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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Mengensystem</span></h1>
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<p>Ein <b>Mengensystem</b> ist in der <a href="Mathematik" title="Mathematik">Mathematik</a> eine <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a>, deren Elemente selbst Mengen sind.
</p><p>Im Kontext der <a href="Graphentheorie" title="Graphentheorie">Graphentheorie</a> wird ein Mengensystem als <a href="Hypergraph" class="mw-redirect" title="Hypergraph">Hypergraph</a> bezeichnet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Formale_Definition">Formale Definition</h2></div>
<p>Eine Menge <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 {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> heißt <i>Mengensystem</i>, wenn jedes Element von <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 {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> eine Menge ist.
</p><p>Ist eine <a href="Grundmenge" title="Grundmenge">Grundmenge</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 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_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> gegeben, so heißt jede <a href="Teilmenge" title="Teilmenge">Teilmenge</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 {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> der <a href="Potenzmenge" title="Potenzmenge">Potenzmenge</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 {\mathcal {P}}(X)=\{A\mid A\subseteq X\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>A</mi>
<mo>∣<!-- ∣ --></mo>
<mi>A</mi>
<mo>⊆<!-- ⊆ --></mo>
<mi>X</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(X)=\{A\mid A\subseteq X\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/291b6f8c222d651d1b2b4a6d080e92049249bc18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.418ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(X)=\{A\mid A\subseteq X\}}" loading="lazy"></span> ein <i>Mengensystem über</i> <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_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. Anders ausgedrückt: <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 {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> ist eine Menge von Mengen (also ein Mengensystem), und jedes Element von <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 {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> ist eine Teilmenge von <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_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>.
</p><p>Umgekehrt ist jedes Mengensystem <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 {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> auch ein Mengensystem über seiner <a href="Vereinigungsmenge" class="mw-redirect" title="Vereinigungsmenge">Vereinigungsmenge</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="{\textstyle \bigcup {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mo>⋃<!-- ⋃ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \bigcup {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a12714e4e13aba58ab8178f01034991fd73d15da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.815ex; height:2.843ex;" alt="{\textstyle \bigcup {\mathcal {S}}}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Stabilität"><span id="Stabilit.C3.A4t"></span>Stabilität</h2></div>
<p>Ein Mengensystem <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 S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S}</annotation>
</semantics>
</math></span><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"></span> heißt <i><a href="Abgeschlossenheit_(algebraische_Struktur)" title="Abgeschlossenheit (algebraische Struktur)">abgeschlossen</a></i> oder <i>stabil</i> bezüglich einer Mengenoperation (<a href="Schnittmenge" class="mw-redirect" title="Schnittmenge">Durchschnitt</a>, <a href="Vereinigungsmenge" class="mw-redirect" title="Vereinigungsmenge">Vereinigung</a>, <a href="Komplement_(Mengenlehre)" title="Komplement (Mengenlehre)">Komplement</a> etc.), wenn die Anwendung der Operation auf Elemente von <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 {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> wieder ein Element von <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 {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> liefert. Mengensysteme werden oftmals bezüglich der stabilen Operationen benannt. So heißt ein Mengensystem zum Beispiel
</p>
<ul><li><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 \cap }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∩<!-- ∩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cap }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d4e886e6f5a28a33e073fb108440c152ecfe2d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cap }" loading="lazy"></span>-stabil (durchschnittsstabil) oder auch ein <a href="%CE%A0-System" title="Π-System">π-System</a>, wenn <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 A,B\in {\mathcal {S}}\Rightarrow A\cap B\in {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>,</mo>
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mi>A</mi>
<mo>∩<!-- ∩ --></mo>
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A,B\in {\mathcal {S}}\Rightarrow A\cap B\in {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/590a314704826afec9bf4b4786f2659839c7c7e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.911ex; height:2.509ex;" alt="{\displaystyle A,B\in {\mathcal {S}}\Rightarrow A\cap B\in {\mathcal {S}}}" loading="lazy"></span> gilt;</li>
<li><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 \cup }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∪<!-- ∪ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cup }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8ff7d0293ad19b43524a133ae5129f3d71f2040.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cup }" loading="lazy"></span>-stabil (vereinigungsstabil), wenn <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 A,B\in {\mathcal {S}}\Rightarrow A\cup B\in {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>,</mo>
