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<title>Transitive Relation</title>

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

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Eine transitive Relation ist in der <a href="Mengenlehre" title="Mengenlehre">Mengenlehre</a> eine zweistellige <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>, die die Eigenschaft hat, dass für drei 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">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> dieser Menge aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xRy}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/324aab4e2674bb19cc073ea887888b98f0fc63d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.249ex; height:2.509ex;" alt="{\displaystyle xRy}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle yRz}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b3a8922539e112f785ba6ec0f4a899b386d5e59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.008ex; height:2.509ex;" alt="{\displaystyle yRz}" loading="lazy"> stets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xRz}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8af3e5e77a92627b4b1120a3167a2c28d7a343b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.182ex; height:2.176ex;" alt="{\displaystyle xRz}" loading="lazy"> folgt. Beispiele für transitive Relationen sind die Gleich- und die Kleiner-Relationen auf den <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a>, denn für drei reelle Zahlen <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">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/409a91214d63eabe46ec10ff3cbba689ab687366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3de8f5225200d63f8f4678bbe6d07c4d384e73ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.342ex; height:2.009ex;" alt="{\displaystyle y=z}" loading="lazy"> gilt immer auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79c798ab79074bda962dc350b24e4b742d3a9b96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.516ex; height:1.676ex;" alt="{\displaystyle x=z}" loading="lazy">, und aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aeb239de6fee56ea8b6a65f7858d95b87632069f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y<z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29fe72701bcf411046e7d2431f0cb5850ec523de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.342ex; height:2.176ex;" alt="{\displaystyle y<z}" loading="lazy"> folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fef9d28e745a05b0f1d25ac92860e9185d8d8514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.516ex; height:1.843ex;" alt="{\displaystyle x<z}" loading="lazy">.

Eine nicht transitive Relation heißt <a href="Intransitive_Relation" title="Intransitive Relation">intransitiv</a> (nicht zu verwechseln mit <a href="Negative_Transitivit%C3%A4t" title="Negative Transitivität">negativer Transitivität</a>). Die Transitivität ist eine der Voraussetzungen für eine <a href="%C3%84quivalenzrelation" title="Äquivalenzrelation">Äquivalenzrelation</a> oder eine <a href="Ordnungsrelation" title="Ordnungsrelation">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 heißt <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"> transitiv, wenn (unter Verwendung der <a href="Infixnotation" title="Infixnotation">Infixnotation</a>) gilt:<a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \forall x,y,z\in M\colon xRy\land yRz\Rightarrow xRz.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/914e327f63227cfc532a641975586c110a95cb9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.533ex; height:2.509ex;" alt="{\displaystyle \forall x,y,z\in M\colon xRy\land yRz\Rightarrow xRz.}" loading="lazy"></dd></dl>

<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>

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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 Menge <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"> kann als <a href="Gerichteter_Graph" title="Gerichteter Graph">gerichteter Graph</a> aufgefasst werden (Beispiel siehe oben). Die Knoten des Graphen sind dabei die Elemente von <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">. Vom Knoten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</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}">

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

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</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}">

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

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</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}">

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

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</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 Transitivität von <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"> lässt sich im Graphen nun so charakterisieren: Wann immer zwei Pfeile aufeinanderfolgen (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\longrightarrow b\longrightarrow c}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb1ecc5b25226053340fbb2463dcc7e20e7c5cf7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.426ex; height:2.176ex;" alt="{\displaystyle a\longrightarrow b\longrightarrow c}" loading="lazy">), gibt es auch einen Pfeil, der Anfangs- und Endknoten direkt verbindet (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\longrightarrow c}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08d1a126ea68e286cb6886a6dc678ede5f81a563.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle a\longrightarrow c}" loading="lazy">).

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

<ul><li>Die Transitivität einer 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>

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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"> erlaubt auch Schlüsse über mehrere Schritte hinweg (wie man leicht durch <a href="Vollst%C3%A4ndige_Induktion" title="Vollständige Induktion">vollständige Induktion</a> zeigt):<a href="#cite_note-:0-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle a\,R\,b_{1}\,R\,b_{2}\,R\,\dots \,R\,b_{n}\,R\,c\implies a\,R\,c}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/281be692d4d07d131f77f9c0b35c547d48589f4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.906ex; height:2.509ex;" alt="{\displaystyle a\,R\,b_{1}\,R\,b_{2}\,R\,\dots \,R\,b_{n}\,R\,c\implies a\,R\,c}" loading="lazy"></dd></dl></li>

