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Please help <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Logical_biconditional&action=edit">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">introducing citations to additional sources</a>.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Logical+biconditional%22">"Logical biconditional"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Logical+biconditional%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Logical+biconditional%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Logical+biconditional%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" 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.sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .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"><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="font-size: 130%; margin: 6px 0px 6px 0px; background: #ddf;"><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></th></tr><tr><td class="sidebar-content"> <table style="width:100%;border-collapse:collapse;border-spacing:0px 0px;border:none;line-height:1.3em;"><tbody><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Negation" title="Negation">NOT</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -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 -A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6daf6db742ace65252b589963f7e7a07603ccb56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.551ex; height:2.343ex;" alt="{\displaystyle -A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:3.009ex;" alt="{\displaystyle {\overline {A}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim 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 \sim A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79bf96c247833282e773fae43602343150c1665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.196ex; height:2.176ex;" alt="{\displaystyle \sim A}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_conjunction" title="Logical conjunction">AND</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\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></span><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}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot 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\cdot B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a90e903f21f11a0f4ab3caca1e6943ba7a9849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.186ex; height:2.176ex;" alt="{\displaystyle A\cdot B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\&B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi mathvariant="normal">&</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\&B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f65ee34a8896390e0d1f193c137d9eb64815c1a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.315ex; height:2.176ex;" alt="{\displaystyle A\&B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\&\&B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi mathvariant="normal">&</mi> <mi mathvariant="normal">&</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\&\&B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab3517dd859d7eaa8f3f1656c8125f99ada1470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.123ex; height:2.176ex;" alt="{\displaystyle A\&\&B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Sheffer_stroke" title="Sheffer stroke">NAND</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\overline {\land }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∧</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\overline {\land }}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0167d56141342887a74d56a036e6fbbad7172b0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.172ex; height:2.843ex;" alt="{\displaystyle A{\overline {\land }}B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\uparrow 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\uparrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5723e9ef44d446f4410c273b056d7c7c8e6f2564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.96ex; height:2.509ex;" alt="{\displaystyle A\uparrow B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mid 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\mid B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1400973074b4691cc0638a68118716a2b218fce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.444ex; height:2.843ex;" alt="{\displaystyle A\mid B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A\cdot B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A\cdot B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/225f35bb78e90b9126458f1bc6bf1ed3f0724bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.301ex; height:3.009ex;" alt="{\displaystyle {\overline {A\cdot B}}}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_disjunction" title="Logical disjunction">OR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\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></span><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}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+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+B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4279cdbd3cb8ec4c3423065d9a7d83a82cfc89e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A+B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mid 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\mid B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1400973074b4691cc0638a68118716a2b218fce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.444ex; height:2.843ex;" alt="{\displaystyle A\mid B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\parallel 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\parallel B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd0239f9b74f7ef1520ba4e30454b06e695289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.96ex; height:2.843ex;" alt="{\displaystyle A\parallel B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_NOR" title="Logical NOR">NOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\overline {\lor }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∨</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\overline {\lor }}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591afbca3e984765b18abb189f4bb1b88116c400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.172ex; height:2.843ex;" alt="{\displaystyle A{\overline {\lor }}B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\downarrow 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\downarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c5e77260e67880093dafe958880ea02f5026164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.96ex; height:2.509ex;" alt="{\displaystyle A\downarrow B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A+B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A+B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08840f8e2022f127fc459d801a8f8ce93f65f55a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.462ex; height:3.176ex;" alt="{\displaystyle {\overline {A+B}}}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/XNOR_gate" title="XNOR gate">XNOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\ {\text{XNOR}}\ B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>XNOR</mtext> </mrow> <mtext> </mtext> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\ {\text{XNOR}}\ B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54fbdc2cf9fad58dda134c0c9baa55a4712b3955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.673ex; height:2.176ex;" alt="{\displaystyle A\ {\text{XNOR}}\ B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> └ <a class="mw-selflink selflink">equivalent</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\equiv 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\equiv B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b933daba3ef47ec3b4f3097ea6e741b85149707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\equiv B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\Leftrightarrow 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\Leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce08fffb4d36ba12921b8b3e06228887015b2b8e" 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\Leftrightarrow B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightharpoons 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\leftrightharpoons B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8639312e9e120cd65c98fc48a6d5256d57288c6" 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\leftrightharpoons B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Exclusive_or" title="Exclusive or">XOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\underline {\lor }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∨</mo> <mo>_</mo> </munder> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\underline {\lor }}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803e1413692912954b90e99694e10c728e27a153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:5.06ex; height:3.176ex;" alt="{\displaystyle A{\underline {\lor }}B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\oplus 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\oplus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0512d6bdd29ff000dea0bf68b853618dcaabc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\oplus B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> └nonequivalent</td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\not \equiv 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\not \equiv B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0339ed15cd58f7263c5eec8e5628168aa6006200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.676ex;" alt="{\displaystyle A\not \equiv B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\not \Leftrightarrow 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\not \Leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47bfd97641b05a7f7fc0bcd02e83fa6532c62bb4" 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\not \Leftrightarrow B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\nleftrightarrow 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\nleftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926467de6fd4709bdbc59c3168a21298cdf0d26c" 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\nleftrightarrow B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Material_conditional" title="Material conditional">implies</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e560143d45c97e6387c7c3aa90e9d7745002228" 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}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\supset 