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<title>Distributive property</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Distributive property</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Distributivity" redirects here; not to be confused with <a href="Distributivism" class="mw-redirect" title="Distributivism">Distributivism</a>.</div>

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</style><table class="infobox vcard"><caption class="infobox-title fn" style="padding-bottom:0.2em;">Distributive property</caption><tbody><tr><td colspan="2" class="infobox-image"><span class="skin-invert-image" typeof="mw:File"></span><div class="infobox-caption">Visualization of distributive law for positive numbers</div></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="Principle" title="Principle">Law</a>, <a href="Rule_of_replacement" title="Rule of replacement">rule of replacement</a></td></tr><tr><th scope="row" class="infobox-label">Field</th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">

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<ul><li><a href="Elementary_algebra" title="Elementary algebra">Elementary algebra</a></li>

<li><a href="Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li>

<li><a href="Abstract_algebra" title="Abstract algebra">Abstract algebra</a></li>

<li><a href="Set_theory" title="Set theory">Set theory</a></li>

<li><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="infobox-label">Symbolic statement</th><td class="infobox-data"><div class="plainlist">

<ol><li>Elementary algebra

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<annotation encoding="application/x-tex">{\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z}</annotation>

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<li>Propositional calculus:

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<annotation encoding="application/x-tex">{\displaystyle (P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))}</annotation>

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</math></span><img src="./f6e44e66770303cef4cf29e118b8e6966ccd6fbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.014ex; height:2.843ex;" alt="{\displaystyle (P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle (P\lor (Q\land R))\Leftrightarrow ((P\lor Q)\land (P\lor R))}</annotation>

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</math></span><img src="./b320f1fbd3369ea11f930908122d66148d3073f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.014ex; height:2.843ex;" alt="{\displaystyle (P\lor (Q\land R))\Leftrightarrow ((P\lor Q)\land (P\lor R))}" loading="lazy"></span></li></ol></li></ol>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>distributive property</b> of <a href="Binary_operation" title="Binary operation">binary operations</a> is a generalization of the <b>distributive law</b>, which asserts that the equality

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<annotation encoding="application/x-tex">{\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z}</annotation>

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is always true in <a href="Elementary_algebra" title="Elementary algebra">elementary algebra</a>.

For example, in <a href="Elementary_arithmetic" title="Elementary arithmetic">elementary arithmetic</a>, one has

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<annotation encoding="application/x-tex">{\displaystyle 2\cdot (1+3)=(2\cdot 1)+(2\cdot 3).}</annotation>

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Therefore, one would say that <a href="Multiplication" title="Multiplication">multiplication</a> <i>distributes</i> over <a href="Addition" title="Addition">addition</a>.

</p><p>This basic property of numbers is part of the definition of most <a href="Algebraic_structure" title="Algebraic structure">algebraic structures</a> that have two operations called addition and multiplication, such as <a href="Complex_number" title="Complex number">complex numbers</a>, <a href="Polynomial" title="Polynomial">polynomials</a>, <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a>, and <a href="Field_(mathematics)" title="Field (mathematics)">fields</a>. It is also encountered in <a href="Boolean_algebra" title="Boolean algebra">Boolean algebra</a> and <a href="Mathematical_logic" title="Mathematical logic">mathematical logic</a>, where each of the <a href="Logical_and" class="mw-redirect" title="Logical and">logical and</a> (denoted <span class="mwe-math-element mwe-math-element-inline"><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 \,}">

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<annotation encoding="application/x-tex">{\displaystyle \,\land \,}</annotation>

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</math></span><img src="./81fa9a1bf2fd8fe99a8f955e4c2b2cce22fc3b71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.009ex;" alt="{\displaystyle \,\land \,}" loading="lazy"></span>) and the <a href="Logical_or" class="mw-redirect" title="Logical or">logical or</a> (denoted <span class="mwe-math-element mwe-math-element-inline"><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 \,}">

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<annotation encoding="application/x-tex">{\displaystyle \,\lor \,}</annotation>

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</math></span><img src="./11b9e8d57e725d846e685feb42fc447fd094fbf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.009ex;" alt="{\displaystyle \,\lor \,}" loading="lazy"></span>) distributes over the other.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Given a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math></span><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span> and two <a href="Binary_operator" class="mw-redirect" title="Binary operator">binary operators</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,*\,}">

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<annotation encoding="application/x-tex">{\displaystyle \,*\,}</annotation>

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</math></span><img src="./52256498cbe97c2c8db8dfdaacff53b620d9ca2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.937ex; height:1.509ex;" alt="{\displaystyle \,*\,}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,}">

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<annotation encoding="application/x-tex">{\displaystyle \,+\,}</annotation>

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</math></span><img src="./31b3c23fb2245f1f8bc8dde4d8c42f582ce4d3e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,+\,}" loading="lazy"></span> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S,}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle S,}</annotation>

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</math></span><img src="./e0bcd8516b165aaacb234616d7d2d23478a35be7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.146ex; height:2.509ex;" alt="{\displaystyle S,}" loading="lazy"></span>

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<ul><li>the operation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,*\,}">

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<annotation encoding="application/x-tex">{\displaystyle \,*\,}</annotation>

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</math></span><img src="./52256498cbe97c2c8db8dfdaacff53b620d9ca2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.937ex; height:1.509ex;" alt="{\displaystyle \,*\,}" loading="lazy"></span> is <em>left-distributive</em> over (or with respect to) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,}">

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<annotation encoding="application/x-tex">{\displaystyle \,+\,}</annotation>

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</math></span><img src="./31b3c23fb2245f1f8bc8dde4d8c42f582ce4d3e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,+\,}" loading="lazy"></span> if, <a href="Given_any" class="mw-redirect" title="Given any">given any</a> elements <span class="mwe-math-element mwe-math-element-inline"><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,{\text{ and }}z}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y,{\text{ and }}z}</annotation>

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</math></span><img src="./23bd270db6b290e4ed3e2e2dd6452dbee3b210a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.55ex; height:2.509ex;" alt="{\displaystyle x,y,{\text{ and }}z}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S,}">

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<annotation encoding="application/x-tex">{\displaystyle S,}</annotation>

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</math></span><img src="./e0bcd8516b165aaacb234616d7d2d23478a35be7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.146ex; height:2.509ex;" alt="{\displaystyle S,}" loading="lazy"></span></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle x*(y+z)=(x*y)+(x*z);}</annotation>