<li>Mit Hilfe der <a href="Komposition_(Mathematik)#Komposition_von_Relationen" title="Komposition (Mathematik)">Verkettung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99add39d2b681e2de7ff62422c32704a05c7ec31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }" loading="lazy"> von Relationen lässt sich die Transitivität auch durch die folgende Bedingung charakterisieren:<a href="#cite_note-:0-2">[2]</a>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bec874926f84c336b6bd0c2ae7fc8de35a4abe17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.585ex; height:2.343ex;" alt="{\displaystyle R\circ R\subseteq R}" loading="lazy"></dd></dl></li>

<li>Ist die 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>

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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"> transitiv, dann gilt dies auch für die <a href="Umkehrrelation" class="mw-redirect" title="Umkehrrelation">konverse Relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{-1}}">

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</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">.<a href="#cite_note-3">[3]</a> 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/d318cde0f890facf49a1441b29e3cf9f9920fcff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.034ex; 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/d7d02c6142f2143950bf6669fa38b12a5b587cd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.034ex; height:1.843ex;" alt="{\displaystyle >\ }" loading="lazy">.</li>

<li>Sind die Relationen <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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> transitiv, dann gilt dies auch für ihre <a href="Schnittmenge" class="mw-redirect" title="Schnittmenge">Schnittmenge</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\cap S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8d2b657f017f3b3fc08ff59bf8f3aaf3c5e28ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.846ex; height:2.176ex;" alt="{\displaystyle R\cap S}" loading="lazy">. Diese Aussage lässt sich von zwei Relationen auf den Durchschnitt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cap _{i\in I}R_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cap _{i\in I}R_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32c0e999b0db8200ef91950bd83fe0086b7e7184.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.838ex; height:2.509ex;" alt="{\displaystyle \cap _{i\in I}R_{i}}" loading="lazy"> einer beliebigen <a href="Familie_(Mathematik)" title="Familie (Mathematik)">Familie</a> von transitiven Relationen verallgemeinern.</li>

<li>Zu jeder beliebigen 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"> gibt es eine kleinste transitive Relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, die <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"> enthält, die sogenannte <a href="Transitive_H%C3%BClle_(Relation)" title="Transitive Hülle (Relation)">transitive Hülle</a> 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">.<a href="#cite_note-4">[4]</a> Beispiel: <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"> sei die Vorgängerrelation auf der Menge der natürlichen Zahlen, es gelte also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,R\,b:\Longleftrightarrow a=b-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\,R\,b:\Longleftrightarrow a=b-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d1bc3b36e530dc5d2e4db7f8d32508bc2bd2214.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.348ex; height:2.343ex;" alt="{\displaystyle a\,R\,b:\Longleftrightarrow a=b-1}" loading="lazy">. 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"> selbst ist nicht transitiv. Als transitive Hülle 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"> ergibt sich die Kleiner-Relation <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/d318cde0f890facf49a1441b29e3cf9f9920fcff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.034ex; height:1.843ex;" alt="{\displaystyle <\ }" loading="lazy">.</li>

<li>Jede Relation 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"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |M|<3}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9de1dca874210284619bb2f93d423c0aa6f9c53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.997ex; height:2.843ex;" alt="{\displaystyle |M|<3}" loading="lazy"> ist transitiv oder symmetrisch.</li></ul>

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

Wichtiges Beispiel aus der <a href="Volkswirtschaftslehre" title="Volkswirtschaftslehre">Volkswirtschaftslehre</a> ist das <a href="Nichts%C3%A4ttigungsaxiom" title="Nichtsättigungsaxiom">Nichtsättigungsaxiom</a>.

<div class="mw-heading mw-heading3"><h3 id="Ordnung_der_reellen_Zahlen">Ordnung der reellen Zahlen</h3></div>

Die Kleiner-Relation <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/d318cde0f890facf49a1441b29e3cf9f9920fcff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.034ex; height:1.843ex;" alt="{\displaystyle <\ }" loading="lazy"> auf den <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a> ist transitiv, denn aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aeb239de6fee56ea8b6a65f7858d95b87632069f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y<z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29fe72701bcf411046e7d2431f0cb5850ec523de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.342ex; height:2.176ex;" alt="{\displaystyle y<z}" loading="lazy"> folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fef9d28e745a05b0f1d25ac92860e9185d8d8514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.516ex; height:1.843ex;" alt="{\displaystyle x<z}" loading="lazy">. Sie ist darüber hinaus eine <a href="Strenge_Totalordnung" class="mw-redirect" title="Strenge Totalordnung">strenge Totalordnung</a>.

