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Ein anderes Wort für Teilmenge ist <b>Untermenge</b>. </p><p>Für die mathematische Abbildung der Einbettung einer Teilmenge in ihre <a href="/wiki/Grundmenge" title="Grundmenge">Grundmenge</a>, die <a href="/wiki/Mathematische_Funktion" class="mw-redirect" title="Mathematische Funktion">mathematische Funktion</a> der <i>Teilmengenbeziehung</i>, wird die <a href="/wiki/Inklusionsabbildung" title="Inklusionsabbildung">Inklusionsabbildung</a> verwendet. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> ist eine Teilmenge von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> ist eine Obermenge von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, wenn jedes <a href="/wiki/Element_(Mathematik)" title="Element (Mathematik)">Element</a> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> auch in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> enthalten ist. Wenn <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> zudem weitere Elemente enthält, die nicht in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> enthalten sind, so ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> eine <b>echte Teilmenge</b> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> ist eine <b>echte Obermenge</b> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. Die Menge <i>aller</i> Teilmengen einer gegebenen Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> heißt die <a href="/wiki/Potenzmenge" title="Potenzmenge">Potenzmenge</a> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. </p><p>Den Begriff <i>Teilmenge</i> prägte <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> – der „Erfinder“ der <a href="/wiki/Mengenlehre" title="Mengenlehre">Mengenlehre</a> – ab 1884; das Symbol der Teilmengenrelation wurde von <a href="/wiki/Ernst_Schr%C3%B6der_(Mathematiker)" title="Ernst Schröder (Mathematiker)">Ernst Schröder</a> 1890 in seiner „<a href="/wiki/Algebra" title="Algebra">Algebra</a> der <a href="/wiki/Logik" title="Logik">Logik</a>“ eingeführt.<sup id="cite_ref-deiser_1-0" class="reference"><a href="#cite_note-deiser-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Definition"><span class="tocnumber">1</span> <span class="toctext">Definition</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Weitere_Notationen"><span class="tocnumber">2</span> <span class="toctext">Weitere Notationen</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Sprechweisen"><span class="tocnumber">3</span> <span class="toctext">Sprechweisen</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Beispiele"><span class="tocnumber">4</span> <span class="toctext">Beispiele</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Eigenschaften"><span class="tocnumber">5</span> <span class="toctext">Eigenschaften</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Inklusion_als_Ordnungsrelation"><span class="tocnumber">6</span> <span class="toctext">Inklusion als Ordnungsrelation</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#Inklusionsketten"><span class="tocnumber">7</span> <span class="toctext">Inklusionsketten</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#Größe_und_Anzahl_von_Teilmengen"><span class="tocnumber">8</span> <span class="toctext">Größe und Anzahl von Teilmengen</span></a></li> <li class="toclevel-1 tocsection-9"><a href="#Literatur"><span class="tocnumber">9</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Weblinks"><span class="tocnumber">10</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#Einzelnachweise"><span class="tocnumber">11</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teilmenge&veaction=edit&section=1" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teilmenge&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Definition"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Wenn <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> Mengen sind und jedes Element von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> auch ein Element von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> ist, nennt man <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> eine Teilmenge oder Untermenge von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B:\Longleftrightarrow \forall x\in A\colon x\in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo>:⟺</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B:\Longleftrightarrow \forall x\in A\colon x\in B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b24df8734c7ab07f54605a8b9ff2f413895c2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:27.034ex; height:2.343ex;" alt="{\displaystyle A\subseteq B:\Longleftrightarrow \forall x\in A\colon x\in B}"></span></dd></dl> <p>Umgekehrt nennt man <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> die Obermenge von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> genau dann, wenn <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> Teilmenge von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> ist: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\supseteq A:\Longleftrightarrow A\subseteq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊇</mo> <mi>A</mi> <mo>:⟺</mo> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\supseteq A:\Longleftrightarrow A\subseteq B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e526d208ed789da18ac43aae0b991d5e05b73a22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.465ex; height:2.343ex;" alt="{\displaystyle B\supseteq A:\Longleftrightarrow A\subseteq B}"></span></dd></dl> <p>Weiterhin gibt es den Begriff der echten Teilmenge. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> ist eine echte Teilmenge von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> genau dann, wenn <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> eine Teilmenge von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> nicht identisch mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> ist. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subsetneq B:\Longleftrightarrow A\subseteq B\land A\neq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊊</mo> <mi>B</mi> <mo>:⟺</mo> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo>∧</mo> <mi>A</mi> <mo>≠</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subsetneq B:\Longleftrightarrow A\subseteq B\land A\neq B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd243e285de3f10de6eea90c60c5fbb28086be73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.653ex; height:2.676ex;" alt="{\displaystyle A\subsetneq B:\Longleftrightarrow A\subseteq B\land A\neq B}"></span></dd></dl> <p>Wieder schreibt man auch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\supsetneq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊋</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\supsetneq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee6f9bc7f3f838bcc1b8136530626324a40bad17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.606ex; height:2.676ex;" alt="{\displaystyle B\supsetneq A}"></span>, wenn <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subsetneq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊊</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subsetneq