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mi>A</mi>
<mo>∪<!-- ∪ --></mo>
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A,B\in {\mathcal {S}}\Rightarrow A\cup B\in {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4edcd21280745380ca848cd0375b552ee21ca8b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.911ex; height:2.509ex;" alt="{\displaystyle A,B\in {\mathcal {S}}\Rightarrow A\cup B\in {\mathcal {S}}}" loading="lazy"></span> gilt;</li>
<li>σ-<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 \cap }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∩<!-- ∩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cap }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d4e886e6f5a28a33e073fb108440c152ecfe2d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cap }" loading="lazy"></span>-stabil oder auch ein <a href="%CE%94-System_(Ma%C3%9Ftheorie)" title="Δ-System (Maßtheorie)">δ-System</a>, wenn für abzählbar unendlich viele Mengen <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 A_{i}\in {\mathcal {S}},\;i\in \mathbb {N} ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
<mo>,</mo>
<mspace width="thickmathspace"></mspace>
<mi>i</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{i}\in {\mathcal {S}},\;i\in \mathbb {N} ,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/056ce801307adfba8651b01990f32aa49ad02c4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.523ex; height:2.509ex;" alt="{\displaystyle A_{i}\in {\mathcal {S}},\;i\in \mathbb {N} ,}" loading="lazy"></span> auch deren Schnittmenge <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="{\textstyle \bigcap _{i\in \mathbb {N} }A_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<munder>
<mo>⋂<!-- ⋂ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
</mrow>
</munder>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \bigcap _{i\in \mathbb {N} }A_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d62556d6a2fa353dc617ddba7a37437e3aae1897.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.948ex; height:3.009ex;" alt="{\textstyle \bigcap _{i\in \mathbb {N} }A_{i}}" loading="lazy"></span> wieder in <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 {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> ist;</li>
<li>σ-<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 \cup }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∪<!-- ∪ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cup }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8ff7d0293ad19b43524a133ae5129f3d71f2040.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cup }" loading="lazy"></span>-stabil oder auch kurz ein σ-System, wenn für abzählbar unendlich viele Mengen <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 A_{i}\in {\mathcal {S}},\;i\in \mathbb {N} ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
<mo>,</mo>
<mspace width="thickmathspace"></mspace>
<mi>i</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{i}\in {\mathcal {S}},\;i\in \mathbb {N} ,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/056ce801307adfba8651b01990f32aa49ad02c4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.523ex; height:2.509ex;" alt="{\displaystyle A_{i}\in {\mathcal {S}},\;i\in \mathbb {N} ,}" loading="lazy"></span> auch deren Vereinigungsmenge <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="{\textstyle \bigcup _{i\in \mathbb {N} }A_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<munder>
<mo>⋃<!-- ⋃ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
</mrow>
</munder>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \bigcup _{i\in \mathbb {N} }A_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48677598a1bd9f5cec961472b337dc588902f349.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.948ex; height:3.009ex;" alt="{\textstyle \bigcup _{i\in \mathbb {N} }A_{i}}" loading="lazy"></span> wieder in <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 {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"></span> ist;</li>
<li><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 \setminus }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo class="MJX-variant">∖<!-- ∖ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \setminus }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0e20e45087a97f0448fc3d4bc27b060084830f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle \setminus }" loading="lazy"></span>-stabil (differenzstabil), wenn <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 A,B\in {\mathcal {S}}\Rightarrow A\setminus B\in {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>,</mo>
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mi>A</mi>
<mo class="MJX-variant">∖<!-- ∖ --></mo>
<mi>B</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A,B\in {\mathcal {S}}\Rightarrow A\setminus B\in {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86fd58363de47f059848455f695fe30f0053863d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.523ex; height:2.843ex;" alt="{\displaystyle A,B\in {\mathcal {S}}\Rightarrow A\setminus B\in {\mathcal {S}}}" loading="lazy"></span> gilt;</li>
<li>komplementstabil, wenn <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 A\in {\mathcal {S}}\Rightarrow A^{c}\in {\mathcal {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>c</mi>
</mrow>
</msup>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {S}}\Rightarrow A^{c}\in {\mathcal {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5a1361b7fbb3faeba571c0f57b1f55e7120694b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.71ex; height:2.343ex;" alt="{\displaystyle A\in {\mathcal {S}}\Rightarrow A^{c}\in {\mathcal {S}}}" loading="lazy"></span> gilt.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>
<p>Die folgenden mathematischen Objekte sind Mengensysteme mit zusätzlichen Eigenschaften. Bei der Formulierung dieser Eigenschaften spielt oft die Stabilität bezüglich bestimmter Mengenoperationen eine Rolle.
