@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-GJ7QW40K-0
  skos:prefLabel "fonction de Dirichlet"@fr, "Dirichlet function"@en ;
  a skos:Concept ;
  skos:broader psr:-DMTN3C45-S .

psr: a skos:ConceptScheme .
psr:-T88XBMNP-M
  skos:prefLabel "set theory"@en, "théorie des ensembles"@fr ;
  a skos:Concept ;
  skos:narrower psr:-DMTN3C45-S .

psr:-L2BN0W1T-P
  skos:prefLabel "fonction"@fr, "function"@en ;
  a skos:Concept ;
  skos:narrower psr:-DMTN3C45-S .

psr:-DMTN3C45-S
  skos:prefLabel "fonction caractéristique"@fr, "characteristic function"@en ;
  skos:definition """In mathematics, an <b>indicator function</b> or a <b>characteristic function</b> of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if <span class="texhtml mvar" style="font-style:italic;">A</span> is a subset of some set <span class="texhtml mvar" style="font-style:italic;">X</span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbf {1} _{A}(x)=1}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn mathvariant="bold">1</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>A</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mn>1</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbf {1} _{A}(x)=1}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9394f68a77dda0cf344969bc54ab234bb18f3ae6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:10.201ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mathbf {1} _{A}(x)=1}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle x\\\\in A,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mi>A</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x\\\\in A,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c649021f8a56d0234bfa3eae952bcab3edad556" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:6.56ex; height:2.509ex;" alt="{\\\\displaystyle x\\\\in A,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbf {1} _{A}(x)=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn mathvariant="bold">1</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>A</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbf {1} _{A}(x)=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc54d93d8cda629151f17bb6f2ef8efa42c6046e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:10.201ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mathbf {1} _{A}(x)=0}"></span> otherwise, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbf {1} _{A}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn mathvariant="bold">1</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>A</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbf {1} _{A}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4a6cdf0babfdaa63fa048de3d9b8d0a4c06522" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.801ex; height:2.509ex;" alt="\\\\mathbf {1} _{A}"></span> is a common notation for the indicator function. Other common notations are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle I_{A},}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>I</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>A</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle I_{A},}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/137ebc08aa81899db8a8a2899f501652a18f7647" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.135ex; height:2.509ex;" alt="{\\\\displaystyle I_{A},}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\chi _{A}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>χ<!-- χ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>A</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\chi _{A}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6131ab890635772613e44154dfe21408f815c0e2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.567ex; height:2.009ex;" alt="{\\\\displaystyle \\\\chi _{A}.}"></span>
<br/>The indicator function  of <span class="texhtml mvar" style="font-style:italic;">A</span> is the Iverson bracket of the property of belonging to <span class="texhtml mvar" style="font-style:italic;">A</span>; that is, 
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbf {1} _{A}(x)=[x\\\\in A].}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn mathvariant="bold">1</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>A</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mi>x</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mi>A</mi>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbf {1} _{A}(x)=[x\\\\in A].}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00af42be0e623a2d4254bcc50c72239c4d5dece0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:16.893ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mathbf {1} _{A}(x)=[x\\\\in A].}"></span></dd></dl>
<br/>For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Indicator_function">https://en.wikipedia.org/wiki/Indicator_function</a>)"""@en, """En mathématiques, une <b>fonction caractéristique</b>, ou <b>fonction indicatrice</b>, est une fonction définie sur un ensemble <i>E</i> qui explicite l’appartenance ou non à un sous-ensemble <i>F</i> de <i>E</i> de tout élément de <i>E</i>. 
<br/>Formellement, la fonction caractéristique d’un sous-ensemble <i>F</i> d’un ensemble <i>E</i> est une fonction&nbsp;:
<br/>
