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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</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">Euler 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 28 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-28" 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">28 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D8%B3%D9%85_%D8%A3%D9%88%D9%8A%D9%84%D8%B1_%D8%A7%D9%84%D8%A8%D9%8A%D8%A7%D9%86%D9%8A" title="رسم أويلر البياني – Arabic" lang="ar" hreflang="ar" data-title="رسم أويلر البياني" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AF%E0%A6%BC%E0%A6%B2%E0%A6%BE%E0%A6%B0_%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="অয়লার রেখাচিত্র – Bangla" lang="bn" hreflang="bn" data-title="অয়লার 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%9A%D1%80%D1%8A%D0%B3%D0%BE%D0%B2%D0%B5_%D0%BD%D0%B0_%D0%9E%D0%B9%D0%BB%D0%B5%D1%80" title="Кръгове на Ойлер – Bulgarian" lang="bg" hreflang="bg" data-title="Кръгове на Ойлер" 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_d%27Euler" title="Diagrama d'Euler – Catalan" lang="ca" hreflang="ca" data-title="Diagrama d'Euler" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Diagram_Euler" title="Diagram Euler – Welsh" lang="cy" hreflang="cy" data-title="Diagram Euler" 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/Euler-diagram" title="Euler-diagram – Danish" lang="da" hreflang="da" data-title="Euler-diagram" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/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-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Diagrama_de_Euler" title="Diagrama de Euler – Spanish" lang="es" hreflang="es" data-title="Diagrama de Euler" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Euler_diagrama" title="Euler diagrama – Basque" lang="eu" hreflang="eu" data-title="Euler 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/%D8%AF%DB%8C%D8%A7%DA%AF%D8%B1%D8%A7%D9%85_%D8%A7%D9%88%DB%8C%D9%84%D8%B1" title="دیاگرام اویلر – Persian" lang="fa" hreflang="fa" data-title="دیاگرام اویلر" 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_d%27Euler" title="Diagramme d'Euler – French" lang="fr" hreflang="fr" data-title="Diagramme d'Euler" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Diagrama_de_Euler" title="Diagrama de Euler – Galician" lang="gl" hreflang="gl" data-title="Diagrama de Euler" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Diagram_Euler" title="Diagram Euler – Indonesian" lang="id" hreflang="id" data-title="Diagram Euler" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Diagramma_di_Eulero-Venn" title="Diagramma di Eulero-Venn – Italian" lang="it" hreflang="it" data-title="Diagramma di Eulero-Venn" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Eulerdiagram" title="Eulerdiagram – Dutch" lang="nl" hreflang="nl" data-title="Eulerdiagram" 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%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E5%9B%B3" title="オイラー図 – Japanese" lang="ja" hreflang="ja" data-title="オイラー図" 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/Eulerdiagram" title="Eulerdiagram – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Eulerdiagram" 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-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%8A%E1%9F%92%E1%9E%99%E1%9E%B6%E1%9E%80%E1%9F%92%E1%9E%9A%E1%9E%B6%E1%9E%98%E1%9E%A2%E1%9E%99%E1%9E%9B%E1%9F%90%E1%9E%9A" title="ដ្យាក្រាមអយល័រ – Khmer" lang="km" hreflang="km" data-title="ដ្យាក្រាមអយល័រ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Diagrama_de_Euler" title="Diagrama de Euler – Portuguese" lang="pt" hreflang="pt" data-title="Diagrama de Euler" 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_Euler" title="Diagramă Euler – Romanian" lang="ro" hreflang="ro" data-title="Diagramă Euler" 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%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%B0" title="Диаграмма Эйлера – Russian" lang="ru" hreflang="ru" data-title="Диаграмма Эйлера" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Eulerjev_diagram" title="Eulerjev diagram – Slovenian" lang="sl" hreflang="sl" data-title="Eulerjev diagram" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%BE%DB%8E%DA%B5%DA%A9%D8%A7%D8%B1%DB%8C%DB%8C_%D8%A6%DB%86%DB%8C%D9%84%DB%95%D8%B1" title="ھێڵکاریی ئۆیلەر – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ھێڵکاریی ئۆیلەر" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Eulerdiagram" title="Eulerdiagram – Swedish" lang="sv" hreflang="sv" data-title="Eulerdiagram" 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%86%E0%AE%AF%E0%AF%8D%E0%AE%B2%E0%AE%B0%E0%AF%8D_%E0%AE%B5%E0%AE%B0%E0%AF%88%E0%AE%AA%E0%AE%9F%E0%AE%AE%E0%AF%8D" title="ஆய்லர் வரைபடம் – Tamil" lang="ta" hreflang="ta" data-title="ஆய்லர் வரைபடம்" 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%B8%AD%E0%B9%87%E0%B8%AD%E0%B8%A2%E0%B9%80%E0%B8%A5%E0%B8%AD%E0%B8%A3%E0%B9%8C" title="แผนภาพอ็อยเลอร์ – Thai" lang="th" hreflang="th" data-title="แผนภาพอ็อยเลอร์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BB%D0%B0_%D0%95%D0%B9%D0%BB%D0%B5%D1%80%D0%B0" title="Кола Ейлера – Ukrainian" lang="uk" hreflang="uk" data-title="Кола Ейлера" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AC%A7%E6%8B%89%E5%9B%BE" title="欧拉图 – Chinese" lang="zh" hreflang="zh" data-title="欧拉图" 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 href="https://www.wikidata.org/wiki/Special:EntityPage/Q2501020#sitelinks-wikipedia" 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Graphical set representation involving overlapping circles</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">This article is about Eulerian circles of set theory and logic. For the geometric Euler circle, see <a href="/wiki/Nine-point_circle" title="Nine-point circle">Nine-point circle</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:EulerDiagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/EulerDiagram.svg/220px-EulerDiagram.svg.png" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/EulerDiagram.svg/330px-EulerDiagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/EulerDiagram.svg/440px-EulerDiagram.svg.png 2x" data-file-width="304" data-file-height="172" /></a><figcaption>Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is a disjoint set (it has no members in common) with "animals"</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Euler_diagram_of_solar_system_bodies.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Euler_diagram_of_solar_system_bodies.svg/300px-Euler_diagram_of_solar_system_bodies.svg.png" decoding="async" width="300" height="231" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Euler_diagram_of_solar_system_bodies.svg/450px-Euler_diagram_of_solar_system_bodies.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Euler_diagram_of_solar_system_bodies.svg/600px-Euler_diagram_of_solar_system_bodies.svg.png 2x" data-file-width="635" data-file-height="490" /></a><figcaption>Euler diagram showing the relationships between different <a href="/wiki/Solar_System" title="Solar System">Solar System</a> objects</figcaption></figure> <p>An <b>Euler diagram</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="/ɔɪ/: 'oi' in 'choice'">ɔɪ</span><span title="'l' in 'lie'">l</span><span title="/ər/: 'er' in 'letter'">ər</span></span>/</a></span></span>, <a href="/wiki/Help:Pronunciation_respelling_key" title="Help:Pronunciation respelling key"><i title="English pronunciation respelling"><span style="font-size:90%">OY</span>-lər</i></a>) is a <a href="/wiki/Diagram" title="Diagram">diagrammatic</a> means of representing <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> and their relationships. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are similar to another set diagramming technique, <a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagrams</a>. Unlike Venn diagrams, which show all possible relations between different sets, the Euler diagram shows only relevant relationships. </p><p>The first use of "Eulerian circles" is commonly attributed to Swiss mathematician <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> (1707–1783). In the United States, both Venn and Euler diagrams were incorporated as part of instruction in <a href="/wiki/Set_theory" title="Set theory">set theory</a> as part of the <a href="/wiki/New_math" class="mw-redirect" title="New math">new math</a> movement of the 1960s. Since then, they have also been adopted by other curriculum fields such as reading<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> as well as organizations and businesses. </p><p>Euler diagrams consist of simple closed shapes in a two-dimensional plane that each depict a set or category. How or whether these shapes overlap demonstrates the relationships between the sets. Each curve divides the plane into two regions or "zones": the interior, which symbolically represents the <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">elements</a> of the set, and the exterior, which represents all elements that are not members of the set. Curves which do not overlap represent <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint sets</a>, which have no elements in common. Two curves that overlap represent sets that <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersect</a>, that have common elements; the zone inside both curves represents the set of elements common to both sets (the <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of the sets). A curve completely within the interior of another is a <a href="/wiki/Subset" title="Subset">subset</a> of it. </p><p><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagrams</a> are a more restrictive form of Euler diagrams. A Venn diagram must contain all 2<sup><i>n</i></sup> logically possible zones of overlap between its <i>n</i> curves, representing all combinations of inclusion/exclusion of its constituent sets. Regions not part of the set are indicated by coloring them black, in contrast to Euler diagrams, where membership in the set is indicated by overlap as well as color. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler_diagram&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Hamilton_Lectures_on_Logic_1874_Euler_Diagrams.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Hamilton_Lectures_on_Logic_1874_Euler_Diagrams.png/300px-Hamilton_Lectures_on_Logic_1874_Euler_Diagrams.png" decoding="async" width="300" height="344" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Hamilton_Lectures_on_Logic_1874_Euler_Diagrams.png/450px-Hamilton_Lectures_on_Logic_1874_Euler_Diagrams.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Hamilton_Lectures_on_Logic_1874_Euler_Diagrams.png/600px-Hamilton_Lectures_on_Logic_1874_Euler_Diagrams.png 2x" data-file-width="1993" data-file-height="2286" /></a><figcaption>A page from Hamilton's <i>Lectures on Logic;</i> the symbols <b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">A</span></span></b>, <b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">E</span></span></b>, <b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">I</span></span></b>, and <b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">O</span></span></b> refer to four types of categorical statement which can occur in a <a href="/wiki/Syllogism" title="Syllogism">syllogism</a> (see <a href="#a_e_i_o_u_descrs_anchor">descriptions, left</a>) The small text to the left erroneously says: "The first employment of circular diagrams in logic improperly ascribed to Euler. To be found in Christian Weise", a book which was actually written by Johann Christian Lange.