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<title>Reflexive Relation</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Reflexive Relation</h1>

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Die Reflexivität einer zweistelligen <a href="Relation_(Mathematik)" title="Relation (Mathematik)">Relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> auf einer <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a> ist gegeben, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xRx}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09b1aa93638c6cdd68b2e5f177dfa0221a8d288c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.423ex; height:2.176ex;" alt="{\displaystyle xRx}" loading="lazy"> für alle Elemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</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"> der Menge gilt, also jedes Element in Relation zu sich selbst steht. Man nennt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> dann reflexiv.

Eine Relation heißt irreflexiv, wenn die Beziehung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xRx}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09b1aa93638c6cdd68b2e5f177dfa0221a8d288c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.423ex; height:2.176ex;" alt="{\displaystyle xRx}" loading="lazy"> für kein Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</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"> der Menge gilt, also kein Element in Relation zu sich selbst steht. Es gibt auch Relationen, die weder reflexiv noch irreflexiv sind, wenn die Beziehung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xRx}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09b1aa93638c6cdd68b2e5f177dfa0221a8d288c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.423ex; height:2.176ex;" alt="{\displaystyle xRx}" loading="lazy"> für einige Elemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</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"> der Menge gilt, doch nicht für alle.

Die Reflexivität ist eine der Voraussetzungen für eine <a href="%C3%84quivalenzrelation" title="Äquivalenzrelation">Äquivalenzrelation</a> oder eine <a href="Ordnungsrelation" title="Ordnungsrelation">Ordnungsrelation</a>; die Irreflexivität ist eine der Voraussetzungen für eine <a href="Strenge_Halbordnung" class="mw-redirect" title="Strenge Halbordnung">strikte Ordnungsrelation</a>.

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

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> eine Menge und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\subseteq M\times M}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8575e1044516a2e1b54090f748c87d59fbdba42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.587ex; height:2.343ex;" alt="{\displaystyle R\subseteq M\times M}" loading="lazy"> eine zweistellige Relation auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, dann definiert man (unter Verwendung der <a href="Infixnotation" title="Infixnotation">Infixnotation</a>):

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> ist reflexiv :<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Longleftrightarrow \forall x\in M:xRx}">

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<annotation encoding="application/x-tex">{\displaystyle \Longleftrightarrow \forall x\in M:xRx}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26e1c5a0a7f98008ec0fc1df5b86c6a6ff5fb495.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.227ex; height:2.176ex;" alt="{\displaystyle \Longleftrightarrow \forall x\in M:xRx}" loading="lazy"></dd>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> ist irreflexiv :<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Longleftrightarrow \forall x\in M:\neg \ xRx}">

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<annotation encoding="application/x-tex">{\displaystyle \Longleftrightarrow \forall x\in M:\neg \ xRx}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fe1ceffd9e294b68dcedc78df8d3ff77e2eda4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.358ex; height:2.176ex;" alt="{\displaystyle \Longleftrightarrow \forall x\in M:\neg \ xRx}" loading="lazy"></dd></dl>

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

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

<ul><li>Die <a href="Vergleich_(Zahlen)" title="Vergleich (Zahlen)">Kleiner-Gleich-Relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/440568a09c3bfdf0e1278bfa79eb137c04e94035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }" loading="lazy"> auf den <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a> ist reflexiv, da stets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq x}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96a5929024fa38cfa940f872cbaba86a7a3b2097.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.758ex; height:2.176ex;" alt="{\displaystyle x\leq x}" loading="lazy"> gilt. Sie ist darüber hinaus eine <a href="Totalordnung" class="mw-redirect" title="Totalordnung">Totalordnung</a>. Gleiches gilt für die Relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \geq }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcef7c0e95bb77a35fd1a874ca91f425215f3c26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \geq }" loading="lazy">.</li></ul>

<ul><li>Die gewöhnliche Gleichheit <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"> auf den reellen Zahlen ist reflexiv, da stets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38d5b1d45c7aa8346eacace86bbb5cb0226f8fb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.758ex; height:1.676ex;" alt="{\displaystyle x=x}" loading="lazy"> gilt. Sie ist darüber hinaus eine <a href="%C3%84quivalenzrelation" title="Äquivalenzrelation">Äquivalenzrelation</a>.</li></ul>