<br/><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\egin{array}{rcl}\\\\chi _{F}:E&amp;\\\\longrightarrow &amp;\\\\{0,1\\\\}\\\\\\\\x&amp;\\\\longmapsto &amp;\\\\left\\\\{{\\egin{matrix}1\\\\ {\\\\mbox{si}}\\\\ x\\\\ \\\\in \\\\ F\\\\\\\\0\\\\ {\\\\mbox{si}}\\\\ x\\\\ \\
otin \\\\ F\\\\end{matrix}}\\
ight.\\\\end{array}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em">
<br/>            <mtr>
<br/>              <mtd>
<br/>                <msub>
<br/>                  <mi>χ<!-- χ --></mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi>F</mi>
<br/>                  </mrow>
<br/>                </msub>
<br/>                <mo>:</mo>
<br/>                <mi>E</mi>
<br/>              </mtd>
<br/>              <mtd>
<br/>                <mo stretchy="false">⟶<!-- ⟶ --></mo>
<br/>              </mtd>
<br/>              <mtd>
<br/>                <mo fence="false" stretchy="false">{</mo>
<br/>                <mn>0</mn>
<br/>                <mo>,</mo>
<br/>                <mn>1</mn>
<br/>                <mo fence="false" stretchy="false">}</mo>
<br/>              </mtd>
<br/>            </mtr>
<br/>            <mtr>
<br/>              <mtd>
<br/>                <mi>x</mi>
<br/>              </mtd>
<br/>              <mtd>
<br/>                <mo stretchy="false">⟼<!-- ⟼ --></mo>
<br/>              </mtd>
<br/>              <mtd>
<br/>                <mrow>
<br/>                  <mo>{</mo>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mtable rowspacing="4pt" columnspacing="1em">
<br/>                      <mtr>
<br/>                        <mtd>
<br/>                          <mn>1</mn>
<br/>                          <mtext>&nbsp;</mtext>
<br/>                          <mrow class="MJX-TeXAtom-ORD">
<br/>                            <mstyle displaystyle="false" scriptlevel="0">
<br/>                              <mtext>si</mtext>
<br/>                            </mstyle>
<br/>                          </mrow>
<br/>                          <mtext>&nbsp;</mtext>
<br/>                          <mi>x</mi>
<br/>                          <mtext>&nbsp;</mtext>
<br/>                          <mo>∈<!-- ∈ --></mo>
<br/>                          <mtext>&nbsp;</mtext>
<br/>                          <mi>F</mi>
<br/>                        </mtd>
<br/>                      </mtr>
<br/>                      <mtr>
<br/>                        <mtd>
<br/>                          <mn>0</mn>
<br/>                          <mtext>&nbsp;</mtext>
<br/>                          <mrow class="MJX-TeXAtom-ORD">
<br/>                            <mstyle displaystyle="false" scriptlevel="0">
<br/>                              <mtext>si</mtext>
<br/>                            </mstyle>
<br/>                          </mrow>
<br/>                          <mtext>&nbsp;</mtext>
<br/>                          <mi>x</mi>
<br/>                          <mtext>&nbsp;</mtext>
<br/>                          <mo>∉<!-- ∉ --></mo>
<br/>                          <mtext>&nbsp;</mtext>
<br/>                          <mi>F</mi>
<br/>                        </mtd>
<br/>                      </mtr>
<br/>                    </mtable>
<br/>                  </mrow>
<br/>                  <mo fence="true" stretchy="true" symmetric="true"></mo>
<br/>                </mrow>
<br/>              </mtd>
<br/>            </mtr>
<br/>          </mtable>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\egin{array}{rcl}\\\\chi _{F}:E&amp;\\\\longrightarrow &amp;\\\\{0,1\\\\}\\\\\\\\x&amp;\\\\longmapsto &amp;\\\\left\\\\{{\\egin{matrix}1\\\\ {\\\\mbox{si}}\\\\ x\\\\ \\\\in \\\\ F\\\\\\\\0\\\\ {\\\\mbox{si}}\\\\ x\\\\ \\
otin \\\\ F\\\\end{matrix}}\\
ight.\\\\end{array}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/823cbe5613ea4048b01246f01a5ef50432c30590" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.171ex; width:29.287ex; height:9.509ex;" alt="{\\egin{array}{rcl}\\\\chi _{F}:E&amp;\\\\longrightarrow &amp;\\\\{0,1\\\\}\\\\\\\\x&amp;\\\\longmapsto &amp;\\\\left\\\\{{\\egin{matrix}1\\\\ {\\\\mbox{si}}\\\\ x\\\\ \\\\in \\\\ F\\\\\\\\0\\\\ {\\\\mbox{si}}\\\\ x\\\\ \\
otin \\\\ F\\\\end{matrix}}\\
ight.\\\\end{array}}"></span></center>
<br/>D'autres notations souvent employées pour la fonction caractéristique de <span class="texhtml mvar" style="font-style:italic;">F</span> sont <span class="texhtml"><b>1</b><sub><i>F</i></sub></span> et <span class="texhtml">𝟙<sub><i>F</i></sub></span>, voire <span class="texhtml">I</span> (<i>i</i> majuscule).
<br/>Le terme de <i>fonction indicatrice</i> est parfois utilisé pour fonction caractéristique. Cette dénomination évite la confusion avec la fonction caractéristique utilisée en probabilité mais en induit une autre, avec la fonction indicatrice en analyse convexe. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_caract%C3%A9ristique_(th%C3%A9orie_des_ensembles)">https://fr.wikipedia.org/wiki/Fonction_caract%C3%A9ristique_(th%C3%A9orie_des_ensembles)</a>)"""@fr ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:narrower psr:-GJ7QW40K-0 ;
  skos:broader psr:-T88XBMNP-M, psr:-L2BN0W1T-P ;
  skos:altLabel "fonction indicatrice"@fr, "indicator function"@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Fonction_caract%C3%A9ristique_(th%C3%A9orie_des_ensembles)>, <https://en.wikipedia.org/wiki/Indicator_function> ;
  dc:modified "2023-07-26"^^xsd:date .