</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Penn+Engineering&rft.atitle=On+the+Diagrammatic+and+Mechanical+Representation+of+Propositions+and+Reasonings&rft.aulast=Venn&rft.aufirst=John&rft_id=https%3A%2F%2Fwww.cis.upenn.edu%2F~bhusnur4%2Fcit592_fall2014%2Fvenn%2520diagrams.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-PM-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-PM_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://www.tandfonline.com/toc/tphm18/current">"The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics"</a>. <i><a href="/wiki/Taylor_%26_Francis" title="Taylor & Francis">Taylor & Francis</a></i><span class="reference-accessdate">. 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On the Diagrammatic and Mechanical Representation of Propositions and Reasonings"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/The_London,_Edinburgh,_and_Dublin_Philosophical_Magazine_and_Journal_of_Science" class="mw-redirect" title="The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science">The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science</a></i>. 5. <b>10</b> (59): 1–18. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786448008626877">10.1080/14786448008626877</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170516204620/https://www.cis.upenn.edu/~bhusnur4/cit592_fall2014/venn%20diagrams.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2017-05-16.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+London%2C+Edinburgh%2C+and+Dublin+Philosophical+Magazine+and+Journal+of+Science&rft.atitle=I.+On+the+Diagrammatic+and+Mechanical+Representation+of+Propositions+and+Reasonings&rft.volume=10&rft.issue=59&rft.pages=1-18&rft.date=1880-07&rft_id=info%3Adoi%2F10.1080%2F14786448008626877&rft.aulast=Venn&rft.aufirst=John&rft_id=https%3A%2F%2Fwww.cis.upenn.edu%2F~bhusnur4%2Fcit592_fall2014%2Fvenn%2520diagrams.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span> <a rel="nofollow" class="external autonumber" href="http://www.tandfonline.com/doi/abs/10.1080/14786448008626877">[1]</a> <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=k68vAQAAIAAJ&pg=PA1">[2]</a></span> </li> <li id="cite_note-Venn1880_2-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Venn1880_2_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Venn1880_2_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVenn1880" class="citation journal cs1"><a href="/wiki/John_Venn" title="John Venn">Venn, John</a> (1880). <a rel="nofollow" class="external text" href="https://archive.org/stream/proceedingsofcam4188083camb#page/47/mode/1up">"On the employment of geometrical diagrams for the sensible representations of logical propositions"</a>. <i><a href="/wiki/Proceedings_of_the_Cambridge_Philosophical_Society" class="mw-redirect" title="Proceedings of the Cambridge Philosophical Society">Proceedings of the Cambridge Philosophical Society</a></i>. <b>4</b>: 47–59.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Cambridge+Philosophical+Society&rft.atitle=On+the+employment+of+geometrical+diagrams+for+the+sensible+representations+of+logical+propositions&rft.volume=4&rft.pages=47-59&rft.date=1880&rft.aulast=Venn&rft.aufirst=John&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fproceedingsofcam4188083camb%23page%2F47%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Sandifer2003-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Sandifer2003_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Sandifer2003_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSandifer2003" class="citation web cs1">Sandifer, Ed (2003). <a rel="nofollow" class="external text" href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2003%20Venn%20Diagrams.pdf">"How Euler Did It"</a> <span class="cs1-format">(PDF)</span>. <i>MAA Online</i>. <a href="/wiki/The_Mathematical_Association_of_America" class="mw-redirect" title="The Mathematical Association of America">The Mathematical Association of America</a> (MAA)<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-10-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MAA+Online&rft.atitle=How+Euler+Did+It&rft.date=2003&rft.aulast=Sandifer&rft.aufirst=Ed&rft_id=http%3A%2F%2Fwww.maa.org%2Feditorial%2Feuler%2FHow%2520Euler%2520Did%2520It%252003%2520Venn%2520Diagrams.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Ruskey2005-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ruskey2005_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ruskey2005_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRuskeyWeston2005" class="citation journal cs1"><a href="/wiki/Frank_Ruskey" title="Frank Ruskey">Ruskey, Frank</a>; Weston, Mark (2005-06-18). <a rel="nofollow" class="external text" href="http://www.combinatorics.org/files/Surveys/ds5/VennEJC.html">"A Survey of Venn Diagrams"</a>. <i><a href="/wiki/The_Electronic_Journal_of_Combinatorics" class="mw-redirect" title="The Electronic Journal of Combinatorics">The Electronic Journal of Combinatorics</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Electronic+Journal+of+Combinatorics&rft.atitle=A+Survey+of+Venn+Diagrams&rft.date=2005-06-18&rft.aulast=Ruskey&rft.aufirst=Frank&rft.au=Weston%2C+Mark&rft_id=http%3A%2F%2Fwww.combinatorics.org%2Ffiles%2FSurveys%2Fds5%2FVennEJC.