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</a> </div> <form action="/w/index.php" method="get" class="minerva-search-form"> <div class="search-box"> <input type="hidden" name="title" value="Special:Search"/> <input class="search skin-minerva-search-trigger" id="searchInput" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f"> </div> <button id="searchIcon" class="cdx-button cdx-button--size-large cdx-button--icon-only cdx-button--weight-quiet skin-minerva-search-trigger"> Search </button> </form> <nav class="minerva-user-navigation" aria-label="User navigation"> </nav> </div> </header> <main id="content" class="mw-body"> <div class="banner-container"> <div id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading">Logical conjunction</h1> <div class="tagline"></div> </div> <ul id="p-associated-pages" 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class="mf-section-0" id="mf-section-0"> <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">Not to be confused with <a href="/wiki/Caret" title="Caret">Circumflex Agent (^)</a>, <a href="/wiki/Lambda" title="Lambda">Capital Lambda (Λ)</a>, <a href="/wiki/Turned_v" title="Turned v">Turned V (Λ)</a>, or <a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior Product (∧)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style>In <a href="/wiki/Logic" title="Logic">logic</a>, <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Linguistics" title="Linguistics">linguistics</a>, and (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }">) is the <a href="/wiki/Truth_function" title="Truth function">truth-functional</a> operator of conjunction or logical conjunction. The <a href="/wiki/Logical_connective" title="Logical connective">logical connective</a> of this operator is typically represented as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"><a href="#cite_note-:2-1">[1]</a> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \&}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \&}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7e8142574f96000e13827167bdfdd69e38076f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \&}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> (prefix) or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"><a href="#cite_note-:1-2">[2]</a> in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"> is the most modern and widely used. <table class="infobox"><caption class="infobox-title" style="background:navy; color:white;">Logical conjunction</caption><tbody><tr><th colspan="2" class="infobox-above">AND</th></tr><tr><td colspan="2" class="infobox-image"><a href="/wiki/File:Venn0001.svg" class="mw-file-description" title="Venn diagram of Logical conjunction"><img alt="Venn diagram of Logical conjunction" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/150px-Venn0001.svg.png" decoding="async" width="150" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/225px-Venn0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/300px-Venn0001.svg.png 2x" data-file-width="410" data-file-height="299"></a></td></tr><tr><th scope="row" class="infobox-label">Definition</th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Truth_table" title="Truth table">Truth table</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1000)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1000</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1000)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecf7c4e0fa1a9b5eb52422144ee5ae0f79d039fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.459ex; height:2.843ex;" alt="{\displaystyle (1000)}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Logic_gate" title="Logic gate">Logic gate</a></th><td class="infobox-data"><a href="/wiki/File:AND_ANSI.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/AND_ANSI.svg/70px-AND_ANSI.svg.png" decoding="async" width="70" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/AND_ANSI.svg/105px-AND_ANSI.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/AND_ANSI.svg/140px-AND_ANSI.svg.png 2x" data-file-width="100" data-file-height="50"></a></td></tr><tr><th colspan="2" class="infobox-header" style="background:navy; color:white;">Normal forms</th></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Disjunctive_normal_form" title="Disjunctive normal form">Disjunctive</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Conjunctive_normal_form" title="Conjunctive normal form">Conjunctive</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Zhegalkin_polynomial" title="Zhegalkin polynomial">Zhegalkin polynomial</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></td></tr><tr><th colspan="2" class="infobox-header" style="background:navy; color:white;"><a href="/wiki/Post%27s_lattice" title="Post's lattice">Post's lattices</a></th></tr><tr><th scope="row" class="infobox-label">0-preserving</th><td class="infobox-data">yes</td></tr><tr><th scope="row" class="infobox-label">1-preserving</th><td class="infobox-data">yes</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Monotonic_function" title="Monotonic function">Monotone</a></th><td class="infobox-data">no</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Affine_transformation" title="Affine transformation">Affine</a></th><td class="infobox-data">no</td></tr><tr><th scope="row" class="infobox-label">Self-dual</th><td class="infobox-data">no</td></tr><tr><td colspan="2" class="infobox-navbar"><style 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.sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Venn_0000_0001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Venn_0000_0001.svg/220px-Venn_0000_0001.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Venn_0000_0001.svg/330px-Venn_0000_0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Venn_0000_0001.svg/440px-Venn_0000_0001.svg.png 2x" data-file-width="200" data-file-height="200"></a><figcaption><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\wedge B\land C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo>∧</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\wedge B\land C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3565b79459dcf8d4555f23660a1811669f1131c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.439ex; height:2.176ex;" alt="{\displaystyle A\wedge B\land C}"></figcaption></figure> The and of a set of operands is true if and only if all of its operands are true, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"> is true if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is true and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is true. An operand of a conjunction is a conjunct.<a href="#cite_note-:21-3">[3]</a> Beyond logic, the term "conjunction" also refers to similar concepts in other fields: <ul><li>In <a href="/wiki/Natural_language" title="Natural language">natural language</a>, the <a href="/wiki/Denotation" title="Denotation">denotation</a> of expressions such as <a href="/wiki/English_language" title="English language">English</a> "<a href="/wiki/Conjunction_(grammar)" title="Conjunction (grammar)">and</a>";</li> <li>In <a href="/wiki/Programming_language" title="Programming language">programming languages</a>, the <a href="/wiki/Short-circuit_evaluation" title="Short-circuit evaluation">short-circuit and</a> <a href="/wiki/Control_flow" title="Control flow">control structure</a>;</li> <li>In <a href="/wiki/Set_theory" title="Set theory">set theory</a>, <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a>.</li> <li>In <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice theory</a>, logical conjunction (<a href="/wiki/Infimum_and_supremum" title="Infimum and supremum">greatest lower bound</a>).</li></ul> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Notation">1 Notation</a></li> <li class="toclevel-1 tocsection-2"><a href="#Definition">2 Definition</a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Truth_table">2.1 Truth table</a></li> <li class="toclevel-2 tocsection-4"><a href="#Defined_by_other_operators">2.2 Defined by other operators</a></li> </ul> </li> <li class="toclevel-1 tocsection-5"><a href="#Introduction_and_elimination_rules">3 Introduction and elimination rules</a></li> <li class="toclevel-1 tocsection-6"><a href="#Negation">4 Negation</a> <ul> <li class="toclevel-2 tocsection-7"><a href="#Definition_2">4.1 Definition</a></li> <li class="toclevel-2 tocsection-8"><a href="#Other_proof_strategies">4.2 Other proof strategies</a></li> </ul> </li> <li class="toclevel-1 tocsection-9"><a href="#Properties">5 Properties</a></li> <li class="toclevel-1 tocsection-10"><a href="#Applications_in_computer_engineering">6 Applications in computer engineering</a></li> <li class="toclevel-1 tocsection-11"><a href="#Set-theoretic_correspondence">7 Set-theoretic correspondence</a></li> <li class="toclevel-1 tocsection-12"><a href="#Natural_language">8 Natural language</a></li> <li class="toclevel-1 tocsection-13"><a href="#See_also">9 See also</a></li> <li class="toclevel-1 tocsection-14"><a href="#References">10 References</a></li> <li class="toclevel-1 tocsection-15"><a href="#External_links">11 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="Notation">Notation</h2> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=1" title="Edit section: Notation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> And is usually denoted by an infix operator: in mathematics and logic, it is denoted by a "wedge" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" data-alt="{\displaystyle \wedge }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (Unicode <style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style>U+2227 ∧ LOGICAL AND),<a href="#cite_note-:2-1">[1]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \&}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \&}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7e8142574f96000e13827167bdfdd69e38076f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \&}"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7e8142574f96000e13827167bdfdd69e38076f" data-alt="{\displaystyle \&}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 1.509ex;vertical-align: 0.019ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" data-alt="{\displaystyle \times }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ; in electronics, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"></noscript><span class="lazy-image-placeholder" style="width: 0.647ex;height: 1.176ex;vertical-align: 0.439ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" data-alt="{\displaystyle \cdot }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ; and in programming languages, <code>&</code>, <code>&&</code>, or <code>and</code>. In <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Łukasiewicz</a>'s <a href="/wiki/Polish_notation#Polish_notation_for_logic" title="Polish notation">prefix notation for logic</a>, the operator is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></noscript><span class="lazy-image-placeholder" style="width: 2.066ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" data-alt="{\displaystyle K}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , for Polish koniunkcja.