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<ul><li>the operation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,*\,}">

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<annotation encoding="application/x-tex">{\displaystyle \,*\,}</annotation>

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</math></span><img src="./52256498cbe97c2c8db8dfdaacff53b620d9ca2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.937ex; height:1.509ex;" alt="{\displaystyle \,*\,}" loading="lazy"></span> is <em>right-distributive</em> over <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,}">

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<annotation encoding="application/x-tex">{\displaystyle \,+\,}</annotation>

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</math></span><img src="./31b3c23fb2245f1f8bc8dde4d8c42f582ce4d3e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,+\,}" loading="lazy"></span> if, given any elements <span class="mwe-math-element mwe-math-element-inline"><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,{\text{ and }}z}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y,{\text{ and }}z}</annotation>

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</math></span><img src="./23bd270db6b290e4ed3e2e2dd6452dbee3b210a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.55ex; height:2.509ex;" alt="{\displaystyle x,y,{\text{ and }}z}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S,}">

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<annotation encoding="application/x-tex">{\displaystyle S,}</annotation>

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</math></span><img src="./e0bcd8516b165aaacb234616d7d2d23478a35be7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.146ex; height:2.509ex;" alt="{\displaystyle S,}" loading="lazy"></span></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle (y+z)*x=(y*x)+(z*x);}</annotation>

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</math></span></span>

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<ul><li>and the operation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,*\,}">

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<annotation encoding="application/x-tex">{\displaystyle \,*\,}</annotation>

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</math></span><img src="./52256498cbe97c2c8db8dfdaacff53b620d9ca2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.937ex; height:1.509ex;" alt="{\displaystyle \,*\,}" loading="lazy"></span> is <em>distributive</em> over <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,}">

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<annotation encoding="application/x-tex">{\displaystyle \,+\,}</annotation>

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</math></span><img src="./31b3c23fb2245f1f8bc8dde4d8c42f582ce4d3e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,+\,}" loading="lazy"></span> if it is left- and right-distributive.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \,*\,}</annotation>

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</math></span><img src="./52256498cbe97c2c8db8dfdaacff53b620d9ca2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.937ex; height:1.509ex;" alt="{\displaystyle \,*\,}" loading="lazy"></span> is <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a>, the three conditions above are <a href="Logical_equivalence" title="Logical equivalence">logically equivalent</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Meaning">Meaning</h2></div>

<p>The operators used for examples in this section are those of the usual <a href="Addition" title="Addition">addition</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,}">

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<annotation encoding="application/x-tex">{\displaystyle \,+\,}</annotation>

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</math></span><img src="./31b3c23fb2245f1f8bc8dde4d8c42f582ce4d3e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,+\,}" loading="lazy"></span> and <a href="Multiplication" title="Multiplication">multiplication</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\cdot .\,}">

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<annotation encoding="application/x-tex">{\displaystyle \,\cdot .\,}</annotation>

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</math></span><img src="./95b537ae7218b42b3af6d07796fbaac8658029a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.068ex; height:1.343ex;" alt="{\displaystyle \,\cdot .\,}" loading="lazy"></span>

</p><p>If the operation denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }">

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<annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation>

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</math></span><img src="./ba2c023bad1bd39ed49080f729cbf26bc448c9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" loading="lazy"></span> is not commutative, there is a distinction between left-distributivity and right-distributivity:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot \left(b\pm c\right)=a\cdot b\pm a\cdot c\qquad {\text{ (left-distributive) }}}">

<mrow>

<mrow>

</mrow>

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<mtext> (left-distributive) </mtext>

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<annotation encoding="application/x-tex">{\displaystyle a\cdot \left(b\pm c\right)=a\cdot b\pm a\cdot c\qquad {\text{ (left-distributive) }}}</annotation>

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<mtext> (right-distributive) </mtext>

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<annotation encoding="application/x-tex">{\displaystyle (a\pm b)\cdot c=a\cdot c\pm b\cdot c\qquad {\text{ (right-distributive) }}.}</annotation>

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</math></span></span>

</p><p>In either case, the distributive property can be described in words as:

</p><p>To multiply a <a href="Summation" title="Summation">sum</a> (or <a href="Difference_(mathematics)" class="mw-redirect" title="Difference (mathematics)">difference</a>) by a factor, each summand (or <a href="Minuend" class="mw-redirect" title="Minuend">minuend</a> and <a href="Subtrahend" class="mw-redirect" title="Subtrahend">subtrahend</a>) is multiplied by this factor and the resulting products are added (or subtracted).

</p><p>If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity implies right-distributivity and vice versa, and one talks simply of <em>distributivity</em>.

</p><p>One example of an operation that is "only" right-distributive is division, which is not commutative:

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<annotation encoding="application/x-tex">{\displaystyle (a\pm b)\div c=a\div c\pm b\div c.}</annotation>

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In this case, left-distributivity does not apply:

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<annotation encoding="application/x-tex">{\displaystyle a\div (b\pm c)\neq a\div b\pm a\div c}</annotation>

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</math></span></span>

</p><p>The distributive laws are among the axioms for <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a> (like the ring of <a href="Integer" title="Integer">integers</a>) and <a href="Field_(mathematics)" title="Field (mathematics)">fields</a> (like the field of <a href="Rational_number" title="Rational number">rational numbers</a>). Here multiplication is distributive over addition, but addition is not distributive over multiplication. Examples of structures with two operations that are each distributive over the other are <a href="Boolean_algebras" class="mw-redirect" title="Boolean algebras">Boolean algebras</a> such as the <a href="Algebra_of_sets" title="Algebra of sets">algebra of sets</a> or the <a href="Switching_algebra" class="mw-redirect" title="Switching algebra">switching algebra</a>.

</p><p>Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of a sum with each summand of the other sum (keeping track of signs) then add up all of the resulting products.

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Real_numbers">Real numbers</h3></div>

<p>In the following examples, the use of the distributive law on the set of real numbers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span> is illustrated. When multiplication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From the point of view of algebra, the real numbers form a <a href="Field_(mathematics)" title="Field (mathematics)">field</a>, which ensures the validity of the distributive law.