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\supset B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee952838d8b3e67045072a8f2b71e7fc0467dea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\supset B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Converse_(logic)" title="Converse (logic)">converse</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\Leftarrow 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\Leftarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8569d34530d97f701080546ca0f20c0defadf8f" 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\Leftarrow B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset 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\subset B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftarrow 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\leftarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456da65c891c438fea04d7e40283b67d600fe92d" 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\leftarrow B}"></span></td></tr></tbody></table></td> </tr><tr><th class="sidebar-heading" style="background: #eef; text-align: center;"> Related concepts</th></tr><tr><td class="sidebar-content"> <div class="hlist" style="line-height:1.3em;"><ul><li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li><li><a href="/wiki/First-order_logic" title="First-order logic">Predicate logic</a></li><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li><li><a href="/wiki/Truth_table" title="Truth table">Truth table</a></li><li><a href="/wiki/Truth_function" title="Truth function">Truth function</a></li><li><a href="/wiki/Boolean_function" title="Boolean function">Boolean function</a></li><li><a href="/wiki/Functional_completeness" title="Functional completeness">Functional completeness</a></li><li><a href="/wiki/Scope_(logic)" title="Scope (logic)">Scope (logic)</a></li></ul></div></td> </tr><tr><th class="sidebar-heading" style="background: #eef; text-align: center;"> Applications</th></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/Logic_gate" title="Logic gate">Digital logic</a></li><li><a href="/wiki/Programming_language" title="Programming language">Programming languages</a></li><li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li><li><a href="/wiki/Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li></ul></div></td> </tr><tr><td class="sidebar-below hlist" style="background: #eef; text-align: center;"> <span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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href="/w/index.php?title=Template_talk:Logical_connectives_sidebar&action=edit&redlink=1" class="new" title="Template talk:Logical connectives sidebar (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Logical_connectives_sidebar" title="Special:EditPage/Template:Logical connectives sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Venn1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/220px-Venn1001.svg.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/330px-Venn1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/440px-Venn1001.svg.png 2x" data-file-width="410" data-file-height="299" /></a><figcaption><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\leftrightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">↔</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\leftrightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e9c3ed4d9717db81fc3794218f377bea4f6eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\leftrightarrow Q}"></span><br />(true part in red)</figcaption></figure> <p>In <a href="/wiki/Logic" title="Logic">logic</a> and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>logical biconditional</b>, also known as <b>material biconditional</b> or <b>equivalence</b> or <b>biimplication</b> or <b>bientailment</b>, is the <a href="/wiki/Logical_connective" title="Logical connective">logical connective</a> used to conjoin two statements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> to form the statement "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span>" (often abbreviated as "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> iff <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span>"<sup id="cite_ref-:2_1-0" class="reference"><a href="#cite_note-:2-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>), where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is known as the <i><a href="/wiki/Antecedent_(logic)" title="Antecedent (logic)">antecedent</a></i>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> the <i><a href="/wiki/Consequent" title="Consequent">consequent</a></i>.<sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Nowadays, notations to represent equivalence include <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftrightarrow ,\Leftrightarrow ,\equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> <mo>,</mo> <mo stretchy="false">⇔</mo> <mo>,</mo> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow ,\Leftrightarrow ,\equiv }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e5a2012c9d789f3dd98a6adcba218e91ed3e60b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.524ex; height:2.176ex;" alt="{\displaystyle \leftrightarrow ,\Leftrightarrow ,\equiv }"></span>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\leftrightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">↔</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\leftrightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e9c3ed4d9717db81fc3794218f377bea4f6eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\leftrightarrow Q}"></span> is logically equivalent to both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\rightarrow Q)\land (Q\rightarrow P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">→</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\rightarrow Q)\land (Q\rightarrow P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f32671a03a64d47feee37e84edf78d094ecce0be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.597ex; height:2.843ex;" alt="{\displaystyle (P\rightarrow Q)\land (Q\rightarrow P)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\land Q)\lor (\neg P\land \neg Q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>Q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\land Q)\lor (\neg P\land \neg Q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acba17910384f15b1f928005ea6311ad59028898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.635ex; height:2.843ex;" alt="{\displaystyle (P\land Q)\lor (\neg P\land \neg Q)}"></span>, and the <a href="/wiki/XNOR_gate" title="XNOR gate">XNOR</a> (exclusive nor) <a href="/wiki/Logical_connective" title="Logical connective">Boolean operator</a>, which means "both or neither". </p><p>Semantically, the only case where a logical biconditional is different from a <a href="/wiki/Material_conditional" title="Material conditional">material conditional</a> is the case where the hypothesis (antecedent) is false but the conclusion (consequent) is true. In this case, the result is true for the conditional, but false for the biconditional.<sup id="cite_ref-:1_2-1" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In the conceptual interpretation, <span class="texhtml"><i>P</i> = <i>Q</i></span> means "All <span class="texhtml mvar" style="font-style:italic;">P</span>'s are <span class="texhtml mvar" style="font-style:italic;">Q</span>'s and all <span class="texhtml mvar" style="font-style:italic;">Q</span>'s are <span class="texhtml mvar" style="font-style:italic;">P</span>'s". In other words, the sets <span class="texhtml mvar" style="font-style:italic;">P</span> and <span class="texhtml mvar" style="font-style:italic;">Q</span> coincide: they are identical. However, this does not mean that <span class="texhtml mvar" style="font-style:italic;">P</span> and <span class="texhtml mvar" style="font-style:italic;">Q</span> need to have the same meaning (e.g., <span class="texhtml mvar" style="font-style:italic;">P</span> could be "equiangular trilateral" and <span class="texhtml mvar" style="font-style:italic;">Q</span> could be "equilateral triangle"). When phrased as a sentence, the antecedent is the <i>subject</i> and the consequent is the <i>predicate</i> of a <a href="/wiki/Universal_affirmative" class="mw-redirect" title="Universal affirmative">universal affirmative</a> proposition (e.g., in the phrase "all men are mortal", "men" is the subject and "mortal" is the predicate). </p><p>In the propositional interpretation, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\leftrightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">↔</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\leftrightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e9c3ed4d9717db81fc3794218f377bea4f6eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\leftrightarrow Q}"></span> means that <span class="texhtml mvar" style="font-style:italic;">P</span> implies <span class="texhtml mvar" style="font-style:italic;">Q</span> and <span class="texhtml mvar" style="font-style:italic;">Q</span> implies <span class="texhtml mvar" style="font-style:italic;">P</span>; in other words, the propositions are <a href="/wiki/Logically_equivalent" class="mw-redirect" title="Logically equivalent">logically equivalent</a>, in the sense that both are either jointly true or jointly false. Again, this does not mean that they need to have the same meaning, as <span class="texhtml mvar" style="font-style:italic;">P</span> could be "the triangle ABC has two equal sides" and <span class="texhtml mvar" style="font-style:italic;">Q</span> could be "the triangle ABC has two equal angles". In general, the antecedent is the <i>premise</i>, or the <i>cause</i>, and the consequent is the <i>consequence</i>. When an implication is translated by a <i>hypothetical</i> (or <i>conditional</i>) judgment, the antecedent is called the <i>hypothesis</i> (or the <i>condition</i>) and the consequent is called the <i>thesis</i>. </p><p>A common way of demonstrating a biconditional of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\leftrightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">↔</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\leftrightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e9c3ed4d9717db81fc3794218f377bea4f6eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\leftrightarrow Q}"></span> is to demonstrate that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86439ea857adc8eaec93c4d14270b8ba6bd2a6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q\rightarrow P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">→</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q\rightarrow