B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf81e9a4a81df2d596b4db1cde6b9bdf82c73db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.606ex; height:2.676ex;" alt="{\displaystyle A\subsetneq B}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Weitere_Notationen">Weitere Notationen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teilmenge&veaction=edit&section=2" title="Abschnitt bearbeiten: Weitere Notationen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teilmenge&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Weitere Notationen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="float:right; margin: 0 0 0.2em 0.2em;padding: 0.2em 0.6em; font-size: 400%; line-height: 1.1em; background-color: #ddddff; border: 1px solid #aaaaff">⊂⊊⊆⊇⊋⊃</div> <p>Einige Autoren benutzen auch die Zeichen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊂</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f51f0eeff0c2a9dcb9c856f87ca0359e701ef01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \subset }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \supset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bfe0828a2ed4c9c6b70987a85c02a1f005843c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \supset }"></span> für Teilmenge und Obermenge anstatt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subseteq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subseteq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a924f8dcb2847bb8871edfdbf4c6b5cca0669228" 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 }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \supseteq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊇</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supseteq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8a31abf11074afa03a75eba80bfce6b98020e4" 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 \supseteq }"></span>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Meistens definiert der Autor dann den Begriff „echte Teilmenge“ nicht. </p><p>Andere Autoren bevorzugen die Zeichen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊂</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f51f0eeff0c2a9dcb9c856f87ca0359e701ef01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \subset }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \supset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bfe0828a2ed4c9c6b70987a85c02a1f005843c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \supset }"></span> für <i>echte</i> Teilmenge und Obermenge also statt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subsetneq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊊</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subsetneq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccda11a9c5eb10088602771f7c0f05ffea7f41c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \subsetneq }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \supsetneq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supsetneq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84cda99291cdfefb94e6f580e14da12d3f43009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \supsetneq }"></span>.<sup id="cite_ref-deiser_1-1" class="reference"><a href="#cite_note-deiser-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Dieser Gebrauch erinnert passenderweise an die Zeichen für <a href="/wiki/Ungleichung" title="Ungleichung">Ungleichheit</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" 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 }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}"></span>. Da diese Notation meistens benutzt wird, wenn der Unterschied zwischen echter und nicht echter Teilmenge wichtig ist, werden die Zeichen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subsetneq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊊</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subsetneq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccda11a9c5eb10088602771f7c0f05ffea7f41c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \subsetneq }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \supsetneq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supsetneq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84cda99291cdfefb94e6f580e14da12d3f43009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \supsetneq }"></span> eher selten benutzt. </p><p>Varianten des Zeichens <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subsetneq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊊</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subsetneq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccda11a9c5eb10088602771f7c0f05ffea7f41c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \subsetneq }"></span> sind außerdem <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varsubsetneq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo class="MJX-variant">⊊</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsubsetneq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e100b112aa2eb37e6e21e4df95a4ec31ac286c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \varsubsetneq }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subsetneqq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⫋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subsetneqq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59b1428bbc8b9aba770fbe531003a669d798b13a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.808ex; height:3.176ex;" alt="{\displaystyle \subsetneqq }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varsubsetneqq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo class="MJX-variant">⫋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsubsetneqq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a211547c8ca3e0915def5c5027f9058770c3674" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.808ex; height:3.009ex;" alt="{\displaystyle \varsubsetneqq }"></span>. Falls <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> keine Teilmenge von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> ist, kann auch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\nsubseteq B:\Longleftrightarrow \lnot \left(A\subseteq B\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊈</mo> <mi>B</mi> <mo>:⟺</mo> <mi mathvariant="normal">¬</mi> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\nsubseteq B:\Longleftrightarrow \lnot \left(A\subseteq B\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6063f247166231298e305ba839d201e170963656" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.212ex; height:3.176ex;" alt="{\displaystyle A\nsubseteq B:\Longleftrightarrow \lnot \left(A\subseteq B\right)}"></span> benutzt werden. Entsprechende Schreibweisen sind <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varsupsetneq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo class="MJX-variant">⊋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsupsetneq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7d70c276d8057ebb602db04bed3614dcbb6068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \varsupsetneq }"></span> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \supsetneq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supsetneq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84cda99291cdfefb94e6f580e14da12d3f43009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \supsetneq }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \supsetneqq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⫌</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supsetneqq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fc01cabd94a8f63d60800a8b4839da77b765078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.808ex; height:3.176ex;" alt="{\displaystyle \supsetneqq }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varsupsetneqq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo class="MJX-variant">⫌</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varsupsetneqq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b91be41bb9942234dfef27c5904814a750c457b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.808ex; height:3.009ex;" alt="{\displaystyle \varsupsetneqq }"></span> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \supsetneq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supsetneq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84cda99291cdfefb94e6f580e14da12d3f43009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \supsetneq }"></span>, sowie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\nsupseteq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊉</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\nsupseteq B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20c4b4e221f16f2fdad0ec46f7238debd31293d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.606ex; height:3.176ex;" alt="{\displaystyle A\nsupseteq B}"></span> (keine Obermenge). </p><p>Die entsprechenden <a href="/wiki/Unicode" title="Unicode">Unicode</a>-Symbole sind: ⊂, ⊃, ⊆, ⊇, ⊄, ⊅, ⊈, ⊉, ⊊, ⊋ (siehe: <a href="/wiki/Unicode-Block_Mathematische_Operatoren" class="mw-redirect" title="Unicode-Block Mathematische Operatoren">Unicode-Block Mathematische Operatoren</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Sprechweisen">Sprechweisen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teilmenge&veaction=edit&section=3" title="Abschnitt bearbeiten: Sprechweisen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teilmenge&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Sprechweisen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Statt „<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> ist eine Teilmenge von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span>“ wird auch „Die Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> ist in der Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> enthalten“ oder „Die Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> wird von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> umfasst“ gesagt. Genauso wird statt „<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> ist eine Obermenge von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>“ auch „Die Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> enthält die Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>“ oder „Die Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> umfasst die Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>“ gesagt. Wenn es nicht zu Missverständnissen kommen kann, wird auch „<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> enthält <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>“ usw. gesagt. Missverständnisse können insbesondere mit „Die Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" 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 B}"></span> enthält das <a href="/wiki/Element_(Mathematik)" title="Element (Mathematik)">Element</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>“ entstehen. </p> <div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teilmenge&veaction=edit&section=4" title="Abschnitt bearbeiten: Beispiele" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teilmenge&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Beispiele"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:PolygonsSet_DE.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/PolygonsSet_DE.svg/300px-PolygonsSet_DE.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/PolygonsSet_DE.svg/450px-PolygonsSet_DE.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/PolygonsSet_DE.svg/600px-PolygonsSet_DE.svg.png 2x" data-file-width="600" data-file-height="450" /></a><figcaption>Die regulären Polygone bilden eine Teilmenge der Menge aller Polygone.</figcaption></figure> <ul><li>{1, 2} ist eine <i>(echte)</i> Teilmenge von {1, 2, 3}.</li> <li>{1, 2, 3} ist eine <i>(unechte)</i> Teilmenge von {1, 2, 3}.</li> <li>{1, 2, 3, 4} ist keine Teilmenge von {1, 2, 3}.</li> <li>{1, 2, 3} ist keine Teilmenge von {2, 3, 4}.</li> <li>{} ist eine <i>(echte)</i> Teilmenge von {1, 2}.</li> <li>{1, 2, 3} ist eine <i>(echte)</i> Obermenge von {1, 2}.</li> <li>{1, 2} ist eine <i>(unechte)</i> Obermenge von {1, 2}.</li> <li>{1} ist keine Obermenge von {1, 2}.</li> <li>Die Menge der <a href="/wiki/Primzahl" title="Primzahl">Primzahlen</a> ist eine echte Teilmenge der Menge der <a href="/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürlichen Zahlen</a>.</li> <li>Die Menge der <a href="/wiki/Rationale_Zahl" title="Rationale Zahl">rationalen Zahlen</a> ist eine echte Teilmenge der Menge der <a href="/wiki/Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a>.</li></ul> <p>Weitere Beispiele als Mengendiagramme: </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Datei:Example_of_A_is_a_proper_subset_of_B.svg" class="mw-file-description" title="A ist eine echte Teilmenge von B"><img alt="A ist eine echte Teilmenge von B" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Example_of_A_is_a_proper_subset_of_B.svg/158px-Example_of_A_is_a_proper_subset_of_B.svg.png" decoding="async" width="158" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Example_of_A_is_a_proper_subset_of_B.svg/236px-Example_of_A_is_a_proper_subset_of_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Example_of_A_is_a_proper_subset_of_B.svg/315px-Example_of_A_is_a_proper_subset_of_B.svg.png 2x" data-file-width="391" data-file-height="298" /></a></span></div> <div class="gallerytext">A ist eine echte Teilmenge von B</div> </li> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Datei:Example_of_C_is_no_proper_subset_of_B.svg" class="mw-file-description" title="C ist zwar eine Teilmenge von B, aber keine echte Teilmenge von B"><img alt="C ist zwar eine Teilmenge von B, aber keine echte Teilmenge von B" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Example_of_C_is_no_proper_subset_of_B.svg/153px-Example_of_C_is_no_proper_subset_of_B.svg.png" decoding="async" width="153" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Example_of_C_is_no_proper_subset_of_B.svg/230px-Example_of_C_is_no_proper_subset_of_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Example_of_C_is_no_proper_subset_of_B.svg/306px-Example_of_C_is_no_proper_subset_of_B.svg.png 2x" data-file-width="384" data-file-height="301" /></a></span></div> <div