</p>

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<dt id="first_example_(mental_and_written_multiplication)"><dfn>First example (mental and written multiplication)</dfn></dt><dd>During mental arithmetic, distributivity is often used unconsciously:

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<annotation encoding="application/x-tex">{\displaystyle 6\cdot 16=6\cdot (10+6)=6\cdot 10+6\cdot 6=60+36=96}</annotation>

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</math></span></span>

Thus, to calculate <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6\cdot 16}">

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<annotation encoding="application/x-tex">{\displaystyle 6\cdot 16}</annotation>

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</math></span><img src="./fd89adb8d6132a95129b7783671be1d3e482457a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.166ex; height:2.176ex;" alt="{\displaystyle 6\cdot 16}" loading="lazy"></span> in one's head, one first multiplies <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6\cdot 10}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 6\cdot 10}</annotation>

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</math></span><img src="./0ff59738536cf087e57bf7f2db9abd6230ef70a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.166ex; height:2.176ex;" alt="{\displaystyle 6\cdot 10}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6\cdot 6}">

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<annotation encoding="application/x-tex">{\displaystyle 6\cdot 6}</annotation>

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</math></span><img src="./f42e2271e617e7a3efef658b37bcf12188c5bf16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.004ex; height:2.176ex;" alt="{\displaystyle 6\cdot 6}" loading="lazy"></span> and add the intermediate results. Written multiplication is also based on the distributive law.

</dd>

<dt id="second_example_(with_variables)"><dfn>Second example (with variables)</dfn></dt><dd>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 3a^{2}b\cdot (4a-5b)=3a^{2}b\cdot 4a-3a^{2}b\cdot 5b=12a^{3}b-15a^{2}b^{2}}</annotation>

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</math></span></span>

</dd>

<dt id="third_example_(with_two_sums)"><dfn>Third example (with two sums)</dfn></dt><dd>

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<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(a+b)\cdot (a-b)&=a\cdot (a-b)+b\cdot (a-b)=a^{2}-ab+ba-b^{2}=a^{2}-b^{2}\\&=(a+b)\cdot a-(a+b)\cdot b=a^{2}+ba-ab-b^{2}=a^{2}-b^{2}\\\end{aligned}}}</annotation>

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</math></span></span>

Here the distributive law was applied twice, and it does not matter which bracket is first multiplied out.

</dd>

<dt id="fourth_example"><dfn>Fourth example</dfn></dt><dd>Here the distributive law is applied the other way around compared to the previous examples. Consider

<msup>

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</msup>

<msup>

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</msup>

<msup>

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</msup>

<msup>

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</msup>

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle 12a^{3}b^{2}-30a^{4}bc+18a^{2}b^{3}c^{2}\,.}</annotation>

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</math></span></span>

Since the factor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6a^{2}b}">

<msup>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle 6a^{2}b}</annotation>

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</math></span><img src="./6af50da0fe3d6990bbbc1629994f085ae41f725d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.444ex; height:2.676ex;" alt="{\displaystyle 6a^{2}b}" loading="lazy"></span> occurs in all summands, it can be factored out. That is, due to the distributive law one obtains

<msup>

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</msup>

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle 12a^{3}b^{2}-30a^{4}bc+18a^{2}b^{3}c^{2}=6a^{2}b\left(2ab-5a^{2}c+3b^{2}c^{2}\right).}</annotation>

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</math></span></span>

</dd>

</dl>

<div class="mw-heading mw-heading3"><h3 id="Matrices">Matrices</h3></div>

<p>The distributive law is valid for <a href="Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>. More precisely,

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<annotation encoding="application/x-tex">{\displaystyle (A+B)\cdot C=A\cdot C+B\cdot C}</annotation>

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</math></span></span>

for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\times m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l\times m}</annotation>

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</math></span><img src="./6ddaae109aa28b89decddf99f3f487c27e3cc42e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.574ex; height:2.176ex;" alt="{\displaystyle l\times m}" loading="lazy"></span>-matrices <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A,B}</annotation>

</semantics>

</math></span><img src="./96c3298ea9aa77c226be56a7d8515baaa517b90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation>

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</math></span><img src="./12b23d207d23dd430b93320539abbb0bde84870d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}" loading="lazy"></span>-matrices <span class="mwe-math-element mwe-math-element-inline"><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,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C,}</annotation>

</semantics>

</math></span><img src="./64528f031cdbe1f52bdaf4ba7a8401108c0d2dc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.413ex; height:2.509ex;" alt="{\displaystyle C,}" loading="lazy"></span> as well as

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle A\cdot (B+C)=A\cdot B+A\cdot C}</annotation>

</semantics>

</math></span></span>

for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\times m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l\times m}</annotation>

</semantics>

</math></span><img src="./6ddaae109aa28b89decddf99f3f487c27e3cc42e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.574ex; height:2.176ex;" alt="{\displaystyle l\times m}" loading="lazy"></span>-matrices <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" 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}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation>

</semantics>

</math></span><img src="./12b23d207d23dd430b93320539abbb0bde84870d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}" loading="lazy"></span>-matrices <span class="mwe-math-element mwe-math-element-inline"><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,C.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B,C.}</annotation>

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</math></span><img src="./056927c4c024800fafba1e9b926b428655189e56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.211ex; height:2.509ex;" alt="{\displaystyle B,C.}" loading="lazy"></span>

Because the commutative property does not hold for matrix multiplication, the second law does not follow from the first law. In this case, they are two different laws.