<a href="#cite_note-4">[4]</a> In mathematics, the conjunction of an arbitrary number of elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\ldots ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\ldots ,a_{n}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451345cc97e2ed923dd4656fcc400c3f37119cca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\ldots ,a_{n}}"></noscript><span class="lazy-image-placeholder" style="width: 9.911ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451345cc97e2ed923dd4656fcc400c3f37119cca" data-alt="{\displaystyle a_{1},\ldots ,a_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  can be denoted as an <a href="/wiki/Iterated_binary_operation" title="Iterated binary operation">iterated binary operation</a> using a "big wedge" ⋀ (Unicode <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">U+22C0 ⋀ N-ARY LOGICAL AND):<a href="#cite_note-5">[5]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge _{i=1}^{n}a_{i}=a_{1}\wedge a_{2}\wedge \ldots a_{n-1}\wedge a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∧</mo> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge _{i=1}^{n}a_{i}=a_{1}\wedge a_{2}\wedge \ldots a_{n-1}\wedge a_{n}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1feb79ba86ee76f756f9571b753a42793633dd71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.606ex; height:6.843ex;" alt="{\displaystyle \bigwedge _{i=1}^{n}a_{i}=a_{1}\wedge a_{2}\wedge \ldots a_{n-1}\wedge a_{n}}"></noscript><span class="lazy-image-placeholder" style="width: 30.606ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1feb79ba86ee76f756f9571b753a42793633dd71" data-alt="{\displaystyle \bigwedge _{i=1}^{n}a_{i}=a_{1}\wedge a_{2}\wedge \ldots a_{n-1}\wedge a_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="Definition">Definition</h2> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=2" title="Edit section: Definition" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> In <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>, logical conjunction is an <a href="/wiki/Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="/wiki/Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="/wiki/Proposition" title="Proposition">propositions</a>, that produces a value of true <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> (also known as iff) both of its operands are true.<a href="#cite_note-:1-2">[2]</a><a href="#cite_note-:2-1">[1]</a> The conjunctive <a href="/wiki/Identity_element" title="Identity element">identity</a> is true, which is to say that AND-ing an expression with true will never change the value of the expression. In keeping with the concept of <a href="/wiki/Vacuous_truth" title="Vacuous truth">vacuous truth</a>, when conjunction is defined as an operator or function of arbitrary <a href="/wiki/Arity" title="Arity">arity</a>, the empty conjunction (AND-ing over an empty set of operands) is often defined as having the result true. <div class="mw-heading mw-heading3"><h3 id="Truth_table">Truth table</h3> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=3" title="Edit section: Truth table" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Variadic_logical_AND.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Variadic_logical_AND.svg/220px-Variadic_logical_AND.svg.png" decoding="async" width="220" height="177" class="mw-file-element" data-file-width="1807" data-file-height="1452"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 177px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Variadic_logical_AND.svg/220px-Variadic_logical_AND.svg.png" data-width="220" data-height="177" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Variadic_logical_AND.svg/330px-Variadic_logical_AND.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Variadic_logical_AND.svg/440px-Variadic_logical_AND.svg.png 2x" data-class="mw-file-element"> </a><figcaption>Conjunctions of the arguments on the left — The <a href="/wiki/Truth_value" title="Truth value">true</a> <a href="/wiki/Bit" title="Bit">bit</a>s form a <a href="/wiki/Sierpinski_triangle" class="mw-redirect" title="Sierpinski triangle">Sierpinski triangle</a>.</figcaption></figure> The <a href="/wiki/Truth_table" title="Truth table">truth table</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></noscript><span class="lazy-image-placeholder" style="width: 6.09ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" data-alt="{\displaystyle A\land B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> :<a href="#cite_note-:2-1">[1]</a><a href="#cite_note-:1-2">[2]</a> <style data-mw-deduplicate="TemplateStyles:r1211505603">.mw-parser-output table.two-ary-tt th{font-weight:normal}.mw-parser-output table.two-ary-tt td.bold{font-weight:bold;color:#444}.mw-parser-output table.two-ary-tt td{text-align:center;padding-left:14px;padding-right:14px}.mw-parser-output table.two-ary-tt td.f{background-color:#fff}.mw-parser-output table.two-ary-tt td.t{background-color:#fcc}.mw-parser-output td.border,.mw-parser-output th.border{border-left:2px solid #777}</style><table class="wikitable sortable two-ary-tt"><tbody><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" data-alt="{\displaystyle B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th class="unsortable border"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></noscript><span class="lazy-image-placeholder" style="width: 6.09ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" data-alt="{\displaystyle A\land B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th></tr><tr><td class="bold f">F</td><td class="bold f">F</td><td class="border f">F</td></tr><tr><td class="bold f">F</td><td class="bold t">T</td><td class="border f">F</td></tr><tr><td class="bold t">T</td><td class="bold f">F</td><td class="border f">F</td></tr><tr><td class="bold t">T</td><td class="bold t">T</td><td class="border t">T</td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Defined_by_other_operators">Defined by other operators</h3> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=4" title="Edit section: Defined by other operators" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> In systems where logical conjunction is not a primitive, it may be defined as<a href="#cite_note-6">[6]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B=\neg (A\to \neg B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo>=</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B=\neg (A\to \neg B)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/173fa875cd41f46afd356d88f54ebc4ec982bed1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.219ex; height:2.843ex;" alt="{\displaystyle A\land B=\neg (A\to \neg B)}"></noscript><span class="lazy-image-placeholder" style="width: 21.219ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/173fa875cd41f46afd356d88f54ebc4ec982bed1" data-alt="{\displaystyle A\land B=\neg (A\to \neg B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> It can be checked by the following truth table (compare the last two columns): <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1211505603"><table class="wikitable sortable two-ary-tt"><tbody><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" data-alt="{\displaystyle B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th class="unsortable border"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8cf55d88686624cd054232a7cf1a6b7e6e84210" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.314ex; height:2.176ex;" alt="{\displaystyle \neg B}"></noscript><span class="lazy-image-placeholder" style="width: 3.314ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8cf55d88686624cd054232a7cf1a6b7e6e84210" data-alt="{\displaystyle \neg B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th class="unsortable border"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\rightarrow \neg B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\rightarrow \neg B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa4203fc2ac31a8d56717a6d81924606a4127382" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.672ex; height:2.176ex;" alt="{\displaystyle A\rightarrow \neg B}"></noscript><span class="lazy-image-placeholder" style="width: 8.672ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa4203fc2ac31a8d56717a6d81924606a4127382" data-alt="{\displaystyle A\rightarrow \neg B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th class="unsortable"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (A\rightarrow \neg B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (A\rightarrow \neg B)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb075b6d78dece6f3df0ca841397915db287984" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.031ex; height:2.843ex;" alt="{\displaystyle \neg (A\rightarrow \neg B)}"></noscript><span class="lazy-image-placeholder" style="width: 12.031ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb075b6d78dece6f3df0ca841397915db287984" data-alt="{\displaystyle \neg (A\rightarrow \neg B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th class="unsortable"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></noscript><span class="lazy-image-placeholder" style="width: 6.09ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" data-alt="{\displaystyle A\land B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th></tr><tr><td class="bold f">F</td><td class="bold f">F</td><td class="border t">T</td><td class="border t">T</td><td class="f">F</td><td class="f">F</td></tr><tr><td class="bold f">F</td><td class="bold t">T</td><td class="border f">F</td><td class="border t">T</td><td class="f">F</td><td class="f">F</td></tr><tr><td class="bold t">T</td><td class="bold f">F</td><td class="border t">T</td><td class="border t">T</td><td class="f">F</td><td class="f">F</td></tr><tr><td class="bold t">T</td><td class="bold t">T</td><td class="border f">F</td><td class="border f">F</td><td class="t">T</td><td class="t">T</td></tr></tbody></table> or <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B=\neg (\neg A\lor \neg B).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo>=</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B=\neg (\neg A\lor \neg B).}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05eb197c28ea0aed77613688b475b36f092c063a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.385ex; height:2.843ex;" alt="{\displaystyle A\land B=\neg (\neg A\lor \neg B).}"></noscript><span class="lazy-image-placeholder" style="width: 22.385ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05eb197c28ea0aed77613688b475b36f092c063a" data-alt="{\displaystyle A\land B=\neg (\neg A\lor \neg B).