Ebenso sind die Relationen <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/d7d02c6142f2143950bf6669fa38b12a5b587cd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.034ex; height:1.843ex;" alt="{\displaystyle >\ }" loading="lazy">, <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_/eb734a37dd21ce173a46342d1cc64c92/11a2affb2bbcce694509721b698a3c4e52fb81ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.034ex; height:2.176ex;" alt="{\displaystyle \leq \ }" loading="lazy"> und <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_/eb734a37dd21ce173a46342d1cc64c92/43ce517b18ce2e7c177baead34a6edb1bee8c881.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.034ex; height:2.176ex;" alt="{\displaystyle \geq \ }" loading="lazy"> transitiv.

<div class="mw-heading mw-heading3"><h3 id="Gleichheit_der_reellen_Zahlen">Gleichheit der reellen Zahlen</h3></div>

Die gewöhnliche Gleichheit <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_/eb734a37dd21ce173a46342d1cc64c92/3be9469eef930cfa27f7fd9cace29705dfcee45a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.034ex; height:1.343ex;" alt="{\displaystyle =\ }" loading="lazy"> auf den reellen Zahlen ist transitiv, denn aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/409a91214d63eabe46ec10ff3cbba689ab687366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3de8f5225200d63f8f4678bbe6d07c4d384e73ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.342ex; height:2.009ex;" alt="{\displaystyle y=z}" loading="lazy"> folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79c798ab79074bda962dc350b24e4b742d3a9b96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.516ex; height:1.676ex;" alt="{\displaystyle x=z}" loading="lazy">. Sie ist darüber hinaus eine <a href="%C3%84quivalenzrelation" title="Äquivalenzrelation">Äquivalenzrelation</a>.

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

</mstyle>

</mrow>

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

</semantics>

</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 hingegen nicht transitiv:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3\neq 5}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6516c93c933d63b45a933e8ad0fb3e5e1f0f86c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.676ex;" alt="{\displaystyle 3\neq 5}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\neq 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 5\neq 3}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d0138c71f879a6c9e6437cdcbf0251f50df2413.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.676ex;" alt="{\displaystyle 5\neq 3}" loading="lazy">, aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\neq 3}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e53c37ab2930faa79b80792f60732ec845b43139.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.676ex;" alt="{\displaystyle 3\neq 3}" loading="lazy"> gilt natürlich nicht.

<div class="mw-heading mw-heading3"><h3 id="Teilbarkeit_der_ganzen_Zahlen">Teilbarkeit der ganzen Zahlen</h3></div>

Die <a href="Teilbarkeit#Definition" title="Teilbarkeit">Teilt-Relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddcc2e0b63522a705c3c691777c9f342806b19d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:0.647ex; height:2.843ex;" alt="{\displaystyle |}" loading="lazy"> für <a href="Ganze_Zahl" title="Ganze Zahl">ganze Zahlen</a> ist transitiv, denn aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a|b}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e801f5cb365c0f6ecd505ba404220ee7c1c43a11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.874ex; height:2.843ex;" alt="{\displaystyle a|b}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b|c}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fae1abb2571997b1834b074a9c7046cd210f943c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.651ex; height:2.843ex;" alt="{\displaystyle b|c}" loading="lazy"> folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a|c}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffa97c9c68c9c48029e78cca0272ec93c6dc42d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.883ex; height:2.843ex;" alt="{\displaystyle a|c}" loading="lazy">. Sie ist darüber hinaus eine <a href="Quasiordnung" title="Quasiordnung">Quasiordnung</a>. Bei der Einschränkung auf die Menge der <a href="Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürlichen Zahlen</a> erhält man eine <a href="Halbordnung" class="mw-redirect" title="Halbordnung">Halbordnung</a>.

Nicht transitiv ist zum Beispiel die <a href="Teilerfremdheit" title="Teilerfremdheit">Teilerfremdheit</a> in der Menge der natürlichen oder ganzen Zahlen. So sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 12}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a522d3aa5812a136a69f06e1b909d809e849be39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 12}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 5}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29483407999b8763f0ea335cf715a6a5e809f44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}" loading="lazy"> teilerfremd, ebenso <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 5}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29483407999b8763f0ea335cf715a6a5e809f44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 9}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32d3d1e1f9dfe0254c628379e69a69711fe4eabd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 9}" loading="lazy">, jedoch haben <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 12}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a522d3aa5812a136a69f06e1b909d809e849be39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 12}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 9}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32d3d1e1f9dfe0254c628379e69a69711fe4eabd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 9}" loading="lazy"> den <a href="Gemeinsamer_Teiler" class="mw-redirect" title="Gemeinsamer Teiler">gemeinsamen Teiler</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/991e33c6e207b12546f15bdfee8b5726eafbbb2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" loading="lazy">.

