SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Image directe</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading">Image directe</h1>

</header>

</div>

L'image directe d'un <a href="Sous-ensemble" class="mw-redirect" title="Sous-ensemble">sous-ensemble</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.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"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> par une <a href="Application_(math%C3%A9matiques)" title="Application (mathématiques)">application</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\,\colon X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\,\colon X\to Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91a2bbbb0c03d63f6622609e391951237f0826ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.067ex; height:2.509ex;" alt="{\displaystyle f\,\colon X\to Y}" loading="lazy"> est le sous-ensemble de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> formé des <a href="%C3%89l%C3%A9ment_(math%C3%A9matiques)" class="mw-redirect" title="Élément (mathématiques)">éléments</a> qui ont, par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, au moins un <a href="Ant%C3%A9c%C3%A9dent_(math%C3%A9matiques)" title="Antécédent (mathématiques)">antécédent</a> appartenant à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.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"> :

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(A)=\{f(x)\mid x\in A\}=\{y\in Y\mid \exists a\in A,y=f(a)\}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91206bd48cb886d85cc95da5f94249ad0bafe7f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.011ex; height:2.843ex;" alt="{\displaystyle f(A)=\{f(x)\mid x\in A\}=\{y\in Y\mid \exists a\in A,y=f(a)\}.}" loading="lazy"></center>

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

<ul><li>On définit en particulier l'<a href="Image_d'une_application" title="Image d'une application">image d'une application</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> <a href="Ensemble_de_d%C3%A9finition" title="Ensemble de définition">définie sur</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> :<center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Im} (f)=f(X).}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Im} (f)=f(X).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86bdf4084ba4ce3913e2be6a05c3085f5ebf92f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.677ex; height:2.843ex;" alt="{\displaystyle \mathrm {Im} (f)=f(X).}" loading="lazy"></center></li>

<li>On se gardera bien de confondre l'image directe par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> d'une partie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.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"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, avec l'<a href="Image_(math%C3%A9matiques)" title="Image (mathématiques)">image par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> d'un élément</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, ou avec l'image de l'application <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"><a href="#cite_note-1">[1]</a>.</li>

<li>Considérons l'application <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/959905040c4110bae682eae9db986227c5506dcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}}" loading="lazy"> dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a,b,c,d\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b3bc922e9ec9581fb5c7274739ed50e0fdd19b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.877ex; height:2.843ex;" alt="{\displaystyle \{a,b,c,d\}}" loading="lazy"> définie par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1)=a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2667b997ed0ca0c967d4227319a8b426c7d51314.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.579ex; height:2.843ex;" alt="{\displaystyle f(1)=a}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(2)=c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(2)=c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fb5184fe323d54fbf233aedcc446018739380c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.356ex; height:2.843ex;" alt="{\displaystyle f(2)=c}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(3)=d}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a96e76bf95640a83cdb82bc99313e363ee479a1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.565ex; height:2.843ex;" alt="{\displaystyle f(3)=d}" loading="lazy">. L'image directe de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2,3\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{2,3\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d118b286f8bcc9dad1b4f5f37d5a7342d95f2e2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{2,3\}}" loading="lazy"> par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(\{2,3\}\right)=\{c,d\}}">

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(\{2,3\}\right)=\{c,d\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84458ec3812bb00a422e8f60850a31028afb7407.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.839ex; height:2.843ex;" alt="{\displaystyle f\left(\{2,3\}\right)=\{c,d\}}" loading="lazy"> tandis que l'image de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a,c,d\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3f36b4e2981a5a93918b420bfeeeb3281cbd78c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.845ex; height:2.843ex;" alt="{\displaystyle \{a,c,d\}}" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Propriétés_élémentaires">Propriétés élémentaires</h2></div>

<ul><li>Pour toutes parties <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bc2435b217c1a0f46f8a517ffa225c6f9440e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ec73b8bc9abc3efb934f5a6ec2803713771f4bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{2}}" loading="lazy"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">,<center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(A_{1}\cup A_{2}\right)=f(A_{1})\cup f(A_{2}).}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(A_{1}\cup A_{2}\right)=f(A_{1})\cup f(A_{2}).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/efbf19b66916a2ebdee59abd93d8e1a79a03102e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.751ex; height:2.843ex;" alt="{\displaystyle f\left(A_{1}\cup A_{2}\right)=f(A_{1})\cup f(A_{2}).}" loading="lazy"></center>Plus généralement, pour toute <a href="Famille_(math%C3%A9matiques)" title="Famille (mathématiques)">famille</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(A_{i}\right)_{i\in I}}">

<msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(A_{i}\right)_{i\in I}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df7ea3baedc29fbe774f225fffcd7d7a98eee878.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.077ex; height:3.009ex;" alt="{\displaystyle \left(A_{i}\right)_{i\in I}}" loading="lazy"> de parties de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">,<center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(\bigcup _{i\in I}A_{i}\right)=\bigcup _{i\in I}f(A_{i}).}">

<mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(\bigcup _{i\in I}A_{i}\right)=\bigcup _{i\in I}f(A_{i}).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7aab110272c5e7b8bf4fb01c8122d7561c25241.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.203ex; height:7.509ex;" alt="{\displaystyle f\left(\bigcup _{i\in I}A_{i}\right)=\bigcup _{i\in I}f(A_{i}).}" loading="lazy"></center></li>

<li>Pour toutes parties <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bc2435b217c1a0f46f8a517ffa225c6f9440e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ec73b8bc9abc3efb934f5a6ec2803713771f4bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{2}}" loading="lazy"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">,<center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(A_{1}\cap A_{2}\right)\subset f(A_{1})\cap f(A_{2})}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(A_{1}\cap A_{2}\right)\subset f(A_{1})\cap f(A_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e32d315146838203926b6c006573da738bbaf82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.104ex; height:2.843ex;" alt="{\displaystyle f\left(A_{1}\cap A_{2}\right)\subset f(A_{1})\cap f(A_{2})}" loading="lazy"></center>et cette inclusion peut être stricte, sauf si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est <a href="Injection_(math%C3%A9matiques)" title="Injection (mathématiques)">injective</a><a href="#cite_note-Wikiversité-2">[2]</a>. On peut même prouver que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est injective si et seulement si pour toutes parties <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bc2435b217c1a0f46f8a517ffa225c6f9440e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ec73b8bc9abc3efb934f5a6ec2803713771f4bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{2}}" loading="lazy"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, on a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(A_{1}\cap A_{2}\right)=f(A_{1})\cap f(A_{2})}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(A_{1}\cap A_{2}\right)=f(A_{1})\cap f(A_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f57313ed19e6e3ab463655910ea79ec41cd35a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.104ex; height:2.843ex;" alt="{\displaystyle f\left(A_{1}\cap A_{2}\right)=f(A_{1})\cap f(A_{2})}" loading="lazy">.</li></ul>

Plus généralement, pour toute famille non vide <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(A_{i}\right)_{i\in I}}">

<msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(A_{i}\right)_{i\in I}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df7ea3baedc29fbe774f225fffcd7d7a98eee878.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.077ex; height:3.009ex;" alt="{\displaystyle \left(A_{i}\right)_{i\in I}}" loading="lazy"> de parties de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">,<center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(\bigcap _{i\in I}A_{i}\right)\subset \bigcap _{i\in I}f(A_{i})}">

<mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(\bigcap _{i\in I}A_{i}\right)\subset \bigcap _{i\in I}f(A_{i})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00e74e652b476d90c154386c23fad124f956dd95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.556ex; height:7.509ex;" alt="{\displaystyle f\left(\bigcap _{i\in I}A_{i}\right)\subset \bigcap _{i\in I}f(A_{i})}" loading="lazy">.</center>

<ul><li>Toute partie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" 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}" loading="lazy"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> <a href="Inclusion_(math%C3%A9matiques)" title="Inclusion (mathématiques)">contient</a> l'image directe de son <a href="Image_r%C3%A9ciproque" title="Image réciproque">image réciproque</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(B)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(B)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/572ddad8cd0a0758fb98e1c94c432dc2f7a06636.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.227ex; height:3.176ex;" alt="{\displaystyle f^{-1}(B)}" loading="lazy"> ; plus précisément<a href="#cite_note-Wikiversité-2">[2]</a> :<center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(f^{-1}(B)\right)=B\cap \mathrm {Im} (f).}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(f^{-1}(B)\right)=B\cap \mathrm {Im} (f).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d3d681ec45cc9b67d6345c09a7c0f379928730.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.977ex; height:3.343ex;" alt="{\displaystyle f\left(f^{-1}(B)\right)=B\cap \mathrm {Im} (f).}" loading="lazy"></center>En particulier, si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est <a href="Surjective" class="mw-redirect" title="Surjective">surjective</a> alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(f^{-1}(B))=B}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(f^{-1}(B))=B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bbdeaaa86e32f977099f6bcf766c0f64c431113.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.177ex; height:3.176ex;" alt="{\displaystyle f(f^{-1}(B))=B}" loading="lazy">.</li></ul>