<sup id="cite_ref-Venn_1881_2-0" class="reference"><a href="#cite_note-Venn_1881-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gailand_1967_3-0" class="reference"><a href="#cite_note-Gailand_1967-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Couturat_1914_and_Venn_assignments1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Couturat_1914_and_Venn_assignments1.jpg/400px-Couturat_1914_and_Venn_assignments1.jpg" decoding="async" width="400" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Couturat_1914_and_Venn_assignments1.jpg/600px-Couturat_1914_and_Venn_assignments1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Couturat_1914_and_Venn_assignments1.jpg/800px-Couturat_1914_and_Venn_assignments1.jpg 2x" data-file-width="1700" data-file-height="800" /></a><figcaption>The diagram to the right is from Couturat<sup id="cite_ref-Courant-1914_4-0" class="reference"><a href="#cite_note-Courant-1914-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page:  74">(p 74)</span></sup> in which he labels the 8 regions of the Venn diagram. The modern name for the "regions" is <i><a href="/wiki/Minterm" class="mw-redirect" title="Minterm">minterms</a></i>. They are shown in the diagram with the variables <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span>, and <span class="texhtml mvar" style="font-style:italic;">z</span> per Venn's drawing. The symbolism is as follows: logical <span class="nowrap"> <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">AND</span></span> [ <b>&</b> ] </span> is represented by arithmetic multiplication, and the logical <span class="nowrap"> <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">NOT</span></span> [ <span style="font-size:120%"><b>¬</b></span> ] </span> is represented by " ' " after the variable, e.g. the region <span class="texhtml mvar" style="font-style:italic;">x</span>'<span class="texhtml mvar" style="font-style:italic;">y</span>'<span class="texhtml mvar" style="font-style:italic;">z</span> is read as "(<span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">NOT</span></span> <span class="texhtml mvar" style="font-style:italic;">x</span>) <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">AND</span></span> (<span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">NOT</span></span> <span class="texhtml mvar" style="font-style:italic;">y</span>) <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">AND</span></span> <span class="texhtml mvar" style="font-style:italic;">z</span>" i.e. <span class="nowrap"> (¬ <span class="texhtml mvar" style="font-style:italic;">x</span>) & (¬ <span class="texhtml mvar" style="font-style:italic;">y</span>) & <span class="texhtml mvar" style="font-style:italic;">z</span> .</span></figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Veitch_and_Karnaugh_3.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Veitch_and_Karnaugh_3.jpg/300px-Veitch_and_Karnaugh_3.jpg" decoding="async" width="300" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Veitch_and_Karnaugh_3.jpg/450px-Veitch_and_Karnaugh_3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/7/7a/Veitch_and_Karnaugh_3.jpg 2x" data-file-width="500" data-file-height="320" /></a><figcaption>Both the Veitch diagram and Karnaugh map show all the <a href="/wiki/Minterms" class="mw-redirect" title="Minterms">minterms</a>, but the Veitch is not particularly useful for reduction of formulas. Observe the strong resemblance between the Venn and Karnaugh diagrams; the colors and the variables <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span>, and <span class="texhtml mvar" style="font-style:italic;">z</span> are per Venn's example.</figcaption></figure> <p>As shown in the illustration to the right, <a href="/wiki/Sir_William_Hamilton,_9th_Baronet" title="Sir William Hamilton, 9th Baronet">Sir William Hamilton</a> erroneously asserted that the original use of the circles to "sensualize... the abstractions of logic"<sup id="cite_ref-Hamilton-1858-1860_5-0" class="reference"><a href="#cite_note-Hamilton-1858-1860-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> was not <a href="/wiki/Leonhard_Paul_Euler" class="mw-redirect" title="Leonhard Paul Euler">Euler</a> (1707–1783) but rather <a href="/wiki/Christian_Weise" title="Christian Weise">Weise</a> (1642–1708);<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> however the latter book was actually written by Johann Christian Lange, rather than Weise.<sup id="cite_ref-Venn_1881_2-1" class="reference"><a href="#cite_note-Venn_1881-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gailand_1967_3-1" class="reference"><a href="#cite_note-Gailand_1967-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> He references Euler's <i><a href="/wiki/Letters_to_a_German_Princess" title="Letters to a German Princess">Letters to a German Princess</a></i>.<sup id="cite_ref-Euler-1791-1842_7-0" class="reference"><a href="#cite_note-Euler-1791-1842-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Mansel_Veitch_1860_credit_note_8-0" class="reference"><a href="#cite_note-Mansel_Veitch_1860_credit_note-8"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> </p><p><span class="anchor" id="a_e_i_o_u_descrs_anchor"></span> In Hamilton's illustration of the four <a href="/wiki/Categorical_proposition" title="Categorical proposition">categorical propositions</a><sup id="cite_ref-Hamilton-1860-Jevons-1881_9-0" class="reference"><a href="#cite_note-Hamilton-1860-Jevons-1881-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> which can occur in a <a href="/wiki/Syllogism" title="Syllogism">syllogism</a> as symbolized by the drawings <b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">A</span></span></b>, <b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">E</span></span></b>, <b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">I</span></span></b>, and <b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">O</span></span></b> are: </p> <dl><dd><b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">A</span></span></b>: The <i>Universal Affirmative</i> <dl><dd>Example: <i>"All metals are elements."</i></dd></dl></dd> <dd><b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">E</span></span></b>: The <i>Universal Negative</i> <dl><dd>Example: <i>"No metals are compound substances."</i></dd></dl></dd> <dd><b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">I</span></span></b>: The <i>Particular Affirmative</i> <dl><dd>Example: "Some metals are brittle."<i></i></dd></dl></dd> <dd><b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">O</span></span></b>: The <i>Particular Negative</i> <dl><dd>Example: <i>"Some metals are not brittle."</i><sup id="cite_ref-Hamilton-1860-Jevons-1881_9-1" class="reference"><a href="#cite_note-Hamilton-1860-Jevons-1881-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></dd></dl></dd></dl> <p><a href="/wiki/John_Venn" title="John Venn">Venn</a> (1834–1923) comments on the remarkable prevalence of the Euler diagram: </p> <dl><dd>"... of the first sixty logical treatises, published during the last century or so, which were consulted for this purpose–somewhat at random, as they happened to be most accessible–it appeared that thirty four appealed to the aid of diagrams, nearly all of these making use of the <a href="/wiki/Leonhard_Paul_Euler" class="mw-redirect" title="Leonhard Paul Euler">Eulerian</a> scheme."<sup id="cite_ref-Venn-1881a-§V_10-0" class="reference"><a href="#cite_note-Venn-1881a-§V-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></dd></dl> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Venn_1881_p_115-116_pasteup.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Venn_1881_p_115-116_pasteup.jpg/300px-Venn_1881_p_115-116_pasteup.jpg" decoding="async" width="300" height="385" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Venn_1881_p_115-116_pasteup.jpg/450px-Venn_1881_p_115-116_pasteup.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/1/1a/Venn_1881_p_115-116_pasteup.jpg 2x" data-file-width="565" data-file-height="725" /></a><figcaption>Composite of two pages from <a href="#CITEREFVenn1881a">Venn (1881a)</a>, pp. 115–116 showing his example of how to convert a syllogism of three parts into his type of diagram; Venn calls the circles "Eulerian circles"<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>But nevertheless, he contended, "the inapplicability of this scheme for the purposes of a really general logic"<sup id="cite_ref-Venn-1881a-§V_10-4" class="reference"><a href="#cite_note-Venn-1881a-§V-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page:  100">(p 100)</span></sup> and then noted that, </p> <dl><dd>“It fits in, but badly, even with the four propositions of the common logic to which it is normally applied.”<sup id="cite_ref-Venn-1881a-§V_10-5" class="reference"><a href="#cite_note-Venn-1881a-§V-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page:  101">(p 101)</span></sup></dd></dl> <p>Venn ends his chapter with the observation illustrated in the examples below—that their use is based on practice and intuition, not on a strict <a href="/wiki/Algorithm" title="Algorithm">algorithmic</a> practice: </p> <dl><dd>“In fact ... those diagrams not only do not fit in with the ordinary scheme of propositions which they are employed to illustrate, but do not seem to have any recognized scheme of propositions to which they could be consistently affiliated.”<sup id="cite_ref-Venn-1881a-§V_10-6" class="reference"><a href="#cite_note-Venn-1881a-§V-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages:  124–125">(pp 124–125)</span></sup></dd></dl> <p>Finally, in his Venn gets to a crucial criticism (italicized in the quote below); observe in Hamilton's illustration that the <b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">O</span></span></b> (<i>Particular Negative</i>) and <b><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">I</span></span></b> (<i>Particular Affirmative</i>) are simply rotated: </p> <dl><dd>“We now come to Euler's well-known circles which were first described in his <i>Lettres a une Princesse d'Allemagne</i> (<i>Letters</i> 102–105).<sup id="cite_ref-Euler-1791-1842_7-1" class="reference"><a href="#cite_note-Euler-1791-1842-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages:  102–105">(pp 102–105)</span></sup> The weak point about these consists in the fact that they only illustrate in strictness the actual relations of classes to one another, rather than the imperfect knowledge of these relations which we may possess, or wish to convey, by means of the proposition. Accordingly they will not fit in with the propositions of common logic, but demand the constitution of a new group of appropriate elementary propositions. ... This defect must have been noticed from the first <i>in the case of the particular affirmative and negative, for the same diagram is commonly employed to stand for them both, which it does indifferently well</i>”.