P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02fdef5455a1bfa49458c8d9821b1211b1e6e2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle Q\rightarrow P}"></span> separately (due to its equivalence to the conjunction of the two converse <a href="/wiki/Material_conditional" title="Material conditional">conditionals</a><sup id="cite_ref-:1_2-2" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>). Yet another way of demonstrating the same biconditional is by demonstrating that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86439ea857adc8eaec93c4d14270b8ba6bd2a6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P\rightarrow \neg Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P\rightarrow \neg Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dddf975e2d6626e24fbb8f41e7a2b16c592adcca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.298ex; height:2.509ex;" alt="{\displaystyle \neg P\rightarrow \neg Q}"></span>. </p><p>When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called a <i>theorem</i> and the other its <i>reciprocal</i>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2008)">citation needed</span></a></i>]</sup> Thus whenever a theorem and its reciprocal are true, we have a biconditional. A simple theorem gives rise to an implication, whose antecedent is the <i>hypothesis</i> and whose consequent is the <i>thesis</i> of the theorem. </p><p>It is often said that the hypothesis is the <i><a href="/wiki/Sufficient_condition" class="mw-redirect" title="Sufficient condition">sufficient condition</a></i> of the thesis, and that the thesis is the <i><a href="/wiki/Necessary_condition" class="mw-redirect" title="Necessary condition">necessary condition</a></i> of the hypothesis. That is, it is sufficient that the hypothesis be true for the thesis to be true, while it is necessary that the thesis be true if the hypothesis were true. When a theorem and its reciprocal are true, its hypothesis is said to be the <a href="/wiki/Necessary_and_sufficient_condition" class="mw-redirect" title="Necessary and sufficient condition">necessary and sufficient condition</a> of the thesis. That is, the hypothesis is both the cause and the consequence of the thesis at the same time. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notations">Notations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_biconditional&action=edit&section=1" title="Edit section: Notations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Notations to represent equivalence used in history include: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> in <a href="/wiki/George_Boole" title="George Boole">George Boole</a> in 1847.<sup id="cite_ref-boole1847_4-0" class="reference"><a href="#cite_note-boole1847-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Although Boole used <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> mainly on classes, he also considered the case that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></span> are propositions in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/409a91214d63eabe46ec10ff3cbba689ab687366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}"></span>, and at the time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> is equivalence.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"></span> in <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a> in 1879;<sup id="cite_ref-frege1879b_5-0" class="reference"><a href="#cite_note-frege1879b-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span> in <a href="/wiki/Paul_Bernays" title="Paul Bernays">Bernays</a> in 1918;<sup id="cite_ref-bernays1918_6-0" class="reference"><a href="#cite_note-bernays1918-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightleftarrows }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇄</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightleftarrows }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbaea31afa0894aa82079b04d6a5e7566da7195" 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 \rightleftarrows }"></span> in <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a> in 1927 (while he used <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span> as the main symbol in the article);<sup id="cite_ref-hilbert1927_7-0" class="reference"><a href="#cite_note-hilbert1927-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" 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 }"></span> in <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a> and <a href="/wiki/Wilhelm_Ackermann" title="Wilhelm Ackermann">Ackermann</a> in 1928<sup id="cite_ref-hilbert-ackermann1928_8-0" class="reference"><a href="#cite_note-hilbert-ackermann1928-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> (they also introduced <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightleftarrows ,\sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇄</mo> <mo>,</mo> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightleftarrows ,\sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8836b8da4243c7fb791d4e2e138bf4a7608e5a73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.166ex; height:2.509ex;" alt="{\displaystyle \rightleftarrows ,\sim }"></span> while they use <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span> as the main symbol in the whole book; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" 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 }"></span> is adopted by many followers such as Becker in 1933<sup id="cite_ref-becker1933_9-0" class="reference"><a href="#cite_note-becker1933-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>);</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> (prefix) in <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Łukasiewicz</a> in 1929<sup id="cite_ref-lukasiewicz1929_10-0" class="reference"><a href="#cite_note-lukasiewicz1929-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> (prefix) in <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Łukasiewicz</a> in 1951;<sup id="cite_ref-lukasiewicz1951_11-0" class="reference"><a href="#cite_note-lukasiewicz1951-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \supset \subset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊃⊂</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supset \subset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e744935a1dc666f388c853967cf5f812c2ace5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:1.843ex;" alt="{\displaystyle \supset \subset }"></span> in <a href="/wiki/Arend_Heyting" title="Arend Heyting">Heyting</a> in 1930;<sup id="cite_ref-heyting1929_12-0" class="reference"><a href="#cite_note-heyting1929-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span> in <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a> in 1954;<sup id="cite_ref-bourbaki1954b_13-0" class="reference"><a href="#cite_note-bourbaki1954b-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subset \supset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊂⊃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subset \supset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9dc56fa9c4547807a5ad92bec8bd59c82a34aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:1.843ex;" alt="{\displaystyle \subset \supset }"></span> in Chazal in 1996;<sup id="cite_ref-chazal1996_14-0" class="reference"><a href="#cite_note-chazal1996-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li></ul> <p>and so on. Somebody else also use <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {EQ} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>EQ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {EQ} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec11c874f38f6e0c6852816d706d212d880075b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.391ex; height:2.509ex;" alt="{\displaystyle \operatorname {EQ} }"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {EQV} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>EQV</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {EQV} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3902fb3f423ae55ec0eb65d2c541eca0077d3830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.134ex; height:2.509ex;" alt="{\displaystyle \operatorname {EQV} }"></span> occasionally.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2023)">citation needed</span></a></i>]</sup><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Vagueness" title="Wikipedia:Vagueness"><span title="This information is too vague. (September 2023)">vague</span></a></i>]</sup><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (September 2023)">clarification needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_biconditional&action=edit&section=2" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Logical_equality" title="Logical equality">Logical equality</a> (also known as biconditional) 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 <i>true</i> if and only if both operands are false or both operands are true.<sup id="cite_ref-:1_2-3" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Truth_table">Truth table</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_biconditional&action=edit&section=3" title="Edit section: Truth table"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following is a truth table for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow 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\leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" 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\leftrightarrow B}"></span>: </p> <style data-mw-deduplicate="TemplateStyles:r1259796377">.mw-parser-output .two-ary-truth-table th{font-weight:normal}.mw-parser-output .two-ary-truth-table abbr{text-decoration:none}.mw-parser-output .two-ary-truth-table tr td:nth-child(1),.mw-parser-output .two-ary-truth-table tr td:nth-child(2){font-weight:bold}.mw-parser-output .two-ary-truth-table td{text-align:center;padding-left:14px;padding-right:14px}.mw-parser-output .two-ary-truth-table-false{background-color:var(--background-color-base,#fff)}.mw-parser-output .two-ary-truth-table-true{background-color:hsl(0,100%,90%)}.mw-parser-output .two-ary-truth-table-border{border-left:2px solid var(--border-color-interactive,#72777d)}@media screen{html.skin-theme-clientpref-night .mw-parser-output .two-ary-truth-table-true{background-color:hsl(0,100%,10%)}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .two-ary-truth-table-true{background-color:hsl(0,100%,10%)}}</style><table class="wikitable sortable two-ary-truth-table"><tbody><tr><th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></th><th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span></th><th