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> It can be checked by the following truth table (compare the last two columns): <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1211505603"><table class="wikitable sortable two-ary-tt"><tbody><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" data-alt="{\displaystyle B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th class="unsortable border"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"></noscript><span class="lazy-image-placeholder" style="width: 3.293ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" data-alt="{\displaystyle \neg A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th class="unsortable"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8cf55d88686624cd054232a7cf1a6b7e6e84210" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.314ex; height:2.176ex;" alt="{\displaystyle \neg B}"></noscript><span class="lazy-image-placeholder" style="width: 3.314ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8cf55d88686624cd054232a7cf1a6b7e6e84210" data-alt="{\displaystyle \neg B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th class="unsortable border"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A\lor \neg B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A\lor \neg B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1020d4923bd093b4d10a73c88d3db0b3211b4ec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.19ex; height:2.176ex;" alt="{\displaystyle \neg A\lor \neg B}"></noscript><span class="lazy-image-placeholder" style="width: 9.19ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1020d4923bd093b4d10a73c88d3db0b3211b4ec0" data-alt="{\displaystyle \neg A\lor \neg B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th class="unsortable"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (\neg A\lor \neg B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (\neg A\lor \neg B)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdffb25cda99ba7533af46896d8471612e831b53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.55ex; height:2.843ex;" alt="{\displaystyle \neg (\neg A\lor \neg B)}"></noscript><span class="lazy-image-placeholder" style="width: 12.55ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdffb25cda99ba7533af46896d8471612e831b53" data-alt="{\displaystyle \neg (\neg A\lor \neg B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th><th class="unsortable"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></noscript><span class="lazy-image-placeholder" style="width: 6.09ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" data-alt="{\displaystyle A\land B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </th></tr><tr><td class="bold f">F</td><td class="bold f">F</td><td class="border t">T</td><td class="t">T</td><td class="border t">T</td><td class="f">F</td><td class="f">F</td></tr><tr><td class="bold f">F</td><td class="bold t">T</td><td class="border t">T</td><td class="f">F</td><td class="border t">T</td><td class="f">F</td><td class="f">F</td></tr><tr><td class="bold t">T</td><td class="bold f">F</td><td class="border f">F</td><td class="t">T</td><td class="border t">T</td><td class="f">F</td><td class="f">F</td></tr><tr><td class="bold t">T</td><td class="bold t">T</td><td class="border f">F</td><td class="f">F</td><td class="border f">F</td><td class="t">T</td><td class="t">T</td></tr></tbody></table> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="Introduction_and_elimination_rules">Introduction and elimination rules</h2> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=5" title="Edit section: Introduction and elimination rules" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> As a rule of inference, <a href="/wiki/Conjunction_introduction" title="Conjunction introduction">conjunction introduction</a> is a classically <a href="/wiki/Validity_(logic)" title="Validity (logic)">valid</a>, simple <a href="/wiki/Argument_form" class="mw-redirect" title="Argument form">argument form</a>. The argument form has two premises, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" data-alt="{\displaystyle B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Intuitively, it permits the inference of their conjunction. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ,</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" data-alt="{\displaystyle B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</dd> <dd>Therefore, A and B.</dd></dl> or in <a href="/wiki/Logical_operator" class="mw-redirect" title="Logical operator">logical operator</a> notation, where \vdash expresses provability: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash A,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b684f523e5fd8c0dfb07cfb6a5696132e21cad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.455ex; height:2.509ex;" alt="{\displaystyle \vdash A,}"></noscript><span class="lazy-image-placeholder" style="width: 4.455ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b684f523e5fd8c0dfb07cfb6a5696132e21cad" data-alt="{\displaystyle \vdash A,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa02acb341837055be929c605bd7a9edf73b6de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.829ex; height:2.176ex;" alt="{\displaystyle \vdash B}"></noscript><span class="lazy-image-placeholder" style="width: 3.829ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa02acb341837055be929c605bd7a9edf73b6de" data-alt="{\displaystyle \vdash B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash A\land B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d536ef29846f62a11cb72d74bcfcba2fce901b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.155ex; height:2.176ex;" alt="{\displaystyle \vdash A\land B}"></noscript><span class="lazy-image-placeholder" style="width: 8.155ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d536ef29846f62a11cb72d74bcfcba2fce901b7f" data-alt="{\displaystyle \vdash A\land B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> Here is an example of an argument that fits the form <a href="/wiki/Conjunction_introduction" title="Conjunction introduction">conjunction introduction</a>: <dl><dd>Bob likes apples.</dd> <dd>Bob likes oranges.</dd> <dd>Therefore, Bob likes apples and Bob likes oranges.</dd></dl> <a href="/wiki/Conjunction_elimination" title="Conjunction elimination">Conjunction elimination</a> is another classically <a href="/wiki/Validity_(logic)" title="Validity (logic)">valid</a>, simple <a href="/wiki/Argument_form" class="mw-redirect" title="Argument form">argument form</a>. Intuitively, it permits the inference from any conjunction of either element of that conjunction. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" data-alt="{\displaystyle B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</dd> <dd>Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</dd></dl> ...or alternatively, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" data-alt="{\displaystyle B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</dd> <dd>Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" data-alt="{\displaystyle B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</dd></dl> In <a href="/wiki/Logical_operator" class="mw-redirect" title="Logical operator">logical operator</a> notation: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash A\land B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d536ef29846f62a11cb72d74bcfcba2fce901b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.155ex; height:2.176ex;" alt="{\displaystyle \vdash A\land B}"></noscript><span class="lazy-image-placeholder" style="width: 8.155ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d536ef29846f62a11cb72d74bcfcba2fce901b7f" data-alt="{\displaystyle \vdash A\land B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b145d194f0785ff840b76ace797a7e077a18f544" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle \vdash A}"></noscript><span class="lazy-image-placeholder" style="width: 3.809ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b145d194f0785ff840b76ace797a7e077a18f544" data-alt="{\displaystyle \vdash A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> ...or alternatively, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash A\land B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d536ef29846f62a11cb72d74bcfcba2fce901b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.155ex; height:2.176ex;" alt="{\displaystyle \vdash A\land B}"></noscript><span class="lazy-image-placeholder" style="width: 8.155ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d536ef29846f62a11cb72d74bcfcba2fce901b7f" data-alt="{\displaystyle \vdash A\land B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa02acb341837055be929c605bd7a9edf73b6de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.829ex; height:2.176ex;" alt="{\displaystyle \vdash B}"></noscript><span class="lazy-image-placeholder" style="width: 3.829ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa02acb341837055be929c605bd7a9edf73b6de" data-alt="{\displaystyle \vdash B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="Negation">Negation</h2> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=6" title="Edit section: Negation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <div class="mw-heading mw-heading3"><h3 id="Definition_2">Definition</h3> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=7" title="Edit section: Definition" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> A conjunction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></noscript><span class="lazy-image-placeholder" style="width: 6.09ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" data-alt="{\displaystyle A\land B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is proven false by establishing either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"></noscript><span class="lazy-image-placeholder" style="width: 3.293ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" data-alt="{\displaystyle \neg A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8cf55d88686624cd054232a7cf1a6b7e6e84210" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.314ex; height:2.176ex;" alt="{\displaystyle \neg B}"></noscript><span class="lazy-image-placeholder" style="width: 3.314ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8cf55d88686624cd054232a7cf1a6b7e6e84210" data-alt="{\displaystyle \neg B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . In terms of the object language, this reads <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A\to \neg (A\land B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A\to \neg (A\land B)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6411846635732799332fb8b49f66ee081f476bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.357ex; height:2.843ex;" alt="{\displaystyle \neg A\to \neg (A\land B)}"></noscript><span class="lazy-image-placeholder" style="width: 16.357ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6411846635732799332fb8b49f66ee081f476bae" data-alt="{\displaystyle \neg A\to \neg (A\land B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> This formula can be seen as a special case of <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\to C)\to ((A\land B)\to C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\to C)\to ((A\land B)\to C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6751af7d8bc26b530f7779c5317b128092b8c933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.636ex; height:2.843ex;" alt="{\displaystyle (A\to C)\to ((A\land B)\to C)}"></noscript><span class="lazy-image-placeholder" style="width: 27.636ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6751af7d8bc26b530f7779c5317b128092b8c933" data-alt="{\displaystyle (A\to C)\to ((A\land B)\to C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></noscript><span class="lazy-image-placeholder" style="width: 1.766ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" data-alt="{\displaystyle C}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a false proposition. <div class="mw-heading mw-heading3"><h3 id="Other_proof_strategies">Other proof strategies</h3> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=8" title="Edit section: Other proof strategies" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8cf55d88686624cd054232a7cf1a6b7e6e84210" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.314ex; height:2.176ex;" alt="{\displaystyle \neg B}"></noscript><span class="lazy-image-placeholder" style="width: 3.314ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8cf55d88686624cd054232a7cf1a6b7e6e84210" data-alt="{\displaystyle \neg B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , then both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"></noscript><span class="lazy-image-placeholder" style="width: 3.293ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" data-alt="{\displaystyle \neg A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  as well as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  prove the conjunction false: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\to \neg {}B)\to \neg (A\land B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\to \neg {}B)\to \neg (A\land B)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5fe2ddde389bb41067e67cbadc4c62fcab7793c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.544ex; height:2.843ex;" alt="{\displaystyle (A\to \neg {}B)\to \neg (A\land B)}"></noscript><span class="lazy-image-placeholder" style="width: 23.544ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5fe2ddde389bb41067e67cbadc4c62fcab7793c" data-alt="{\displaystyle (A\to \neg {}B)\to \neg (A\land B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> In other words, a conjunction can actually be proven false just by knowing about the relation of its conjuncts, and not necessary about their truth values. This formula can be seen as a special case of <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\to (B\to C))\to ((A\land B)\to C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\to (B\to C))\to ((A\land B)\to C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b1f2e7d1056f3be372e1e896484c1b1ac5fd29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.823ex; height:2.843ex;" alt="{\displaystyle (A\to (B\to C))\to ((A\land B)\to C)}"></noscript><span class="lazy-image-placeholder" style="width: 34.823ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b1f2e7d1056f3be372e1e896484c1b1ac5fd29" data-alt="{\displaystyle (A\to (B\to C))\to ((A\land B)\to C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></noscript><span class="lazy-image-placeholder" style="width: 1.766ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" data-alt="{\displaystyle C}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a false proposition. Either of the above are constructively valid proofs by contradiction. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="Properties">Properties</h2> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=9" title="Edit section: Properties" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <a href="/wiki/Commutative_property" title="Commutative property">commutativity</a>: yes <table style="text-align: center; border: 1px solid darkgray;"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></noscript><span class="lazy-image-placeholder" style="width: 6.09ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" data-alt="{\displaystyle A\land B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\land A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>∧</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\land A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5920298dbda4592b71e53822ce01829cd77f4190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle B\land A}"></noscript><span class="lazy-image-placeholder" style="width: 6.09ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5920298dbda4592b71e53822ce01829cd77f4190" data-alt="{\displaystyle B\land A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/File:Venn0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/50px-Venn0001.svg.png" decoding="async" width="50" height="36" class="mw-file-element" data-file-width="410" data-file-height="299"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 36px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/50px-Venn0001.svg.png" data-width="50" data-height="36" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/75px-Venn0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/100px-Venn0001.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/50px-Venn0001.svg.png" decoding="async" width="50" height="36" class="mw-file-element" data-file-width="410" data-file-height="299"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 36px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/50px-Venn0001.svg.png" data-width="50" data-height="36" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/75px-Venn0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/100px-Venn0001.svg.png 2x" data-class="mw-file-element"> </a> </td></tr></tbody></table> <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a>: yes<a href="#cite_note-:13-7">[7]</a> <table style="text-align: center; border: 1px solid darkgray;"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17327d088840ce291c8db59b592489ef8e6e94bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~A}"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17327d088840ce291c8db59b592489ef8e6e94bd" data-alt="{\displaystyle ~A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~~\land ~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo>∧</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~~\land ~~~}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296d4bd5f22dac2701fa42f57e9c5b65d1dd63f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.066ex; height:2.009ex;" alt="{\displaystyle ~~~\land ~~~}"></noscript><span class="lazy-image-placeholder" style="width: 6.066ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296d4bd5f22dac2701fa42f57e9c5b65d1dd63f9" data-alt="{\displaystyle ~~~\land ~~~}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\land C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\land C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78cc0188b905ef850ed33a9a4068e49794712b8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.922ex; height:2.843ex;" alt="{\displaystyle (B\land C)}"></noscript><span class="lazy-image-placeholder" style="width: 7.922ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78cc0188b905ef850ed33a9a4068e49794712b8d" data-alt="{\displaystyle (B\land C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land B)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdcd2d1d13bc1f915aa141415965509a4e2b4f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.899ex; height:2.843ex;" alt="{\displaystyle (A\land B)}"></noscript><span class="lazy-image-placeholder" style="width: 7.899ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdcd2d1d13bc1f915aa141415965509a4e2b4f1" data-alt="{\displaystyle (A\land B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~~\land ~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo>∧</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~~\land ~~~}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296d4bd5f22dac2701fa42f57e9c5b65d1dd63f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.066ex; height:2.009ex;" alt="{\displaystyle ~~~\land ~~~}"></noscript><span class="lazy-image-placeholder" style="width: 6.066ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296d4bd5f22dac2701fa42f57e9c5b65d1dd63f9" data-alt="{\displaystyle ~~~\land ~~~}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~C}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f52ed2496dc4077efa433abb4685684a158d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.347ex; height:2.176ex;" alt="{\displaystyle ~C}"></noscript><span class="lazy-image-placeholder" style="width: 2.347ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f52ed2496dc4077efa433abb4685684a158d7e" data-alt="{\displaystyle ~C}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/File:Venn_0101_0101.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/50px-Venn_0101_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/50px-Venn_0101_0101.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/75px-Venn_0101_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/100px-Venn_0101_0101.svg.png 2x" data-class="mw-file-element"> </a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~~\land ~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo>∧</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~~\land ~~~}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296d4bd5f22dac2701fa42f57e9c5b65d1dd63f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.066ex; height:2.009ex;" alt="{\displaystyle ~~~\land ~~~}"></noscript><span class="lazy-image-placeholder" style="width: 6.066ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296d4bd5f22dac2701fa42f57e9c5b65d1dd63f9" data-alt="{\displaystyle ~~~\land ~~~}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0000_0011.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Venn_0000_0011.svg/50px-Venn_0000_0011.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Venn_0000_0011.svg/50px-Venn_0000_0011.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Venn_0000_0011.svg/75px-Venn_0000_0011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Venn_0000_0011.svg/100px-Venn_0000_0011.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0000_0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Venn_0000_0001.svg/50px-Venn_0000_0001.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Venn_0000_0001.svg/50px-Venn_0000_0001.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Venn_0000_0001.svg/75px-Venn_0000_0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Venn_0000_0001.svg/100px-Venn_0000_0001.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0001_0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/50px-Venn_0001_0001.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/50px-Venn_0001_0001.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/75px-Venn_0001_0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/100px-Venn_0001_0001.svg.png 2x" data-class="mw-file-element"> </a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~~\land ~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo>∧</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~~\land ~~~}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296d4bd5f22dac2701fa42f57e9c5b65d1dd63f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.066ex; height:2.009ex;" alt="{\displaystyle ~~~\land ~~~}"></noscript><span class="lazy-image-placeholder" style="width: 6.066ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296d4bd5f22dac2701fa42f57e9c5b65d1dd63f9" data-alt="{\displaystyle ~~~\land ~~~}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0000_1111.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/50px-Venn_0000_1111.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/50px-Venn_0000_1111.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/75px-Venn_0000_1111.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/100px-Venn_0000_1111.svg.png 2x" data-class="mw-file-element"> </a> </td></tr></tbody></table> <a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distributivity</a>: with various operations, especially with <a href="/wiki/Logical_disjunction" title="Logical disjunction">or</a> <table style="text-align: center; border: 1px solid darkgray;"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17327d088840ce291c8db59b592489ef8e6e94bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~A}"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17327d088840ce291c8db59b592489ef8e6e94bd" data-alt="{\displaystyle ~A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" data-alt="{\displaystyle \land }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\lor C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∨</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\lor C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c18aad468eb6ae0354f697dd4035fb970946d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.922ex; height:2.843ex;" alt="{\displaystyle (B\lor C)}"></noscript><span class="lazy-image-placeholder" style="width: 7.922ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c18aad468eb6ae0354f697dd4035fb970946d2" data-alt="{\displaystyle (B\lor C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land B)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdcd2d1d13bc1f915aa141415965509a4e2b4f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.899ex; height:2.843ex;" alt="{\displaystyle (A\land B)}"></noscript><span class="lazy-image-placeholder" style="width: 7.899ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdcd2d1d13bc1f915aa141415965509a4e2b4f1" data-alt="{\displaystyle (A\land B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" data-alt="{\displaystyle \lor }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887cc87745fae3b8e066d3d41dce3e430063844e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.901ex; height:2.843ex;" alt="{\displaystyle (A\land C)}"></noscript><span class="lazy-image-placeholder" style="width: 7.901ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887cc87745fae3b8e066d3d41dce3e430063844e" data-alt="{\displaystyle (A\land C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/File:Venn_0101_0101.