[italics added]<sup id="cite_ref-Venn-1881b-§XX_12-0" class="reference"><a href="#cite_note-Venn-1881b-§XX-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Venn-1881a-§V_10-7" class="reference"><a href="#cite_note-Venn-1881a-§V-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page:  100, Footnote 1">(p 100, Footnote 1)</span></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Whatever the case, armed with these observations and criticisms, Venn<sup id="cite_ref-Venn-1881a-§V_10-8" class="reference"><a href="#cite_note-Venn-1881a-§V-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages:  100–125">(pp 100–125)</span></sup> then demonstrates how he derived what has become known as his <a href="/wiki/Venn_diagrams" class="mw-redirect" title="Venn diagrams">Venn diagrams</a> from the “... old-fashioned Euler diagrams.” In particular Venn gives an example, shown at the left. </p><p>By 1914, <a href="/wiki/Louis_Couturat" title="Louis Couturat">Couturat</a> (1868–1914) had labeled the terms as shown on the drawing at the right.<sup id="cite_ref-Courant-1914_4-1" class="reference"><a href="#cite_note-Courant-1914-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Moreover, he had labeled the <i>exterior region</i> (shown as <span class="texhtml mvar" style="font-style:italic;">a</span>'<span class="texhtml mvar" style="font-style:italic;">b</span>'<span class="texhtml mvar" style="font-style:italic;">c</span>') as well. He succinctly explains how to use the diagram – one must <i>strike out</i> the regions that are to vanish: </p> <dl><dd>"Venn's method is translated in geometrical diagrams which represent all the constituents, so that, in order to obtain the result, we need only <i>strike out (by shading)</i> those which are made to vanish by the data of the problem."[italics added]<sup id="cite_ref-Courant-1914_4-2" class="reference"><a href="#cite_note-Courant-1914-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page:  73">(p 73)</span></sup></dd></dl> <p>Given the Venn's assignments, then, the unshaded areas <i>inside</i> the circles can be summed to yield the following equation for Venn's example: </p> <dl><dd>"<span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">NO</span></span> <span class="texhtml mvar" style="font-style:italic;">y</span> is <span class="texhtml mvar" style="font-style:italic;">z</span> and <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">ALL</span></span> <span class="texhtml mvar" style="font-style:italic;">x</span> is <span class="texhtml mvar" style="font-style:italic;">y</span>: therefore <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">NO</span></span> <span class="texhtml mvar" style="font-style:italic;">x</span> is <span class="texhtml mvar" style="font-style:italic;">z</span>" has the equation <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;">x</span>'<span class="texhtml mvar" style="font-style:italic;">y</span><span class="texhtml mvar" style="font-style:italic;">z</span>' + <span class="texhtml mvar" style="font-style:italic;">x</span><span class="texhtml mvar" style="font-style:italic;">y</span><span class="texhtml mvar" style="font-style:italic;">z</span>' + <span class="texhtml mvar" style="font-style:italic;">x</span>'<span class="texhtml mvar" style="font-style:italic;">y</span>'<span class="texhtml mvar" style="font-style:italic;">z</span> </span> for the unshaded area <i>inside</i> the circles (but this is not entirely correct; see the next paragraph).</dd></dl> <p>In Venn the background surrounding the circles, does not appear: That is, the term marked "0", <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;">x</span>'<span class="texhtml mvar" style="font-style:italic;">y</span>'<span class="texhtml mvar" style="font-style:italic;">z</span>' .</span> Nowhere is it discussed or labeled, but Couturat corrects this in his drawing.<sup id="cite_ref-Courant-1914_4-3" class="reference"><a href="#cite_note-Courant-1914-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The correct equation must include this unshaded area shown in boldface: </p> <dl><dd>"<span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">NO</span></span> <span class="texhtml mvar" style="font-style:italic;">y</span> is <span class="texhtml mvar" style="font-style:italic;">z</span> and <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">ALL</span></span> <span class="texhtml mvar" style="font-style:italic;">x</span> is <span class="texhtml mvar" style="font-style:italic;">y</span>: therefore <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">NO</span></span> <span class="texhtml mvar" style="font-style:italic;">x</span> is <span class="texhtml mvar" style="font-style:italic;">z</span>" has the equation <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;">x</span>'<span class="texhtml mvar" style="font-style:italic;">y</span><span class="texhtml mvar" style="font-style:italic;">z</span>' + <span class="texhtml mvar" style="font-style:italic;">x</span><span class="texhtml mvar" style="font-style:italic;">y</span><span class="texhtml mvar" style="font-style:italic;">z</span>' + <span class="texhtml mvar" style="font-style:italic;">x</span>'<span class="texhtml mvar" style="font-style:italic;">y</span>'<span class="texhtml mvar" style="font-style:italic;">z</span> + <b> <span class="texhtml mvar" style="font-style:italic;">x</span>'<span class="texhtml mvar" style="font-style:italic;">y</span>'<span class="texhtml mvar" style="font-style:italic;">z</span>'</b>.</span></dd></dl> <p>In modern use, the Venn diagram includes a "box" that surrounds all the circles; this is called the universe of discourse or the <a href="/wiki/Domain_of_discourse" title="Domain of discourse">domain of discourse</a>. </p><p>Couturat<sup id="cite_ref-Courant-1914_4-4" class="reference"><a href="#cite_note-Courant-1914-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> observed that, in a direct <a href="/wiki/Algorithm" title="Algorithm">algorithmic</a> (formal, systematic) manner, one cannot derive reduced Boolean equations, nor does it show how to arrive at the conclusion "<span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">NO</span></span> <span class="texhtml mvar" style="font-style:italic;">x</span> is <span class="texhtml mvar" style="font-style:italic;">z</span>". Couturat concluded that the process "has ... serious inconveniences as a method for solving logical problems": </p> <dl><dd>"It does not show how the data are exhibited by canceling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i. e., "throwing of the problem into an equation" and the transformation of the premises, nor with the subsequent steps, i. e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form."<sup id="cite_ref-Courant-1914_4-5" class="reference"><a href="#cite_note-Courant-1914-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page:  75">(p 75)</span></sup></dd></dl> <p>Thus the matter would rest until 1952 when <a href="/wiki/Maurice_Karnaugh" title="Maurice Karnaugh">Maurice Karnaugh</a> (1924–2022) would adapt and expand a method proposed by <a href="/wiki/Edward_W._Veitch" title="Edward W. Veitch">Edward W. Veitch</a>; this work would rely on the <a href="/wiki/Truth_table" title="Truth table">truth table</a> method precisely defined by <a href="/wiki/Emil_Post" class="mw-redirect" title="Emil Post">Emil Post</a><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and the application of propositional logic to switching logic by (among others) <a href="/wiki/Claude_Shannon" title="Claude Shannon">Shannon</a>, <a href="/wiki/George_Stibitz" title="George Stibitz">Stibitz</a>, and <a href="/wiki/Alan_Turing" title="Alan Turing">Turing</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> For example, Hill & Peterson (1968)<sup id="cite_ref-Hill-Peterson-1964-1968_16-0" class="reference"><a href="#cite_note-Hill-Peterson-1964-1968-16"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> present the Venn diagram with shading and all. They give examples of Venn diagrams to solve example switching-circuit problems, but end up with this statement: </p> <dl><dd><dl><dd>"For more than three variables, the basic illustrative form of the Venn diagram is inadequate. Extensions are possible, however, the most convenient of which is the Karnaugh map, to be discussed in Chapter 6."<sup id="cite_ref-Hill-Peterson-1964-1968_16-1" class="reference"><a href="#cite_note-Hill-Peterson-1964-1968-16"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page:  64">(p 64)</span></sup></dd></dl></dd></dl> <p>In Chapter 6, section 6.4 "Karnaugh map representation of Boolean functions" they begin with: </p> <dl><dd><dl><dd>"The Karnaugh map<sup>1</sup> [<sup>1</sup>Karnaugh 1953] is one of the most powerful tools in the repertory of the logic designer. ... A Karnaugh map may be regarded either as a pictorial form of a truth table or as an extension of the Venn diagram."<sup id="cite_ref-Hill-Peterson-1964-1968_16-2" class="reference"><a href="#cite_note-Hill-Peterson-1964-1968-16"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages:  103–104">(pp 103–104)</span></sup></dd></dl></dd></dl> <p>The history of Karnaugh's development of his "chart" or "map" method is obscure. The chain of citations becomes an academic game of "credit, credit; ¿who's got the credit?": <a href="#CITEREFKarnaugh1953">Karnaugh (1953)</a> referenced <a href="#CITEREFVeitch1952">Veitch (1952)</a>, Veitch, referenced <a href="#CITEREFShannon1938">Shannon (1938)</a>,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> and <a href="#CITEREFShannon1938">Shannon (1938)</a>, in turn referenced (among other authors of logic texts) <a href="#CITEREFCouturat1914">Couturat (1914)</a>. In Veitch's method the variables are arranged in a rectangle or square; as described in <a href="/wiki/Karnaugh_map" title="Karnaugh map">Karnaugh map</a>, Karnaugh in his method changed the order of the variables to correspond to what has become known as (the vertices of) a <a href="/wiki/Hypercube" title="Hypercube">hypercube</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_between_Euler_and_Venn_diagrams">Relation between Euler and Venn diagrams</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler_diagram&action=edit&section=2" title="Edit section: Relation between Euler and Venn diagrams"><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:Syllogism-Set-Diagrams.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Syllogism-Set-Diagrams.svg/300px-Syllogism-Set-Diagrams.svg.png" decoding="async" width="300" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Syllogism-Set-Diagrams.svg/450px-Syllogism-Set-Diagrams.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Syllogism-Set-Diagrams.svg/600px-Syllogism-Set-Diagrams.svg.png 2x" data-file-width="800" data-file-height="566" /></a><figcaption>Examples of small <a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagrams</a> <i>(on left)</i> with shaded regions representing <a href="/wiki/Empty_set" title="Empty set">empty sets</a>, showing how they can be easily transformed into equivalent Euler diagrams <i>(right)</i></figcaption></figure> <p><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagrams</a> are a more restrictive form of Euler diagrams. A Venn diagram must contain all 2<sup><i>n</i></sup> logically possible zones of overlap between its <i>n</i> curves, representing all combinations of inclusion/exclusion of its constituent sets. Regions not part of the set are indicated by coloring them black, in contrast to Euler diagrams, where membership in the set is indicated by overlap as well as color. When the number of sets grows beyond 3 a Venn diagram becomes visually complex, especially compared to the corresponding Euler diagram. 