class="unsortable two-ary-truth-table-border"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow 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\leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" 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\leftrightarrow B}"></span></th></tr><tr><td class="two-ary-truth-table-false"><abbr title="false">F</abbr></td><td class="two-ary-truth-table-false"><abbr title="false">F</abbr></td><td class="two-ary-truth-table-border two-ary-truth-table-true"><abbr title="true">T</abbr></td></tr><tr><td class="two-ary-truth-table-false"><abbr title="false">F</abbr></td><td class="two-ary-truth-table-true"><abbr title="true">T</abbr></td><td class="two-ary-truth-table-border two-ary-truth-table-false"><abbr title="false">F</abbr></td></tr><tr><td class="two-ary-truth-table-true"><abbr title="true">T</abbr></td><td class="two-ary-truth-table-false"><abbr title="false">F</abbr></td><td class="two-ary-truth-table-border two-ary-truth-table-false"><abbr title="false">F</abbr></td></tr><tr><td class="two-ary-truth-table-true"><abbr title="true">T</abbr></td><td class="two-ary-truth-table-true"><abbr title="true">T</abbr></td><td class="two-ary-truth-table-border two-ary-truth-table-true"><abbr title="true">T</abbr></td></tr></tbody></table> <div style="clear:both;" class=""></div> <p>When more than two statements are involved, combining them with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" 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 }"></span> might be ambiguous. For example, the statement </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}\leftrightarrow x_{2}\leftrightarrow x_{3}\leftrightarrow \cdots \leftrightarrow x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">↔</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">↔</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">↔</mo> <mo>⋯</mo> <mo stretchy="false">↔</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}\leftrightarrow x_{2}\leftrightarrow x_{3}\leftrightarrow \cdots \leftrightarrow x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7c5f468c31c11cad0ebca97c0caab996229a3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.879ex; height:2.176ex;" alt="{\displaystyle x_{1}\leftrightarrow x_{2}\leftrightarrow x_{3}\leftrightarrow \cdots \leftrightarrow x_{n}}"></span></dd></dl> <p>may be interpreted as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (((x_{1}\leftrightarrow x_{2})\leftrightarrow x_{3})\leftrightarrow \cdots )\leftrightarrow x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">↔</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo>⋯</mo> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (((x_{1}\leftrightarrow x_{2})\leftrightarrow x_{3})\leftrightarrow \cdots )\leftrightarrow x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e5169b6993f5fcdd297747e3565f850afb1d37f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.307ex; height:2.843ex;" alt="{\displaystyle (((x_{1}\leftrightarrow x_{2})\leftrightarrow x_{3})\leftrightarrow \cdots )\leftrightarrow x_{n}}"></span>,</dd></dl> <p>or may be interpreted as saying that all <span class="texhtml"><i>x<sub>i</sub></i></span> are <i>jointly true or jointly false</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1}\land \cdots \land x_{n})\lor (\neg x_{1}\land \cdots \land \neg x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1}\land \cdots \land x_{n})\lor (\neg x_{1}\land \cdots \land \neg x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33a6687f6655f1609a0370af236e801ea884747c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.943ex; height:2.843ex;" alt="{\displaystyle (x_{1}\land \cdots \land x_{n})\lor (\neg x_{1}\land \cdots \land \neg x_{n})}"></span></dd></dl> <p>As it turns out, these two statements are only the same when zero or two arguments are involved. In fact, the following truth tables only show the same bit pattern in the line with no argument and in the lines with two arguments: </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Variadic_logical_XAND.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Variadic_logical_XAND.svg/220px-Variadic_logical_XAND.svg.png" decoding="async" width="220" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Variadic_logical_XAND.svg/330px-Variadic_logical_XAND.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Variadic_logical_XAND.svg/440px-Variadic_logical_XAND.svg.png 2x" data-file-width="1807" data-file-height="1452" /></a><figcaption><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~x_{1}\leftrightarrow \cdots \leftrightarrow x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">↔</mo> <mo>⋯</mo> <mo stretchy="false">↔</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~x_{1}\leftrightarrow \cdots \leftrightarrow x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c4edb2ff1e1fa91f34cfa184d3a66c7cd9a84e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.464ex; height:2.176ex;" alt="{\displaystyle ~x_{1}\leftrightarrow \cdots \leftrightarrow x_{n}}"></span><br />meant as equivalent to<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg ~(\neg x_{1}\oplus \cdots \oplus \neg x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <mo>⋯</mo> <mo>⊕</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg ~(\neg x_{1}\oplus \cdots \oplus \neg x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e04d657c4f822dde5bbbf0972c1d619c0245fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.377ex; height:2.843ex;" alt="{\displaystyle \neg ~(\neg x_{1}\oplus \cdots \oplus \neg x_{n})}"></span><br /><br />The central Venn diagram below,<br />and line <i>(ABC  )</i> in this matrix<br />represent the same operation.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Variadic_logical_all_or_nothing.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Variadic_logical_all_or_nothing.svg/220px-Variadic_logical_all_or_nothing.svg.png" decoding="async" width="220" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Variadic_logical_all_or_nothing.svg/330px-Variadic_logical_all_or_nothing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Variadic_logical_all_or_nothing.svg/440px-Variadic_logical_all_or_nothing.svg.png 2x" data-file-width="1807" data-file-height="1452" /></a><figcaption><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~x_{1}\leftrightarrow \cdots \leftrightarrow x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">↔</mo> <mo>⋯</mo> <mo stretchy="false">↔</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~x_{1}\leftrightarrow \cdots \leftrightarrow x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c4edb2ff1e1fa91f34cfa184d3a66c7cd9a84e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.464ex; height:2.176ex;" alt="{\displaystyle ~x_{1}\leftrightarrow \cdots \leftrightarrow x_{n}}"></span><br />meant as shorthand for<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (~x_{1}\land \cdots \land x_{n}~)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (~x_{1}\land \cdots \land x_{n}~)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55635b8fd03cf4664c9c26e5e23a2564fdf86fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.791ex; height:2.843ex;" alt="{\displaystyle (~x_{1}\land \cdots \land x_{n}~)}"></span><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor ~(\neg x_{1}\land \cdots \land \neg x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor ~(\neg x_{1}\land \cdots \land \neg x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/551d41e22df1dabe316a2c0169a2c7234b64c535" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.861ex; height:2.843ex;" alt="{\displaystyle \lor ~(\neg x_{1}\land \cdots \land \neg x_{n})}"></span><br /><br />The Venn diagram directly below,<br />and line <i>(ABC  )</i> in this matrix<br />represent the same operation.</figcaption></figure> <div style="clear:both;" class=""></div> <p>The left Venn diagram below, and the lines <i>(AB    )</i> in these matrices represent the same operation. </p> <div class="mw-heading mw-heading3"><h3 id="Venn_diagrams">Venn diagrams</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_biconditional&action=edit&section=4" title="Edit section: Venn diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Red areas stand for true (as in <span typeof="mw:File"><a href="/wiki/File:Venn0001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/40px-Venn0001.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/60px-Venn0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/80px-Venn0001.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span> for <i><a href="/wiki/Logical_conjunction" title="Logical conjunction">and</a></i>). </p> <table border="0" style="width:100%"> <tbody><tr> <td style="vertical-align:top;"> <table style="background:#f9f9f9; border:1px solid #ccc; float:left;"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Venn1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/220px-Venn1001.svg.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/330px-Venn1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/440px-Venn1001.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span> </td></tr> <tr> <td>The biconditional of two statements<br />is the <a href="/wiki/Negation" title="Negation">negation</a> of the <a href="/wiki/Exclusive_or" title="Exclusive or">exclusive or</a>: </td></tr> <tr style="text-align:center;"> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A\leftrightarrow B~~\Leftrightarrow ~~\neg (A\oplus B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">⇔</mo> <mtext> </mtext> <mtext> </mtext> <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\leftrightarrow B~~\Leftrightarrow ~~\neg (A\oplus B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6867f5b0dda0ff788d2db0ce999bdd9952e73a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.346ex; height:2.843ex;" alt="{\displaystyle ~A\leftrightarrow B~~\Leftrightarrow ~~\neg (A\oplus B)}"></span> <p><span typeof="mw:File"><a href="/wiki/File:Venn1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/40px-Venn1001.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/60px-Venn1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/80px-Venn1001.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8267a60d507fc47c33b38ddf5b4ad83e9ed39114" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.519ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow \neg }"></span> <span typeof="mw:File"><a href="/wiki/File:Venn0110.