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Baron_1969-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Baron_1969_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Baron_1969_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaron1969" class="citation journal cs1"><a href="/wiki/Margaret_Baron" title="Margaret Baron">Baron, Margaret E.</a> (May 1969). "A Note on The Historical Development of Logic Diagrams". <i><a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a></i>. <b>53</b> (384): 113–125. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3614533">10.2307/3614533</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3614533">3614533</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125364002">125364002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=A+Note+on+The+Historical+Development+of+Logic+Diagrams&rft.volume=53&rft.issue=384&rft.pages=113-125&rft.date=1969-05&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125364002%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3614533%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3614533&rft.aulast=Baron&rft.aufirst=Margaret+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Leibniz_1690-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Leibniz_1690_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeibniz1903" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz, Gottfried Wilhelm</a> (1903) [ca. 1690]. 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In <a href="/wiki/Louis_Couturat" title="Louis Couturat">Couturat, Louis</a> (ed.). <i>Opuscules et fragmentes inedits de Leibniz</i> (in Latin). pp. 292–321.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=De+Formae+Logicae+per+linearum+ductus&rft.btitle=Opuscules+et+fragmentes+inedits+de+Leibniz&rft.pages=292-321&rft.date=1903&rft.aulast=Leibniz&rft.aufirst=Gottfried+Wilhelm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Venn1881-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-Venn1881_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Venn1881_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVenn1881" class="citation book cs1"><a href="/wiki/John_Venn" title="John Venn">Venn, John</a> (1881). <a rel="nofollow" class="external text" href="https://archive.org/details/symboliclogic00venngoog"><i>Symbolic logic</i></a>. <a href="/wiki/Macmillan_(publisher)" class="mw-redirect" title="Macmillan (publisher)">Macmillan</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/symboliclogic00venngoog/page/n150">108</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2013-04-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symbolic+logic&rft.pages=108&rft.pub=Macmillan&rft.date=1881&rft.aulast=Venn&rft.aufirst=John&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsymboliclogic00venngoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Gailand_1967-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gailand_1967_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_Queen1967" class="citation book cs1">Mac Queen, Gailand (October 1967). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170414163921/https://macsphere.mcmaster.ca/bitstream/11375/10794/1/fulltext.pdf"><i>The Logic Diagram</i></a> <span class="cs1-format">(PDF)</span> (Thesis). <a href="/wiki/McMaster_University" title="McMaster University">McMaster University</a>. Archived from <a rel="nofollow" class="external text" href="https://macsphere.mcmaster.ca/bitstream/11375/10794/1/fulltext.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-04-14<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-04-14</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Logic+Diagram&rft.pub=McMaster+University&rft.date=1967-10&rft.aulast=Mac+Queen&rft.aufirst=Gailand&rft_id=https%3A%2F%2Fmacsphere.mcmaster.ca%2Fbitstream%2F11375%2F10794%2F1%2Ffulltext.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span> (NB. Has a detailed history of the evolution of logic diagrams including but not limited to the Venn diagram.)</span> </li> <li id="cite_note-Maths_Today-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Maths_Today_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Maths_Today_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVerburgt2023" class="citation magazine cs1">Verburgt, Lukas M. (April 2023). "The Venn Behind the Diagram". <i>Mathematics Today</i>. Vol. 59, no. 2. <a href="/wiki/Institute_of_Mathematics_and_its_Applications" title="Institute of Mathematics and its Applications">Institute of Mathematics and its Applications</a>. pp. 53–55.