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 diagrams 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> <p>In a logical setting, one can use model-theoretic semantics to interpret Euler diagrams, within a <a href="/wiki/Universe_of_discourse" class="mw-redirect" title="Universe of discourse">universe of discourse</a>. In the examples below, the Euler diagram depicts that the sets <i>Animal</i> and <i>Mineral</i> are disjoint since the corresponding curves are disjoint, and also that the set <i>Four Legs</i> is a subset of the set of <i>Animal</i>s. The Venn diagram, which uses the same categories of <i>Animal</i>, <i>Mineral</i>, and <i>Four Legs</i>, does not encapsulate these relationships. Traditionally the <i>emptiness</i> of a set in Venn diagrams is depicted by shading in the region. Euler diagrams represent <i>emptiness</i> either by shading or by the absence of a region. </p><p>Often a set of well-formedness conditions are imposed; these are topological or geometric constraints imposed on the structure of the diagram. For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves. However, this sort of transformation of a Venn diagram with shading into an Euler diagram without shading is not always possible. There are examples of Euler diagrams with 9 sets that are not drawable using simple closed curves without the creation of unwanted zones since they would have to have non-planar dual graphs. </p> <div class="mw-heading mw-heading3"><h3 id="Example:_Euler-_to_Venn-diagram_and_Karnaugh_map">Example: Euler- to Venn-diagram and Karnaugh map</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler_diagram&action=edit&section=3" title="Edit section: Example: Euler- to Venn-diagram and Karnaugh map"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This example shows the Euler and Venn diagrams and Karnaugh map deriving and verifying the deduction "No <i>X</i>s are <i>Z</i>s". In the illustration and table the following logical symbols are used: </p> <ul><li>1 can be read as "true", 0 as "false"</li> <li>~ for NOT and abbreviated to ' when illustrating the minterms e.g. x' =<sub>defined</sub> NOT x,</li> <li>+ for Boolean OR (from <a href="/wiki/Boolean_algebra_(logic)" class="mw-redirect" title="Boolean algebra (logic)">Boolean algebra</a>: 0 + 0 = 0, 0 + 1 = 1 + 0 = 1, 1 + 1 = 1)</li> <li>& (logical AND) between propositions; in the minterms AND is omitted in a manner similar to arithmetic multiplication: e.g. x'y'z =<sub>defined</sub> ~x & ~y & z (From Boolean algebra: 0⋅0 = 0, 0⋅1 = 1⋅0 = 0, 1⋅1 = 1, where "⋅" is shown for clarity)</li> <li>→ (logical IMPLICATION): read as IF ... THEN ..., or " IMPLIES ", <i>P</i> → <i>Q</i> = <sub>defined</sub> NOT <i>P</i> OR <i>Q</i></li></ul> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Veitch_and_Karnaugh_truth_table_4.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Veitch_and_Karnaugh_truth_table_4.jpg/921px-Veitch_and_Karnaugh_truth_table_4.jpg" decoding="async" width="921" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Veitch_and_Karnaugh_truth_table_4.jpg/1382px-Veitch_and_Karnaugh_truth_table_4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Veitch_and_Karnaugh_truth_table_4.jpg/1842px-Veitch_and_Karnaugh_truth_table_4.jpg 2x" data-file-width="1843" data-file-height="450" /></a><figcaption>Before it can be presented in a Venn diagram or Karnaugh Map, the Euler diagram's syllogism "No <i>Y</i> is <i>Z</i>, All <i>X</i> is <i>Y</i>" must first be reworded into the more formal language of the <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional calculus</a>: " 'It is not the case that: <var style="padding-right: 1px;">Y</var> AND <var style="padding-right: 1px;">Z'</var> AND 'If an <var style="padding-right: 1px;">X</var> then a <var style="padding-right: 1px;">Y'</var> ". Once the propositions are reduced to symbols and a propositional formula ( ~(y & z) & (x → y) ), one can construct the formula's <a href="/wiki/Truth_table" title="Truth table">truth table</a>; from this table the Venn and/or the Karnaugh map are readily produced. By use of the adjacency of "1"s in the Karnaugh map (indicated by the grey ovals around terms 0 and 1 and around terms 2 and 6) one can "reduce" the example's <a href="/wiki/Boolean_equation" class="mw-redirect" title="Boolean equation">Boolean equation</a> i.e. (x'y'z' + x'y'z) + (x'yz' + xyz') to just two terms: x'y' + yz'. But the means for deducing the notion that "No X is Z", and just how the reduction relates to this deduction, is not forthcoming from this example.</figcaption></figure> <p>Given a proposed conclusion such as "No <i>X</i> is a <i>Z</i>", one can test whether or not it is a correct <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deduction</a> by use of a <a href="/wiki/Truth_table" title="Truth table">truth table</a>. The easiest method is put the starting formula on the left (abbreviate it as <i>P</i>) and put the (possible) deduction on the right (abbreviate it as <i>Q</i>) and connect the two with <a href="/wiki/Logical_implication" class="mw-redirect" title="Logical implication">logical implication</a> i.e. <i>P</i> → <i>Q</i>, read as IF <i>P</i> THEN <i>Q</i>. If the evaluation of the truth table produces all 1s under the implication-sign (→, the so-called <i>major connective</i>) then <i>P</i> → <i>Q</i> is a <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautology</a>. Given this fact, one can "detach" the formula on the right (abbreviated as <i>Q</i>) in the manner described below the truth table. </p><p>Given the example above, the formula for the Euler and Venn diagrams is: </p> <dl><dd>"No <i>Y</i>s are <i>Z</i>s" and "All <i>X</i>s are <i>Y</i>s": ( ~(y & z) & (x → y) ) =<sub>defined</sub> <i>P</i></dd></dl> <p>And the proposed deduction is: </p> <dl><dd>"No <i>X</i>s are <i>Z</i>s": ( ~ (x & z) ) =<sub>defined</sub> <i>Q</i></dd></dl> <p>So now the formula to be evaluated can be abbreviated to: </p> <dl><dd>( ~(y & z) & (x → y) ) → ( ~ (x & z) ): <i>P</i> → <i>Q</i></dd> <dd>IF ( "No <i>Y</i>s are <i>Z</i>s" and "All <i>X</i>s are <i>Y</i>s" ) THEN ( "No <i>X</i>s are <i>Z</i>s" )</dd></dl> <table style="width:auto; border:1px solid darkgray; border-collapse: collapse; margin-left: auto; margin-right: auto; text-align:center"> <caption>The Truth Table demonstrates that the formula ( ~(y & z) & (x → y) ) → ( ~ (x & z) ) is a tautology as shown by all 1s in yellow column. </caption> <tbody><tr> <th style="width:80pt">Square no. </th> <th style="width:160pt">Venn, Karnaugh region </th> <th style="width:13pt"> </th> <th style="width:20pt">x </th> <th style="width:20pt">y </th> <th style="width:20pt">z </th> <th style="width:13pt"> </th> <th style="width:30pt">(~ </th> <th style="width:30pt">(y </th> <th style="width:30pt">& </th> <th style="width:30pt">z) </th> <th style="width:30pt">& </th> <th style="width:30pt">(x </th> <th style="width:30pt">→ </th> <th style="width:30pt">y)) </th> <th style="width:30pt">→ </th> <th style="width:30pt">(~ </th> <th style="width:30pt">(x </th> <th style="width:30pt">& </th> <th style="width:30pt">z)) </th></tr> <tr style="font-size:12pt; height:25pt"> <td><span id="math_0" class="reference nourlexpansion" style="font-weight:bold;">0</span> </td> <td>x'y'z' </td> <td>  </td> <td>0 </td> <td>0 </td> <td>0 </td> <td>  </td> <td>1 </td> <td>0 </td> <td>0 </td> <td>0 </td> <th>1 </th> <td>0 </td> <td>1 </td> <td>0 </td> <td style="background-color:#F5FF93">1 </td> <th>1 </th> <td>0 </td> <td>0 </td> <td>0 </td></tr> <tr style="font-size:12pt; height:25pt"> <td><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span> </td> <td>x'y'z </td> <td>  </td> <td>0 </td> <td>0 </td> <td>1 </td> <td>  </td> <td>1 </td> <td>0 </td> <td>0 </td> <td>1 </td> <th>1 </th> <td>0 </td> <td>1 </td> <td>0 </td> <td style="background-color:#F5FF93">1 </td> <th>1 </th> <td>0 </td> <td>0 </td> <td>1 </td></tr> <tr style="font-size:12pt; height:25pt"> <td><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span> </td> <td>x'yz' </td> <td>  </td> <td>0 </td> <td>1 </td> <td>0 </td> <td>  </td> <td>1 </td> <td>1 </td> <td>0 </td> <td>0 </td> <th>1 </th> <td>0 </td> <td>1 </td> <td>1 </td> <td style="background-color:#F5FF93">1 </td> <th>1 </th> <td>0 </td> <td>0 </td> <td>0 </td></tr> <tr style="font-size:12pt; height:25pt"> <td style="background-color:#EDB9B9"><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span> </td> <td style="background-color:#EDB9B9">x'yz </td> <td>  </td> <td style="background-color:#EDB9B9">0 </td> <td style="background-color:#EDB9B9">1 </td> <td style="background-color:#EDB9B9">1 </td> <td>  </td> <td style="background-color:#EDB9B9">0 </td> <td style="background-color:#EDB9B9">1 </td> <td style="background-color:#EDB9B9">1 </td> <td style="background-color:#EDB9B9">1 </td> <th style="background-color:#EDB9B9;">0 </th> <td style="background-color:#EDB9B9">0 </td> <td style="background-color:#EDB9B9">1 </td> <td style="background-color:#EDB9B9">1 </td> <td style="background-color:#F5FF93">1 </td> <th style="background-color:#EDB9B9;">1 </th> <td style="background-color:#EDB9B9">0 </td> <td style="background-color:#EDB9B9">0 </td> <td style="background-color:#EDB9B9">1 </td></tr> <tr style="font-size:12pt; height:25pt"> <td style="background-color:#DBE5F1"><span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span> </td> <td style="background-color:#DBE5F1">xy'z' </td> <td>  </td> <td style="background-color:#DBE5F1">1 </td> <td style="background-color:#DBE5F1">0 </td> <td style="background-color:#DBE5F1">0 </td> <td>  </td> <td style="background-color:#DBE5F1">1 </td> <td style="background-color:#DBE5F1">0 </td> <td style="background-color:#DBE5F1">0 </td> <td style="background-color:#DBE5F1">0 </td> <th style="background-color:#DBE5F1;">0 </th> <td style="background-color:#DBE5F1">1 </td> <td style="background-color:#DBE5F1">0 </td> <td style="background-color:#DBE5F1">0 </td> <td style="background-color:#F5FF93">1 </td> <th style="background-color:#DBE5F1;">1 </th> <td style="background-color:#DBE5F1">1 </td> <td style="background-color:#DBE5F1">0 </td> <td style="background-color:#DBE5F1">0 </td></tr> <tr style="font-size:12pt; height:25pt"> <td style="background-color:#FFCCFF"><span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span> </td> <td style="background-color:#FFCCFF">xy'z </td> <td>  </td> <td style="background-color:#FFCCFF">1 </td> <td style="background-color:#FFCCFF">0 </td> <td style="background-color:#FFCCFF">1 </td> <td>  </td> <td style="background-color:#FFCCFF">1 </td> <td style="background-color:#FFCCFF">0 </td> <td style="background-color:#FFCCFF">0 </td> <td style="background-color:#FFCCFF">1 </td> <th style="background-color:#FFCCFF;">0 </th> <td style="background-color:#FFCCFF">1 </td> <td style="background-color:#FFCCFF">0 </td> <td style="background-color:#FFCCFF">0 </td> <td style="background-color:#F5FF93">1 </td> <th style="background-color:#FFCCFF;">0 </th> <td style="background-color:#FFCCFF">1 </td> <td style="background-color:#FFCCFF">1 </td> <td style="background-color:#FFCCFF">1 </td></tr> <tr style="font-size:12pt; height:25pt"> <td><span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span> </td> <td>xyz' </td> <td>  </td> <td>1 </td> <td>1 </td> <td>0 </td> <td>  </td> <td>1 </td> <td>1 </td> <td>0 </td> <td>0 </td> <th>1 </th> <td>1 </td> <td>1 </td> <td>1 </td> <td style="background-color:#F5FF93">1 </td> <th>1 </th> <td>1 </td> <td>0 </td> <td>0 </td></tr> <tr style="font-size:12pt; height:25pt"> <td style="background-color:#D6B4D0"><span id="math_7" class="reference nourlexpansion" style="font-weight:bold;">7</span> </td> <td style="background-color:#D6B4D0">xyz </td> <td>  </td> <td style="background-color:#D6B4D0">1 </td> <td style="background-color:#D6B4D0">1 </td> <td style="background-color:#D6B4D0">1 </td> <td>  </td> <td style="background-color:#D6B4D0">0 </td> <td style="background-color:#D6B4D0">1 </td> <td style="background-color:#D6B4D0">1 </td> <td style="background-color:#D6B4D0">1 </td> <th style="background-color:#D6B4D0;">0 </th> <td style="background-color:#D6B4D0">1 </td> <td style="background-color:#D6B4D0">1 </td> <td style="background-color:#D6B4D0">1 </td> <td style="background-color:#F5FF93">1 </td> <th style="background-color:#D6B4D0;">0 </th> <td style="background-color:#D6B4D0">1 </td> <td style="background-color:#D6B4D0">1 </td> <td style="background-color:#D6B4D0">1 </td></tr></tbody></table> <p>At this point the above implication <i>P</i> → <i>Q</i> (i.e. ~(y & z) & (x → y) ) → ~(x & z) ) is still a formula, and the deduction – the "detachment" of <i>Q</i> out of <i>P</i> → <i>Q</i> – has not occurred. But given the demonstration that <i>P</i> → <i>Q</i> is tautology, the stage is now set for the use of the procedure of <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a> to "detach" Q: "No <i>X</i>s are <i>Z</i>s" and dispense with the terms on the left.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>nb 1<span class="cite-bracket">]</span></a></sup> </p><p><i>Modus ponens</i> (or "the fundamental rule of inference"<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>) is often written as follows: The two terms on the left, <i>P</i> → <i>Q</i> and <i>P</i>, are called <i>premises</i> (by convention linked by a comma), the symbol ⊢ means "yields" (in the sense of logical deduction), and the term on the right is called the <i>conclusion</i>: </p> <dl><dd><i>P</i> → <i>Q</i>, <i>P</i> ⊢ <i>Q</i></dd></dl> <p>For the modus ponens to succeed, both premises <i>P</i> → <i>Q</i> and <i>P</i> must be <i>true</i>. Because, as demonstrated above the premise <i>P</i> → <i>Q</i> is a tautology, "truth" is always the case no matter how x, y and z are valued, but "truth" is only the case for <i>P</i> in those circumstances when <i>P</i> evaluates as "true" (e.g. rows <b><a href="#math_0">0</a></b> OR <b><a href="#math_1">1</a></b> OR <b><a href="#math_2">2</a></b> OR <b><a href="#math_6">6</a></b>: x'y'z' + x'y'z + x'yz' + xyz' = x'y' + yz').<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><i>P</i> → <i>Q</i> , <i>P</i> ⊢ <i>Q</i> <ul><li>i.e.: ( ~(y & z) & (x → y) ) → ( ~ (x & z) ) , ( ~(y & z) & (x → y) ) ⊢ ( ~ (x & z) )</li> <li>i.e.: IF "No <i>Y</i>s are <i>Z</i>s" and "All <i>X</i>s are <i>Y</i>s" <i>THEN</i> "No <i>X</i>s are <i>Z</i>s", "No <i>Y</i>s are <i>Z</i>s" and "All <i>X</i>s are <i>Y</i>s" ⊢ "No <i>X</i>s are <i>Z</i>s"</li></ul></dd></dl> <p>One is now free to "detach" the conclusion "No <i>X</i>s are <i>Z</i>s", perhaps to use it in a subsequent deduction (or as a topic of conversation). </p><p>The use of tautological implication means that other possible deductions exist besides "No <i>X</i>s are <i>Z</i>s"; the criterion for a successful deduction is that the 1s under the sub-major connective on the right <i>include</i> all the 1s under the sub-major connective on the left (the <i>major</i> connective being the implication that results in the tautology). For example, in the truth table, on the right side of the implication (→, the major connective symbol) the bold-face column under the sub-major connective symbol " <b>~</b> " has all the same 1s that appear in the bold-faced column under the left-side sub-major connective <b>&</b> (rows <b><a href="#math_0">0</a></b>, <b><a href="#math_1">1</a></b>, <b><a href="#math_2">2</a></b> and <b><a href="#math_6">6</a></b>), plus two more (rows <b><a href="#math_3">3</a></b> and <b><a href="#math_4">4</a></b>). </p> <div class="mw-heading mw-heading2"><h2 id="Gallery">Gallery</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler_diagram&action=edit&section=4" title="Edit section: Gallery"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="thumb tright" style=""><div class="thumbinner" style="width:352px"><div class="thumbimage noresize" style="width:350px;"> <figure class="mw-halign-none noresize mw-ext-imagemap-desc-top-right" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Supranational_European_Bodies.svg/350px-Supranational_European_Bodies.svg.png" decoding="async" width="350" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Supranational_European_Bodies.svg/525px-Supranational_European_Bodies.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Supranational_European_Bodies.svg/700px-Supranational_European_Bodies.svg.png 2x" data-file-width="1205" data-file-height="780" usemap="#ImageMap_d82a643fc8cd28a3" resource="/wiki/File:Supranational_European_Bodies-en.svg" /></span><map name="ImageMap_d82a643fc8cd28a3"><area href="/wiki/European_Political_Community" shape="rect" coords="17,6,126,15" alt="European Political Community" title="European Political Community" /><area href="/wiki/Schengen_Area" shape="rect" coords="13,22,67,29" alt="Schengen Area" title="Schengen Area" /><area href="/wiki/Council_of_Europe" shape="rect" coords="219,20,285,28" alt="Council of Europe" title="Council of Europe" /><area href="/wiki/European_Union" shape="rect" coords="45,55,103,64" alt="European Union" title="European Union" /><area href="/wiki/European_Economic_Area" shape="rect" coords="16,158,33,166" alt="European Economic Area" title="European Economic Area" /><area href="/wiki/Eurozone" shape="rect" coords="48,177,84,185" alt="Eurozone" title="Eurozone" /><area href="/wiki/European_Union_Customs_Union" shape="rect" coords="48,205,116,214" alt="European Union Customs Union" title="European Union Customs Union" /><area href="/wiki/European_Free_Trade_Association" shape="rect" coords="15,106,36,113" alt="European Free Trade Association" title="European Free Trade Association" /><area href="/wiki/Nordic_Council" shape="rect" coords="65,87,94,103" alt="Nordic Council" title="Nordic Council" /><area href="/wiki/Visegr%C3%A1d_Group" shape="rect" coords="103,87,138,105" alt="Visegrád Group" title="Visegrád Group" /><area href="/wiki/Baltic_Assembly" shape="rect" coords="55,125,113,134" alt="Baltic Assembly" title="Baltic Assembly" /><area href="/wiki/Benelux" shape="rect" coords="68,157,100,164" alt="Benelux" title="Benelux" /><area href="/wiki/GUAM_Organization_for_Democracy_and_Economic_Development" shape="rect" coords="231,42,256,51" alt="GUAM Organization for Democracy and Economic Development" title="GUAM Organization for Democracy and Economic Development" /><area href="/wiki/Central_European_Free_Trade_Agreement" shape="rect" coords="269,71,295,80" alt="Central European Free Trade Agreement" title="Central European Free Trade Agreement" /><area href="/wiki/Open_Balkan" shape="rect" coords="247,90,273,106" alt="Open Balkan" title="Open Balkan" /><area href="/wiki/Organization_of_the_Black_Sea_Economic_Cooperation" shape="rect" coords="323,36,346,44" alt="Organization of the Black Sea Economic Cooperation" title="Organization of the Black Sea Economic Cooperation" /><area href="/wiki/Union_State" shape="rect" coords="323,144,346,158" alt="Union State" title="Union State" /><area href="/wiki/Common_Travel_Area" shape="rect" coords="243,150,285,164" alt="Common Travel Area" title="Common Travel Area" /><area href="/wiki/International_status_and_usage_of_the_euro#Sovereign_states" shape="rect" coords="218,201,292,215" alt="International status and usage of the euro#Sovereign states" title="International status and usage of the euro#Sovereign states" /><area href="/wiki/Switzerland" shape="rect" coords="17,35,35,47" alt="Switzerland" title="Switzerland" /><area href="/wiki/Liechtenstein" shape="rect" coords="17,52,35,64" alt="Liechtenstein" title="Liechtenstein" /><area href="/wiki/Iceland" shape="rect" coords="17,73,35,84" alt="Iceland" title="Iceland" /><area href="/wiki/Norway" shape="rect" coords="17,90,35,102" alt="Norway" title="Norway" /><area href="/wiki/Sweden" shape="rect" coords="47,73,62,84" alt="Sweden" title="Sweden" /><area href="/wiki/Denmark" shape="rect" coords="71,73,87,84" alt="Denmark" title="Denmark" /><area href="/wiki/Finland" shape="rect" coords="47,90,62,102" alt="Finland" title="Finland" /><area href="/wiki/Poland" shape="rect" coords="103,73,121,84" alt="Poland" title="Poland" /><area href="/wiki/Czech_Republic" shape="rect" coords="122,73,140,84" alt="Czech Republic" title="Czech Republic" /><area href="/wiki/Hungary" shape="rect" coords="141,73,158,84" alt="Hungary" title="Hungary" /><area href="/wiki/Slovakia" shape="rect" coords="141,90,158,102" alt="Slovakia" title="Slovakia" /><area href="/wiki/Bulgaria" shape="rect" coords="169,55,186,67" alt="Bulgaria" title="Bulgaria" /><area href="/wiki/Romania" shape="rect" coords="169,73,186,84" alt="Romania" title="Romania" /><area href="/wiki/Greece" shape="rect" coords="169,110,186,122" alt="Greece" title="Greece" /><area href="/wiki/Estonia" shape="rect" coords="57,112,74,124" alt="Estonia" title="Estonia" /><area href="/wiki/Latvia" shape="rect" coords="76,112,93,124" alt="Latvia" title="Latvia" /><area href="/wiki/Lithuania" shape="rect" coords="96,112,112,124" alt="Lithuania" title="Lithuania" /><area href="/wiki/Belgium" shape="rect" coords="57,142,74,154" alt="Belgium" title="Belgium" /><area href="/wiki/Netherlands" shape="rect" coords="76,142,93,154" alt="Netherlands" title="Netherlands" /><area href="/wiki/Luxembourg" shape="rect" coords="96,142,112,154" alt="Luxembourg" title="Luxembourg" /><area href="/wiki/Italy" shape="rect" coords="124,113,141,126" alt="Italy" title="Italy" /><area href="/wiki/France" shape="rect" coords="124,131,141,142" alt="France" title="France" /><area href="/wiki/Spain" shape="rect" coords="124,147,141,158" alt="Spain" title="Spain" /><area href="/wiki/Austria" shape="rect" coords="145,113,163,126" alt="Austria" title="Austria" /><area href="/wiki/Germany" shape="rect" coords="145,131,163,142" alt="Germany" title="Germany" /><area href="/wiki/Portugal" shape="rect" coords="145,147,163,158" alt="Portugal" title="Portugal" /><area href="/wiki/Slovenia" shape="rect" coords="167,131,185,142" alt="Slovenia" title="Slovenia" /><area href="/wiki/Malta" shape="rect" coords="167,147,185,158" alt="Malta" title="Malta" /><area