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/40px-Venn0110.svg.png" decoding="async" width="40" height="29" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/60px-Venn0110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/80px-Venn0110.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span> </p> </td></tr></tbody></table> </td> <td style="width: 100px"> </td> <td style="vertical-align:top;"> <table style="background:#f9f9f9; border:1px solid #ccc; margin:auto;"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Venn_0110_1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/220px-Venn_0110_1001.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/330px-Venn_0110_1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/440px-Venn_0110_1001.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </td></tr> <tr> <td>The biconditional and the<br />exclusive or of three statements<br />give the same result:<br /> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A\leftrightarrow B\leftrightarrow C~~\Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> <mo stretchy="false">↔</mo> <mi>C</mi> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A\leftrightarrow B\leftrightarrow C~~\Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae61c1784544263137c6c4bd42f641efa9b15e2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.212ex; height:2.176ex;" alt="{\displaystyle ~A\leftrightarrow B\leftrightarrow C~~\Leftrightarrow }"></span><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A\oplus B\oplus C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> <mo>⊕</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A\oplus B\oplus C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f6a3c5959fdbf878afeffb60eff4001543bcd9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.535ex; height:2.343ex;" alt="{\displaystyle ~A\oplus B\oplus C}"></span> </p><p><span typeof="mw:File"><a href="/wiki/File:Venn_1001_1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Venn_1001_1001.svg/40px-Venn_1001_1001.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Venn_1001_1001.svg/60px-Venn_1001_1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Venn_1001_1001.svg/80px-Venn_1001_1001.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" 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 }"></span> <span typeof="mw:File"><a href="/wiki/File:Venn_0000_1111.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/40px-Venn_0000_1111.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/60px-Venn_0000_1111.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/80px-Venn_0000_1111.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~\Leftrightarrow ~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">⇔</mo> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~\Leftrightarrow ~~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae72374044e6fde6bc190024377c814ed6a0fba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.937ex; height:1.843ex;" alt="{\displaystyle ~~\Leftrightarrow ~~}"></span> </p><p><span typeof="mw:File"><a href="/wiki/File:Venn_0110_0110.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Venn_0110_0110.svg/40px-Venn_0110_0110.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Venn_0110_0110.svg/60px-Venn_0110_0110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Venn_0110_0110.svg/80px-Venn_0110_0110.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span> <span typeof="mw:File"><a href="/wiki/File:Venn_0000_1111.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/40px-Venn_0000_1111.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/60px-Venn_0000_1111.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Venn_0000_1111.svg/80px-Venn_0000_1111.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~\Leftrightarrow ~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">⇔</mo> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~\Leftrightarrow ~~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae72374044e6fde6bc190024377c814ed6a0fba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.937ex; height:1.843ex;" alt="{\displaystyle ~~\Leftrightarrow ~~}"></span> <span typeof="mw:File"><a href="/wiki/File:Venn_0110_1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/40px-Venn_0110_1001.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/60px-Venn_0110_1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/80px-Venn_0110_1001.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </p> </td></tr></tbody></table> </td> <td style="width: 100px"> </td> <td style="vertical-align:top;"> <table style="background:#f9f9f9; border:1px solid #ccc; float:right;"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Venn_1000_0001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Venn_1000_0001.svg/220px-Venn_1000_0001.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Venn_1000_0001.svg/330px-Venn_1000_0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Venn_1000_0001.svg/440px-Venn_1000_0001.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </td></tr> <tr> <td>But <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A\leftrightarrow B\leftrightarrow C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> <mo stretchy="false">↔</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A\leftrightarrow B\leftrightarrow C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a3e2b7dcae3446f13e8a34d0ed87623ddfafa16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.082ex; height:2.176ex;" alt="{\displaystyle ~A\leftrightarrow B\leftrightarrow C}"></span><br />may also be used as an abbreviation<br />for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\leftrightarrow B)\land (B\leftrightarrow C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">↔</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\leftrightarrow B)\land (B\leftrightarrow C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9106e5ccbc0e00904454ba568f14d2287bd79ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.467ex; height:2.843ex;" alt="{\displaystyle (A\leftrightarrow B)\land (B\leftrightarrow C)}"></span> <p><span typeof="mw:File"><a href="/wiki/File:Venn_1001_1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Venn_1001_1001.svg/40px-Venn_1001_1001.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Venn_1001_1001.svg/60px-Venn_1001_1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Venn_1001_1001.svg/80px-Venn_1001_1001.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span> <span typeof="mw:File"><a href="/wiki/File:Venn_1100_0011.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Venn_1100_0011.svg/40px-Venn_1100_0011.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Venn_1100_0011.svg/60px-Venn_1100_0011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Venn_1100_0011.svg/80px-Venn_1100_0011.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~\Leftrightarrow ~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">⇔</mo> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~\Leftrightarrow ~~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae72374044e6fde6bc190024377c814ed6a0fba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.937ex; height:1.843ex;" alt="{\displaystyle ~~\Leftrightarrow ~~}"></span> <span typeof="mw:File"><a href="/wiki/File:Venn_1000_0001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Venn_1000_0001.svg/40px-Venn_1000_0001.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Venn_1000_0001.svg/60px-Venn_1000_0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Venn_1000_0001.svg/80px-Venn_1000_0001.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </p> </td></tr></tbody></table> </td></tr></tbody></table> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_biconditional&action=edit&section=5" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">Commutativity</a>: Yes</b> </p> <table style="text-align:center; border:1px solid darkgrey;"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow 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\leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" 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\leftrightarrow B}"></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span>     </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\leftrightarrow A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">↔</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\leftrightarrow A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7af99ff4fc686b0e512ce505333c779a8b639ba" 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 B\leftrightarrow A}"></span> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Venn1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/50px-Venn1001.svg.png" decoding="async" width="50" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/75px-Venn1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/100px-Venn1001.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span>     </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/50px-Venn1001.svg.png" decoding="async" width="50" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/75px-Venn1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/100px-Venn1001.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span> </td></tr></tbody></table> <p><b><a href="/wiki/Associativity" class="mw-redirect" title="Associativity">Associativity</a>: Yes</b> </p> <table style="text-align:center; border:1px solid darkgrey;"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~~\leftrightarrow ~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">↔</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~~\leftrightarrow ~~~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baea3f92a48d748936a8f7e89ea640135896686b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.098ex; height:1.843ex;" alt="{\displaystyle ~~~\leftrightarrow ~~~}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\leftrightarrow C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">↔</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\leftrightarrow C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47284c8cd91f50f54b76c5d3d7591c4d2848a047" 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\leftrightarrow C)}"></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span>     </td> <td> </td> <td> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\leftrightarrow B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\leftrightarrow B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377a507f43f614500c9c326f3135fbb062cd5fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.93ex; height:2.843ex;" alt="{\displaystyle (A\leftrightarrow B)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~~\leftrightarrow ~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">↔</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~~\leftrightarrow ~~~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baea3f92a48d748936a8f7e89ea640135896686b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.098ex; height:1.843ex;" alt="{\displaystyle ~~~\leftrightarrow ~~~}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~C}"> <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></span><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}"></span> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Venn_0101_0101.svg" class="mw-file-description"><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" 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-file-width="200" data-file-height="200" /></a></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~~\leftrightarrow ~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">↔</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~~\leftrightarrow ~~~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baea3f92a48d748936a8f7e89ea640135896686b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.098ex; height:1.843ex;" alt="{\displaystyle ~~~\leftrightarrow ~~~}"></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn_1100_0011.