<dl><dd>On peut même prouver que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est surjective si et seulement si pour toute partie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" 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}" loading="lazy"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> on a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(f^{-1}(B)\right)=B}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(f^{-1}(B)\right)=B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/985307a785d2000a8cb3ec1b0af06ab82102ae86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.885ex; height:3.343ex;" alt="{\displaystyle f\left(f^{-1}(B)\right)=B}" loading="lazy">.</dd>

<dd>(Une démonstration est proposée dans l'article <a href="Surjection" title="Surjection">Surjection</a>.)</dd></dl>

<ul><li>Toute partie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.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"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> est contenue dans l'image réciproque de son image directe :<center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset f^{-1}\left(f(A)\right)}">

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subset f^{-1}\left(f(A)\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ccb5b6186a058977298f768c5cf3b275e03af2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.522ex; height:3.176ex;" alt="{\displaystyle A\subset f^{-1}\left(f(A)\right)}" loading="lazy"></center>et cette inclusion peut être stricte, sauf si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est injective<a href="#cite_note-Wikiversité-2">[2]</a>. On peut même prouver que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> est injective si et seulement si pour toutes parties <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.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"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, on a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=f^{-1}\left(f(A)\right)}">

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=f^{-1}\left(f(A)\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a9024d1cbdae6ffafeb70cfeaaf4443036fe2a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.522ex; height:3.176ex;" alt="{\displaystyle A=f^{-1}\left(f(A)\right)}" loading="lazy">.</li>

<li>Si l'on considère de plus une application <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:Y\rightarrow Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g:Y\rightarrow Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a976067ecdfce0310e7d05b555ea816197f0410.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.121ex; height:2.509ex;" alt="{\displaystyle g:Y\rightarrow Z}" loading="lazy">, alors l'image directe d'une partie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.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"> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> par la <a href="Composition_de_fonctions" title="Composition de fonctions">composée</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f:X\to Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f:X\to Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e5cfd853da7199884ac0460cecd0b281f9846d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.801ex; height:2.509ex;" alt="{\displaystyle g\circ f:X\to Z}" loading="lazy"> est :</li></ul>

<mrow>

</mrow>

<mrow>

<mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (g\circ f)\left(A\right)=g\left(f(A)\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2e5443dcb63dcdc687f39d23d8032dbdef39d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.58ex; height:2.843ex;" alt="{\displaystyle (g\circ f)\left(A\right)=g\left(f(A)\right)}" loading="lazy"></center>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Pour éviter toute confusion, <a href="Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a> et <a href="Garrett_Birkhoff" title="Garrett Birkhoff">Garrett Birkhoff</a>, <cite class="italique">Algèbre</cite> [détail des éditions]<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alg%C3%A8bre&rft.au=Saunders+Mac+Lane&rft.au=Garrett+Birkhoff&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AImage+directe">, vol. 1, <abbr class="abbr" title="page">p.</abbr> 8, parlent d'une application ensembliste, qu'ils notent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">*.

</li>

<li id="cite_note-Wikiversité-2">Pour une démonstration, voir par exemple le <a class="external text" href="https://fr.wikiversity.org/wiki/fr:Application_(math%C3%A9matiques)/Exercices/Images_directes_et_r%C3%A9ciproques">corrigé de l'exercice correspondant</a> sur Wikiversité.

</li>

</ol></div>

</div>

<div class="mw-heading mw-heading2"><h2 id="Articles_connexes">Articles connexes</h2></div>

<ul><li><a href="Th%C3%A9orie_na%C3%AFve_des_ensembles" title="Théorie naïve des ensembles">Théorie naïve des ensembles</a></li>

<li><a href="Fonction_multivalu%C3%A9e#Domaine,_image,_sélection" title="Fonction multivaluée">Image d'une partie par une fonction multivaluée</a> (autrement dit : par une <a href="Relation_binaire" title="Relation binaire">relation binaire</a>)</li></ul>

<ul id="bandeau-portail" class="bandeau-portail"><li> Portail des mathématiques </li> </ul></div><div><div class="zim-footer">

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