</p>
<table>
<tbody><tr>
<td>
<ul><li><a href="%CE%94-Ring_(Mengensystem)" title="Δ-Ring (Mengensystem)">δ-Ring</a></li>
<li><a href="Dynkin-System" title="Dynkin-System">Dynkin-System</a></li>
<li><a href="Fr%C3%A9chet-Filter" title="Fréchet-Filter">Fréchet-Filter</a></li>
<li><a href="H%C3%BCllensystem" class="mw-redirect" title="Hüllensystem">Hüllensystem</a></li>
<li><a href="Kernsystem" class="mw-redirect" title="Kernsystem">Kernsystem</a></li>
<li><a href="Matroid" title="Matroid">Matroid</a></li>
<li><a href="Algebra_(Mengensystem)" title="Algebra (Mengensystem)">Mengenalgebra</a></li>
<li><a href="Filter_(Mathematik)" title="Filter (Mathematik)">Mengenfilter</a></li>
<li><a href="Halbring_(Mengensystem)" title="Halbring (Mengensystem)">Mengenhalbring</a></li>
<li><a href="Ring_(Mengensystem)" title="Ring (Mengensystem)">Mengenring</a></li></ul>
</td>
<td>
</td>
<td>
<ul><li><a href="Mengenverband" title="Mengenverband">Mengenverband</a></li>
<li><a href="Monotone_Klasse" title="Monotone Klasse">Monotone Klasse</a></li>
<li><a href="Partition_(Mengenlehre)" title="Partition (Mengenlehre)">Partition</a></li>
<li><a href="Potenzmenge" title="Potenzmenge">Potenzmenge</a></li>
<li><a href="%CE%A3-Algebra" title="Σ-Algebra">σ-Algebra</a></li>
<li><a href="%CE%A3-Ring" title="Σ-Ring">σ-Ring</a></li>
<li>Topologie (System der offenen Mengen eines <a href="Topologischer_Raum" title="Topologischer Raum">topologischen Raums</a>)</li>
<li><a href="Ungerichteter_Graph" class="mw-redirect" title="Ungerichteter Graph">Ungerichteter Graph</a></li>
<li><a href="Zermelosystem" title="Zermelosystem">Zermelosystem</a></li></ul>
<dl><dd> </dd></dl>
</td></tr></tbody></table>
<div class="mw-heading mw-heading2"><h2 id="Hypergraphen">Hypergraphen</h2></div>
<p>Im Kontext der Graphentheorie wird ein Mengensystem auch als Hypergraph bezeichnet. Die Elemente der Grundmenge heißen dann <a href="Knoten_(Graphentheorie)" title="Knoten (Graphentheorie)">Knoten</a> und die Elemente des Mengensystems heißen <a href="Hyperkante" class="mw-redirect" title="Hyperkante">Hyperkanten</a>. Man kann sich eine Hyperkante als Verallgemeinerung einer <a href="Kante_(Graphentheorie)" title="Kante (Graphentheorie)">Kante</a> in einem gewöhnlichen <a href="Graph_(Graphentheorie)" title="Graph (Graphentheorie)">Graphen</a> vorstellen, die eben nicht zwei, sondern mehrere Knoten gleichzeitig miteinander „verbindet“. Im nebenstehenden Beispiel gilt:
</p>
<dl><dd>Menge der Knoten <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 =\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle =\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa24c49a6a58a314ad15a92b0dab82c0d405e469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.255ex; height:2.843ex;" alt="{\displaystyle =\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\}}" loading="lazy"></span>.</dd>
<dd>Menge der Hyperkanten <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 =\{e_{1},e_{2},e_{3},e_{4}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle =\{e_{1},e_{2},e_{3},e_{4}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d9d7e604819f31c6c5a84ee6ba0d973a331c08c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.431ex; height:2.843ex;" alt="{\displaystyle =\{e_{1},e_{2},e_{3},e_{4}\}}" loading="lazy"></span>, wobei
<dl><dd>Hyperkante <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 e_{1}=\{v_{1},v_{2},v_{3}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{1}=\{v_{1},v_{2},v_{3}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e8613d967a83c59342011a0fc33df049ed6ede1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.175ex; height:2.843ex;" alt="{\displaystyle e_{1}=\{v_{1},v_{2},v_{3}\}}" loading="lazy"></span>,</dd>