class="gallerytext">C ist zwar eine Teilmenge von B, aber keine echte Teilmenge von B</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teilmenge&veaction=edit&section=5" title="Abschnitt bearbeiten: Eigenschaften" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teilmenge&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Eigenschaften"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Die <a href="/wiki/Leere_Menge" title="Leere Menge">leere Menge</a> ist Teilmenge jeder Menge: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing \subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing \subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b947b700faaf478c312531745e80bf69ed50d493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.65ex; height:2.343ex;" alt="{\displaystyle \varnothing \subseteq A}"></span></dd></dl></li> <li>Jede Menge ist Teilmenge von sich selbst: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ce5093be9e30238b83393aed738eafd3a43030" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.585ex; height:2.343ex;" alt="{\displaystyle A\subseteq A}"></span></dd></dl></li> <li>Charakterisierung der Inklusion mit Hilfe der <a href="/wiki/Vereinigungsmenge" class="mw-redirect" title="Vereinigungsmenge">Vereinigung</a>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\Leftrightarrow A\cup B=B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo stretchy="false">⇔</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo>=</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\Leftrightarrow A\cup B=B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8d5041785e02e3298cd6ae005cb1104978d994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.172ex; height:2.343ex;" alt="{\displaystyle A\subseteq B\Leftrightarrow A\cup B=B}"></span></dd></dl></li> <li>Charakterisierung der Inklusion mit Hilfe des <a href="/wiki/Schnittmenge" class="mw-redirect" title="Schnittmenge">Durchschnitts</a>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\Leftrightarrow A\cap B=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo stretchy="false">⇔</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\Leftrightarrow A\cap B=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/008c6e8ae34d3e83d618f58933492c663051d892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.151ex; height:2.343ex;" alt="{\displaystyle A\subseteq B\Leftrightarrow A\cap B=A}"></span></dd></dl></li> <li>Charakterisierung der Inklusion mit Hilfe der <a href="/wiki/Differenzmenge" class="mw-redirect" title="Differenzmenge">Differenzmenge</a>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\Leftrightarrow A\setminus B=\varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo stretchy="false">⇔</mo> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> <mo>=</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\Leftrightarrow A\setminus B=\varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c9a560c0b0e299224410ed0f945b94b6c118cb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.828ex; height:2.843ex;" alt="{\displaystyle A\subseteq B\Leftrightarrow A\setminus B=\varnothing }"></span></dd></dl></li> <li>Charakterisierung der Inklusion mit Hilfe der <a href="/wiki/Indikatorfunktion" title="Indikatorfunktion">charakteristischen Funktion</a>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\Leftrightarrow \chi _{A}\leq \chi _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo stretchy="false">⇔</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\Leftrightarrow \chi _{A}\leq \chi _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37f6f2354e7d647609a56edfd7cefcc5f99bbfe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.173ex; height:2.509ex;" alt="{\displaystyle A\subseteq B\Leftrightarrow \chi _{A}\leq \chi _{B}}"></span></dd></dl></li> <li>Zwei Mengen sind genau dann gleich, wenn jede eine Teilmenge der anderen ist: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=B\Leftrightarrow A\subseteq B\land B\subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>B</mi> <mo stretchy="false">⇔</mo> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo>∧</mo> <mi>B</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=B\Leftrightarrow A\subseteq B\land B\subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80a4492f9eefc48f22938e226ce457c02dfe2fab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.013ex; height:2.343ex;" alt="{\displaystyle A=B\Leftrightarrow A\subseteq B\land B\subseteq A}"></span></dd> <dd>Diese Regel wird oft beim Nachweis der Gleichheit zweier Mengen verwendet, indem man die gegenseitige Inklusion (in zwei Arbeitsschritten) zeigt.</dd></dl></li> <li>Beim Übergang zum <a href="/wiki/Komplement_(Mengenlehre)" title="Komplement (Mengenlehre)">Komplement</a> dreht sich die Richtung der Inklusion um: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\Rightarrow A^{\rm {c}}\supseteq B^{\rm {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo stretchy="false">⇒</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msup> <mo>⊇</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\Rightarrow A^{\rm {c}}\supseteq B^{\rm {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83df1a8168889d4b9a2591da7245dde71d6e7c5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.75ex; height:2.509ex;" alt="{\displaystyle A\subseteq B\Rightarrow A^{\rm {c}}\supseteq B^{\rm {c}}}"></span></dd></dl></li> <li>Bei der Bildung der Schnittmenge erhält man stets eine Teilmenge: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B\subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cap B\subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7bd25ce29ca352002ab4f7e70da86f7221ef33e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.931ex; height:2.343ex;" alt="{\displaystyle A\cap B\subseteq A}"></span></dd></dl></li> <li>Bei der Bildung der Vereinigungsmenge erhält man stets eine Obermenge: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cup B\supseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo>⊇</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cup B\supseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1a27432616e7f3b813b846ca6ffbc235b10ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.931ex; height:2.343ex;" alt="{\displaystyle A\cup B\supseteq A}"></span></dd></dl></li></ul> <div class="mw-heading mw-heading2"><h2 id="Inklusion_als_Ordnungsrelation">Inklusion als Ordnungsrelation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teilmenge&veaction=edit&section=6" title="Abschnitt bearbeiten: Inklusion als Ordnungsrelation" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teilmenge&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Inklusion als Ordnungsrelation"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Subset_with_expansion.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Subset_with_expansion.svg/220px-Subset_with_expansion.svg.png" decoding="async" width="220" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Subset_with_expansion.svg/330px-Subset_with_expansion.