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Today&rft.atitle=The+Venn+Behind+the+Diagram&rft.volume=59&rft.issue=2&rft.pages=53-55&rft.date=2023-04&rft.aulast=Verburgt&rft.aufirst=Lukas+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Lewis1918-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lewis1918_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lewis1918_14-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lewis1918_14-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLewis1918" class="citation book cs1"><a href="/wiki/Clarence_Irving_Lewis" class="mw-redirect" title="Clarence Irving Lewis">Lewis, Clarence Irving</a> (1918). <a rel="nofollow" class="external text" href="https://archive.org/details/asurveyofsymboli00lewiuoft"><i>A Survey of Symbolic Logic</i></a>. Berkeley: <a href="/wiki/University_of_California_Press" title="University of California Press">University of California Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Survey+of+Symbolic+Logic&rft.place=Berkeley&rft.pub=University+of+California+Press&rft.date=1918&rft.aulast=Lewis&rft.aufirst=Clarence+Irving&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fasurveyofsymboli00lewiuoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Henderson_1963-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Henderson_1963_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenderson1963" class="citation journal cs1"><a href="/wiki/David_Wilson_Henderson" class="mw-redirect" title="David Wilson Henderson">Henderson, David Wilson</a> (April 1963). 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Archived from <a rel="nofollow" class="external text" href="http://www.readingquest.org/strat/venn.html">the original</a> on 2009-04-29<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-06-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Strategies+for+Reading+Comprehension+Venn+Diagrams&rft_id=http%3A%2F%2Fwww.readingquest.org%2Fstrat%2Fvenn.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeo2010" class="citation news cs1">Leo, Alex (2010-03-18). <a rel="nofollow" class="external text" href="https://www.huffpost.com/entry/funniest-venn-diagrams-th_n_347552">"Jesus, Karaoke, And Serial Killers: The Funniest Venn Diagrams The Web Has To Offer"</a>. <i>Huffpost</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Huffpost&rft.atitle=Jesus%2C+Karaoke%2C+And+Serial+Killers%3A+The+Funniest+Venn+Diagrams+The+Web+Has+To+Offer&rft.date=2010-03-18&rft.aulast=Leo&rft.aufirst=Alex&rft_id=https%3A%2F%2Fwww.huffpost.com%2Fentry%2Ffunniest-venn-diagrams-th_n_347552&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoran2018" class="citation news cs1">Moran, Lee (2018-12-15). <a rel="nofollow" class="external text" href="https://www.huffpost.com/entry/scott-walker-venn-diagram-meme_n_5c14e6d5e4b05d7e5d8258cc">"Scott Walker Gets Mercilessly Mocked By Twitter Users Over Venn Diagram Fail"</a>. <i>HuffPost</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=HuffPost&rft.atitle=Scott+Walker+Gets+Mercilessly+Mocked+By+Twitter+Users+Over+Venn+Diagram+Fail&rft.date=2018-12-15&rft.aulast=Moran&rft.aufirst=Lee&rft_id=https%3A%2F%2Fwww.huffpost.com%2Fentry%2Fscott-walker-venn-diagram-meme_n_5c14e6d5e4b05d7e5d8258cc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Weisstein_2020-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-Weisstein_2020_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/VennDiagram.html">"Venn Diagram"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Venn+Diagram&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FVennDiagram.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Kent_2004-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kent_2004_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.cs.kent.ac.uk/events/conf/2004/euler/eulerdiagrams.html">"Euler Diagrams 2004: Brighton, UK: September 22–23"</a>. Reasoning with Diagrams project, University of Kent. 2004<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-08-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Euler+Diagrams+2004%3A+Brighton%2C+UK%3A+September+22%E2%80%9323&rft.pub=Reasoning+with+Diagrams+project%2C+University+of+Kent&rft.date=2004&rft_id=http%3A%2F%2Fwww.cs.kent.ac.uk%2Fevents%2Fconf%2F2004%2Feuler%2Feulerdiagrams.