<dd>Hyperkante <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 e_{2}=\{v_{2},v_{3}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{2}=\{v_{2},v_{3}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0d44f710a9ceebde5e2e26c48a68142cc1a473f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.959ex; height:2.843ex;" alt="{\displaystyle e_{2}=\{v_{2},v_{3}\}}" loading="lazy"></span>,</dd>
<dd>Hyperkante <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 e_{3}=\{v_{3},v_{5},v_{6}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{3}=\{v_{3},v_{5},v_{6}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58aab6fd51c6fec5bcd23dd2b24cae298d3804d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.175ex; height:2.843ex;" alt="{\displaystyle e_{3}=\{v_{3},v_{5},v_{6}\}}" loading="lazy"></span>,</dd>
<dd>Hyperkante <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 e_{4}=\{v_{4}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{4}=\{v_{4}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9e602ee706c40437db70df43ab66ca442a173c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.743ex; height:2.843ex;" alt="{\displaystyle e_{4}=\{v_{4}\}}" loading="lazy"></span>.</dd></dl></dd></dl>
<p>In vielen Anwendungsfällen von Hypergraphen wird die Knotenmenge als endlich festgelegt und die leere Hyperkante ausgeschlossen.
</p><p>Verbindet jede Hyperkante genau 2 Knoten, liegt ein <a href="Ungerichteter_Graph" class="mw-redirect" title="Ungerichteter Graph">ungerichteter Graph</a> vor (genauer: ein <i>ungerichteter Graph ohne Mehrfachkanten und ohne Schleifen</i>). Das Mengensystem besteht dann also nur aus 2-elementigen Teilmengen der Grundmenge. Im nebenstehenden Beispiel gilt:
</p>
<dl><dd>Grundmenge = <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 \{1,2,3,4,5,6\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
<mo>,</mo>
<mn>4</mn>
<mo>,</mo>
<mn>5</mn>
<mo>,</mo>
<mn>6</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{1,2,3,4,5,6\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc427c339db79f243cb79154253ff8151a31c23e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.469ex; height:2.843ex;" alt="{\displaystyle \{1,2,3,4,5,6\}}" loading="lazy"></span>,</dd>
<dd>Mengensystem = <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 \{\{1,2\},\{1,5\},\{2,3\},\{2,5\},\{3,4\},\{4,5\},\{4,6\}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo>,</mo>
<mn>5</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>2</mn>
<mo>,</mo>
<mn>5</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>3</mn>
<mo>,</mo>
<mn>4</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>4</mn>
<mo>,</mo>
<mn>5</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>4</mn>
<mo>,</mo>
<mn>6</mn>
<mo fence="false" stretchy="false">}</mo>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{\{1,2\},\{1,5\},\{2,3\},\{2,5\},\{3,4\},\{4,5\},\{4,6\}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fed9949018c1cc3961b81c34d57fb531786d05b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.315ex; height:2.843ex;" alt="{\displaystyle \{\{1,2\},\{1,5\},\{2,3\},\{2,5\},\{3,4\},\{4,5\},\{4,6\}\}}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Axiomatische_Mengenlehre">Axiomatische Mengenlehre</h2></div>
<p>In der <a href="Zermelo-Fraenkel-Mengenlehre" title="Zermelo-Fraenkel-Mengenlehre">Zermelo-Fraenkel-Mengenlehre</a> gibt es nur einen Typ von Objekten, nämlich Mengen. Damit sind alle Elemente einer Menge selbst wieder Mengen, und die Begriffe <i>Menge</i> und <i>Mengensystem</i> stimmen überein.