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Subset_with_expansion.svg/440px-Subset_with_expansion.svg.png 2x" data-file-width="470" data-file-height="247" /></a><figcaption>Wenn A ⊆ B und B ⊆ C ist, dann ist auch A ⊆ C</figcaption></figure> <p>Die Inklusion als Beziehung zwischen Mengen erfüllt die drei Eigenschaften einer <a href="/wiki/Partielle_Ordnung" class="mw-redirect" title="Partielle Ordnung">partiellen Ordnungsrelation</a>, sie ist nämlich <a href="/wiki/Reflexive_Relation" title="Reflexive Relation">reflexiv</a>, <a href="/wiki/Antisymmetrische_Relation" title="Antisymmetrische Relation">antisymmetrisch</a> und <a href="/wiki/Transitive_Relation" title="Transitive Relation">transitiv</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ce5093be9e30238b83393aed738eafd3a43030" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.585ex; height:2.343ex;" alt="{\displaystyle A\subseteq A}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\subseteq A\Rightarrow A=B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo>⊆</mo> <mi>A</mi> <mo stretchy="false">⇒</mo> <mi>A</mi> <mo>=</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\subseteq A\Rightarrow A=B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/826b23232ea2cfdb6b7fc4ed79e7e0460f96e1df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.667ex; height:2.343ex;" alt="{\displaystyle A\subseteq B\subseteq A\Rightarrow A=B}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\subseteq C\Rightarrow A\subseteq C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo>⊆</mo> <mi>C</mi> <mo stretchy="false">⇒</mo> <mi>A</mi> <mo>⊆</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\subseteq C\Rightarrow A\subseteq C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b357e8f0264d0c85a8a2578e8661ab73a82e6f2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.692ex; height:2.343ex;" alt="{\displaystyle A\subseteq B\subseteq C\Rightarrow A\subseteq C}"></span></dd></dl> <p>(Dabei ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\subseteq C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo>⊆</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\subseteq C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/740882e1b9f59d59a9058acb6844a4fecc56719b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.47ex; height:2.343ex;" alt="{\displaystyle A\subseteq B\subseteq C}"></span> eine Kurzschreibweise für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b09068bd2f7ba899aeb883ebe670b2ad07b0c851" 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}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subseteq C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87dc1864578bb8f696bbfb7ba1c134c0f1f20458" 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}"></span>.) </p><p>Ist also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa66ea8156203e93f2dca12aca24538c2bdce761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.829ex; height:2.176ex;" alt="{\displaystyle M\,}"></span> eine Menge von Mengen (ein <a href="/wiki/Mengensystem" title="Mengensystem">Mengensystem</a>), dann ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,\subseteq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mo>⊆</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,\subseteq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b53acff657735014717557582c781f3d016998c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.094ex; height:2.843ex;" alt="{\displaystyle (M,\subseteq )}"></span> eine <a href="/wiki/Halbordnung" class="mw-redirect" title="Halbordnung">Halbordnung</a>. Insbesondere gilt dies für die <a href="/wiki/Potenzmenge" title="Potenzmenge">Potenzmenge</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ed5b6b7f1ad70cba0f7b3cf4603bf627321b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.493ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(X)}"></span> einer gegebenen Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <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="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Inklusionsketten">Inklusionsketten</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teilmenge&veaction=edit&section=7" title="Abschnitt bearbeiten: Inklusionsketten" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teilmenge&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Inklusionsketten"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa66ea8156203e93f2dca12aca24538c2bdce761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.829ex; height:2.176ex;" alt="{\displaystyle M\,}"></span> ein <a href="/wiki/Mengensystem" title="Mengensystem">Mengensystem</a>, so dass von je zwei der in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa66ea8156203e93f2dca12aca24538c2bdce761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.829ex; height:2.176ex;" alt="{\displaystyle M\,}"></span> vorkommenden Mengen die eine die andere umfasst oder von der anderen umfasst wird, so nennt man ein solches Mengensystem eine <b>Inklusionskette</b>. Ein Beispiel hierfür liefert das System <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{]{-\infty ,x}[}\mid x\in \mathbb {R} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>x</mi> </mrow> <mo stretchy="false">[</mo> </mrow> <mo>∣</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{]{-\infty ,x}[}\mid x\in \mathbb {R} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/859ccfd71aa073323c9c536313a7e253b1cd8178" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.9ex; height:2.843ex;" alt="{\displaystyle \{{]{-\infty ,x}[}\mid x\in \mathbb {R} \}}"></span> der <a href="/wiki/Intervall_(Mathematik)" title="Intervall (Mathematik)">linksseitig unbeschränkten offenen Intervalle</a> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>. </p><p>Ein spezieller Fall einer Inklusionskette liegt vor, wenn eine (endliche oder unendliche) <a href="/wiki/Mengenfolge" title="Mengenfolge">Mengenfolge</a> gegeben ist, welche vermöge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subseteq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subseteq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a924f8dcb2847bb8871edfdbf4c6b5cca0669228" 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 }"></span> <a href="/wiki/Monoton_wachsende_Mengenfolge" class="mw-redirect" title="Monoton wachsende Mengenfolge"><i>aufsteigend</i></a> oder vermöge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \supseteq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊇</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \supseteq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8a31abf11074afa03a75eba80bfce6b98020e4" 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 \supseteq }"></span> <a href="/wiki/Monoton_fallende_Mengenfolge" class="mw-redirect" title="Monoton fallende Mengenfolge"><i>absteigend</i></a> angeordnet ist. Man schreibt dann kurz: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}\subseteq A_{2}\subseteq A_{3}\subseteq \ ...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊆</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊆</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⊆</mo> <mtext> </mtext> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}\subseteq A_{2}\subseteq A_{3}\subseteq \ ...