</p>

<div class="mw-heading mw-heading3"><h3 id="Other_examples">Other examples</h3></div>

<ul><li><a href="Ordinal_arithmetic#Multiplication" title="Ordinal arithmetic">Multiplication</a> of <a href="Ordinal_number" title="Ordinal number">ordinal numbers</a>, in contrast, is only left-distributive, not right-distributive.</li>

<li>The <a href="Cross_product" title="Cross product">cross product</a> is left- and right-distributive over <a href="Vector_addition" class="mw-redirect" title="Vector addition">vector addition</a>, though not commutative.</li>

<li>The <a href="Union_(set_theory)" title="Union (set theory)">union</a> of sets is distributive over <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a>, and intersection is distributive over union.</li>

<li><a href="Logical_disjunction" title="Logical disjunction">Logical disjunction</a> ("or") is distributive over <a href="Logical_conjunction" title="Logical conjunction">logical conjunction</a> ("and"), and vice versa.</li>

<li>For <a href="Real_number" title="Real number">real numbers</a> (and for any <a href="Totally_ordered_set" class="mw-redirect" title="Totally ordered set">totally ordered set</a>), the <a href="Maximum" class="mw-redirect" title="Maximum">maximum</a> operation is distributive over the <a href="Minimum" class="mw-redirect" title="Minimum">minimum</a> operation, and vice versa: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(a,\min(b,c))=\min(\max(a,b),\max(a,c))\quad {\text{ and }}\quad \min(a,\max(b,c))=\max(\min(a,b),\min(a,c)).}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \max(a,\min(b,c))=\min(\max(a,b),\max(a,c))\quad {\text{ and }}\quad \min(a,\max(b,c))=\max(\min(a,b),\min(a,c)).}</annotation>

</semantics>

</math></span></span></li>

<li>For <a href="Integer" title="Integer">integers</a>, the <a href="Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> is distributive over the <a href="Least_common_multiple" title="Least common multiple">least common multiple</a>, and vice versa: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(a,\operatorname {lcm} (b,c))=\operatorname {lcm} (\gcd(a,b),\gcd(a,c))\quad {\text{ and }}\quad \operatorname {lcm} (a,\gcd(b,c))=\gcd(\operatorname {lcm} (a,b),\operatorname {lcm} (a,c)).}">

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<annotation encoding="application/x-tex">{\displaystyle \gcd(a,\operatorname {lcm} (b,c))=\operatorname {lcm} (\gcd(a,b),\gcd(a,c))\quad {\text{ and }}\quad \operatorname {lcm} (a,\gcd(b,c))=\gcd(\operatorname {lcm} (a,b),\operatorname {lcm} (a,c)).}</annotation>

</semantics>

</math></span></span></li>

<li>For real numbers, addition distributes over the maximum operation, and also over the minimum operation: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+\max(b,c)=\max(a+b,a+c)\quad {\text{ and }}\quad a+\min(b,c)=\min(a+b,a+c).}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a+\max(b,c)=\max(a+b,a+c)\quad {\text{ and }}\quad a+\min(b,c)=\min(a+b,a+c).}</annotation>

</semantics>

</math></span></span></li>

<li>For <a href="Binomial_(polynomial)" title="Binomial (polynomial)">binomial</a> multiplication, distribution is sometimes referred to as the <a href="FOIL_Method" class="mw-redirect" title="FOIL Method">FOIL Method</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> (First terms <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ac,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ac,}</annotation>

</semantics>

</math></span><img src="./7bdf6672c06648e01c79e7682e85fe7b7db858a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.883ex; height:2.009ex;" alt="{\displaystyle ac,}" loading="lazy"></span> Outer <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ad,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ad,}</annotation>

</semantics>

</math></span><img src="./cda03365da0d88959e9823c213c70b11c91906c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.093ex; height:2.509ex;" alt="{\displaystyle ad,}" loading="lazy"></span> Inner <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle bc,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle bc,}</annotation>

</semantics>

</math></span><img src="./43377610065af24bc7f7c8a8111627e568e91f30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.651ex; height:2.509ex;" alt="{\displaystyle bc,}" loading="lazy"></span> and Last <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle bd}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle bd}</annotation>

</semantics>

</math></span><img src="./3bc109ffc2966e361a2018a1dc7301c2f193a0f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.213ex; height:2.176ex;" alt="{\displaystyle bd}" loading="lazy"></span>) such as: <span class="mwe-math-element mwe-math-element-inline"><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)\cdot (c+d)=ac+ad+bc+bd.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a+b)\cdot (c+d)=ac+ad+bc+bd.}</annotation>

</semantics>

</math></span><img src="./3db8ca8a119bf4bea0f76213995c8f9c26c3ba78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.595ex; height:2.843ex;" alt="{\displaystyle (a+b)\cdot (c+d)=ac+ad+bc+bd.}" loading="lazy"></span></li>

<li>In all <a href="Semirings" class="mw-redirect" title="Semirings">semirings</a>, including the <a href="Complex_number" title="Complex number">complex numbers</a>, the <a href="Quaternion" title="Quaternion">quaternions</a>, <a href="Polynomial" title="Polynomial">polynomials</a>, and <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, multiplication distributes over addition: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(v+w)=uv+uw,(u+v)w=uw+vw.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(v+w)=uv+uw,(u+v)w=uw+vw.}</annotation>

</semantics>

</math></span><img src="./381f127e90f17160bceb3caf8e9a7caacf865ba1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.337ex; height:2.843ex;" alt="{\displaystyle u(v+w)=uv+uw,(u+v)w=uw+vw.}" loading="lazy"></span></li>

<li>In all <a href="Algebra_over_a_field" title="Algebra over a field">algebras over a field</a>, including the <a href="Octonion" title="Octonion">octonions</a> and other <a href="Non-associative_algebra" title="Non-associative algebra">non-associative algebras</a>, multiplication distributes over addition.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Propositional_logic">Propositional logic</h2></div>

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</style><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="Rule_of_inference" title="Rule of inference">Transformation rules</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;;background:#ddf;font-size:110%; border-bottom:1px #fefefe solid;">

<a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional calculus</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;">

<a href="Rule_of_inference" title="Rule of inference">Rules of inference</a> (<a href="List_of_rules_of_inference" title="List of rules of inference">List</a>)</th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;">

<ul><li><a href="Conditional_proof" title="Conditional proof"><span>Implication introduction</span></a> / <a href="Modus_ponens" title="Modus ponens"><span title="A→B,   A   ⊢   B">elimination (<i>modus ponens</i>)</span></a></li>

<li><a href="Biconditional_introduction" title="Biconditional introduction"><span title="A→B,   B→A   ⊢   A↔B">Biconditional introduction</span></a> / <a href="Biconditional_elimination" title="Biconditional elimination"><span title="A↔B   ⊢   A→B">elimination</span></a></li>

<li><a href="Conjunction_introduction" title="Conjunction introduction"><span title="A,   B   ⊢   A∧B">Conjunction introduction</span></a> / <a href="Conjunction_elimination" title="Conjunction elimination"><span title="A∧B   ⊢   A">elimination</span></a></li>