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/50px-Venn_0101_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/50px-Venn_0101_0101.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/75px-Venn_0101_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/100px-Venn_0101_0101.svg.png 2x" data-class="mw-file-element"> </a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" data-alt="{\displaystyle \land }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0011_1111.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Venn_0011_1111.svg/50px-Venn_0011_1111.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Venn_0011_1111.svg/50px-Venn_0011_1111.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Venn_0011_1111.svg/75px-Venn_0011_1111.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Venn_0011_1111.svg/100px-Venn_0011_1111.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0001_0101.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Venn_0001_0101.svg/50px-Venn_0001_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Venn_0001_0101.svg/50px-Venn_0001_0101.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Venn_0001_0101.svg/75px-Venn_0001_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Venn_0001_0101.svg/100px-Venn_0001_0101.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0001_0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/50px-Venn_0001_0001.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/50px-Venn_0001_0001.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/75px-Venn_0001_0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/100px-Venn_0001_0001.svg.png 2x" data-class="mw-file-element"> </a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" data-alt="{\displaystyle \lor }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0000_0101.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/50px-Venn_0000_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/50px-Venn_0000_0101.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/75px-Venn_0000_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/100px-Venn_0000_0101.svg.png 2x" data-class="mw-file-element"> </a> </td></tr></tbody></table> <table class="collapsible collapsed" style="width: 100%; border: 1px solid #aaaaaa;"> <tbody><tr> <th bgcolor="#ccccff">others </th></tr> <tr> <td> with <a href="/wiki/Exclusive_or" title="Exclusive or">exclusive or</a>: <table style="text-align: center; border: 1px solid darkgray;"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17327d088840ce291c8db59b592489ef8e6e94bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~A}"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17327d088840ce291c8db59b592489ef8e6e94bd" data-alt="{\displaystyle ~A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" data-alt="{\displaystyle \land }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\oplus C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>⊕</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\oplus C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d4800638f9d7524cd268e8e9443f12bf67afac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.18ex; height:2.843ex;" alt="{\displaystyle (B\oplus C)}"></noscript><span class="lazy-image-placeholder" style="width: 8.18ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d4800638f9d7524cd268e8e9443f12bf67afac" data-alt="{\displaystyle (B\oplus C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land B)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdcd2d1d13bc1f915aa141415965509a4e2b4f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.899ex; height:2.843ex;" alt="{\displaystyle (A\land B)}"></noscript><span class="lazy-image-placeholder" style="width: 7.899ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdcd2d1d13bc1f915aa141415965509a4e2b4f1" data-alt="{\displaystyle (A\land B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.176ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" data-alt="{\displaystyle \oplus }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887cc87745fae3b8e066d3d41dce3e430063844e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.901ex; height:2.843ex;" alt="{\displaystyle (A\land C)}"></noscript><span class="lazy-image-placeholder" style="width: 7.901ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887cc87745fae3b8e066d3d41dce3e430063844e" data-alt="{\displaystyle (A\land C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/File:Venn_0101_0101.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/50px-Venn_0101_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/50px-Venn_0101_0101.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/75px-Venn_0101_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/100px-Venn_0101_0101.svg.png 2x" data-class="mw-file-element"> </a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" data-alt="{\displaystyle \land }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0011_1100.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Venn_0011_1100.svg/50px-Venn_0011_1100.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Venn_0011_1100.svg/50px-Venn_0011_1100.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Venn_0011_1100.svg/75px-Venn_0011_1100.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Venn_0011_1100.svg/100px-Venn_0011_1100.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0001_0100.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Venn_0001_0100.svg/50px-Venn_0001_0100.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Venn_0001_0100.svg/50px-Venn_0001_0100.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Venn_0001_0100.svg/75px-Venn_0001_0100.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Venn_0001_0100.svg/100px-Venn_0001_0100.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0001_0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/50px-Venn_0001_0001.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/50px-Venn_0001_0001.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/75px-Venn_0001_0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/100px-Venn_0001_0001.svg.png 2x" data-class="mw-file-element"> </a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.176ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" data-alt="{\displaystyle \oplus }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0000_0101.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/50px-Venn_0000_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/50px-Venn_0000_0101.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/75px-Venn_0000_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/100px-Venn_0000_0101.svg.png 2x" data-class="mw-file-element"> </a> </td></tr></tbody></table> with <a href="/wiki/Material_nonimplication" title="Material nonimplication">material nonimplication</a>: <table style="text-align: center; border: 1px solid darkgray;"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17327d088840ce291c8db59b592489ef8e6e94bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~A}"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17327d088840ce291c8db59b592489ef8e6e94bd" data-alt="{\displaystyle ~A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" data-alt="{\displaystyle \land }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\nrightarrow C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>↛</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\nrightarrow C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d881f71105d0587a2b95607cc353b2021b5b345" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.954ex; height:2.843ex;" alt="{\displaystyle (B\nrightarrow C)}"></noscript><span class="lazy-image-placeholder" style="width: 8.954ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d881f71105d0587a2b95607cc353b2021b5b345" data-alt="{\displaystyle (B\nrightarrow C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land B)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdcd2d1d13bc1f915aa141415965509a4e2b4f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.899ex; height:2.843ex;" alt="{\displaystyle (A\land B)}"></noscript><span class="lazy-image-placeholder" style="width: 7.899ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdcd2d1d13bc1f915aa141415965509a4e2b4f1" data-alt="{\displaystyle (A\land B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↛</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c458d67617e028ed10948d2dbcfef80e9e060a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.509ex;vertical-align: 0.137ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c458d67617e028ed10948d2dbcfef80e9e060a2" data-alt="{\displaystyle \nrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887cc87745fae3b8e066d3d41dce3e430063844e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.901ex; height:2.843ex;" alt="{\displaystyle (A\land C)}"></noscript><span class="lazy-image-placeholder" style="width: 7.901ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887cc87745fae3b8e066d3d41dce3e430063844e" data-alt="{\displaystyle (A\land C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/File:Venn_0101_0101.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/50px-Venn_0101_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/50px-Venn_0101_0101.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/75px-Venn_0101_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/100px-Venn_0101_0101.svg.png 2x" data-class="mw-file-element"> </a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" data-alt="{\displaystyle \land }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0011_0000.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Venn_0011_0000.svg/50px-Venn_0011_0000.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Venn_0011_0000.svg/50px-Venn_0011_0000.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Venn_0011_0000.svg/75px-Venn_0011_0000.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Venn_0011_0000.svg/100px-Venn_0011_0000.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0001_0000.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Venn_0001_0000.svg/50px-Venn_0001_0000.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Venn_0001_0000.svg/50px-Venn_0001_0000.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Venn_0001_0000.svg/75px-Venn_0001_0000.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Venn_0001_0000.svg/100px-Venn_0001_0000.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0001_0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/50px-Venn_0001_0001.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/50px-Venn_0001_0001.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/75px-Venn_0001_0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/100px-Venn_0001_0001.svg.png 2x" data-class="mw-file-element"> </a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↛</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c458d67617e028ed10948d2dbcfef80e9e060a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.137ex; margin-bottom: -0.308ex; width:2.324ex; height:1.509ex;" alt="{\displaystyle \nrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.509ex;vertical-align: 0.137ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c458d67617e028ed10948d2dbcfef80e9e060a2" data-alt="{\displaystyle \nrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0000_0101.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/50px-Venn_0000_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/50px-Venn_0000_0101.