href="/wiki/Croatia" shape="rect" coords="145,172,163,183" alt="Croatia" title="Croatia" /><area href="/wiki/Cyprus" shape="rect" coords="192,131,209,142" alt="Cyprus" title="Cyprus" /><area href="/wiki/Republic_of_Ireland" shape="rect" coords="193,151,211,163" alt="Republic of Ireland" title="Republic of Ireland" /><area href="/wiki/United_Kingdom" shape="rect" coords="221,151,238,163" alt="United Kingdom" title="United Kingdom" /><area href="/wiki/Turkey" shape="rect" coords="192,35,209,47" alt="Turkey" title="Turkey" /><area href="/wiki/Monaco" shape="rect" coords="156,202,172,214" alt="Monaco" title="Monaco" /><area href="/wiki/Andorra" shape="rect" coords="174,202,192,214" alt="Andorra" title="Andorra" /><area href="/wiki/San_Marino" shape="rect" coords="193,202,211,214" alt="San Marino" title="San Marino" /><area href="/wiki/Vatican_City" shape="rect" coords="326,202,343,214" alt="Vatican City" title="Vatican City" /><area href="/wiki/Georgia_(country)" shape="rect" coords="224,54,240,65" alt="Georgia (country)" title="Georgia (country)" /><area href="/wiki/Ukraine" shape="rect" coords="224,71,240,84" alt="Ukraine" title="Ukraine" /><area href="/wiki/Azerbaijan" shape="rect" coords="246,54,263,65" alt="Azerbaijan" title="Azerbaijan" /><area href="/wiki/Moldova" shape="rect" coords="246,71,263,84" alt="Moldova" title="Moldova" /><area href="/wiki/Bosnia_and_Herzegovina" shape="rect" coords="250,129,267,141" alt="Bosnia and Herzegovina" title="Bosnia and Herzegovina" /><area href="/wiki/Armenia" shape="rect" coords="273,54,291,65" alt="Armenia" title="Armenia" /><area href="/wiki/Montenegro" shape="rect" coords="273,129,291,141" alt="Montenegro" title="Montenegro" /><area href="/wiki/North_Macedonia" shape="rect" coords="273,108,291,119" alt="North Macedonia" title="North Macedonia" /><area href="/wiki/Albania" shape="rect" coords="273,92,291,103" alt="Albania" title="Albania" /><area href="/wiki/Serbia" shape="rect" coords="250,108,267,119" alt="Serbia" title="Serbia" /><area href="/wiki/Kosovo" shape="rect" coords="299,129,317,141" alt="Kosovo" title="Kosovo" /><area href="/wiki/Russia" shape="rect" coords="326,110,343,122" alt="Russia" title="Russia" /><area href="/wiki/Belarus" shape="rect" coords="326,129,343,141" alt="Belarus" title="Belarus" /></map><figcaption></figcaption></figure></div><div class="thumbcaption">An <a class="mw-selflink selflink">Euler diagram</a> showing the relationships between various multinational European organisations and agreements <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 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li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><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" style="float:right"><ul><li class="nv-view"><a href="/wiki/Template:Supranational_European_Bodies" title="Template:Supranational European Bodies"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Supranational_European_Bodies" title="Template talk:Supranational European Bodies"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Supranational_European_Bodies" title="Special:EditPage/Template:Supranational European Bodies"><abbr title="Edit this template">e</abbr></a></li></ul></div></div></div></div> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:VennDiagram.svg" class="mw-file-description" title="A Venn diagram showing all possible intersections"><img alt="A Venn diagram showing all possible intersections" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/VennDiagram.svg/120px-VennDiagram.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/VennDiagram.svg/180px-VennDiagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/VennDiagram.svg/240px-VennDiagram.svg.png 2x" data-file-width="304" data-file-height="296" /></a></span></div> <div class="gallerytext">A <a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a> showing all possible intersections</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Supranational_European_Bodies.svg" class="mw-file-description" title="Euler diagram visualizing a real situation, the relationships between various supranational European organizations (clickable version)"><img alt="Euler diagram visualizing a real situation, the relationships between various supranational European organizations (clickable version)" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Supranational_European_Bodies.svg/120px-Supranational_European_Bodies.svg.png" decoding="async" width="120" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Supranational_European_Bodies.svg/180px-Supranational_European_Bodies.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Supranational_European_Bodies.svg/240px-Supranational_European_Bodies.svg.png 2x" data-file-width="1205" data-file-height="780" /></a></span></div> <div class="gallerytext">Euler diagram visualizing a real situation, the relationships between various <a href="/wiki/International_organizations_in_Europe" class="mw-redirect" title="International organizations in Europe">supranational European organizations</a> (<a href="/wiki/Template:Supranational_European_Bodies" title="Template:Supranational European Bodies">clickable version</a>)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Euler_and_Venn_diagrams.svg" class="mw-file-description" title="Humorous diagram comparing Euler and Venn diagrams"><img alt="Humorous diagram comparing Euler and Venn diagrams" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Euler_and_Venn_diagrams.svg/120px-Euler_and_Venn_diagrams.svg.png" decoding="async" width="120" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Euler_and_Venn_diagrams.svg/180px-Euler_and_Venn_diagrams.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Euler_and_Venn_diagrams.svg/240px-Euler_and_Venn_diagrams.svg.png 2x" data-file-width="660" data-file-height="600" /></a></span></div> <div class="gallerytext">Humorous diagram comparing Euler and <a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagrams</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Euler_diagram_of_triangle_types.svg" class="mw-file-description" title="Euler diagram of types of triangles, using the definition that isosceles triangles have at least (rather than exactly) 2 equal sides"><img alt="Euler diagram of types of triangles, using the definition that isosceles triangles have at least (rather than exactly) 2 equal sides" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Euler_diagram_of_triangle_types.svg/120px-Euler_diagram_of_triangle_types.svg.png" decoding="async" width="120" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Euler_diagram_of_triangle_types.svg/180px-Euler_diagram_of_triangle_types.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Euler_diagram_of_triangle_types.svg/240px-Euler_diagram_of_triangle_types.svg.png 2x" data-file-width="512" data-file-height="205" /></a></span></div> <div class="gallerytext">Euler diagram of types of <a href="/wiki/Triangle" title="Triangle">triangles</a>, using the definition that isosceles triangles have at least (rather than exactly) 2 equal sides</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:British_Isles_Euler_diagram_15.svg" class="mw-file-description" title="Euler diagram of terminology of the British Isles"><img alt="Euler diagram of terminology of the British Isles" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/British_Isles_Euler_diagram_15.svg/120px-British_Isles_Euler_diagram_15.svg.png" decoding="async" width="120" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/British_Isles_Euler_diagram_15.svg/180px-British_Isles_Euler_diagram_15.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/British_Isles_Euler_diagram_15.svg/240px-British_Isles_Euler_diagram_15.svg.png 2x" data-file-width="512" data-file-height="416" /></a></span></div> <div class="gallerytext">Euler diagram of terminology of the <a href="/wiki/British_Isles" title="British Isles">British Isles</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:An_Euler_diagram_of_Eulerian_circles_Showing_Different_Types_of_Metaheuristics.jpg" class="mw-file-description" title="Euler diagram categorizing different types of metaheuristics"><img alt="Euler diagram categorizing different types of metaheuristics" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/An_Euler_diagram_of_Eulerian_circles_Showing_Different_Types_of_Metaheuristics.jpg/120px-An_Euler_diagram_of_Eulerian_circles_Showing_Different_Types_of_Metaheuristics.jpg" decoding="async" width="120" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/An_Euler_diagram_of_Eulerian_circles_Showing_Different_Types_of_Metaheuristics.jpg/180px-An_Euler_diagram_of_Eulerian_circles_Showing_Different_Types_of_Metaheuristics.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/An_Euler_diagram_of_Eulerian_circles_Showing_Different_Types_of_Metaheuristics.jpg/240px-An_Euler_diagram_of_Eulerian_circles_Showing_Different_Types_of_Metaheuristics.jpg 2x" data-file-width="1270" data-file-height="921" /></a></span></div> <div class="gallerytext">Euler diagram categorizing different types of <a href="/wiki/Metaheuristic" title="Metaheuristic">metaheuristics</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Homograph_homophone_venn_diagram.svg" class="mw-file-description" title="Euler Diagram displaying the relationship between Homographs, homophones, and synonyms"><img alt="Euler Diagram displaying the relationship between Homographs, homophones, and synonyms" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Homograph_homophone_venn_diagram.svg/107px-Homograph_homophone_venn_diagram.svg.png" decoding="async" width="107" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Homograph_homophone_venn_diagram.svg/160px-Homograph_homophone_venn_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Homograph_homophone_venn_diagram.svg/213px-Homograph_homophone_venn_diagram.svg.png 2x" data-file-width="512" data-file-height="576" /></a></span></div> <div class="gallerytext">Euler Diagram displaying the relationship between Homographs, homophones, and synonyms</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Venn_and_Euler_diagrams_of_3-ary_Boolean_relations.svg" class="mw-file-description" title="The 22 (of 256) essentially different Venn diagrams with 3 circles (top) and their corresponding Euler diagrams.(bottom) Some of the Euler diagrams are not typical; some are even equivalent to Venn diagrams. Areas are shaded to indicate that they contain no elements."><img alt="The 22 (of 256) essentially different Venn diagrams with 3 circles (top) and their corresponding Euler diagrams.(bottom) Some of the Euler diagrams are not typical; some are even equivalent to Venn diagrams. Areas are shaded to indicate that they contain no elements." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Venn_and_Euler_diagrams_of_3-ary_Boolean_relations.svg/118px-Venn_and_Euler_diagrams_of_3-ary_Boolean_relations.svg.png" decoding="async" width="118" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Venn_and_Euler_diagrams_of_3-ary_Boolean_relations.svg/176px-Venn_and_Euler_diagrams_of_3-ary_Boolean_relations.