Leere Menge – Wikipedia

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href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Spezial:Anmelden&returnto=Leere+Menge&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.watch" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> <span class="minerva-icon minerva-icon--star"></span> <span>Beobachten</span> </a></li> <li id="page-actions-edit" class="page-actions-menu__list-item"><a role="button" id="ca-edit" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Leere_Menge&action=edit&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a></li> </ul> </nav> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <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> <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>Die <b>leere Menge</b> ist ein grundlegender Begriff aus der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Mengenlehre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mengenlehre">Mengenlehre</a>. Man bezeichnet damit die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Menge_(Mathematik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Menge (Mathematik)">Menge</a>, die keine Elemente enthält. Da Mengen über ihre Elemente charakterisiert werden und zwei Mengen genau dann gleich sind, wenn sie dieselben Elemente haben (siehe <a href="https://de-m-wikipedia-org.translate.goog/wiki/Extensionalit%C3%A4tsaxiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Extensionalitätsaxiom">Extensionalitätsaxiom</a> der Mengenlehre), gibt es nur eine einzige leere Menge.</p> <p>Die leere Menge ist nicht mit einer <a href="https://de-m-wikipedia-org.translate.goog/wiki/Nullmenge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nullmenge">Nullmenge</a> zu verwechseln, welche eine Menge mit dem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Ma%C3%9F_(Mathematik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Maß (Mathematik)">Maß</a> null ist. Eine solche Menge kann sogar unendlich viele Elemente enthalten.</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="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Notation_und_Codierung"><span class="tocnumber">1</span> <span class="toctext">Notation und Codierung</span></a></li> <li class="toclevel-1 tocsection-2"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Leermengenaxiom"><span class="tocnumber">2</span> <span class="toctext">Leermengenaxiom</span></a></li> <li class="toclevel-1 tocsection-3"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Eigenschaften"><span class="tocnumber">3</span> <span class="toctext">Eigenschaften</span></a></li> <li class="toclevel-1 tocsection-4"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Die_leere_Funktion"><span class="tocnumber">4</span> <span class="toctext">Die leere Funktion</span></a></li> <li class="toclevel-1 tocsection-5"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Kardinalit%C3%A4t_der_leeren_Menge"><span class="tocnumber">5</span> <span class="toctext">Kardinalität der leeren Menge</span></a></li> <li class="toclevel-1 tocsection-6"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Literatur"><span class="tocnumber">6</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-7"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Weblinks"><span class="tocnumber">7</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-8"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Einzelnachweise"><span class="tocnumber">8</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Notation_und_Codierung">Notation und Codierung</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Leere_Menge&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Notation und Codierung" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>Als Zeichen für die leere Menge hat sich das von <a href="https://de-m-wikipedia-org.translate.goog/wiki/Andr%C3%A9_Weil?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="André Weil">André Weil</a> eingeführte<sup id="cite_ref-1" class="reference"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> und von <a href="https://de-m-wikipedia-org.translate.goog/wiki/Nicolas_Bourbaki?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nicolas Bourbaki">Nicolas Bourbaki</a> verwendete 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 \varnothing }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant"> ∅ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \varnothing } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"> </noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.009ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" data-alt="{\displaystyle \varnothing }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (ein durchgestrichener Kreis) weitgehend gegenüber anderen Notationen (wie <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 \Lambda }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Λ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Lambda } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"> </noscript><span class="lazy-image-placeholder" style="width: 1.613ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" data-alt="{\displaystyle \Lambda }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> oder <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 {\mathit {\Phi }}}"><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-mathit" mathvariant="italic"> Φ </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathit {\Phi }}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e7e1e2631d79b19ccb28a083a97f38ff719307d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.03ex; width:1.694ex; height:2.176ex;" alt="{\displaystyle {\mathit {\Phi }}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.694ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e7e1e2631d79b19ccb28a083a97f38ff719307d" data-alt="{\displaystyle {\mathit {\Phi }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>)<sup id="cite_ref-2" class="reference"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> durchgesetzt. Eine typographische Variante davon 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 \emptyset }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∅ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \emptyset } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"> </noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 2.