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Edwards_2004-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-Edwards_2004_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Edwards_2004_22-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Edwards_2004_22-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdwards2004" class="citation book cs1"><a href="/wiki/Anthony_William_Fairbank_Edwards" class="mw-redirect" title="Anthony William Fairbank Edwards">Edwards, Anthony William Fairbank</a> (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7_0Thy4V3JIC&pg=PA65"><i>Cogwheels of the Mind: The Story of Venn Diagrams</i></a>. Baltimore, Maryland, USA: <a href="/wiki/Johns_Hopkins_University_Press" title="Johns Hopkins University Press">Johns Hopkins University Press</a>. p. 65. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-7434-5" title="Special:BookSources/978-0-8018-7434-5"><bdi>978-0-8018-7434-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cogwheels+of+the+Mind%3A+The+Story+of+Venn+Diagrams&rft.place=Baltimore%2C+Maryland%2C+USA&rft.pages=65&rft.pub=Johns+Hopkins+University+Press&rft.date=2004&rft.isbn=978-0-8018-7434-5&rft.aulast=Edwards&rft.aufirst=Anthony+William+Fairbank&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7_0Thy4V3JIC%26pg%3DPA65&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span>.</span> </li> <li id="cite_note-Joaquin_2017-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-Joaquin_2017_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoaquinBoyles2017" class="citation journal cs1">Joaquin, Jeremiah Joven; Boyles, Robert James M. (June 2017). <a rel="nofollow" class="external text" href="https://www.pdcnet.org/teachphil/content/teachphil_2017_0040_0002_0161_0180">"Teaching Syllogistic Logic via a Retooled Venn Diagrammatical Technique"</a>. <i><a href="/wiki/Teaching_Philosophy" title="Teaching Philosophy">Teaching Philosophy</a></i>. <b>40</b> (2): 161–180. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5840%2Fteachphil201771767">10.5840/teachphil201771767</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181121120401/https://www.pdcnet.org/teachphil/content/teachphil_2017_0040_0002_0161_0180">Archived</a> from the original on 2018-11-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-05-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Teaching+Philosophy&rft.atitle=Teaching+Syllogistic+Logic+via+a+Retooled+Venn+Diagrammatical+Technique&rft.volume=40&rft.issue=2&rft.pages=161-180&rft.date=2017-06&rft_id=info%3Adoi%2F10.5840%2Fteachphil201771767&rft.aulast=Joaquin&rft.aufirst=Jeremiah+Joven&rft.au=Boyles%2C+Robert+James+M.&rft_id=https%3A%2F%2Fwww.pdcnet.org%2Fteachphil%2Fcontent%2Fteachphil_2017_0040_0002_0161_0180&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Grimaldi_2004-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-Grimaldi_2004_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrimaldi2004" class="citation book cs1"><a href="/wiki/Ralph_Grimaldi" title="Ralph Grimaldi">Grimaldi, Ralph P.</a> (2004). <i>Discrete and combinatorial mathematics</i>. Boston: <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. p. 143. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-72634-3" title="Special:BookSources/978-0-201-72634-3"><bdi>978-0-201-72634-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+and+combinatorial+mathematics&rft.place=Boston&rft.pages=143&rft.pub=Addison-Wesley&rft.date=2004&rft.isbn=978-0-201-72634-3&rft.aulast=Grimaldi&rft.aufirst=Ralph+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Johnson_2001-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-Johnson_2001_25-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnson2001" class="citation book cs1">Johnson, David L. (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8KtRMofBKc0C&pg=PA62">"3.3 Laws"</a>. <a rel="nofollow" class="external text" href="https://archive.org/details/elementsoflogicv0000john/page/62"><i>Elements of logic via numbers and sets</i></a>. Springer Undergraduate Mathematics Series. Berlin, Germany: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/elementsoflogicv0000john/page/62">62</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-76123-5" title="Special:BookSources/978-3-540-76123-5"><bdi>978-3-540-76123-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3.3+Laws&rft.btitle=Elements+of+logic+via+numbers+and+sets&rft.place=Berlin%2C+Germany&rft.series=Springer+Undergraduate+Mathematics+Series&rft.pages=62&rft.pub=Springer-Verlag&rft.date=2001&rft.isbn=978-3-540-76123-5&rft.aulast=Johnson&rft.aufirst=David+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8KtRMofBKc0C%26pg%3DPA62&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMahmoodianRezaieVatan1987" class="citation web cs1"><a href="/wiki/Ebadollah_S._Mahmoodian" title="Ebadollah S. Mahmoodian">Mahmoodian, Ebadollah S.</a>; Rezaie, M.; Vatan, F. (March 1987). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170501202223/http://sharif.ir/~emahmood/papers/Generalized-Venn-Diagram1987.pdf">"Generalization of Venn Diagram"</a> <span class="cs1-format">(PDF)</span>. <i>Eighteenth Annual Iranian Mathematics Conference</i>. Tehran and Isfahan, Iran. Archived from <a rel="nofollow" class="external text" href="http://sharif.ir/~emahmood/papers/Generalized-Venn-Diagram1987.