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

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

</mstyle>

</mrow>

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

</semantics>

</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 transitiv, denn aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b09068bd2f7ba899aeb883ebe670b2ad07b0c851.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle A\subseteq B}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq C}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87dc1864578bb8f696bbfb7ba1c134c0f1f20458.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.629ex; height:2.343ex;" alt="{\displaystyle B\subseteq C}" loading="lazy"> folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq C}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c52a0a9fb646e916904c85763d980be597191ad2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.608ex; height:2.343ex;" alt="{\displaystyle A\subseteq C}" loading="lazy">. Darüber hinaus ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subseteq }">

</mstyle>

</mrow>

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

</semantics>

</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"> eine Halbordnung.

Nicht transitiv ist zum Beispiel die <a href="Disjunkt" title="Disjunkt">Disjunktheit</a> von Mengen in jedem <a href="Mengensystem" title="Mengensystem">Mengensystem</a> <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">, das zwei verschiedene disjunkte Mengen enthält. So sind in der <a href="Potenzmenge" title="Potenzmenge">Potenzmenge</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M={\mathcal {P}}(\mathbb {N} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M={\mathcal {P}}(\mathbb {N} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed2f47be18cd587c088e1974b1be52e345f97bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.732ex; height:2.843ex;" alt="{\displaystyle M={\mathcal {P}}(\mathbb {N} )}" loading="lazy"> die Mengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace 1,2\rbrace }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ead46a0865476b87f775c73095d05520180a621.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 \lbrace 1,2\rbrace }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace 3\rbrace }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4e71502d1d56000933b9ddb9131eab4cbe838fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \lbrace 3\rbrace }" loading="lazy"> disjunkt, ebenso <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace 3\rbrace }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4e71502d1d56000933b9ddb9131eab4cbe838fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \lbrace 3\rbrace }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace 1,4\rbrace }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lbrace 1,4\rbrace }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d52cd1cf260f1e0a0eb24abc9b207d42b81c0397.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 \lbrace 1,4\rbrace }" loading="lazy">, nicht aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace 1,2\rbrace }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ead46a0865476b87f775c73095d05520180a621.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 \lbrace 1,2\rbrace }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace 1,4\rbrace }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lbrace 1,4\rbrace }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d52cd1cf260f1e0a0eb24abc9b207d42b81c0397.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 \lbrace 1,4\rbrace }" loading="lazy"> (da sie das Element 1 gemeinsam haben).

<div class="mw-heading mw-heading3"><h3 id="Parallele_Geraden">Parallele Geraden</h3></div>

In der <a href="Geometrie" title="Geometrie">Geometrie</a> ist die <a href="Parallel_(Geometrie)" class="mw-redirect" title="Parallel (Geometrie)">Parallelität</a> von Geraden transitiv: Sind sowohl die Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3755e3e04ec295992b2b5331655ef83a500a05c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}" loading="lazy"> parallel als auch die Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/207852a9b2705498c65b929b79fdda27b257ec87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{3}}" loading="lazy">, dann sind auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3755e3e04ec295992b2b5331655ef83a500a05c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/207852a9b2705498c65b929b79fdda27b257ec87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{3}}" loading="lazy"> parallel. Darüber hinaus ist die Parallelität eine Äquivalenzrelation.

<div class="mw-heading mw-heading3"><h3 id="Implikation_in_der_Logik">Implikation in der Logik</h3></div>

In der <a href="Logik" title="Logik">Logik</a> gilt die Transitivität bezüglich der <a href="Implikation" title="Implikation">Implikation</a>, wobei dies in der <a href="Pr%C3%A4dikatenlogik" title="Prädikatenlogik">Prädikatenlogik</a> auch als <a href="Modus_barbara" class="mw-redirect" title="Modus barbara">Modus barbara</a> bekannt ist:

Aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\Rightarrow B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e560143d45c97e6387c7c3aa90e9d7745002228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\Rightarrow B}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\Rightarrow C}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0b4266a521fdd07bcbc6e2df36106cb482e7e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.144ex; height:2.176ex;" alt="{\displaystyle B\Rightarrow C}" loading="lazy"> folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\Rightarrow C}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c237f084643e7d3cac97ead01b167dc17cf8701b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.124ex; height:2.176ex;" alt="{\displaystyle A\Rightarrow C}" loading="lazy"> (vergleiche auch: <a href="Schnittregel" title="Schnittregel">Schnittregel</a>).