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Venn_and_Euler_diagrams_of_3-ary_Boolean_relations.svg/235px-Venn_and_Euler_diagrams_of_3-ary_Boolean_relations.svg.png 2x" data-file-width="2253" data-file-height="2299" /></a></span></div> <div class="gallerytext">The 22 (of 256) essentially different Venn diagrams with 3 circles <i>(top)</i> and their corresponding Euler diagrams.<i>(bottom)</i><small><br />Some of the Euler diagrams are not typical; some are even equivalent to Venn diagrams. Areas are shaded to indicate that they contain no elements.</small></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Milne-Edwards_diagram.pdf" class="mw-file-description" title="Henri Milne - Edwards's (1844) diagram of relationships of vertebrate animals, illustrated as a series of nested sets"><img alt="Henri Milne - Edwards's (1844) diagram of relationships of vertebrate animals, illustrated as a series of nested sets" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Milne-Edwards_diagram.pdf/page1-120px-Milne-Edwards_diagram.pdf.jpg" decoding="async" width="120" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Milne-Edwards_diagram.pdf/page1-180px-Milne-Edwards_diagram.pdf.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Milne-Edwards_diagram.pdf/page1-240px-Milne-Edwards_diagram.pdf.jpg 2x" data-file-width="1650" data-file-height="1275" /></a></span></div> <div class="gallerytext">Henri Milne - Edwards's (1844) diagram of relationships of vertebrate animals, illustrated as a series of nested sets</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Euler_diagram_numbers_with_many_divisors.svg" class="mw-file-description" title="Euler diagram of numbers under 100"><img alt="Euler diagram of numbers under 100" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Euler_diagram_numbers_with_many_divisors.svg/120px-Euler_diagram_numbers_with_many_divisors.svg.png" decoding="async" width="120" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Euler_diagram_numbers_with_many_divisors.svg/180px-Euler_diagram_numbers_with_many_divisors.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Euler_diagram_numbers_with_many_divisors.svg/240px-Euler_diagram_numbers_with_many_divisors.svg.png 2x" data-file-width="512" data-file-height="410" /></a></span></div> <div class="gallerytext">Euler diagram of numbers under 100</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler_diagram&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Intersectionality" title="Intersectionality">Intersectionality</a></li> <li><a href="/wiki/Rainbow_box" title="Rainbow box">Rainbow box</a></li> <li><a href="/wiki/Spider_diagram" title="Spider diagram">Spider diagram</a> – an extension of Euler diagrams adding existence to contour intersections</li> <li><a href="/wiki/Three_circles_model" title="Three circles model">Three circles model</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=Euler_diagram&action=edit&section=6" 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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Mansel_Veitch_1860_credit_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mansel_Veitch_1860_credit_note_8-0">^</a></b></span> <span class="reference-text"> By the time these lectures of Hamilton were published, Hamilton had died. His editors (marked by <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">ED</span></span>.), responsible for most of the footnote text, were the logicians <a href="/wiki/Henry_Longueville_Mansel" title="Henry Longueville Mansel">Henry Longueville Mansel</a> and <a href="/wiki/John_Veitch_(poet)" title="John Veitch (poet)">John Veitch</a>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"> <a href="#CITEREFSandifer2004">Sandifer (2004)</a> points out that <a href="/wiki/Leonhard_Paul_Euler" class="mw-redirect" title="Leonhard Paul Euler">Euler</a> himself also makes such observations: Euler reports that his figure 45 (a simple intersection of two circles) has 4 different interpretations.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"> See footnote in <a href="/wiki/George_Stibitz" title="George Stibitz">George Stibitz</a> article.</span> </li> </ol></div></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">This is a sophisticated concept. Russell and Whitehead (2nd edition 1927) in their <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> describe it this way: "The trust in inference is the belief that if the two former assertions [the premises P, P→Q ] are not in error, the final assertion is not in error . . . An inference is the dropping of a true premiss [sic]; it is the dissolution of an implication" (p. 9). Further discussion of this appears in "Primitive Ideas and Propositions" as the first of their "primitive propositions" (axioms): *1.1 Anything implied by a true elementary proposition is true" (p. 94). In a footnote the authors refer the reader back to Russell's 1903 <i>Principles of Mathematics</i> §38.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Reichenbach discusses the fact that the implication <i>P</i> → <i>Q</i> need not be a tautology (a so-called "tautological implication"). Even "simple" implication (connective or adjunctive) work, but only for those rows of the truth table that evaluate as true, cf Reichenbach 1947:64–66.</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=Euler_diagram&action=edit&section=7" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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/20090429093334/http://readingquest.org/strat/venn.html">"Strategies for Reading Comprehension Venn Diagrams"</a>. 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%3AEuler+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Venn_1881-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Venn_1881_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Venn_1881_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite 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/symboliclogic00vennuoft"><i>Symbolic Logic</i></a>. London: <a href="/wiki/MacMillan_and_Co." class="mw-redirect" title="MacMillan and Co.">MacMillan and Co.</a> p. 509.</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.place=London&rft.pages=509&rft.pub=MacMillan+and+Co.&rft.date=1881&rft.aulast=Venn&rft.aufirst=John&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsymboliclogic00vennuoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Gailand_1967-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Gailand_1967_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Gailand_1967_3-1"><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="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>. p. 5. 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.pages=5&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%3AEuler+diagram" class="Z3988"></span> (NB. Has a detailed history of the evolution of logic diagrams including but not limited to the Euler diagram.)</span> </li> <li id="cite_note-Courant-1914-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Courant-1914_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Courant-1914_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Courant-1914_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Courant-1914_4-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Courant-1914_4-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Courant-1914_4-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFCouturat1914">Couturat (1914)</a>, pp.  73, 75</span> </li> <li id="cite_note-Hamilton-1858-1860-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hamilton-1858-1860_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamilton1858–1860" class="citation book cs1"><a href="/wiki/Sir_William_Hamilton,_9th_Baronet" title="Sir William Hamilton, 9th Baronet">Hamilton, W.R.</a> (1858–1860). <i>Lectures on Metaphysics and Logic</i>. p. 180.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Metaphysics+and+Logic&rft.pages=180&rft.date=1858%2F1860&rft.aulast=Hamilton&rft.aufirst=W.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeise1712" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Christian_Weise" title="Christian Weise">Weise, C.</a> (1712). <i>Nucleus Logicae Weisianae</i> [<i>Weissian core of logic</i>] (in Latin).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nucleus+Logicae+Weisianae&rft.date=1712&rft.aulast=Weise&rft.aufirst=C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span> — Published 4 years after Weise's death.</span> </li> <li id="cite_note-Euler-1791-1842-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Euler-1791-1842_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Euler-1791-1842_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="CITEREFEuler1842" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Leonhard_Paul_Euler" class="mw-redirect" title="Leonhard Paul Euler">Euler, L.P.</a> (1842) [17 February 1791]. "Partie II, Lettre XXXV". In <a href="/wiki/Antoine_Augustin_Cournot" title="Antoine Augustin Cournot">Cournot</a> (ed.). <a href="/wiki/Letters_to_a_German_Princess" title="Letters to a German Princess"><i>Lettres a une Princesse d'Allemagne</i></a> [<i>Letters to a German Princess</i>] (in French). pp. 412–417.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Partie+II%2C+Lettre+XXXV&rft.btitle=Lettres+a+une+Princesse+d%27Allemagne&rft.pages=412-417&rft.date=1842&rft.aulast=Euler&rft.aufirst=L.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Hamilton-1860-Jevons-1881-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hamilton-1860-Jevons-1881_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hamilton-1860-Jevons-1881_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFHamilton1860">Hamilton (1860)</a>, p. 179; these examples are from <a href="#CITEREFJevons1880">Jevons (1880)</a>, pp.  71 ff.</span> </li> <li id="cite_note-Venn-1881a-§V-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Venn-1881a-§V_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Venn-1881a-§V_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Venn-1881a-§V_10-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Venn-1881a-§V_10-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Venn-1881a-§V_10-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Venn-1881a-§V_10-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Venn-1881a-§V_10-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Venn-1881a-§V_10-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-Venn-1881a-§V_10-8"><sup><i><b>i</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVenn1881a" class="citation book cs1"><a href="/wiki/John_Venn" title="John Venn">Venn, J.</a> (1881a). "Chapter V – Diagrammatic representation". <i>Symbolic Logic</i>. p. 100, Footnote 1.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+V+%E2%80%93+Diagrammatic+representation&rft.btitle=Symbolic+Logic&rft.pages=100%2C+Footnote+1&rft.date=1881&rft.aulast=Venn&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">cf <a href="#CITEREFSandifer2004">Sandifer (2004)</a> <a href="#CITEREFVenn1881a">Venn (1881a)</a>, pp.  114 ff;<sup id="cite_ref-Venn-1881a-§V_10-1" class="reference"><a href="#cite_note-Venn-1881a-§V-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> in the "Eulerian scheme" <a href="#CITEREFVenn1881a">Venn (1881a)</a>, p.  100<sup id="cite_ref-Venn-1881a-§V_10-2" class="reference"><a href="#cite_note-Venn-1881a-§V-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> of "old-fashioned Eulerian diagrams" <a href="#CITEREFVenn1881a">Venn (1881a)</a>, p.  113<sup id="cite_ref-Venn-1881a-§V_10-3" class="reference"><a href="#cite_note-Venn-1881a-§V-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-Venn-1881b-§XX-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Venn-1881b-§XX_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVenn1881b" class="citation book cs1"><a href="/wiki/John_Venn" title="John Venn">Venn, J.