<ul><li>Die <a href="Teilmenge" title="Teilmenge">Teilmengenbeziehung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subseteq }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a924f8dcb2847bb8871edfdbf4c6b5cca0669228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \subseteq }" loading="lazy"> zwischen <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Mengen</a> ist reflexiv, da stets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1ce5093be9e30238b83393aed738eafd3a43030.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.585ex; height:2.343ex;" alt="{\displaystyle A\subseteq A}" loading="lazy"> gilt. Sie ist darüber hinaus eine <a href="Halbordnung" class="mw-redirect" title="Halbordnung">Halbordnung</a>.</li></ul>

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

<ul><li>Die Kleiner-Relation <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33737c89a17785dacc8638b4d66db3d5c8670de1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}" loading="lazy"> auf den <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a> ist irreflexiv, da nie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<x}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce90a27742f68ed7a58da6fb169367ace6cb8326.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.758ex; height:1.843ex;" alt="{\displaystyle x<x}" loading="lazy"> gilt. Sie ist darüber hinaus eine <a href="Strenge_Totalordnung" class="mw-redirect" title="Strenge Totalordnung">strenge Totalordnung</a>. Gleiches gilt für die Relation <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b27b77ab4e3293ea9ce65cef60fea655c398423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle >}" loading="lazy">.</li></ul>

<ul><li>Die Ungleichheit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neq }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38cc3d8d8c60120bc2f905bae4d5e10d8ad6a3f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.676ex;" alt="{\displaystyle \neq }" loading="lazy"> auf den reellen Zahlen ist irreflexiv, da nie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq x}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39df4cb02d792b90d62a0e502b001434bce719f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.758ex; height:2.676ex;" alt="{\displaystyle x\neq x}" loading="lazy"> gilt.</li></ul>

<ul><li>Die <a href="Echte_Teilmenge" class="mw-redirect" title="Echte Teilmenge">echte Teilmengenbeziehung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subset }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f51f0eeff0c2a9dcb9c856f87ca0359e701ef01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \subset }" loading="lazy"> zwischen <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Mengen</a> ist irreflexiv, da nie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/591a0259a895623c32e639a58dbda27e3c5bd825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.585ex; height:2.176ex;" alt="{\displaystyle A\subset A}" loading="lazy"> gilt. Sie ist darüber hinaus eine <a href="Strenge_Halbordnung" class="mw-redirect" title="Strenge Halbordnung">strenge Halbordnung</a>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Weder_reflexiv_noch_irreflexiv">Weder reflexiv noch irreflexiv</h3></div>

Die folgende Relation auf der Menge der reellen Zahlen ist weder reflexiv noch irreflexiv:

<msup>

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<annotation encoding="application/x-tex">{\displaystyle xRy:\Longleftrightarrow y=x^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/baa270dde23bd000a6a192793b71ee6cdc84f289.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.141ex; height:3.009ex;" alt="{\displaystyle xRy:\Longleftrightarrow y=x^{2}}" loading="lazy"></dd></dl>

Grund: Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:=1}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca0552d5e790e0c39c8e80236463ec5ef345a687.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.237ex; height:2.176ex;" alt="{\displaystyle x:=1}" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xRx}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09b1aa93638c6cdd68b2e5f177dfa0221a8d288c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.423ex; height:2.176ex;" alt="{\displaystyle xRx}" loading="lazy">, für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:=2}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c26fd74994513e464a0ec100c7419a662aa017d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.237ex; height:2.176ex;" alt="{\displaystyle x:=2}" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg xRx}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c94a9ac884e6b6bd80dd88e810e5695b06c6467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.974ex; height:2.176ex;" alt="{\displaystyle \neg xRx}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Darstellung_als_gerichteter_Graph">Darstellung als gerichteter Graph</h2></div>

Jede beliebige Relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> auf einer Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> kann als <a href="Gerichteter_Graph" title="Gerichteter Graph">gerichteter Graph</a> aufgefasst werden (siehe Beispiel im Bild oben). Die Knoten des Graphen sind dabei die Elemente von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. Vom Knoten <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/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"> zum Knoten <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/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> wird genau dann eine <a href="Gerichtete_Kante" class="mw-redirect" title="Gerichtete Kante">gerichtete Kante</a> (ein Pfeil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\longrightarrow b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4889b51107e38779bb26d95f8456d5f9a49b2e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.323ex; height:2.176ex;" alt="{\displaystyle a\longrightarrow b}" loading="lazy">) gezogen, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aRb}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8b0b52168739fd16b254298771ec07b900e5a6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.991ex; height:2.176ex;" alt="{\displaystyle aRb}" loading="lazy"> gilt.