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-05-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-05-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Eighteenth+Annual+Iranian+Mathematics+Conference&rft.atitle=Generalization+of+Venn+Diagram&rft.date=1987-03&rft.aulast=Mahmoodian&rft.aufirst=Ebadollah+S.&rft.au=Rezaie%2C+M.&rft.au=Vatan%2C+F.&rft_id=http%3A%2F%2Fsharif.ir%2F~emahmood%2Fpapers%2FGeneralized-Venn-Diagram1987.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdwards1989" class="citation journal cs1"><a href="/wiki/Anthony_William_Fairbank_Edwards" class="mw-redirect" title="Anthony William Fairbank Edwards">Edwards, Anthony William Fairbank</a> (1989-01-07). "Venn diagrams for many sets". <i><a href="/wiki/New_Scientist" title="New Scientist">New Scientist</a></i>. <b>121</b> (1646): 51–56.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=New+Scientist&rft.atitle=Venn+diagrams+for+many+sets&rft.volume=121&rft.issue=1646&rft.pages=51-56&rft.date=1989-01-07&rft.aulast=Edwards&rft.aufirst=Anthony+William+Fairbank&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWatkinson1990" class="citation book cs1">Watkinson, John (1990). "4.10. Hamming distance". <i>Coding for Digital Recording</i>. Stoneham, MA, USA: <a href="/wiki/Focal_Press" title="Focal Press">Focal Press</a>. pp. 94–99, foldout in backsleeve. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-240-51293-8" title="Special:BookSources/978-0-240-51293-8"><bdi>978-0-240-51293-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=4.10.+Hamming+distance&rft.btitle=Coding+for+Digital+Recording&rft.place=Stoneham%2C+MA%2C+USA&rft.pages=94-99%2C+foldout+in+backsleeve&rft.pub=Focal+Press&rft.date=1990&rft.isbn=978-0-240-51293-8&rft.aulast=Watkinson&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span> (NB. The book comes with a 3-page foldout of a seven-bit cylindrical Venn diagram.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart2003" class="citation book cs1"><a href="/wiki/Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Stewart, Ian</a> (June 2003) [1992]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=u5GPE97-ZhsC&pg=PA51">"Chapter 4. Cogwheels of the Mind"</a>. <i>Another Fine Math You've Got Me Into</i> (reprint of 1st ed.). Mineola, New York, USA: <a href="/wiki/Dover_Publications,_Inc." class="mw-redirect" title="Dover Publications, Inc.">Dover Publications, Inc.</a> (<a href="/wiki/W._H._Freeman" class="mw-redirect" title="W. H. Freeman">W. H. Freeman</a>). pp. 51–64. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43181-9" title="Special:BookSources/978-0-486-43181-9"><bdi>978-0-486-43181-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+4.+Cogwheels+of+the+Mind&rft.btitle=Another+Fine+Math+You%27ve+Got+Me+Into&rft.place=Mineola%2C+New+York%2C+USA&rft.pages=51-64&rft.edition=reprint+of+1st&rft.pub=Dover+Publications%2C+Inc.+%28W.+H.+Freeman%29&rft.date=2003-06&rft.isbn=978-0-486-43181-9&rft.aulast=Stewart&rft.aufirst=Ian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Du5GPE97-ZhsC%26pg%3DPA51&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGlassner2004" class="citation book cs1"><a href="/wiki/Andrew_Glassner" title="Andrew Glassner">Glassner, Andrew</a> (2004). "Venn and Now". <i>Morphs, Mallards, and Montages: Computer-Aided Imagination</i>. Wellesley, MA, USA: <a href="/wiki/A._K._Peters" class="mw-redirect" title="A. K. Peters">A. K. Peters</a>. pp. 161–184. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1568812311" title="Special:BookSources/978-1568812311"><bdi>978-1568812311</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Venn+and+Now&rft.btitle=Morphs%2C+Mallards%2C+and+Montages%3A+Computer-Aided+Imagination&rft.place=Wellesley%2C+MA%2C+USA&rft.pages=161-184&rft.pub=A.+K.+Peters&rft.date=2004&rft.isbn=978-1568812311&rft.aulast=Glassner&rft.aufirst=Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></li> <li><span class="anchor" id="Newroz"></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMamakaniRuskey2012" class="citation web cs1">Mamakani, Khalegh; <a href="/wiki/Frank_Ruskey" title="Frank Ruskey">Ruskey, Frank</a> (2012-07-27). <a rel="nofollow" class="external text" href="http://webhome.cs.uvic.ca/~ruskey/Publications/Venn11/Venn11.html">"A New Rose: The First Simple Symmetric 11-Venn Diagram"</a>. p. 6452. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1207.6452">1207.6452</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012arXiv1207.6452M">2012arXiv1207.6452M</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170501204303/http://webhome.cs.uvic.ca/~ruskey/Publications/Venn11/Venn11.html">Archived</a> from the original on 2017-05-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-05-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+New+Rose%3A+The+First+Simple+Symmetric+11-Venn+Diagram&rft.pages=6452&rft.date=2012-07-27&rft_id=info%3Aarxiv%2F1207.6452&rft_id=info%3Abibcode%2F2012arXiv1207.6452M&rft.aulast=Mamakani&rft.aufirst=Khalegh&rft.au=Ruskey%2C+Frank&rft_id=http%3A%2F%2Fwebhome.cs.uvic.ca%2F~ruskey%2FPublications%2FVenn11%2FVenn11.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Venn_diagram&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox 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//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Venn_diagrams" class="extiw" title="commons:Category:Venn diagrams">Venn diagrams</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Venn_diagram">"Venn diagram"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Venn+diagram&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DVenn_diagram&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVenn+diagram" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/LewisCarroll/dunham.shtml">Lewis Carroll's Logic Game – Venn vs. Euler</a> at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">Cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="http://www.combinatorics.org/Surveys/ds5/VennTriangleEJC.html">Six sets Venn diagrams made from triangles</a></li> <li><a rel="nofollow" class="external text" href="http://moebio.com/research/sevensets/">Interactive seven sets Venn diagram</a></li> <li><a rel="nofollow" class="external text" href="https://www.usgs.gov/software/vbvenn-visual-basic-venn-diagram-software-page">VBVenn, a free open source program for calculating and graphing quantitative two-circle Venn diagrams</a></li> <li><a rel="nofollow" class="external text" href="https://interactivenn.net/index.html">InteractiVenn, a web-based tool for visualizing Venn diagrams</a></li> <li><a rel="nofollow" class="external text" href="https://www.deepvenn.com/">DeepVenn, a tool for creating area-proportional Venn Diagrams</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output 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.navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_logic" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a class="mw-selflink selflink">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Concrete_category" title="Concrete category">Concrete</a>/<a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li> <li><a href="/wiki/Category_of_sets" title="Category of sets">Category of sets</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History of logic</a></li> <li><a href="/wiki/History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a href="/wiki/Logicism" title="Logicism">Logicism</a></li> <li><a href="/wiki/Mathematical_object" title="Mathematical object">Mathematical object</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Supertask" title="Supertask">Supertask</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span 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height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/150px-Venn_A_intersect_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/200px-Venn_A_intersect_B.svg.png 2x" data-file-width="350" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Axiom" title="Axiom">Axioms</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom_of_adjunction" title="Axiom of adjunction">Adjunction</a></li> <li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">Choice</a> <ul><li><a href="/wiki/Axiom_of_countable_choice" title="Axiom of countable choice">countable</a></li> <li><a href="/wiki/Axiom_of_dependent_choice" title="Axiom of dependent choice">dependent</a></li> <li><a href="/wiki/Axiom_of_global_choice" title="Axiom of global choice">global</a></li></ul></li> <li><a href="/wiki/Axiom_of_constructibility" title="Axiom of constructibility">Constructibility (V=L)</a></li> <li><a href="/wiki/Axiom_of_determinacy" title="Axiom of determinacy">Determinacy</a> <ul><li><a href="/wiki/Axiom_of_projective_determinacy" title="Axiom of projective determinacy">projective</a></li></ul></li> <li><a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">Extensionality</a></li> <li><a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">Infinity</a></li> <li><a href="/wiki/Axiom_of_limitation_of_size" title="Axiom of limitation of size">Limitation of size</a></li> <li><a href="/wiki/Axiom_of_pairing" title="Axiom of pairing">Pairing</a></li> <li><a href="/wiki/Axiom_of_power_set" title="Axiom of power set">Power set</a></li> <li><a href="/wiki/Axiom_of_regularity" title="Axiom of regularity">Regularity</a></li> <li><a href="/wiki/Axiom_of_union" title="Axiom of union">Union</a></li> <li><a href="/wiki/Martin%27s_axiom" title="Martin's