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/75px-Venn_0000_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/100px-Venn_0000_0101.svg.png 2x" data-class="mw-file-element"> </a> </td></tr></tbody></table> with itself: <table style="text-align: center; border: 1px solid darkgray;"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17327d088840ce291c8db59b592489ef8e6e94bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~A}"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17327d088840ce291c8db59b592489ef8e6e94bd" data-alt="{\displaystyle ~A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" data-alt="{\displaystyle \land }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\land C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\land C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78cc0188b905ef850ed33a9a4068e49794712b8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.922ex; height:2.843ex;" alt="{\displaystyle (B\land C)}"></noscript><span class="lazy-image-placeholder" style="width: 7.922ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78cc0188b905ef850ed33a9a4068e49794712b8d" data-alt="{\displaystyle (B\land C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land B)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdcd2d1d13bc1f915aa141415965509a4e2b4f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.899ex; height:2.843ex;" alt="{\displaystyle (A\land B)}"></noscript><span class="lazy-image-placeholder" style="width: 7.899ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdcd2d1d13bc1f915aa141415965509a4e2b4f1" data-alt="{\displaystyle (A\land B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" data-alt="{\displaystyle \land }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887cc87745fae3b8e066d3d41dce3e430063844e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.901ex; height:2.843ex;" alt="{\displaystyle (A\land C)}"></noscript><span class="lazy-image-placeholder" style="width: 7.901ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887cc87745fae3b8e066d3d41dce3e430063844e" data-alt="{\displaystyle (A\land C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/File:Venn_0101_0101.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/50px-Venn_0101_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/50px-Venn_0101_0101.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/75px-Venn_0101_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Venn_0101_0101.svg/100px-Venn_0101_0101.svg.png 2x" data-class="mw-file-element"> </a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" data-alt="{\displaystyle \land }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0000_0011.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Venn_0000_0011.svg/50px-Venn_0000_0011.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Venn_0000_0011.svg/50px-Venn_0000_0011.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Venn_0000_0011.svg/75px-Venn_0000_0011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Venn_0000_0011.svg/100px-Venn_0000_0011.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0000_0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Venn_0000_0001.svg/50px-Venn_0000_0001.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Venn_0000_0001.svg/50px-Venn_0000_0001.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Venn_0000_0001.svg/75px-Venn_0000_0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Venn_0000_0001.svg/100px-Venn_0000_0001.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0001_0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/50px-Venn_0001_0001.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/50px-Venn_0001_0001.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/75px-Venn_0001_0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Venn_0001_0001.svg/100px-Venn_0001_0001.svg.png 2x" data-class="mw-file-element"> </a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" data-alt="{\displaystyle \land }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0000_0101.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/50px-Venn_0000_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/50px-Venn_0000_0101.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/75px-Venn_0000_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/100px-Venn_0000_0101.svg.png 2x" data-class="mw-file-element"> </a> </td></tr></tbody></table> </td></tr></tbody></table> <a href="/wiki/Idempotency" class="mw-redirect" title="Idempotency">idempotency</a>: yes <table style="text-align: center; border: 1px solid darkgray;"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A~}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00229fc56bafa7e9b522aedb3bed5dca455bc561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.904ex; height:2.176ex;" alt="{\displaystyle ~A~}"></noscript><span class="lazy-image-placeholder" style="width: 2.904ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00229fc56bafa7e9b522aedb3bed5dca455bc561" data-alt="{\displaystyle ~A~}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\land ~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>∧</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\land ~}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b78b7e7950527f71b3b15b62d8459c636df43065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.744ex; height:2.009ex;" alt="{\displaystyle ~\land ~}"></noscript><span class="lazy-image-placeholder" style="width: 3.744ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b78b7e7950527f71b3b15b62d8459c636df43065" data-alt="{\displaystyle ~\land ~}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A~}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00229fc56bafa7e9b522aedb3bed5dca455bc561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.904ex; height:2.176ex;" alt="{\displaystyle ~A~}"></noscript><span class="lazy-image-placeholder" style="width: 2.904ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00229fc56bafa7e9b522aedb3bed5dca455bc561" data-alt="{\displaystyle ~A~}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A~}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc59051ffaf2eaace4f7b01f440b9067b722fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle A~}"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc59051ffaf2eaace4f7b01f440b9067b722fb0" data-alt="{\displaystyle A~}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/File:Venn01.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/36px-Venn01.svg.png" decoding="async" width="36" height="36" class="mw-file-element" data-file-width="280" data-file-height="280"></noscript><span class="lazy-image-placeholder" style="width: 36px;height: 36px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/36px-Venn01.svg.png" data-width="36" data-height="36" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/54px-Venn01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/72px-Venn01.svg.png 2x" data-class="mw-file-element"> </a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\land ~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>∧</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\land ~}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b78b7e7950527f71b3b15b62d8459c636df43065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.744ex; height:2.009ex;" alt="{\displaystyle ~\land ~}"></noscript><span class="lazy-image-placeholder" style="width: 3.744ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b78b7e7950527f71b3b15b62d8459c636df43065" data-alt="{\displaystyle ~\land ~}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn01.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/36px-Venn01.svg.png" decoding="async" width="36" height="36" class="mw-file-element" data-file-width="280" data-file-height="280"></noscript><span class="lazy-image-placeholder" style="width: 36px;height: 36px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/36px-Venn01.svg.png" data-width="36" data-height="36" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/54px-Venn01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/72px-Venn01.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn01.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/36px-Venn01.svg.png" decoding="async" width="36" height="36" class="mw-file-element" data-file-width="280" data-file-height="280"></noscript><span class="lazy-image-placeholder" style="width: 36px;height: 36px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/36px-Venn01.svg.png" data-width="36" data-height="36" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/54px-Venn01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Venn01.svg/72px-Venn01.svg.png 2x" data-class="mw-file-element"> </a> </td></tr></tbody></table> <a href="/wiki/Monotonic_function#In_Boolean_functions" title="Monotonic function">monotonicity</a>: yes <table style="text-align: center; border: 1px solid darkgray;"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\rightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\rightarrow B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23efef033def56a67de7ded823f14626de26d174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\rightarrow B}"></noscript><span class="lazy-image-placeholder" style="width: 7.121ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23efef033def56a67de7ded823f14626de26d174" data-alt="{\displaystyle A\rightarrow B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" data-alt="{\displaystyle \Rightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> </td> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887cc87745fae3b8e066d3d41dce3e430063844e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.901ex; height:2.843ex;" alt="{\displaystyle (A\land C)}"></noscript><span class="lazy-image-placeholder" style="width: 7.901ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887cc87745fae3b8e066d3d41dce3e430063844e" data-alt="{\displaystyle (A\land C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" data-alt="{\displaystyle \rightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\land C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\land C)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78cc0188b905ef850ed33a9a4068e49794712b8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.922ex; height:2.843ex;" alt="{\displaystyle (B\land C)}"></noscript><span class="lazy-image-placeholder" style="width: 7.922ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78cc0188b905ef850ed33a9a4068e49794712b8d" data-alt="{\displaystyle (B\land C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/File:Venn_1011_1011.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Venn_1011_1011.svg/50px-Venn_1011_1011.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Venn_1011_1011.svg/50px-Venn_1011_1011.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Venn_1011_1011.svg/75px-Venn_1011_1011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Venn_1011_1011.svg/100px-Venn_1011_1011.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" data-alt="{\displaystyle \Rightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_1111_1011.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Venn_1111_1011.svg/50px-Venn_1111_1011.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Venn_1111_1011.svg/50px-Venn_1111_1011.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Venn_1111_1011.svg/75px-Venn_1111_1011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Venn_1111_1011.svg/100px-Venn_1111_1011.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" data-alt="{\displaystyle \Leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0000_0101.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/50px-Venn_0000_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/50px-Venn_0000_0101.