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Venn_1100_0011.svg/50px-Venn_1100_0011.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Venn_1100_0011.svg/75px-Venn_1100_0011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Venn_1100_0011.svg/100px-Venn_1100_0011.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span>     </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn_0110_1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/50px-Venn_0110_1001.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/75px-Venn_0110_1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Venn_0110_1001.svg/100px-Venn_0110_1001.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span>     </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn_1001_1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Venn_1001_1001.svg/50px-Venn_1001_1001.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Venn_1001_1001.svg/75px-Venn_1001_1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Venn_1001_1001.svg/100px-Venn_1001_1001.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~~\leftrightarrow ~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">↔</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~~\leftrightarrow ~~~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baea3f92a48d748936a8f7e89ea640135896686b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.098ex; height:1.843ex;" alt="{\displaystyle ~~~\leftrightarrow ~~~}"></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn_0000_1111.svg" class="mw-file-description"><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" 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-file-width="200" data-file-height="200" /></a></span> </td></tr></tbody></table> <p><b><a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">Distributivity</a>:</b> Biconditional doesn't distribute over any binary function (not even itself), but <a href="/wiki/Logical_disjunction#Properties" title="Logical disjunction">logical disjunction distributes</a> over biconditional. </p><p><b><a href="/wiki/Idempotency" class="mw-redirect" title="Idempotency">Idempotency</a>: No</b><br /> </p> <table style="text-align:center; border:1px solid darkgrey;"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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~}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\leftrightarrow ~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">↔</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\leftrightarrow ~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee3507fd52f15d937cd86ae23b11b30ad61d177e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.775ex; height:1.843ex;" alt="{\displaystyle ~\leftrightarrow ~}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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~}"></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span>     </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~1~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~1~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12c70b7208ccc4ef9a41069267824959699f37f0" 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 ~1~}"></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nLeftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⇎</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nLeftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeebe3d2aa323d1bd18a7c3ba7ff3179f7931471" 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 \nLeftrightarrow }"></span>     </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~A~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~A~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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~}"></span> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Venn01.svg" class="mw-file-description"><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" 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-file-width="280" data-file-height="280" /></a></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\leftrightarrow ~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">↔</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\leftrightarrow ~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee3507fd52f15d937cd86ae23b11b30ad61d177e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.775ex; height:1.843ex;" alt="{\displaystyle ~\leftrightarrow ~}"></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn01.svg" class="mw-file-description"><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" 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-file-width="280" data-file-height="280" /></a></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span>     </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn11.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Venn11.svg/36px-Venn11.svg.png" decoding="async" width="36" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Venn11.svg/54px-Venn11.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Venn11.svg/72px-Venn11.svg.png 2x" data-file-width="280" data-file-height="280" /></a></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nLeftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⇎</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nLeftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeebe3d2aa323d1bd18a7c3ba7ff3179f7931471" 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 \nLeftrightarrow }"></span>     </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn01.svg" class="mw-file-description"><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" 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-file-width="280" data-file-height="280" /></a></span> </td></tr></tbody></table> <p><b><a href="/wiki/Monotone_Boolean_function" class="mw-redirect" title="Monotone Boolean function">Monotonicity</a>: No</b> </p> <table style="text-align:center; border:1px solid darkgrey;"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\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></span><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}"></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e05a42e88f019861cf404b6f982d8f729a4c4ab" 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 \nRightarrow }"></span>     </td> <td> </td> <td> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\leftrightarrow 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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\leftrightarrow C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b880edcbd441046d3aa4da7176c8e9a24920df5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.933ex; height:2.843ex;" alt="{\displaystyle (A\leftrightarrow C)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\leftrightarrow C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">↔</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\leftrightarrow C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47284c8cd91f50f54b76c5d3d7591c4d2848a047" 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\leftrightarrow C)}"></span> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Venn_1011_1011.svg" class="mw-file-description"><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" 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-file-width="200" data-file-height="200" /></a></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e05a42e88f019861cf404b6f982d8f729a4c4ab" 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 \nRightarrow }"></span>     </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn_1101_1011.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Venn_1101_1011.svg/50px-Venn_1101_1011.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Venn_1101_1011.svg/75px-Venn_1101_1011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Venn_1101_1011.svg/100px-Venn_1101_1011.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span>     </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn_1010_0101.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Venn_1010_0101.svg/50px-Venn_1010_0101.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Venn_1010_0101.svg/75px-Venn_1010_0101.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Venn_1010_0101.svg/100px-Venn_1010_0101.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn_1100_0011.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Venn_1100_0011.svg/50px-Venn_1100_0011.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Venn_1100_0011.svg/75px-Venn_1100_0011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Venn_1100_0011.svg/100px-Venn_1100_0011.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </td></tr></tbody></table> <p><b>Truth-preserving: Yes</b><br /> When all inputs are true, the output is true. </p> <table style="text-align:center; border:1px solid darkgrey;"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\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></span><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}"></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span>     </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow 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\leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" 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\leftrightarrow B}"></span> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Venn0001.svg" class="mw-file-description"><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" 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-file-width="410" data-file-height="299" /></a></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span>     </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/60px-Venn1001.svg.png" decoding="async" width="60" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/90px-Venn1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/120px-Venn1001.