Die Reflexivität von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> lässt sich im Graphen nun so charakterisieren: Für jeden Knoten <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/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"> gibt es eine <a href="Schleife_(Graphentheorie)" title="Schleife (Graphentheorie)">Schleife</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {a}{\circlearrowright }}}">

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35710c05e7f2900d153d13dbb6a3e3343515cde3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:3.676ex;" alt="{\displaystyle {\stackrel {a}{\circlearrowright }}}" loading="lazy">. Entsprechend ist die Irreflexivität dadurch gegeben, dass es für keinen Knoten <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/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"> eine Schleife <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {a}{\circlearrowright }}}">

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35710c05e7f2900d153d13dbb6a3e3343515cde3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:3.676ex;" alt="{\displaystyle {\stackrel {a}{\circlearrowright }}}" loading="lazy"> gibt.

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

<ul><li>Mit Hilfe der identischen Relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Id_{M}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/181f1a09c64780b0ddea86cfe5df889629ffc887.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.34ex; height:2.509ex;" alt="{\displaystyle Id_{M}}" loading="lazy"> (die aus allen Paaren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72f9e25892f6d000349b8bb6578a59567efbdd63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle (x,x)}" loading="lazy"> besteht) kann man die Begriffe auch so charakterisieren:

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> ist reflexiv <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Longleftrightarrow Id_{M}\subseteq R}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Longleftrightarrow Id_{M}\subseteq R}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61f404db665f11f0a9b6e4b11c07af4815faa3b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.164ex; height:2.509ex;" alt="{\displaystyle \Longleftrightarrow Id_{M}\subseteq R}" loading="lazy"></dd>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> ist irreflexiv <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Longleftrightarrow Id_{M}\cap R=\varnothing }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Longleftrightarrow Id_{M}\cap R=\varnothing }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee8b6fbf9b8081acf7ded93c94c7f9fd282755b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.555ex; height:2.509ex;" alt="{\displaystyle \Longleftrightarrow Id_{M}\cap R=\varnothing }" loading="lazy"></dd></dl></li></ul>

<ul><li>Ist die Relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> reflexiv bzw. irreflexiv, dann gilt dies auch für die konverse Relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91122db9e35b74f39c755e207749062fffa55e7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.097ex; height:2.676ex;" alt="{\displaystyle R^{-1}}" loading="lazy">. Beispiele: die zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/440568a09c3bfdf0e1278bfa79eb137c04e94035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }" loading="lazy"> konverse Relation ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \geq }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcef7c0e95bb77a35fd1a874ca91f425215f3c26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \geq }" loading="lazy">, die zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33737c89a17785dacc8638b4d66db3d5c8670de1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}" loading="lazy"> konverse ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle >}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b27b77ab4e3293ea9ce65cef60fea655c398423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle >}" loading="lazy">.</li></ul>

<ul><li>Ist die Relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> reflexiv, dann ist die <a href="Komplement_(Mengenlehre)" title="Komplement (Mengenlehre)">komplementäre</a> Relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\rm {c}}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8651196874534990610da40a623feccf386fbb5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.726ex; height:2.343ex;" alt="{\displaystyle R^{\rm {c}}}" loading="lazy"> irreflexiv. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"> irreflexiv, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\rm {c}}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8651196874534990610da40a623feccf386fbb5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.726ex; height:2.343ex;" alt="{\displaystyle R^{\rm {c}}}" loading="lazy"> reflexiv. Dabei ist die komplementäre Relation definiert durch

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xR^{\rm {c}}y:\Longleftrightarrow \neg xRy}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbe1bb405379c0047cbc98f6bb71ce180128422b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.265ex; height:2.676ex;" alt="{\displaystyle xR^{\rm {c}}y:\Longleftrightarrow \neg xRy}" loading="lazy">.</dd></dl></li></ul>

<ul><li>Die Relation auf der <a href="Leere_Menge" title="Leere Menge">leeren Menge</a> ist als einzige Relation sowohl reflexiv als auch irreflexiv.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Nichts%C3%A4ttigungsaxiom" title="Nichtsättigungsaxiom">Nichtsättigungsaxiom</a></li>

<li><a href="Reflexive_H%C3%BClle" title="Reflexive Hülle">Reflexive Hülle</a></li>

<li><a href="Reflexiv-transitive_H%C3%BClle" class="mw-redirect" title="Reflexiv-transitive Hülle">Reflexiv-transitive Hülle</a></li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-12-27" href="https://de.wikipedia.org/wiki/?title=Reflexive_Relation&oldid=262776709">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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