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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Intransitive Relation</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Eine <b>intransitive Relation</b> ist in der <a href="Mathematik" title="Mathematik">Mathematik</a> eine zweistellige <a href="Relation_(Mathematik)" title="Relation (Mathematik)">Relation</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
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</math></span><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"></span> auf einer <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a>, die die Eigenschaft hat, dass es mindestens drei Elemente <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><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"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
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</math></span><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"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
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</math></span><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"></span> aus dieser Menge gibt, für die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xRy}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mi>R</mi>
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle xRy}</annotation>
</semantics>
</math></span><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"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle yRz}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mi>R</mi>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle yRz}</annotation>
</semantics>
</math></span><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"></span> gelten, aber nicht <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xRz}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mi>R</mi>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle xRz}</annotation>
</semantics>
</math></span><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"></span>. Eine Relation ist also intransitiv, wenn sie nicht <a href="Transitive_Relation" title="Transitive Relation">transitiv</a> ist. Ursprünglich wurden intransitive Relationen vom <a href="Marquis_de_Condorcet" class="mw-redirect" title="Marquis de Condorcet">Marquis de Condorcet</a> im Zusammenhang von Wahlen untersucht (siehe auch <a href="Condorcet-Paradoxon" title="Condorcet-Paradoxon">Condorcet-Paradoxon</a>).
</p>
<div class="mw-heading mw-heading2"><h2 id="Formale_Definition">Formale Definition</h2></div>
<p>Ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><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"></span> eine Menge und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\subseteq M\times M}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo>⊆<!-- ⊆ --></mo>
<mi>M</mi>
<mo>×<!-- × --></mo>
<mi>M</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle R\subseteq M\times M}</annotation>
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</math></span><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"></span> eine zweistellige Relation auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><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"></span>, dann heißt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><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"></span> intransitiv, wenn gilt:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x,y,z\in M:xRy\land yRz\land \neg xRz.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∃<!-- ∃ --></mi>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo>,</mo>
<mi>z</mi>
<mo>∈<!-- ∈ --></mo>
<mi>M</mi>
<mo>:</mo>
<mi>x</mi>
<mi>R</mi>
<mi>y</mi>
<mo>∧<!-- ∧ --></mo>
<mi>y</mi>
<mi>R</mi>
<mi>z</mi>
<mo>∧<!-- ∧ --></mo>
<mi mathvariant="normal">¬<!-- ¬ --></mi>
<mi>x</mi>
<mi>R</mi>
<mi>z</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \exists x,y,z\in M:xRy\land yRz\land \neg xRz.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56a32772da9c73f47f1aa105db5d167a93649572.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.955ex; height:2.509ex;" alt="{\displaystyle \exists x,y,z\in M:xRy\land yRz\land \neg xRz.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>
<p>Ein anschauliches Beispiel für eine intransitive <a href="Pr%C3%A4ferenzrelation" title="Präferenzrelation">Präferenzrelation</a> ist das Spiel <a href="Schere%2C_Stein%2C_Papier" title="Schere, Stein, Papier">Schere, Stein, Papier</a>. Hierbei gewinnt die Wahl von Stein gegen Schere, Schere gegen Papier und Papier gegen Stein. Wäre die Relation transitiv, so müsste aus „Stein gewinnt gegen Schere“ und „Schere gewinnt gegen Papier“ folgen: „Stein gewinnt gegen Papier“, was aber den Spielregeln widerspricht. Aus diesem Grund kann die Relation nicht mehr transitiv sein, sie ist intransitiv.
</p><p>Ein weiteres Beispiel einer intransitiven Relation sind <a href="Intransitive_W%C3%BCrfel" title="Intransitive Würfel">intransitive Würfel</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Patrick_Suppes" title="Patrick Suppes">Patrick Suppes</a>: <i>Introduction to Logic</i>, Dover Pubn Inc, 1999, ISBN 0-486-40687-3</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2024-09-24" href="https://de.wikipedia.org/wiki/?title=Intransitive_Relation&oldid=248868183">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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