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/229f154b459e890dc7ec065764e701882e710276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.983ex; height:2.509ex;" alt="{\displaystyle A_{1}\subseteq A_{2}\subseteq A_{3}\subseteq \ ...}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}\supseteq A_{2}\supseteq A_{3}\supseteq \ ...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊇</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊇</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⊇</mo> <mtext> </mtext> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}\supseteq A_{2}\supseteq A_{3}\supseteq \ ...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df56f651126743985c8615c2f07b8eb01ff4e5b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.983ex; height:2.509ex;" alt="{\displaystyle A_{1}\supseteq A_{2}\supseteq A_{3}\supseteq \ ...}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Größe_und_Anzahl_von_Teilmengen"><span id="Gr.C3.B6.C3.9Fe_und_Anzahl_von_Teilmengen"></span>Größe und Anzahl von Teilmengen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teilmenge&veaction=edit&section=8" title="Abschnitt bearbeiten: Größe und Anzahl von Teilmengen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teilmenge&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Größe und Anzahl von Teilmengen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Jede Teilmenge einer <a href="/wiki/Endliche_Menge" title="Endliche Menge">endlichen</a> Menge ist endlich und für die <a href="/wiki/M%C3%A4chtigkeit_(Mathematik)" title="Mächtigkeit (Mathematik)">Mächtigkeiten</a> gilt: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\Rightarrow \left|A\right|\leq \left|B\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo stretchy="false">⇒</mo> <mrow> <mo>|</mo> <mi>A</mi> <mo>|</mo> </mrow> <mo>≤</mo> <mrow> <mo>|</mo> <mi>B</mi> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\Rightarrow \left|A\right|\leq \left|B\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a23f6d08374741185de518bbea58289829046a03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.413ex; height:2.843ex;" alt="{\displaystyle A\subseteq B\Rightarrow \left|A\right|\leq \left|B\right|}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subsetneq B\Rightarrow \left|A\right|<\left|B\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊊</mo> <mi>B</mi> <mo stretchy="false">⇒</mo> <mrow> <mo>|</mo> <mi>A</mi> <mo>|</mo> </mrow> <mo><</mo> <mrow> <mo>|</mo> <mi>B</mi> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subsetneq B\Rightarrow \left|A\right|<\left|B\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ab872aea220016312034553194cc12857341c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.413ex; height:2.843ex;" alt="{\displaystyle A\subsetneq B\Rightarrow \left|A\right|<\left|B\right|}"></span></dd></dl></li> <li>Jede Obermenge einer <a href="/wiki/Unendliche_Menge" title="Unendliche Menge">unendlichen</a> Menge ist unendlich.</li> <li>Auch bei unendlichen Mengen gilt für die Mächtigkeiten: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\Rightarrow \left|A\right|\leq \left|B\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo stretchy="false">⇒</mo> <mrow> <mo>|</mo> <mi>A</mi> <mo>|</mo> </mrow> <mo>≤</mo> <mrow> <mo>|</mo> <mi>B</mi> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\Rightarrow \left|A\right|\leq \left|B\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a23f6d08374741185de518bbea58289829046a03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.413ex; height:2.843ex;" alt="{\displaystyle A\subseteq B\Rightarrow \left|A\right|\leq \left|B\right|}"></span></dd></dl></li> <li>Bei unendlichen Mengen ist es aber möglich, dass eine echte Teilmenge dieselbe Mächtigkeit hat wie ihre <a href="/wiki/Grundmenge" title="Grundmenge">Grundmenge</a>. Zum Beispiel sind die <a href="/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürlichen Zahlen</a> eine echte Teilmenge der <a href="/wiki/Ganze_Zahl" title="Ganze Zahl">ganzen Zahlen</a>, aber die beiden Mengen sind gleich mächtig (nämlich <a href="/wiki/Abz%C3%A4hlbar_unendlich" class="mw-redirect" title="Abzählbar unendlich">abzählbar unendlich</a>).</li> <li>Nach dem <a href="/wiki/Satz_von_Cantor" title="Satz von Cantor">Satz von Cantor</a> ist die <a href="/wiki/Potenzmenge" title="Potenzmenge">Potenzmenge</a> einer Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> stets mächtiger als die Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> selbst: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A|<{\bigl |}{\mathcal {P}}(A){\bigr |}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <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>A</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |A|<{\bigl |}{\mathcal {P}}(A){\bigr |}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b95f35a7f59df03704da2bbe8abefa7805e366d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.685ex; height:3.176ex;" alt="{\displaystyle |A|<{\bigl |}{\mathcal {P}}(A){\bigr |}}"></span>.</li> <li>Eine endliche Menge mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> Elementen hat genau <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}"></span> Teilmengen.</li> <li>Die Anzahl der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>-elementigen Teilmengen einer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-elementigen (endlichen) Menge ist durch den <a href="/wiki/Binomialkoeffizient" title="Binomialkoeffizient">Binomialkoeffizienten</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {n}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {n}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/206415d3742167e319b2e52c2ca7563b799abad7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\displaystyle {\tbinom {n}{k}}}"></span> gegeben.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teilmenge&veaction=edit&section=9" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teilmenge&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Oliver Deiser: <i>Einführung in die Mengenlehre.</i> Springer, 2004, <a href="/wiki/Spezial:ISBN-Suche/9783540204015" class="internal mw-magiclink-isbn">ISBN 978-3-540-20401-5</a>.</li> <li>John L. Kelley: <cite style="font-style:italic">General Topology</cite>. Springer-Verlag, Berlin / Heidelberg / New York 1975, <a href="/wiki/Spezial:ISBN-Suche/3540901256" class="internal mw-magiclink-isbn">ISBN 3-540-90125-6</a> (Reprint der Edition bei Van Nostrand aus dem Jahre 1955).<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:Teilmenge&rft.au=John+L.+Kelley&rft.btitle=General+Topology&rft.date=1975&rft.genre=book&rft.isbn=3540901256&rft.place=Berlin+%2F+Heidelberg+%2F+New+York&rft.pub=Springer-Verlag" style="display:none"> </span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teilmenge&veaction=edit&section=10" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teilmenge&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wikibooks"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span></div><b><a href="https://de.wikibooks.org/wiki/Mathe_f%C3%BCr_Nicht-Freaks:_Teilmenge_und_echte_Teilmenge" class="extiw" title="b:Mathe für Nicht-Freaks: Teilmenge und echte Teilmenge">Wikibooks: Mathe für Nicht-Freaks: Teilmenge und echte Teilmenge</a></b> – Lern- und Lehrmaterialien</div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><span class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wiktionary"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/16px-Wiktfavicon_en.