</p><p>Beispiel: Jede <a href="Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürliche Zahl</a> wird in diesem Zusammenhang mit der Menge ihrer Vorgänger identifiziert. Dies ergibt den folgenden Aufbau:
</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 0=\emptyset }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo>=</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0=\emptyset }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4dc0d5ccb76dc369fda298be3b0fb1447fa05788.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.423ex; height:2.509ex;" alt="{\displaystyle 0=\emptyset }" loading="lazy"></span> (die <a href="Leere_Menge" title="Leere Menge">leere Menge</a>),</dd>
<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 1=\{0\}=\{\emptyset \}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1=\{0\}=\{\emptyset \}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57bb7e0c1bded70cb95371ca8eff9eacf281c904.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.334ex; height:2.843ex;" alt="{\displaystyle 1=\{0\}=\{\emptyset \}}" loading="lazy"></span>,</dd>
<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 2=\{0,1\}=\{\emptyset ,\{\emptyset \}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
<mo fence="false" stretchy="false">}</mo>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2=\{0,1\}=\{\emptyset ,\{\emptyset \}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5819bd05a289cd5d57f85682a6d4a6399bb0055.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.052ex; height:2.843ex;" alt="{\displaystyle 2=\{0,1\}=\{\emptyset ,\{\emptyset \}\}}" loading="lazy"></span>,</dd>
<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 3=\{0,1,2\}=\{\emptyset ,\{\emptyset \},\{\emptyset ,\{\emptyset \}\}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>3</mn>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
<mo fence="false" stretchy="false">}</mo>
<mo fence="false" stretchy="false">}</mo>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 3=\{0,1,2\}=\{\emptyset ,\{\emptyset \},\{\emptyset ,\{\emptyset \}\}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/271232719a21ad52577353e8a52b882a4cdb8546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.291ex; height:2.843ex;" alt="{\displaystyle 3=\{0,1,2\}=\{\emptyset ,\{\emptyset \},\{\emptyset ,\{\emptyset \}\}\}}" loading="lazy"></span>,</dd>
<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 4=\{0,1,2,3\}=\dotsc }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>4</mn>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mo>…<!-- … --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 4=\{0,1,2,3\}=\dotsc }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e01a85801d0d058e2433f07ff77d7fbb746715e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.159ex; height:2.843ex;" alt="{\displaystyle 4=\{0,1,2,3\}=\dotsc }" loading="lazy"></span>,</dd>
<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 \vdots }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>⋮<!-- ⋮ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \vdots }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8039d9feb6596ae092e5305108722975060c083.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:3.676ex;" alt="{\displaystyle \vdots }" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Oliver Deiser: <i>Einführung in die Mengenlehre</i>. Springer, 2004, ISBN 978-3-540-20401-5</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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