<li><a href="Disjunction_introduction" title="Disjunction introduction"><span title="A   ⊢   A∨B">Disjunction introduction</span></a> / <a href="Disjunction_elimination" title="Disjunction elimination"><span title="A∨B,   A→C,   B→C   ⊢   C">elimination</span></a></li>

<li><a href="Disjunctive_syllogism" title="Disjunctive syllogism"><span title="A∨B,   ¬A   ⊢   B">Disjunctive</span></a> / <a href="Hypothetical_syllogism" title="Hypothetical syllogism"><span title="A→B,   B→C   ⊢   A→C">hypothetical syllogism</span></a></li>

<li><a href="Constructive_dilemma" title="Constructive dilemma"><span title="A→P,   B→Q,   A∨B   ⊢   P∨Q">Constructive</span></a> / <a href="Destructive_dilemma" title="Destructive dilemma"><span title="A→P,   B→Q,   ¬P∨¬Q   ⊢   ¬A∨¬B">destructive dilemma</span></a></li>

<li><a href="Absorption_(logic)" title="Absorption (logic)"><span title="A→B   ⊢   A→A∧B">Absorption</span></a> / <a href="Modus_tollens" title="Modus tollens"><span title="A→B,   ¬B   ⊢   ¬A"><i>modus tollens</i></span></a> / <a href="Modus_ponendo_tollens" title="Modus ponendo tollens"><span title="¬(A∧B),   A   ⊢   ¬B"><i>modus ponendo tollens</i></span></a></li>

<li><a href="Modus_non_excipiens" title="Modus non excipiens">Modus non excipiens</a></li>

<li><a href="Negation_introduction" title="Negation introduction">Negation introduction</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="background:#eaeaff;">

<a href="Rule_of_replacement" title="Rule of replacement">Rules of replacement</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;">

<ul><li><a href="Associative_property#Propositional_logic" title="Associative property"><span title="A∨(B∨C)   =   (A∨B)∨C">Associativity</span></a></li>

<li><a href="Commutative_property#Propositional_logic" title="Commutative property"><span title="A∨B   =   B∨A">Commutativity</span></a></li>

<li><a class="mw-selflink-fragment" href="#Propositional_logic"><span title="A∧(B∨C)   =   (A∧B)∨(A∧C)">Distributivity</span></a></li>

<li><a href="Double_negation" title="Double negation"><span title="¬¬A   =   A">Double negation</span></a></li>

<li><a href="De_Morgan's_laws" title="De Morgan's laws">De Morgan's laws</a></li>

<li><a href="Transposition_(logic)" class="mw-redirect" title="Transposition (logic)">Transposition</a></li>

<li><a href="Material_implication_(rule_of_inference)" title="Material implication (rule of inference)"><span title="A→B   ⊢   ¬A∨B">Material implication</span></a></li>

<li><a href="Exportation_(logic)" title="Exportation (logic)"><span title="(A∧B)→C   ⊢   A→(B→C)">Exportation</span></a></li>

<li><a href="Tautology_(rule_of_inference)" title="Tautology (rule of inference)"><span title="A∨A   =   A">Tautology</span></a></li></ul>

</div></td>

</tr><tr><th class="sidebar-heading" style="background:#eaeaff;;background:#ddf;font-size:110%;">

<a href="First-order_logic" title="First-order logic">Predicate logic</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;">

<a href="Rule_of_inference" title="Rule of inference">Rules of inference</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;">

<ul><li><a href="Universal_generalization" title="Universal generalization">Universal generalization</a> / <a href="Universal_instantiation" title="Universal instantiation">instantiation</a></li>

<li><a href="Existential_generalization" title="Existential generalization">Existential generalization</a> / <a href="Existential_instantiation" title="Existential instantiation">instantiation</a></li></ul></td>

</tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

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<div class="mw-heading mw-heading3"><h3 id="Rule_of_replacement">Rule of replacement</h3></div>

<p>In standard truth-functional propositional logic, <em>distribution</em><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><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> in logical proofs uses two valid <a href="Rule_of_replacement" title="Rule of replacement">rules of replacement</a> to expand individual occurrences of certain <a href="Logical_connective" title="Logical connective">logical connectives</a>, within some <a href="Logical_formula" class="mw-redirect" title="Logical formula">formula</a>, into separate applications of those connectives across subformulas of the given formula. The rules are

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<annotation encoding="application/x-tex">{\displaystyle (P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))\qquad {\text{ and }}\qquad (P\lor (Q\land R))\Leftrightarrow ((P\lor Q)\land (P\lor R))}</annotation>

</semantics>

</math></span></span>

where "<span class="mwe-math-element mwe-math-element-inline"><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 }">

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<annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation>

</semantics>

</math></span><img src="./64812e13399c20cf3ce94e049d3bb2d85f26abcf.svg" 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 }" loading="lazy"></span>", also written <span class="mwe-math-element mwe-math-element-inline"><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 ,\,}">

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<annotation encoding="application/x-tex">{\displaystyle \,\equiv ,\,}</annotation>

</semantics>

</math></span><img src="./1b33bda7fbff75407e9373819734938575c1cd1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.229ex; height:2.009ex;" alt="{\displaystyle \,\equiv ,\,}" loading="lazy"></span> is a <a href="Metalogic" title="Metalogic">metalogical</a> <a href="Symbol_(formal)" title="Symbol (formal)">symbol</a> representing "can be replaced in a proof with" or "is <a href="Logical_equivalence" title="Logical equivalence">logically equivalent</a> to".

</p>

<div class="mw-heading mw-heading3"><h3 id="Truth_functional_connectives">Truth functional connectives</h3></div>

<p><em>Distributivity</em> is a property of some logical connectives of truth-functional <a href="Propositional_logic" title="Propositional logic">propositional logic</a>. The following logical equivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functional <a href="Tautology_(logic)" title="Tautology (logic)">tautologies</a>.