</a> (1881b). "Chapter XX – Historic notes". <i>Symbolic Logic</i>. p. 424.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+XX+%E2%80%93+Historic+notes&rft.btitle=Symbolic+Logic&rft.pages=424&rft.date=1881&rft.aulast=Venn&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span>}</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPost1921" class="citation thesis cs1"><a href="/wiki/Emil_Post" class="mw-redirect" title="Emil Post">Post, E.</a> (1921). <i>Introduction to a general theory of elementary propositions</i> (Ph.D. thesis).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Introduction+to+a+general+theory+of+elementary+propositions&rft.degree=Ph.D.&rft.date=1921&rft.aulast=Post&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span></span> </li> <li id="cite_note-Hill-Peterson-1964-1968-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hill-Peterson-1964-1968_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hill-Peterson-1964-1968_16-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Hill-Peterson-1964-1968_16-2"><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="CITEREFHillPeterson1968" class="citation book cs1">Hill & Peterson (1968) [1964]. "Set theory as an example of Boolean algebra". <i>Boolean Algebra</i>. <span class="nowrap">sections 4.5 ff</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Set+theory+as+an+example+of+Boolean+algebra&rft.btitle=Boolean+Algebra&rft.pages=%3Cspan+class%3D%22nowrap%22%3Esections+4.5+ff%3C%2Fspan%3E&rft.date=1968&rft.au=Hill&rft.au=Peterson&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShannon1938" class="citation report cs1"><a href="/wiki/Claude_E._Shannon" class="mw-redirect" title="Claude E. Shannon">Shannon, C.E.</a> (1938). <span style="color:gray">[no title cited]: In effect, Shannon's master's thesis</span> (Report). <a href="/wiki/M.I.T." class="mw-redirect" title="M.I.T.">M.I.T.</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=%3Cspan+style%3D%22color%3Agray%22%3E%5Bno+title+cited%5D%3A+In+effect%2C+Shannon%27s+master%27s+thesis%3C%2Fspan%3E&rft.pub=M.I.T.&rft.date=1938&rft.aulast=Shannon&rft.aufirst=C.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+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">cf Reichenbach 1947:64</span> </li> </ol></div></div> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler_diagram&action=edit&section=8" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCouturat1914" class="citation book cs1"><a href="/wiki/Louis_Couturat" title="Louis Couturat">Couturat, Louis</a> (1914). <i>The Algebra of Logic: Authorized English Translation by Lydia Gillingham Robinson with a Preface by Philip E. B. Jourdain</i>. Chicago and London: <a href="/wiki/The_Open_Court_Publishing_Company" class="mw-redirect" title="The Open Court Publishing Company">The Open Court Publishing Company</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Algebra+of+Logic%3A+Authorized+English+Translation+by+Lydia+Gillingham+Robinson+with+a+Preface+by+Philip+E.+B.+Jourdain&rft.place=Chicago+and+London&rft.pub=The+Open+Court+Publishing+Company&rft.date=1914&rft.aulast=Couturat&rft.aufirst=Louis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamilton1860" class="citation book cs1"><a href="/wiki/Sir_William_Hamilton,_9th_Baronet" title="Sir William Hamilton, 9th Baronet">Sir William Hamilton</a> (1860). <a href="/wiki/Henry_Longueville_Mansel" title="Henry Longueville Mansel">Mansel, Henry Longueville</a>; <a href="/wiki/John_Veitch_(poet)" title="John Veitch (poet)">Veitch, John</a> (eds.). <i>Lectures on Metaphysics and Logic</i>. Edinburgh and London: <a href="/wiki/William_Blackwood_and_Sons" class="mw-redirect" title="William Blackwood and Sons">William Blackwood and Sons</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Metaphysics+and+Logic&rft.place=Edinburgh+and+London&rft.pub=William+Blackwood+and+Sons&rft.date=1860&rft.au=Sir+William+Hamilton&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJevons1880" class="citation book cs1"><a href="/wiki/W._Stanley_Jevons" class="mw-redirect" title="W. Stanley Jevons">Jevons, W. Stanley</a> (1880). <i>Elementary Lessons in Logic: Deductive and Inductive. With Copious Questions and Examples, and a Vocabulary of Logical Terms</i>. London and New York: <a href="/wiki/M._A._MacMillan_and_Co." class="mw-redirect" title="M. A. MacMillan and Co.">M. A. MacMillan and Co.</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Lessons+in+Logic%3A+Deductive+and+Inductive.+With+Copious+Questions+and+Examples%2C+and+a+Vocabulary+of+Logical+Terms&rft.place=London+and+New+York&rft.pub=M.+A.+MacMillan+and+Co.&rft.date=1880&rft.aulast=Jevons&rft.aufirst=W.+Stanley&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarnaugh1953" class="citation journal cs1"><a href="/wiki/Maurice_Karnaugh" title="Maurice Karnaugh">Karnaugh, Maurice</a> (November 1953) [1953-04-23, 1953-03-17]. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170416232229/http://philectrosophy.com/documents/The%20Map%20Method%20For%20Synthesis%20of.pdf">"The Map Method for Synthesis of Combinational Logic Circuits"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Transactions_of_the_American_Institute_of_Electrical_Engineers,_Part_I:_Communication_and_Electronics" class="mw-redirect" title="Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics">Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics</a></i>. <b>72</b> (5): 593–599. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTCE.1953.6371932">10.1109/TCE.1953.6371932</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:51636736">51636736</a>. Paper 53-217. Archived from <a rel="nofollow" class="external text" href="http://philectrosophy.com/documents/The%20Map%20Method%20For%20Synthesis%20of.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-04-16<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-04-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Institute+of+Electrical+Engineers%2C+Part+I%3A+Communication+and+Electronics&rft.atitle=The+Map+Method+for+Synthesis+of+Combinational+Logic+Circuits&rft.volume=72&rft.issue=5&rft.pages=593-599&rft.date=1953-11&rft_id=info%3Adoi%2F10.1109%2FTCE.1953.6371932&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A51636736%23id-name%3DS2CID&rft.aulast=Karnaugh&rft.aufirst=Maurice&rft_id=http%3A%2F%2Fphilectrosophy.com%2Fdocuments%2FThe%2520Map%2520Method%2520For%2520Synthesis%2520of.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSandifer2004" class="citation web cs1">Sandifer, Ed (January 2004). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130126165922/http://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.org</i>. Archived from <a rel="nofollow" class="external text" href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2003%20Venn%20Diagrams.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2013-01-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=maa.org&rft.atitle=How+Euler+Did+It&rft.date=2004-01&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%3AEuler+diagram" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVeitch1952" class="citation journal cs1"><a href="/wiki/Edward_W._Veitch" title="Edward W. Veitch">Veitch, Edward Westbrook</a> (1952-05-03) [1952-05-02]. "A Chart Method for Simplifying Truth Functions". <i>Transactions of the 1952 ACM Annual Meeting</i>. ACM Annual Conference/Annual Meeting: Proceedings of the 1952 ACM Annual Meeting (Pittsburgh, Pennsylvania, USA). New York, USA: <a href="/wiki/Association_for_Computing_Machinery" title="Association for Computing Machinery">Association for Computing Machinery</a> (ACM): 127–133. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F609784.609801">10.1145/609784.609801</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:17284651">17284651</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+1952+ACM+Annual+Meeting&rft.atitle=A+Chart+Method+for+Simplifying+Truth+Functions&rft.pages=127-133&rft.date=1952-05-03&rft_id=info%3Adoi%2F10.1145%2F609784.609801&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17284651%23id-name%3DS2CID&rft.aulast=Veitch&rft.aufirst=Edward+Westbrook&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuler+diagram" class="Z3988"></span></li></ul> <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=Euler_diagram&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By date of publishing: </p> <ul><li><a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> and <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> 1913 1st edition, 1927 2nd edition <i>Principia Mathematica to *56</i> Cambridge At The <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">University Press</a> (1962 edition), UK, no ISBN.</li> <li><a href="/wiki/Emil_Post" class="mw-redirect" title="Emil Post">Emil Post</a> 1921 "Introduction to a general theory of elementary propositions" reprinted with commentary by <a href="/wiki/Jean_van_Heijenoort" title="Jean van Heijenoort">Jean van Heijenoort</a> in Jean van Heijenoort, editor 1967 <i>From Frege to Gödel: A Source Book of Mathematical Logic, 1879–1931</i>, <a href="/wiki/Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>, Cambridge, MA, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-674-32449-8" title="Special:BookSources/0-674-32449-8">0-674-32449-8</a> (pbk.)</li> <li><a href="/wiki/Claude_E._Shannon" class="mw-redirect" title="Claude E. Shannon">Claude E. Shannon</a> 1938 "A Symbolic Analysis of Relay and Switching Circuits", <i>Transactions American Institute of Electrical Engineers</i> vol 57, pp. 471–495. Derived from <i>Claude Elwood Shannon: Collected Papers</i> edited by N.J.A. Solane and Aaron D. Wyner, <a href="/wiki/IEEE_Press" class="mw-redirect" title="IEEE Press">IEEE Press</a>, New York.</li> <li><a href="/wiki/Hans_Reichenbach" title="Hans Reichenbach">Hans Reichenbach</a> 1947 <i>Elements of Symbolic Logic</i> republished 1980 by <a href="/wiki/Dover_Publications,_Inc." class="mw-redirect" title="Dover Publications, Inc.">Dover Publications, Inc.</a>, NY, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-24004-5" title="Special:BookSources/0-486-24004-5">0-486-24004-5</a>.</li> <li>Frederich J. Hill and Gerald R. Peterson 1968, 1974 <i>Introduction to Switching Theory and Logical Design</i>, <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, NY, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-39882-0" title="Special:BookSources/978-0-471-39882-0">978-0-471-39882-0</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euler_diagram&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Euler_diagrams" class="extiw" title="commons:Category:Euler diagrams">Euler diagrams</a></span>.</div></div> </div> <ul><li>Euler Diagrams. 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