Die Implikation definiert eine Quasiordnung auf den Formeln der jeweils betrachteten Logik.

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

<ul><li><cite class="lang" lang="en" dir="auto" style="font-style:italic">Transitivity</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Transitivity">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Transitive+Relation&rft.atitle=Transitivity&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none"> </li></ul>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Seymor Lipschutz, Marc Lipson: <cite style="font-style:italic">Schaum's Outline of Discrete Mathematics</cite>. McGraw Hill Professional, 1997, ISBN 978-0-07-136841-4, S. 33 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=6A5g3RiYiBUC&pg=PA33&dq=Transitive+Relation&hl=de&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjE6fru2_7-AhXFhP0HHZqRA3gQ6AF6BAgBEAI#v=onepage&q=Transitive%20Relation&f=false">google.de</a> [abgerufen am 18. Mai 2023]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Transitive+Relation&rft.au=Seymor+Lipschutz%2C+Marc+Lipson&rft.btitle=Schaum%27s+Outline+of+Discrete+Mathematics&rft.date=1997-06-22&rft.genre=book&rft.isbn=9780071368414&rft.pages=33&rft.pub=McGraw+Hill+Professional" style="display:none">

</li>

<li id="cite_note-:0-2">↑ <a href="#cite_ref-:0_2-0">a</a> <a href="#cite_ref-:0_2-1">b</a> Seymor Lipschutz, Marc Lipson: <cite style="font-style:italic">Schaum's Outline of Discrete Mathematics</cite>. McGraw Hill Professional, 1997, ISBN 978-0-07-136841-4, S. 34 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=6A5g3RiYiBUC&pg=PA33&dq=Transitive+Relation&hl=de&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjE6fru2_7-AhXFhP0HHZqRA3gQ6AF6BAgBEAI#v=onepage&q=Transitive%20Relation&f=false">google.de</a> [abgerufen am 18. Mai 2023]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Transitive+Relation&rft.au=Seymor+Lipschutz%2C+Marc+Lipson&rft.btitle=Schaum%27s+Outline+of+Discrete+Mathematics&rft.date=1997-06-22&rft.genre=book&rft.isbn=9780071368414&rft.pages=34&rft.pub=McGraw+Hill+Professional" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Dov M. Gabbay, John Woods: <cite style="font-style:italic">The Rise of Modern Logic: from Leibniz to Frege</cite>. Elsevier, 2004, ISBN 978-0-08-053287-5, S. 509 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=74-g3vSAbnoC&pg=PA509&dq=convers+relation+transitive&hl=de&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwi9xb_I4_7-AhWrgf0HHSIECpcQ6AF6BAgcEAI#v=onepage&q=convers%20relation%20transitive&f=false">google.de</a> [abgerufen am 18. Mai 2023]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Transitive+Relation&rft.au=Dov+M.+Gabbay%2C+John+Woods&rft.btitle=The+Rise+of+Modern+Logic%3A+from+Leibniz+to+Frege&rft.date=2004-03-08&rft.genre=book&rft.isbn=9780080532875&rft.pages=509&rft.pub=Elsevier" style="display:none">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Seymor Lipschutz, Marc Lipson: <cite style="font-style:italic">Schaum's Outline of Discrete Mathematics</cite>. McGraw Hill Professional, 1997, ISBN 978-0-07-136841-4, S. 35 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=6A5g3RiYiBUC&pg=PA33&dq=Transitive+Relation&hl=de&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjE6fru2_7-AhXFhP0HHZqRA3gQ6AF6BAgBEAI#v=onepage&q=Transitive%20Relation&f=false">google.de</a> [abgerufen am 18. Mai 2023]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Transitive+Relation&rft.au=Seymor+Lipschutz%2C+Marc+Lipson&rft.btitle=Schaum%27s+Outline+of+Discrete+Mathematics&rft.date=1997-06-22&rft.genre=book&rft.isbn=9780071368414&rft.pages=35&rft.pub=McGraw+Hill+Professional" style="display:none">

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