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<mtext> equivalence </mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{13}&(P&&\;\land &&(Q\lor R))&&\;\Leftrightarrow \;&&((P\land Q)&&\;\lor (P\land R))&&\quad {\text{ Distribution of }}&&{\text{ conjunction }}&&{\text{ over }}&&{\text{ disjunction }}\\&(P&&\;\lor &&(Q\land R))&&\;\Leftrightarrow \;&&((P\lor Q)&&\;\land (P\lor R))&&\quad {\text{ Distribution of }}&&{\text{ disjunction }}&&{\text{ over }}&&{\text{ conjunction }}\\&(P&&\;\land &&(Q\land R))&&\;\Leftrightarrow \;&&((P\land Q)&&\;\land (P\land R))&&\quad {\text{ Distribution of }}&&{\text{ conjunction }}&&{\text{ over }}&&{\text{ conjunction }}\\&(P&&\;\lor &&(Q\lor R))&&\;\Leftrightarrow \;&&((P\lor Q)&&\;\lor (P\lor R))&&\quad {\text{ Distribution of }}&&{\text{ disjunction }}&&{\text{ over }}&&{\text{ disjunction }}\\&(P&&\to &&(Q\to R))&&\;\Leftrightarrow \;&&((P\to Q)&&\to (P\to R))&&\quad {\text{ Distribution of }}&&{\text{ implication }}&&{\text{ }}&&{\text{ }}\\&(P&&\to &&(Q\leftrightarrow R))&&\;\Leftrightarrow \;&&((P\to Q)&&\leftrightarrow (P\to R))&&\quad {\text{ Distribution of }}&&{\text{ implication }}&&{\text{ over }}&&{\text{ equivalence }}\\&(P&&\to &&(Q\land R))&&\;\Leftrightarrow \;&&((P\to Q)&&\;\land (P\to R))&&\quad {\text{ Distribution of }}&&{\text{ implication }}&&{\text{ over }}&&{\text{ conjunction }}\\&(P&&\;\lor &&(Q\leftrightarrow R))&&\;\Leftrightarrow \;&&((P\lor Q)&&\leftrightarrow (P\lor R))&&\quad {\text{ Distribution of }}&&{\text{ disjunction }}&&{\text{ over }}&&{\text{ equivalence }}\\\end{alignedat}}}</annotation>

</semantics>

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<dl><dt>Double distribution</dt>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{13}&((P\land Q)&&\;\lor (R\land S))&&\;\Leftrightarrow \;&&(((P\lor R)\land (P\lor S))&&\;\land ((Q\lor R)\land (Q\lor S)))&&\\&((P\lor Q)&&\;\land (R\lor S))&&\;\Leftrightarrow \;&&(((P\land R)\lor (P\land S))&&\;\lor ((Q\land R)\lor (Q\land S)))&&\\\end{alignedat}}}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Distributivity_and_rounding">Distributivity and rounding</h2></div>

<p>In approximate arithmetic, such as <a href="Floating-point_arithmetic" title="Floating-point arithmetic">floating-point arithmetic</a>, the distributive property of multiplication (and division) over addition may fail because of the limitations of <a href="Arithmetic_precision" class="mw-redirect" title="Arithmetic precision">arithmetic precision</a>. For example, the identity <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/3+1/3+1/3=(1+1+1)/3}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/3+1/3+1/3=(1+1+1)/3}</annotation>

</semantics>

</math></span><img src="./51ccb7fdb28bb849d601f94baee3e760e74a0dd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.544ex; height:2.843ex;" alt="{\displaystyle 1/3+1/3+1/3=(1+1+1)/3}" loading="lazy"></span> fails in <a href="Decimal_arithmetic" class="mw-redirect" title="Decimal arithmetic">decimal arithmetic</a>, regardless of the number of <a href="Significant_digit" class="mw-redirect" title="Significant digit">significant digits</a>. Methods such as <a href="Banker's_rounding" class="mw-redirect" title="Banker's rounding">banker's rounding</a> may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.

</p>

<div class="mw-heading mw-heading2"><h2 id="In_rings_and_other_structures">In rings and other structures</h2></div>

<p>Distributivity is most commonly found in <a href="Semiring" title="Semiring">semirings</a>, notably the particular cases of <a href="Ring_(algebra)" class="mw-redirect" title="Ring (algebra)">rings</a> and <a href="Distributive_lattice" title="Distributive lattice">distributive lattices</a>.

</p><p>A semiring has two binary operations, commonly denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,+\,}</annotation>

</semantics>

</math></span><img src="./31b3c23fb2245f1f8bc8dde4d8c42f582ce4d3e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,+\,}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,*,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,*,}</annotation>

</semantics>

</math></span><img src="./363b631548acd06a2ef40d5de331217f00c8c8ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.196ex; height:2.009ex;" alt="{\displaystyle \,*,}" loading="lazy"></span> and requires that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,*\,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,*\,}</annotation>

</semantics>

</math></span><img src="./52256498cbe97c2c8db8dfdaacff53b620d9ca2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.937ex; height:1.509ex;" alt="{\displaystyle \,*\,}" loading="lazy"></span> must distribute over <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,+.}</annotation>

</semantics>

</math></span><img src="./da04993bc384d00e6a8dc409652b29e04a4e3218.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.842ex; height:2.176ex;" alt="{\displaystyle \,+.}" loading="lazy"></span>

</p><p>A ring is a semiring with additive inverses.

</p><p>A <a href="Lattice_(order)" title="Lattice (order)">lattice</a> is another kind of <a href="Algebraic_structure" title="Algebraic structure">algebraic structure</a> with two binary operations, <span class="mwe-math-element mwe-math-element-inline"><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 {\text{ and }}\lor .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\land {\text{ and }}\lor .}</annotation>

</semantics>

</math></span><img src="./66e244ab60111ffeac54f723ce13d45061aa4669.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.043ex; height:2.176ex;" alt="{\displaystyle \,\land {\text{ and }}\lor .}" loading="lazy"></span>

If either of these operations distributes over the other (say <span class="mwe-math-element mwe-math-element-inline"><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 \,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\land \,}</annotation>

</semantics>

</math></span><img src="./81fa9a1bf2fd8fe99a8f955e4c2b2cce22fc3b71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.009ex;" alt="{\displaystyle \,\land \,}" loading="lazy"></span> distributes over <span class="mwe-math-element mwe-math-element-inline"><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 }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\lor }</annotation>

</semantics>

</math></span><img src="./dbe7b3298f2837104b7851f46547fd687b8833c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.937ex; height:2.009ex;" alt="{\displaystyle \,\lor }" loading="lazy"></span>), then the reverse also holds (<span class="mwe-math-element mwe-math-element-inline"><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 \,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\lor \,}</annotation>

</semantics>

</math></span><img src="./11b9e8d57e725d846e685feb42fc447fd094fbf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.009ex;" alt="{\displaystyle \,\lor \,}" loading="lazy"></span> distributes over <span class="mwe-math-element mwe-math-element-inline"><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 \,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\land \,}</annotation>

</semantics>

</math></span><img src="./81fa9a1bf2fd8fe99a8f955e4c2b2cce22fc3b71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.009ex;" alt="{\displaystyle \,\land \,}" loading="lazy"></span>), and the lattice is called distributive. See also <em><a href="Distributivity_(order_theory)" title="Distributivity (order theory)">Distributivity (order theory)</a></em>.