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/24px-Wiktfavicon_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/32px-Wiktfavicon_en.svg.png 2x" data-file-width="16" data-file-height="16" /></span></span></span><b><a href="https://de.wiktionary.org/wiki/Teilmenge" class="extiw" title="wikt:Teilmenge">Wiktionary: Teilmenge</a></b> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teilmenge&veaction=edit&section=11" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teilmenge&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-deiser-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-deiser_1-0">a</a></sup> <sup><a href="#cite_ref-deiser_1-1">b</a></sup></span> <span class="reference-text">Oliver Deiser: <i>Einführung in die Mengenlehre</i>. Springer, 2004, <a href="/wiki/Spezial:ISBN-Suche/9783540204015" class="internal mw-magiclink-isbn">ISBN 978-3-540-20401-5</a>, S. 33 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=94jSPGG1UHkC&pg=PA33#v=onepage">Auszug (Google)</a>).</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Adolf Fraenkel: <i>Einleitung in die Mengenlehre: Eine Elementare Einführung in das Reich des Unendlichgrossen.</i> 2. Auflage. Springer, 2013, <a href="/wiki/Spezial:ISBN-Suche/9783662259009" class="internal mw-magiclink-isbn">ISBN 9783662259009</a>, <a rel="nofollow" class="external text" href="https://books.google.de/books?id=mJXLBgAAQBAJ&pg=PA15">S. 15.</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/index.php?title=Set_theory&oldid=24823"><i>Set theory</i>.</a> In: <i>Encyclopedia of Mathematics</i>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Otto Kerner, Joseph Maurer, Jutta Steffens, Thomas Thode, Rudolf Voller: <i>Vieweg Mathematik Lexikon</i>. Vieweg, 1988, <a href="/wiki/Spezial:ISBN-Suche/3528063084" class="internal mw-magiclink-isbn">ISBN 3-528-06308-4</a>, S. 190.</span> </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Teilmenge&oldid=250483667">https://de.wikipedia.org/w/index.php?title=Teilmenge&oldid=250483667</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorie</a>: <ul><li><a href="/wiki/Kategorie:Mengenlehre" title="Kategorie:Mengenlehre">Mengenlehre</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Teilmenge" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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href="https://de.wikibooks.org/wiki/Mathe_f%C3%BCr_Nicht-Freaks:_Teilmenge_und_echte_Teilmenge" hreflang="de"><span>Wikibooks</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q177646" title="Link zum verbundenen Objekt im Datenrepositorium [g]" accesskey="g"><span>Wikidata-Datenobjekt</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">In anderen Sprachen</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%89%B3%E1%88%85%E1%89%B3%E1%8B%AD_%E1%88%B5%E1%89%A5%E1%88%B5%E1%89%A5" title="ታህታይ ስብስብ – Amharisch" lang="am" hreflang="am" data-title="ታህታይ ስብስብ" data-language-autonym="አማርኛ" data-language-local-name="Amharisch" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AC%D9%85%D9%88%D8%B9%D8%A9_%D8%AC%D8%B2%D8%A6%D9%8A%D8%A9" title="مجموعة جزئية – Arabisch" lang="ar" hreflang="ar" data-title="مجموعة جزئية" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D0%B4%D0%BC%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0" title="Падмноства – Belarussisch" lang="be" hreflang="be" data-title="Падмноства" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D0%B0%D0%B4%D0%BC%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0" title="Падмноства – Weißrussisch (Taraschkewiza)" lang="be-tarask" hreflang="be-tarask" data-title="Падмноства" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Weißrussisch (Taraschkewiza)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Подмножество – Bulgarisch" lang="bg" hreflang="bg" data-title="Подмножество" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%89%E0%A6%AA%E0%A6%B8%E0%A7%87%E0%A6%9F" title="উপসেট – Bengalisch" lang="bn" hreflang="bn" data-title="উপসেট" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Subconjunt" title="Subconjunt – Katalanisch" lang="ca" hreflang="ca" data-title="Subconjunt" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%DB%8E%D8%B1%DA%A9%DB%86%D9%85%DB%95%DA%B5" title="ژێرکۆمەڵ – Zentralkurdisch" lang="ckb" hreflang="ckb" data-title="ژێرکۆمەڵ" data-language-autonym="کوردی" data-language-local-name="Zentralkurdisch" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Podmno%C5%BEina" title="Podmnožina – Tschechisch" lang="cs" hreflang="cs" data-title="Podmnožina" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D0%B9%D0%B9%D1%8B%D1%88" title="Аййыш – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Аййыш" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Is-set" title="Is-set – Walisisch" lang="cy" hreflang="cy" data-title="Is-set" data-language-autonym="Cymraeg" data-language-local-name="Walisisch" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Delm%C3%A6ngde" title="Delmængde – Dänisch" lang="da" hreflang="da" data-title="Delmængde" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A5%CF%80%CE%BF%CF%83%CF%8D%CE%BD%CE%BF%CE%BB%CE%BF" title="Υποσύνολο – Griechisch" lang="el" hreflang="el" data-title="Υποσύνολο" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Subset" title="Subset – Englisch" lang="en" hreflang="en" data-title="Subset" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Subaro" title="Subaro – Esperanto" lang="eo" hreflang="eo" data-title="Subaro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Subconjunto" title="Subconjunto – Spanisch" lang="es" hreflang="es" data-title="Subconjunto" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Alamhulk" title="Alamhulk – Estnisch" lang="et" hreflang="et" data-title="Alamhulk" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Azpimultzo" title="Azpimultzo – Baskisch" lang="eu" hreflang="eu" data-title="Azpimultzo" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B2%DB%8C%D8%B1%D9%85%D8%AC%D9%85%D9%88%D8%B9%D9%87" title="زیرمجموعه – Persisch" lang="fa" hreflang="fa" data-title="زیرمجموعه" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Osajoukko" title="Osajoukko – Finnisch" lang="fi" hreflang="fi" data-title="Osajoukko" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Alambhulk" title="Alambhulk – Võro" lang="vro" hreflang="vro" data-title="Alambhulk" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Subset" title="Subset – Fidschi" lang="fj" hreflang="fj" data-title="Subset" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fidschi" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Inclusion_(math%C3%A9matiques)" title="Inclusion (mathématiques) – Französisch" lang="fr" hreflang="fr" data-title="Inclusion (mathématiques)" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Subconxunto" title="Subconxunto – Galicisch" lang="gl" hreflang="gl" data-title="Subconxunto" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%AA-%D7%A7%D7%91%D7%95%D7%A6%D7%94" title="תת-קבוצה – Hebräisch" lang="he" hreflang="he" data-title="תת-קבוצה" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%89%E0%A4%AA%E0%A4%B8%E0%A4%AE%E0%A5%81%E0%A4%9A%E0%A5%8D%E0%A4%9A%E0%A4%AF" title="उपसमुच्चय – Hindi" lang="hi" hreflang="hi" data-title="उपसमुच्चय" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Podskup" title="Podskup – Kroatisch" lang="hr" hreflang="hr" data-title="Podskup" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/R%C3%A9szhalmaz" title="Részhalmaz – Ungarisch" lang="hu" hreflang="hu" data-title="Részhalmaz" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D5%B6%D5%A9%D5%A1%D5%A2%D5%A1%D5%A6%D5%B4%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Ենթաբազմություն – Armenisch" lang="hy" hreflang="hy" data-title="Ենթաբազմություն" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Subinsimul" title="Subinsimul – Interlingua" lang="ia" hreflang="ia" data-title="Subinsimul" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Himpunan_bagian" title="Himpunan bagian – Indonesisch" lang="id" hreflang="id" data-title="Himpunan bagian" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Subensemblo" title="Subensemblo – Ido" lang="io" hreflang="io" data-title="Subensemblo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hlutmengi" title="Hlutmengi – Isländisch" lang="is" hreflang="is" data-title="Hlutmengi" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Inclusione_(matematica)" title="Inclusione (matematica) – Italienisch" lang="it" hreflang="it" data-title="Inclusione (matematica)" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%83%A8%E5%88%86%E9%9B%86%E5%90%88" title="部分集合 – Japanisch" lang="ja" hreflang="ja" data-title="部分集合" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%80%EB%B6%84%EC%A7%91%ED%95%A9" title="부분집합 – Koreanisch" lang="ko" hreflang="ko" data-title="부분집합" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Binkom" title="Binkom – Kurdisch" lang="ku" hreflang="ku" data-title="Binkom" data-language-autonym="Kurdî" data-language-local-name="Kurdisch" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Inclusion" title="Inclusion – Lombardisch" lang="lmo" hreflang="lmo" data-title="Inclusion" data-language-autonym="Lombard" data-language-local-name="Lombardisch" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Poaibis" title="Poaibis – Litauisch" lang="lt" hreflang="lt" data-title="Poaibis" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Подмножество – Mazedonisch" lang="mk" hreflang="mk" data-title="Подмножество" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%94%D1%8D%D0%B4_%D0%BE%D0%BB%D0%BE%D0%BD%D0%BB%D0%BE%D0%B3" title="Дэд олонлог – Mongolisch" lang="mn" hreflang="mn" data-title="Дэд олонлог" data-language-autonym="Монгол" data-language-local-name="Mongolisch" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Subset" title="Subset – Malaiisch" lang="ms" hreflang="ms" data-title="Subset" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaiisch" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Deelverzameling" title="Deelverzameling – Niederländisch" lang="nl" hreflang="nl" data-title="Deelverzameling" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Delmengd" title="Delmengd – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Delmengd" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Delmengde" title="Delmengde – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Delmengde" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Podzbi%C3%B3r" title="Podzbiór – Polnisch" lang="pl" hreflang="pl" data-title="Podzbiór" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Subconjunto" title="Subconjunto – Portugiesisch" lang="pt" hreflang="pt" data-title="Subconjunto" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Submul%C8%9Bime" title="Submulțime – Rumänisch" lang="ro" hreflang="ro" data-title="Submulțime" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Подмножество – Russisch" lang="ru" hreflang="ru" data-title="Подмножество" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Suttanzemi" title="Suttanzemi – Sizilianisch" lang="scn" hreflang="scn" data-title="Suttanzemi" data-language-autonym="Sicilianu" data-language-local-name="Sizilianisch" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Podskup" title="Podskup – Serbokroatisch" lang="sh" hreflang="sh" data-title="Podskup" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Subset" title="Subset – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Subset" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Podmno%C5%BEina" title="Podmnožina – Slowakisch" lang="sk" hreflang="sk" data-title="Podmnožina" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Podmno%C5%BEica" title="Podmnožica – Slowenisch" lang="sl" hreflang="sl" data-title="Podmnožica" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D1%81%D0%BA%D1%83%D0%BF" title="Подскуп – Serbisch" lang="sr" hreflang="sr" data-title="Подскуп" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Delm%C3%A4ngd" title="Delmängd – Schwedisch" lang="sv" hreflang="sv" data-title="Delmängd" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%89%E0%AE%9F%E0%AF%8D%E0%AE%95%E0%AE%A3%E0%AE%AE%E0%AF%8D" title="உட்கணம் – Tamil" lang="ta" hreflang="ta" data-title="உட்கணம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%8B%E0%B8%95%E0%B8%A2%E0%B9%88%E0%B8%AD%E0%B8%A2" title="เซตย่อย – Thailändisch" lang="th" hreflang="th" data-title="เซตย่อย" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Subpangkat" title="Subpangkat – Tagalog" lang="tl" hreflang="tl" data-title="Subpangkat" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Alt_k%C3%BCme" title="Alt küme – Türkisch" lang="tr" hreflang="tr" data-title="Alt küme" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%96%D0%B4%D0%BC%D0%BD%D0%BE%D0%B6%D0%B8%D0%BD%D0%B0" title="Підмножина – Ukrainisch" lang="uk" hreflang="uk" data-title="Підмножина" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%E1%BA%ADp_h%E1%BB%A3p_con" title="Tập hợp con – Vietnamesisch" lang="vi" hreflang="vi" data-title="Tập hợp con" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/D%C3%AAelverzoamelienge" title="Dêelverzoamelienge – Westflämisch" lang="vls" hreflang="vls" data-title="Dêelverzoamelienge" data-language-autonym="West-Vlams" data-language-local-name="Westflämisch" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%AD%90%E9%9B%86" title="子集 – Wu" lang="wuu" hreflang="wuu" data-title="子集" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AD%90%E9%9B%86" title="子集 – Chinesisch" lang="zh" hreflang="zh" data-title="子集" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%AD%90%E9%9B%86" title="子集 – Klassisches Chinesisch" lang="lzh" hreflang="lzh" data-title="子集" data-language-autonym="文言" data-language-local-name="Klassisches Chinesisch" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%AD%90%E9%9B%86" title="子集 – Kantonesisch" lang="yue" hreflang="yue" data-title="子集" data-language-autonym="粵語" data-language-local-name="Kantonesisch" 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