axiom">Martin's axiom</a></li></ul> <ul><li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a> <ul><li><a href="/wiki/Axiom_schema_of_replacement" title="Axiom schema of replacement">replacement</a></li> <li><a href="/wiki/Axiom_schema_of_specification" title="Axiom schema of specification">specification</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)#Basic_operations" title="Set (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">Complement</a> (i.e. set difference)</li> <li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/Disjoint_union" title="Disjoint union">Disjoint union</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">Identities</a></li> <li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">Intersection</a></li> <li><a href="/wiki/Power_set" title="Power set">Power set</a></li> <li><a href="/wiki/Symmetric_difference" title="Symmetric difference">Symmetric difference</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">Union</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li>Concepts</li><li>Methods</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost" title="Almost">Almost</a></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal number</a> (<a href="/wiki/Large_cardinal" title="Large cardinal">large</a>)</li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li><a href="/wiki/Constructible_universe" title="Constructible universe">Constructible universe</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">Continuum hypothesis</a></li> <li><a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Diagonal argument</a></li> <li><a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a> <ul><li><a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a></li> <li><a href="/wiki/Tuple" title="Tuple">tuple</a></li></ul></li> <li><a href="/wiki/Family_of_sets" title="Family of sets">Family</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Bijection" title="Bijection">One-to-one correspondence</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Set-builder_notation" title="Set-builder notation">Set-builder notation</a></li> <li><a href="/wiki/Transfinite_induction" title="Transfinite induction">Transfinite induction</a></li> <li><a class="mw-selflink selflink">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)" title="Set (mathematics)">Set</a> types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_set" title="Amorphous set">Amorphous</a></li> <li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a> (<a href="/wiki/Hereditarily_finite_set" title="Hereditarily finite set">hereditarily</a>)</li> <li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">Filter</a> <ul><li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">base</a></li> <li><a href="/wiki/Filter_(set_theory)#Filters_and_prefilters" title="Filter (set theory)">subbase</a></li> <li><a href="/wiki/Ultrafilter_on_a_set" title="Ultrafilter on a set">Ultrafilter</a></li></ul></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a> (<a href="/wiki/Dedekind-infinite_set" title="Dedekind-infinite set">Dedekind-infinite</a>)</li> <li><a href="/wiki/Computable_set" title="Computable set">Recursive</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Subset" title="Subset">Subset <b>·</b> Superset</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternative_set_theory" class="mw-redirect" title="Alternative set theory">Alternative</a></li> <li><a href="/wiki/Set_theory#Formalized_set_theory" title="Set theory">Axiomatic</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a></li></ul> <ul><li><a href="/wiki/Zermelo_set_theory" title="Zermelo set theory">Zermelo</a> <ul><li><a href="/wiki/General_set_theory" title="General set theory">General</a></li></ul></li> <li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> <ul><li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li></ul></li> <li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel </a> <ul><li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">von Neumann–Bernays–Gödel </a> <ul><li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li></ul></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes</a></li><li>Problems</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li> <li><a href="/wiki/Suslin%27s_problem" title="Suslin's problem">Suslin's problem</a></li> <li><a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Set_theorists" title="Category:Set theorists">Set theorists</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Paul_Bernays" title="Paul Bernays">Paul Bernays</a></li> <li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Paul_Cohen" title="Paul Cohen">Paul Cohen</a></li> <li><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li> <li><a href="/wiki/Abraham_Fraenkel" 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