</p><p>A <a href="Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a> can be interpreted either as a special kind of ring (a <a href="Boolean_ring" title="Boolean ring">Boolean ring</a>) or a special kind of distributive lattice (a <a href="Boolean_lattice" class="mw-redirect" title="Boolean lattice">Boolean lattice</a>). Each interpretation is responsible for different distributive laws in the Boolean algebra.

</p><p>Similar structures without distributive laws are <a href="Near-ring" title="Near-ring">near-rings</a> and <a href="Near-field_(mathematics)" title="Near-field (mathematics)">near-fields</a> instead of rings and <a href="Division_ring" title="Division ring">division rings</a>. The operations are usually defined to be distributive on the right but not on the left.

</p>

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

<p>

</p><p>In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or the extension to infinitary operations. Especially in <a href="Order_theory" title="Order theory">order theory</a> one finds numerous important variants of distributivity, some of which include infinitary operations, such as the <a href="Infinite_distributive_law" class="mw-redirect" title="Infinite distributive law">infinite distributive law</a>; others being defined in the presence of only <em>one</em> binary operation, such as the according definitions and their relations are given in the article <a href="Distributivity_(order_theory)" title="Distributivity (order theory)">distributivity (order theory)</a>. This also includes the notion of a <a href="Completely_distributive_lattice" title="Completely distributive lattice">completely distributive lattice</a>.

</p><p>In the presence of an ordering relation, one can also weaken the above equalities by replacing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,=\,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,=\,}</annotation>

</semantics>

</math></span><img src="./936ee397fe5ec4b211f75366b8f9c2e9ddf415f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.582ex; height:1.343ex;" alt="{\displaystyle \,=\,}" loading="lazy"></span> by either <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\leq \,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\leq \,}</annotation>

</semantics>

</math></span><img src="./24112548985eab096493f73f838580442780b57f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,\leq \,}" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\geq .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\geq .}</annotation>

</semantics>

</math></span><img src="./c87ac987e87f7d8f480d0bd692b182836a79c2b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.842ex; height:2.176ex;" alt="{\displaystyle \,\geq .}" loading="lazy"></span> Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notion of <b>sub-distributivity</b> as explained in the article on <a href="Interval_arithmetic" title="Interval arithmetic">interval arithmetic</a>.

</p><p>In <a href="Category_theory" title="Category theory">category theory</a>, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\mu ,\nu )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (S,\mu ,\nu )}</annotation>

</semantics>

</math></span><img src="./fa5baf0893a29df010cc54664c1699294a0e2cd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.01ex; height:2.843ex;" alt="{\displaystyle (S,\mu ,\nu )}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(S^{\prime },\mu ^{\prime },\nu ^{\prime }\right)}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(S^{\prime },\mu ^{\prime },\nu ^{\prime }\right)}</annotation>

</semantics>

</math></span><img src="./33ecb4074c83dff20c25558fc7c83771f24bb061.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.112ex; height:3.009ex;" alt="{\displaystyle \left(S^{\prime },\mu ^{\prime },\nu ^{\prime }\right)}" loading="lazy"></span> are <a href="Monad_(category_theory)" title="Monad (category theory)">monads</a> on a <a href="Category_(mathematics)" title="Category (mathematics)">category</a> <span class="mwe-math-element mwe-math-element-inline"><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,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C,}</annotation>

</semantics>

</math></span><img src="./64528f031cdbe1f52bdaf4ba7a8401108c0d2dc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.413ex; height:2.509ex;" alt="{\displaystyle C,}" loading="lazy"></span> a <b>distributive law</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S.S^{\prime }\to S^{\prime }.S}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S.S^{\prime }\to S^{\prime }.S}</annotation>

</semantics>

</math></span><img src="./f9be50ceef0b234f174ace523f897192a7d0648c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.093ex; height:2.509ex;" alt="{\displaystyle S.S^{\prime }\to S^{\prime }.S}" loading="lazy"></span> is a <a href="Natural_transformation" title="Natural transformation">natural transformation</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda :S.S^{\prime }\to S^{\prime }.S}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda :S.S^{\prime }\to S^{\prime }.S}</annotation>

</semantics>

</math></span><img src="./a897a135dd5bcd91cd577043b9ab7ed42beaca85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.386ex; height:2.509ex;" alt="{\displaystyle \lambda :S.S^{\prime }\to S^{\prime }.S}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(S^{\prime },\lambda \right)}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(S^{\prime },\lambda \right)}</annotation>

</semantics>

</math></span><img src="./2ecffc23629d9834920516c024403e789813d732.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.405ex; height:3.009ex;" alt="{\displaystyle \left(S^{\prime },\lambda \right)}" loading="lazy"></span> is a <a href="Lax_map_of_monads" class="mw-redirect" title="Lax map of monads">lax map of monads</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\to S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\to S}</annotation>

</semantics>

</math></span><img src="./bf53e44dfa263077b0945848b83632fe18efcb8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle S\to S}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\lambda )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (S,\lambda )}</annotation>

</semantics>

</math></span><img src="./1f601909396ba6692513b4f0869f5f722a30a6c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.698ex; height:2.843ex;" alt="{\displaystyle (S,\lambda )}" loading="lazy"></span> is a <a href="Colax_map_of_monads" class="mw-redirect" title="Colax map of monads">colax map of monads</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\prime }\to S^{\prime }.}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{\prime }\to S^{\prime }.}</annotation>

</semantics>

</math></span><img src="./0cf2d798e2e0fc17544bda5edd1ddff03c597610.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.674ex; height:2.509ex;" alt="{\displaystyle S^{\prime }\to S^{\prime }.}" loading="lazy"></span> This is exactly the data needed to define a monad structure on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\prime }.S}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{\prime }.S}</annotation>