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/75px-Venn_0000_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Venn_0000_0101.svg/100px-Venn_0000_0101.svg.png 2x" data-class="mw-file-element"> </a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" data-alt="{\displaystyle \rightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn_0000_0011.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Venn_0000_0011.svg/50px-Venn_0000_0011.svg.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="200" data-file-height="200"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 50px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Venn_0000_0011.svg/50px-Venn_0000_0011.svg.png" data-width="50" data-height="50" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Venn_0000_0011.svg/75px-Venn_0000_0011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Venn_0000_0011.svg/100px-Venn_0000_0011.svg.png 2x" data-class="mw-file-element"> </a> </td></tr></tbody></table> truth-preserving: yes When all inputs are true, the output is true. <table style="text-align: center; border: 1px solid darkgray;"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></noscript><span class="lazy-image-placeholder" style="width: 6.09ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" data-alt="{\displaystyle A\land B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" data-alt="{\displaystyle \Rightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></noscript><span class="lazy-image-placeholder" style="width: 6.09ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" data-alt="{\displaystyle A\land B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/File:Venn0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/50px-Venn0001.svg.png" decoding="async" width="50" height="36" class="mw-file-element" data-file-width="410" data-file-height="299"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 36px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/50px-Venn0001.svg.png" data-width="50" data-height="36" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/75px-Venn0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/100px-Venn0001.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" data-alt="{\displaystyle \Rightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/60px-Venn0001.svg.png" decoding="async" width="60" height="44" class="mw-file-element" data-file-width="410" data-file-height="299"></noscript><span class="lazy-image-placeholder" style="width: 60px;height: 44px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/60px-Venn0001.svg.png" data-width="60" data-height="44" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/90px-Venn0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/120px-Venn0001.svg.png 2x" data-class="mw-file-element"> </a> </td></tr> <tr> <td> </td> <td> </td> <td>(to be tested) </td></tr></tbody></table> falsehood-preserving: yes When all inputs are false, the output is false. <table style="text-align: center; border: 1px solid darkgray;"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></noscript><span class="lazy-image-placeholder" style="width: 6.09ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" data-alt="{\displaystyle A\land B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" data-alt="{\displaystyle \Rightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"></noscript><span class="lazy-image-placeholder" style="width: 6.09ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" data-alt="{\displaystyle A\lor B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/File:Venn0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/60px-Venn0001.svg.png" decoding="async" width="60" height="44" class="mw-file-element" data-file-width="410" data-file-height="299"></noscript><span class="lazy-image-placeholder" style="width: 60px;height: 44px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/60px-Venn0001.svg.png" data-width="60" data-height="44" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/90px-Venn0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/120px-Venn0001.svg.png 2x" data-class="mw-file-element"> </a> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" data-alt="{\displaystyle \Rightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><a href="/wiki/File:Venn0111.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/50px-Venn0111.svg.png" decoding="async" width="50" height="37" class="mw-file-element" data-file-width="380" data-file-height="280"></noscript><span class="lazy-image-placeholder" style="width: 50px;height: 37px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/50px-Venn0111.svg.png" data-width="50" data-height="37" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/75px-Venn0111.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/100px-Venn0111.svg.png 2x" data-class="mw-file-element"> </a> </td></tr> <tr> <td>(to be tested) </td> <td> </td> <td> </td></tr></tbody></table> <a href="/wiki/Hadamard_transform" title="Hadamard transform">Walsh spectrum</a>: (1,-1,-1,1) Non<a href="/wiki/Linear#Boolean_functions" class="mw-redirect" title="Linear">linearity</a>: 1 (the function is <a href="/wiki/Bent_function" title="Bent function">bent</a>) If using <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> values for true (1) and false (0), then logical conjunction works exactly like normal arithmetic <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="Applications_in_computer_engineering">Applications in computer engineering</h2> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=10" title="Edit section: Applications in computer engineering" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:AND_Gate_diagram.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/AND_Gate_diagram.svg/220px-AND_Gate_diagram.svg.png" decoding="async" width="220" height="71" class="mw-file-element" data-file-width="721" data-file-height="232"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 71px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/AND_Gate_diagram.svg/220px-AND_Gate_diagram.svg.png" data-width="220" data-height="71" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/AND_Gate_diagram.svg/330px-AND_Gate_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/AND_Gate_diagram.svg/440px-AND_Gate_diagram.svg.png 2x" data-class="mw-file-element"> </a><figcaption>AND <a href="/wiki/Logic_gate" title="Logic gate">logic gate</a></figcaption></figure> In high-level computer programming and <a href="/wiki/Digital_electronics" title="Digital electronics">digital electronics</a>, logical conjunction is commonly represented by an infix operator, usually as a keyword such as "<code>AND</code>", an algebraic multiplication, or the ampersand symbol <code>&</code> (sometimes doubled as in <code>&&</code>). Many languages also provide <a href="/wiki/Short-circuit_evaluation" title="Short-circuit evaluation">short-circuit</a> control structures corresponding to logical conjunction. Logical conjunction is often used for bitwise operations, where <code>0</code> corresponds to false and <code>1</code> to true: <ul><li><code>0 AND 0</code> = <code>0</code>,</li> <li><code>0 AND 1</code> = <code>0</code>,</li> <li><code>1 AND 0</code> = <code>0</code>,</li> <li><code>1 AND 1</code> = <code>1</code>.</li></ul> The operation can also be applied to two binary <a href="/wiki/Words" class="mw-redirect" title="Words">words</a> viewed as <a href="/wiki/Bitstring" class="mw-redirect" title="Bitstring">bitstrings</a> of equal length, by taking the bitwise AND of each pair of bits at corresponding positions. For example: <ul><li><code>11000110 AND 10100011</code> = <code>10000010</code>.</li></ul> This can be used to select part of a bitstring using a <a href="/wiki/Mask_(computing)" title="Mask (computing)">bit mask</a>. For example, <code>10011101 AND 00001000</code> = <code>00001000</code> extracts the fourth bit of an 8-bit bitstring. In <a href="/wiki/Computer_networking" class="mw-redirect" title="Computer networking">computer networking</a>, bit masks are used to derive the network address of a <a href="/wiki/Subnetwork" class="mw-redirect" title="Subnetwork">subnet</a> within an existing network from a given <a href="/wiki/IP_address" title="IP address">IP address</a>, by ANDing the IP address and the <a href="/wiki/Subnetwork#Binary_subnet_masks" class="mw-redirect" title="Subnetwork">subnet mask</a>. Logical conjunction "<code>AND</code>" is also used in <a href="/wiki/SQL" title="SQL">SQL</a> operations to form <a href="/wiki/Database" title="Database">database</a> queries. The <a href="/wiki/Curry%E2%80%93Howard_correspondence" title="Curry–Howard correspondence">Curry–Howard correspondence</a> relates logical conjunction to <a href="/wiki/Product_type" title="Product type">product types</a>. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="Set-theoretic_correspondence">Set-theoretic correspondence</h2> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=11" title="Edit section: Set-theoretic correspondence" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> The membership of an element of an <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection set</a> in <a href="/wiki/Set_theory" title="Set theory">set theory</a> is defined in terms of a logical conjunction: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in A\cap B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in A\cap B}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb0259b3f4a3d584762f9b950f4ad35ce2a4077e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.26ex; height:2.176ex;" alt="{\displaystyle x\in A\cap B}"></noscript><span class="lazy-image-placeholder" style="width: 10.26ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb0259b3f4a3d584762f9b950f4ad35ce2a4077e" data-alt="{\displaystyle x\in A\cap B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x\in A)\wedge (x\in B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x\in A)\wedge (x\in B)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac6a265521ed874b36d1d46ad8eaef0cafcea20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.049ex; height:2.843ex;" alt="{\displaystyle (x\in A)\wedge (x\in B)}"></noscript><span class="lazy-image-placeholder" style="width: 18.049ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac6a265521ed874b36d1d46ad8eaef0cafcea20" data-alt="{\displaystyle (x\in A)\wedge (x\in B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Through this correspondence, set-theoretic intersection shares several properties with logical conjunction, such as <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a>, <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a> and <a href="/wiki/Idempotence" title="Idempotence">idempotence</a>. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><h2 id="Natural_language">Natural language</h2> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=12" title="Edit section: Natural language" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> As with other notions formalized in mathematical logic, the logical conjunction and is related to, but not the same as, the <a href="/wiki/Grammatical_conjunction" class="mw-redirect" title="Grammatical conjunction">grammatical conjunction</a> and in natural languages. English "and" has properties not captured by logical conjunction. For example, "and" sometimes implies order having the sense of "then". For example, "They got married and had a child" in common discourse means that the marriage came before the child. The word "and" can also imply a partition of a thing into parts, as "The American flag is red, white, and blue." Here, it is not meant that the flag is at once red, white, and blue, but rather that it has a part of each color. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><h2 id="See_also">See also</h2> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=13" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-9 collapsible-block" id="mf-section-9"> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/And-inverter_graph" title="And-inverter graph">And-inverter graph</a></li> <li><a href="/wiki/AND_gate" title="AND gate">AND gate</a></li> <li><a href="/wiki/Bitwise_AND" class="mw-redirect" title="Bitwise AND">Bitwise AND</a></li> <li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_conjunctive_query" title="Boolean conjunctive query">Boolean conjunctive query</a></li> <li><a href="/wiki/Boolean_domain" title="Boolean domain">Boolean domain</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean function</a></li> <li><a href="/wiki/Boolean-valued_function" title="Boolean-valued function">Boolean-valued function</a></li> <li><a href="/wiki/Conjunction/disjunction_duality" title="Conjunction/disjunction duality">Conjunction/disjunction duality</a></li> <li><a href="/wiki/Conjunction_elimination" title="Conjunction elimination">Conjunction elimination</a></li> <li><a href="/wiki/Conjunction_(grammar)" title="Conjunction (grammar)">Conjunction (grammar)</a></li> <li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Fr%C3%A9chet_inequalities" title="Fréchet inequalities">Fréchet inequalities</a></li> <li><a href="/wiki/Homogeneity_(linguistics)" class="mw-redirect" title="Homogeneity (linguistics)">Homogeneity (linguistics)</a></li> <li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">List of Boolean algebra topics</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">Logical disjunction</a></li> <li><a href="/wiki/Logical_graph" class="mw-redirect" title="Logical graph">Logical graph</a></li> <li><a href="/wiki/Negation" title="Negation">Negation</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a></li> <li><a href="/wiki/Peano%E2%80%93Russell_notation" title="Peano–Russell notation">Peano–Russell notation</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li></ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"><h2 id="References">References</h2> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=14" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-10 collapsible-block" id="mf-section-10"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-:2-1">^ <a href="#cite_ref-:2_1-0">a</a> <a href="#cite_ref-:2_1-1">b</a> <a href="#cite_ref-:2_1-2">c</a> <a 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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 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Mathematics LibreTexts. 2019-08-13. Retrieved 2020-09-02.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mathematics+LibreTexts&rft.atitle=2.2%3A+Conjunctions+and+Disjunctions&rft.date=2019-08-13&rft_id=https%3A%2F%2Fmath.libretexts.org%2FCourses%2FMonroe_Community_College%2FMTH_220_Discrete_Math%2F2%253A_Logic%2F2.2%253A_Conjunctions_and_Disjunctions&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+conjunction" class="Z3988"> </li> <li id="cite_note-:1-2">^ <a href="#cite_ref-:1_2-0">a</a> <a href="#cite_ref-:1_2-1">b</a> <a href="#cite_ref-:1_2-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://philosophy.lander.edu/logic/conjunct.html">"Conjunction, Negation, and Disjunction"</a>. philosophy.lander.edu. 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London: Routledge. p. 17. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-203-85155-5" title="Special:BookSources/978-0-203-85155-5"><bdi>978-0-203-85155-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic%3A+the+basics&rft.place=London&rft.series=The+basics&rft.pages=17&rft.edition=1.+publ&rft.pub=Routledge&rft.date=2010&rft.isbn=978-0-203-85155-5&rft.aulast=Beall&rft.aufirst=Jeffrey+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+conjunction" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="/wiki/J%C3%B3zef_Maria_Boche%C5%84ski" title="Józef Maria Bocheński">Józef Maria Bocheński</a> (1959), A Précis of Mathematical Logic, translated by Otto Bird from the French and German editions, Dordrecht, South Holland: D. Reidel, passim. </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Conjunction.html">"Conjunction"</a>. MathWorld--A Wolfram Web Resource. Retrieved 24 September 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld--A+Wolfram+Web+Resource&rft.atitle=Conjunction&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FConjunction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+conjunction" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith" class="citation web cs1">Smith, Peter. <a rel="nofollow" class="external text" href="http://www.logicmatters.net/resources/pdfs/ProofSystems.pdf">"Types of proof system"</a> (PDF). p. 4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Types+of+proof+system&rft.pages=4&rft.aulast=Smith&rft.aufirst=Peter&rft_id=http%3A%2F%2Fwww.logicmatters.net%2Fresources%2Fpdfs%2FProofSystems.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+conjunction" class="Z3988"> </li> <li id="cite_note-:13-7"><a href="#cite_ref-:13_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHowson1997" class="citation book cs1">Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. p. 38. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-415-13342-5" title="Special:BookSources/978-0-415-13342-5"><bdi>978-0-415-13342-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic+with+trees%3A+an+introduction+to+symbolic+logic&rft.place=London%3B+New+York&rft.pages=38&rft.pub=Routledge&rft.date=1997&rft.isbn=978-0-415-13342-5&rft.aulast=Howson&rft.aufirst=Colin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+conjunction" class="Z3988"> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"><h2 id="External_links">External links</h2> <a role="button" href="/w/index.php?title=Logical_conjunction&action=edit&section=15" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button 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منطقی – Persian" lang="fa" hreflang="fa" data-title="عطف منطقی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Conjonction_logique" title="Conjonction logique – French" lang="fr" hreflang="fr" data-title="Conjonction logique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%85%BC%EB%A6%AC%EA%B3%B1" title="논리곱 – Korean" lang="ko" hreflang="ko" data-title="논리곱" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%B8%D5%B6%D5%B5%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1" title="Կոնյունկցիա – Armenian" lang="hy" hreflang="hy" data-title="Կոնյունկցիա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Logika_konjungsi" title="Logika konjungsi – Indonesian" lang="id" hreflang="id" data-title="Logika konjungsi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Congiunzione_logica" title="Congiunzione logica – Italian" lang="it" hreflang="it" data-title="Congiunzione logica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%95%D7%92%D7%9D_(%D7%9C%D7%95%D7%92%D7%99%D7%A7%D7%94)" title="וגם (לוגיקה) – Hebrew" lang="he" hreflang="he" data-title="וגם (לוגיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%8A%D1%8E%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Конъюнкция – Kazakh" lang="kk" hreflang="kk" data-title="Конъюнкция" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%8A%D1%8E%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Конъюнкция – Kyrgyz" lang="ky" hreflang="ky" data-title="Конъюнкция" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Konjunkcija_(logika)" title="Konjunkcija (logika) – Lithuanian" lang="lt" hreflang="lt" data-title="Konjunkcija (logika)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Dobi" title="Dobi – Lombard" lang="lmo" hreflang="lmo" data-title="Dobi" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Konjunkci%C3%B3_(logika)" title="Konjunkció (logika) – Hungarian" lang="hu" hreflang="hu" data-title="Konjunkció (logika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D1%87%D0%BA%D0%B0_%D0%BA%D0%BE%D0%BD%D1%98%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Логичка конјункција – Macedonian" lang="mk" hreflang="mk" data-title="Логичка конјункција" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Logische_conjunctie" title="Logische conjunctie – Dutch" lang="nl" hreflang="nl" data-title="Logische conjunctie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%AB%96%E7%90%86%E7%A9%8D" title="論理積 – Japanese" lang="ja" hreflang="ja" data-title="論理積" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Konjunksjon_(logikk)" title="Konjunksjon (logikk) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Konjunksjon (logikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Congionsion" title="Congionsion – Piedmontese" lang="pms" hreflang="pms" data-title="Congionsion" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Koniunkcja_(logika)" title="Koniunkcja (logika) – Polish" lang="pl" hreflang="pl" data-title="Koniunkcja (logika)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Conjun%C3%A7%C3%A3o_l%C3%B3gica" title="Conjunção lógica – Portuguese" lang="pt" hreflang="pt" data-title="Conjunção lógica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%8A%D1%8E%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Конъюнкция – Russian" lang="ru" hreflang="ru" data-title="Конъюнкция" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Konjuksioni" title="Konjuksioni – Albanian" lang="sq" hreflang="sq" data-title="Konjuksioni" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Logical_conjunction" title="Logical conjunction – Simple English" lang="en-simple" hreflang="en-simple" data-title="Logical conjunction" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Konjunkcia_(logika)" title="Konjunkcia (logika) – Slovak" lang="sk" hreflang="sk" data-title="Konjunkcia (logika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Konjunkcija_(logika)" title="Konjunkcija (logika) – Slovenian" lang="sl" hreflang="sl" data-title="Konjunkcija (logika)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D1%87%D0%BA%D0%B0_%D0%BA%D0%BE%D0%BD%D1%98%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Логичка конјункција – Serbian" lang="sr" hreflang="sr" data-title="Логичка конјункција" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Logi%C4%8Dka_konjunkcija" title="Logička konjunkcija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Logička konjunkcija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Konjunktio_(logiikka)" title="Konjunktio (logiikka) – Finnish" lang="fi" hreflang="fi" data-title="Konjunktio (logiikka)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Konjunktion_(logik)" title="Konjunktion (logik) – Swedish" lang="sv" hreflang="sv" data-title="Konjunktion (logik)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%8A%E0%B8%B7%E0%B9%88%E0%B8%AD%E0%B8%A1%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%95%E0%B8%A3%E0%B8%A3%E0%B8%81%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C" title="การเชื่อมเชิงตรรกศาสตร์ – Thai" lang="th" hreflang="th" data-title="การเชื่อมเชิงตรรกศาสตร์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%27%D1%8E%D0%BD%D0%BA%D1%86%D1%96%D1%8F" title="Кон'юнкція – Ukrainian" lang="uk" hreflang="uk" data-title="Кон'юнкція" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%82%8F%E8%BC%AF%E8%88%87" title="邏輯與 – Cantonese" lang="yue" hreflang="yue" data-title="邏輯與" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%80%BB%E8%BE%91%E4%B8%8E" title="逻辑与 – Chinese" lang="zh" hreflang="zh" data-title="逻辑与" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 13 November 2024, at 10:29 (UTC).</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About 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