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span> </td></tr></tbody></table> <p><b>Falsehood-preserving: No</b><br /> When all inputs are false, the output is not false. </p> <table style="text-align:center; border:1px solid darkgrey;"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow 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\leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" 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\leftrightarrow B}"></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e05a42e88f019861cf404b6f982d8f729a4c4ab" 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 \nRightarrow }"></span>     </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\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></span><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}"></span> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Venn1001.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/60px-Venn1001.svg.png" decoding="async" width="60" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/90px-Venn1001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Venn1001.svg/120px-Venn1001.svg.png 2x" data-file-width="410" data-file-height="299" /></a></span> </td> <td>    <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e05a42e88f019861cf404b6f982d8f729a4c4ab" 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 \nRightarrow }"></span>     </td> <td><span typeof="mw:File"><a href="/wiki/File:Venn0111.svg" class="mw-file-description"><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" 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-file-width="380" data-file-height="280" /></a></span> </td></tr></tbody></table> <p><b><a href="/wiki/Hadamard_transform" title="Hadamard transform">Walsh spectrum</a>: (2,0,0,2)</b> </p><p><b>Non<a href="/wiki/Linear#Boolean_functions" class="mw-redirect" title="Linear">linearity</a>: 0</b> (the function is linear) </p> <div class="mw-heading mw-heading2"><h2 id="Rules_of_inference">Rules of inference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_biconditional&action=edit&section=6" title="Edit section: Rules of inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Rules_of_inference" class="mw-redirect" title="Rules of inference">Rules of inference</a></div> <p>Like all connectives in first-order logic, the biconditional has rules of inference that govern its use in formal proofs. </p> <div class="mw-heading mw-heading3"><h3 id="Biconditional_introduction">Biconditional introduction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_biconditional&action=edit&section=7" title="Edit section: Biconditional introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Biconditional_introduction" title="Biconditional introduction">Biconditional introduction</a></div> <p>Biconditional introduction allows one to infer that if B follows from A and A follows from B, then A <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> B. </p><p>For example, from the statements "if I'm breathing, then I'm alive" and "if I'm alive, then I'm breathing", it can be inferred that "I'm breathing if and only if I'm alive" or equivalently, "I'm alive if and only if I'm breathing." Or more schematically: </p> <pre> B → A    <u> A → B   </u> ∴ A ↔ B </pre> <pre> B → A    <u> A → B   </u> ∴ B ↔ A </pre> <div class="mw-heading mw-heading3"><h3 id="Biconditional_elimination">Biconditional elimination</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_biconditional&action=edit&section=8" title="Edit section: Biconditional elimination"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Biconditional elimination allows one to infer a <a href="/wiki/Material_conditional" title="Material conditional">conditional</a> from a biconditional: if A <small>↔</small> B is true, then one may infer either A <small>→</small> B, or B <small>→</small> A. </p><p>For example, if it is true that I'm breathing <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> I'm alive, then it's true that <i>if</i> I'm breathing, then I'm alive; likewise, it's true that <i>if</i> I'm alive, then I'm breathing. Or more schematically: </p> <pre> <u>A ↔ B  </u> ∴ A → B </pre> <pre> <u>A ↔ B  </u> ∴ B → A </pre> <div class="mw-heading mw-heading2"><h2 id="Colloquial_usage">Colloquial usage</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_biconditional&action=edit&section=9" title="Edit section: Colloquial usage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One unambiguous way of stating a biconditional in plain English is to adopt the form "<i>b</i> if <i>a</i> and <i>a</i> if <i>b</i>"—if the standard form "<i>a</i> if and only if <i>b</i>" is not used. Slightly more formally, one could also say that "<i>b</i> implies <i>a</i> and <i>a</i> implies <i>b</i>", or "<i>a</i> is necessary and sufficient for <i>b</i>". The plain English "if'" may sometimes be used as a biconditional (especially in the context of a mathematical definition<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>). In which case, one must take into consideration the surrounding context when interpreting these words. </p><p>For example, the statement "I'll buy you a new wallet if you need one" may be interpreted as a biconditional, since the speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional). However, "it is cloudy if it is raining" is generally not meant as a biconditional, since it can still be cloudy even if it is not raining. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_biconditional&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output 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href="/wiki/Logical_equality" title="Logical equality">Logical equality</a></li> <li><a href="/wiki/XNOR_gate" title="XNOR gate">XNOR gate</a></li> <li><a href="/wiki/Biconditional_elimination" title="Biconditional elimination">Biconditional elimination</a></li> <li><a href="/wiki/Biconditional_introduction" title="Biconditional introduction">Biconditional introduction</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_biconditional&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:2-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-:2_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Iff.html">"Iff"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Iff&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FIff.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-:1-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:1_2-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeil" class="citation web cs1">Peil, Timothy. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201024171606/http://web.mnstate.edu/peil/geometry/Logic/4logic.htm">"Conditionals and Biconditionals"</a>. <i>web.mnstate.edu</i>. Archived from <a rel="nofollow" class="external text" href="http://web.mnstate.edu/peil/geometry/Logic/4logic.htm">the original</a> on 2020-10-24<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=web.mnstate.edu&rft.atitle=Conditionals+and+Biconditionals&rft.aulast=Peil&rft.aufirst=Timothy&rft_id=http%3A%2F%2Fweb.mnstate.edu%2Fpeil%2Fgeometry%2FLogic%2F4logic.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrennan1961" class="citation book cs1">Brennan, Joseph G. (1961). <i>Handbook of Logic</i> (2nd ed.). Harper & Row. p. 81.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Logic&rft.pages=81&rft.edition=2nd&rft.pub=Harper+%26+Row&rft.date=1961&rft.aulast=Brennan&rft.aufirst=Joseph+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-boole1847-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-boole1847_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoole1847" class="citation book cs1">Boole, G. (1847). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalanal00booluoft"><i>The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning</i></a>. Cambridge/London: Macmillan, Barclay, & Macmillan/George Bell. p. 17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Analysis+of+Logic%2C+Being+an+Essay+Towards+a+Calculus+of+Deductive+Reasoning&rft.place=Cambridge%2FLondon&rft.pages=17&rft.pub=Macmillan%2C+Barclay%2C+%26+Macmillan%2FGeorge+Bell&rft.date=1847&rft.aulast=Boole&rft.aufirst=G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalanal00booluoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-frege1879b-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-frege1879b_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrege1879" class="citation book cs1 cs1-prop-foreign-lang-source">Frege, G. (1879). <i>Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens</i> (in German). Halle a/S.: Verlag von Louis Nebert. p. 15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Begriffsschrift%2C+eine+der+arithmetischen+nachgebildete+Formelsprache+des+reinen+Denkens&rft.place=Halle+a%2FS.&rft.pages=15&rft.pub=Verlag+von+Louis+Nebert&rft.date=1879&rft.aulast=Frege&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-bernays1918-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-bernays1918_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernays1918" class="citation book cs1">Bernays, P. (1918). <i>Beiträge zur axiomatischen Behandlung des Logik-Kalküls</i>. Göttingen: Universität Göttingen. p. 3.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Beitr%C3%A4ge+zur+axiomatischen+Behandlung+des+Logik-Kalk%C3%BCls&rft.place=G%C3%B6ttingen&rft.pages=3&rft.pub=Universit%C3%A4t+G%C3%B6ttingen&rft.date=1918&rft.aulast=Bernays&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-hilbert1927-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-hilbert1927_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbert1928" class="citation journal cs1 cs1-prop-foreign-lang-source">Hilbert, D. (1928) [1927]. "Die Grundlagen der Mathematik". <i>Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität</i> (in German). <b>6</b>: 65–85. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02940602">10.1007/BF02940602</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Abhandlungen+aus+dem+mathematischen+Seminar+der+Hamburgischen+Universit%C3%A4t&rft.atitle=Die+Grundlagen+der+Mathematik&rft.volume=6&rft.pages=65-85&rft.date=1928&rft_id=info%3Adoi%2F10.1007%2FBF02940602&rft.aulast=Hilbert&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-hilbert-ackermann1928-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-hilbert-ackermann1928_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbertAckermann1928" class="citation book cs1 cs1-prop-foreign-lang-source">Hilbert, D.; Ackermann, W. (1928). <i>Grundzügen der theoretischen Logik</i> (in German) (1 ed.). Berlin: Verlag von Julius Springer. p. 4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Grundz%C3%BCgen+der+theoretischen+Logik&rft.place=Berlin&rft.pages=4&rft.edition=1&rft.pub=Verlag+von+Julius+Springer&rft.date=1928&rft.aulast=Hilbert&rft.aufirst=D.&rft.au=Ackermann%2C+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-becker1933-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-becker1933_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBecker1933" class="citation book cs1 cs1-prop-foreign-lang-source">Becker, A. (1933). <i>Die Aristotelische Theorie der Möglichkeitsschlösse: Eine logisch-philologische Untersuchung der Kapitel 13-22 von Aristoteles' Analytica priora I</i> (in German). Berlin: Junker und Dünnhaupt Verlag. p. 4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Die+Aristotelische+Theorie+der+M%C3%B6glichkeitsschl%C3%B6sse%3A+Eine+logisch-philologische+Untersuchung+der+Kapitel+13-22+von+Aristoteles%27+Analytica+priora+I&rft.place=Berlin&rft.pages=4&rft.pub=Junker+und+D%C3%BCnnhaupt+Verlag&rft.date=1933&rft.aulast=Becker&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-lukasiewicz1929-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-lukasiewicz1929_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFŁukasiewicz1958" class="citation book cs1 cs1-prop-foreign-lang-source">Łukasiewicz, J. (1958) [1929]. Słupecki, J. (ed.). <i>Elementy logiki matematycznej</i> (in Polish) (2 ed.). Warszawa: Państwowe Wydawnictwo Naukowe.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementy+logiki+matematycznej&rft.place=Warszawa&rft.edition=2&rft.pub=Pa%C5%84stwowe+Wydawnictwo+Naukowe&rft.date=1958&rft.aulast=%C5%81ukasiewicz&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-lukasiewicz1951-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-lukasiewicz1951_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFŁukasiewicz1957" class="citation book cs1 cs1-prop-foreign-lang-source">Łukasiewicz, J. (1957) [1951]. Słupecki, J. (ed.). <i>Aristotle's Syllogistic from the Standpoint of Modern Formal Logic</i> (in Polish) (2 ed.). Glasgow, New York, Toronto, Melbourne, Wellington, Bombay, Calcutta, Madras, Karachi, Lahore, Dacca, Cape Town, Salisbury, Nairobi, Ibadan, Accra, Kuala Lumpur and Hong Kong: Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Aristotle%27s+Syllogistic+from+the+Standpoint+of+Modern+Formal+Logic&rft.place=Glasgow%2C+New+York%2C+Toronto%2C+Melbourne%2C+Wellington%2C+Bombay%2C+Calcutta%2C+Madras%2C+Karachi%2C+Lahore%2C+Dacca%2C+Cape+Town%2C+Salisbury%2C+Nairobi%2C+Ibadan%2C+Accra%2C+Kuala+Lumpur+and+Hong+Kong&rft.edition=2&rft.pub=Oxford+University+Press&rft.date=1957&rft.aulast=%C5%81ukasiewicz&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-heyting1929-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-heyting1929_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeyting1930" class="citation journal cs1 cs1-prop-foreign-lang-source">Heyting, A. (1930). "Die formalen Regeln der intuitionistischen Logik". <i>Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse</i> (in German): 42–56.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sitzungsberichte+der+Preussischen+Akademie+der+Wissenschaften%2C+Physikalisch-mathematische+Klasse&rft.atitle=Die+formalen+Regeln+der+intuitionistischen+Logik&rft.pages=42-56&rft.date=1930&rft.aulast=Heyting&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-bourbaki1954b-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-bourbaki1954b_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1954" class="citation book cs1 cs1-prop-foreign-lang-source">Bourbaki, N. (1954). <i>Théorie des ensembles</i> (in French). Paris: Hermann & Cie, Éditeurs. p. 32.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+des+ensembles&rft.place=Paris&rft.pages=32&rft.pub=Hermann+%26+Cie%2C+%C3%89diteurs&rft.date=1954&rft.aulast=Bourbaki&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-chazal1996-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-chazal1996_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChazal1996" class="citation book cs1">Chazal, G. (1996). <i>Eléments de logique formelle</i>. Paris: Hermes Science Publications.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=El%C3%A9ments+de+logique+formelle&rft.place=Paris&rft.pub=Hermes+Science+Publications&rft.date=1996&rft.aulast=Chazal&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+biconditional" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">In fact, such is the style adopted by <a href="/wiki/Wikipedia:Manual_of_Style/Mathematics" title="Wikipedia:Manual of Style/Mathematics">Wikipedia's manual of style in mathematics</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_biconditional&action=edit&section=12" title="Edit section: External 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.navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Common_logical_connectives" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Logical_connectives" title="Template:Logical connectives"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Logical_connectives" title="Template talk:Logical connectives"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Logical_connectives" title="Special:EditPage/Template:Logical connectives"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Common_logical_connectives" style="font-size:114%;margin:0 4em">Common <a href="/wiki/Logical_connective" title="Logical connective">logical connectives</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a>/<a href="/wiki/Logical_truth" title="Logical truth">True</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" 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 \top }"></span></li></ul> </div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Logical_connectives_Hasse_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/80px-Logical_connectives_Hasse_diagram.svg.png" decoding="async" width="80" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/120px-Logical_connectives_Hasse_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Logical_connectives_Hasse_diagram.svg/160px-Logical_connectives_Hasse_diagram.svg.png 2x" data-file-width="744" data-file-height="1052" /></a></span></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sheffer_stroke" title="Sheffer stroke">Alternative denial</a> (<a href="/wiki/NAND_gate" title="NAND gate">NAND gate</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span></li> <li><a href="/wiki/Converse_(logic)" title="Converse (logic)">Converse implication</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">←</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0fb4bce772117bbaf55b7ca1539ceff9ae218c" 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 \leftarrow }"></span></li> <li><a href="/wiki/Material_conditional" title="Material conditional">Implication</a> (<a href="/wiki/IMPLY_gate" title="IMPLY gate">IMPLY gate</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">Disjunction</a> (<a href="/wiki/OR_gate" title="OR gate">OR gate</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Negation" title="Negation">Negation</a> (<a href="/wiki/Inverter_(logic_gate)" title="Inverter (logic gate)">NOT gate</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></span></li> <li><a href="/wiki/Exclusive_or" title="Exclusive or">Exclusive or</a> (<a href="/wiki/XOR_gate" title="XOR gate">XOR gate</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \not \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not \leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/363ed81fd02da85c658dde9f17737c13b7263e49" 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 \not \leftrightarrow }"></span></li> <li><a class="mw-selflink selflink">Biconditional</a> (<a href="/wiki/XNOR_gate" title="XNOR gate">XNOR gate</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" 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 }"></span></li> <li><a href="/wiki/Statement_(logic)" title="Statement (logic)">Statement</a> (<a href="/wiki/Digital_buffer" title="Digital buffer">Digital buffer</a>)</li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Logical_NOR" title="Logical NOR">Joint denial</a> (<a href="/wiki/NOR_gate" title="NOR gate">NOR gate</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4618f22b0f780805eb94bb407578d9bc9487947a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \downarrow }"></span></li> <li><a href="/wiki/Material_nonimplication" title="Material nonimplication">Nonimplication</a> (<a href="/wiki/NIMPLY_gate" title="NIMPLY gate">NIMPLY gate</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span></li> <li><a href="/wiki/Converse_nonimplication" title="Converse nonimplication">Converse nonimplication</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nleftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>↚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nleftarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7694c9fc8eebe8a57c8156dd3c2caf022a619439" 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 \nleftarrow }"></span></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">Conjunction</a> (<a href="/wiki/AND_gate" title="AND gate">AND gate</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contradiction" title="Contradiction">Contradiction</a>/<a href="/wiki/False_(logic)" title="False (logic)">False</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" 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 \bot }"></span></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><span class="nowrap"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/18px-Socrates.png" decoding="async" width="18" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/27px-Socrates.png 1.5x, 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axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a class="mw-selflink selflink">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Concrete_category" title="Concrete category">Concrete</a>/<a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li> <li><a href="/wiki/Category_of_sets" title="Category of sets">Category of sets</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History of logic</a></li> <li><a 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