</semantics>

</math></span><img src="./a89983e272be09df59bef76cf4b31ee2d0ee9ce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.74ex; height:2.509ex;" alt="{\displaystyle S^{\prime }.S}" loading="lazy"></span>: the multiplication map is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\prime }\mu .\mu ^{\prime }S^{2}.S^{\prime }\lambda S}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{\prime }\mu .\mu ^{\prime }S^{2}.S^{\prime }\lambda S}</annotation>

</semantics>

</math></span><img src="./b4beb4bfece742632c36ef7205ce459c5f7313d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.399ex; height:3.176ex;" alt="{\displaystyle S^{\prime }\mu .\mu ^{\prime }S^{2}.S^{\prime }\lambda S}" loading="lazy"></span> and the unit map is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta ^{\prime }S.\eta .}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta ^{\prime }S.\eta .}</annotation>

</semantics>

</math></span><img src="./6e11f87329493afede1c3b7a7f3b510c54e41ecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.208ex; height:3.009ex;" alt="{\displaystyle \eta ^{\prime }S.\eta .}" loading="lazy"></span> See: <a href="Distributive_law_between_monads" title="Distributive law between monads">distributive law between monads</a>.

</p><p>A <a href="Generalized_distributive_law" title="Generalized distributive law">generalized distributive law</a> has also been proposed in the area of <a href="Information_theory" title="Information theory">information theory</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Antidistributivity">Antidistributivity</h3></div>

<p>The ubiquitous <a href="Identity_(mathematics)" title="Identity (mathematics)">identity</a> that relates inverses to the binary operation in any <a href="Group_(mathematics)" title="Group (mathematics)">group</a>, namely <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (xy)^{-1}=y^{-1}x^{-1},}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (xy)^{-1}=y^{-1}x^{-1},}</annotation>

</semantics>

</math></span><img src="./4dab799aac63d61877a832b3abd7a4f91937526f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.528ex; height:3.176ex;" alt="{\displaystyle (xy)^{-1}=y^{-1}x^{-1},}" loading="lazy"></span> which is taken as an axiom in the more general context of a <a href="Semigroup_with_involution" title="Semigroup with involution">semigroup with involution</a>, has sometimes been called an <b>antidistributive property</b> (of inversion as a <a href="Unary_operation" title="Unary operation">unary operation</a>).<sup id="cite_ref-BrinkKahl1997_5-0" class="reference"><a href="#cite_note-BrinkKahl1997-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>In the context of a <a href="Near-ring" title="Near-ring">near-ring</a>, which removes the commutativity of the additively written group and assumes only one-sided distributivity, one can speak of (two-sided) <b>distributive elements</b> but also of <b>antidistributive elements</b>. The latter reverse the order of (the non-commutative) addition; assuming a left-nearring (i.e. one which all elements distribute when multiplied on the left), then an antidistributive element <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> reverses the order of addition when multiplied to the right: <span class="mwe-math-element mwe-math-element-inline"><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)a=ya+xa.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x+y)a=ya+xa.}</annotation>

</semantics>

</math></span><img src="./2e15bddd84508e144ea2c76078a764bf47136029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.895ex; height:2.843ex;" alt="{\displaystyle (x+y)a=ya+xa.}" loading="lazy"></span><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>In the study of <a href="Propositional_logic" title="Propositional logic">propositional logic</a> and <a href="Boolean_algebra" title="Boolean algebra">Boolean algebra</a>, the term <b>antidistributive law</b> is sometimes used to denote the interchange between conjunction and disjunction when implication factors over them:<sup id="cite_ref-Hehner1993_7-0" class="reference"><a href="#cite_note-Hehner1993-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (a\lor b)\Rightarrow c\equiv (a\Rightarrow c)\land (b\Rightarrow c)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle (a\land b)\Rightarrow c\equiv (a\Rightarrow c)\lor (b\Rightarrow c).}</annotation>

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</math></span></span>

</p><p>These two <a href="Tautology_(logic)" title="Tautology (logic)">tautologies</a> are a direct consequence of the duality in <a href="De_Morgan's_laws" title="De Morgan's laws">De Morgan's laws</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://mathonline.wikidot.com/distributivity-of-binary-operations">Distributivity of Binary Operations</a> from Mathonline</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Kim Steward (2011) <a rel="nofollow" class="external text" href="http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut28_multpoly.htm">Multiplying Polynomials</a> from Virtual Math Lab at <a href="West_Texas_A%26M_University" title="West Texas A&M University">West Texas A&M University</a></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"><a href="Elliott_Mendelson" title="Elliott Mendelson">Elliott Mendelson</a> (1964) <i>Introduction to Mathematical Logic</i>, page 21, D. Van Nostrand Company</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> (1941) <i>Introduction to Logic</i>, page 52, <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a></span>

</li>

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</style><cite id="CITEREFChris_BrinkWolfram_KahlGunther_Schmidt1997" class="citation book cs1">Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/relationalmethod00jips"><i>Relational Methods in Computer Science</i></a></span>. Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/relationalmethod00jips/page/n16">4</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-211-82971-4</bdi>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFCelestina_Cotti_FerreroGiovanni_Ferrero2002" class="citation book cs1">Celestina Cotti Ferrero; Giovanni Ferrero (2002). <i>Nearrings: Some Developments Linked to Semigroups and Groups</i>. Kluwer Academic Publishers. pp. 62 and 67. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4613-0267-4</bdi>.</cite></span>

</li>

<li id="cite_note-Hehner1993-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hehner1993_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFEric_C.R._Hehner1993" class="citation book cs1"><a href="Eric_Hehner" title="Eric Hehner">Eric C.R. Hehner</a> (1993). <i>A Practical Theory of Programming</i>. Springer Science & Business Media. p. 230. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4419-8596-5</bdi>.</cite></span>

</li>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/distributivity" class="extiw external" title="wiktionary:distributivity">distributivity</a></b></i> in Wiktionary, the free dictionary.</div></div>

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<ul><li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Arithmetic/DistributiveLaw.shtml">A demonstration of the Distributive Law</a> for integer arithmetic (from <a href="Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a>)</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-19" href="https://en.wikipedia.org/wiki/?title=Distributive_property&oldid=1301362000">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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