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" data-alt="{\displaystyle \emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (ein durchgestrichenes schmales Oval). Vor allem in der Schulmathematik wird die leere Menge auch gern durch eine leere Mengenklammer dargestellt: <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 \left\{\right\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> { </mo> <mrow></mrow> <mo> } </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left\{\right\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b729acc6c97590fb4a6a891a6f6abaed05c04391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.325ex; height:2.843ex;" alt="{\displaystyle \left\{\right\}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.325ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b729acc6c97590fb4a6a891a6f6abaed05c04391" data-alt="{\displaystyle \left\{\right\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. Dieses Zeichen wirkt einem Missverständnis entgegen: Die leere Menge ist nicht <i>nichts,</i> sondern eine Menge, die nichts enthält.</p> <p>Das ∅ ist in <a href="https://de-m-wikipedia-org.translate.goog/wiki/Hypertext_Markup_Language?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hypertext Markup Language">HTML</a> als <code>&#8709;</code> bzw. als <code>&empty;</code> kodiert; in <a href="https://de-m-wikipedia-org.translate.goog/wiki/Unicode?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Unicode">Unicode</a> als <code>U+2205</code> und in <a href="https://de-m-wikipedia-org.translate.goog/wiki/LaTeX?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="LaTeX">LaTeX</a> als <code>\varnothing</code>. Alternativ gibt es in LaTeX das Symbol <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 \emptyset }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∅ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \emptyset } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"> </noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 2.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" data-alt="{\displaystyle \emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, das durch <code>\emptyset</code> erzeugt wird. Nicht verwechselt werden sollte es mit dem ähnlich aussehenden <a href="https://de-m-wikipedia-org.translate.goog/wiki/Durchmesserzeichen?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Durchmesserzeichen">Durchmesserzeichen</a> ⌀, das als <code>U+2300</code> kodiert ist, oder dem skandinavischen Buchstaben <a href="https://de-m-wikipedia-org.translate.goog/wiki/%C3%98?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ø">Ø</a> (<code>U+00D8</code> bzw. <code>U+00F8</code>).</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Leermengenaxiom">Leermengenaxiom</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Leere_Menge&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Leermengenaxiom" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>Ein Axiom, das die Existenz einer leeren Menge fordert, wurde erstmals 1907 von <a href="https://de-m-wikipedia-org.translate.goog/wiki/Ernst_Zermelo?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ernst Zermelo">Ernst Zermelo</a> in der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Zermelo-Mengenlehre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Zermelo-Mengenlehre">Zermelo-Mengenlehre</a> formuliert. Es wurde später in die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Zermelo-Fraenkel-Mengenlehre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Zermelo-Fraenkel-Mengenlehre">Zermelo-Fraenkel-Mengenlehre</a> ZF und andere axiomatische Mengenlehren übernommen. Dieses <b>Leermengenaxiom</b> lautet verbal: <i>Es gibt eine Menge, die keine Elemente enthält.</i> Die präzise logische Formel lautet:</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 \exists M\colon \forall X\colon \lnot (X\in M)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∃ </mi> <mi> M </mi> <mo> : </mo> <mi mathvariant="normal"> ∀ </mi> <mi> X </mi> <mo> : </mo> <mi mathvariant="normal"> ¬ </mi> <mo stretchy="false"> ( </mo> <mi> X </mi> <mo> ∈ </mo> <mi> M </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \exists M\colon \forall X\colon \lnot (X\in M)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf174e209e851227a3b03db5b51020815234906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.698ex; height:2.843ex;" alt="{\displaystyle \exists M\colon \forall X\colon \lnot (X\in M)}"> </noscript><span class="lazy-image-placeholder" style="width: 19.698ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf174e209e851227a3b03db5b51020815234906" data-alt="{\displaystyle \exists M\colon \forall X\colon \lnot (X\in M)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Die Eindeutigkeit der leeren Menge folgt aus dem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Extensionalit%C3%A4tsaxiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Extensionalitätsaxiom">Extensionalitätsaxiom</a>. Die Existenz der leeren Menge folgt mit dem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Aussonderungsaxiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aussonderungsaxiom">Aussonderungsaxiom</a> aus der Existenz irgendeiner anderen Menge. In ZF, das im <a href="https://de-m-wikipedia-org.translate.goog/wiki/Unendlichkeitsaxiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Unendlichkeitsaxiom">Unendlichkeitsaxiom</a> die Existenz einer Menge fordert, ist das Leermengenaxiom damit entbehrlich.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Eigenschaften">Eigenschaften</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Leere_Menge&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Eigenschaften" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <ul> <li>Die leere Menge ist <a href="https://de-m-wikipedia-org.translate.goog/wiki/Teilmenge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Teilmenge">Teilmenge</a> 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 \emptyset \subseteq A}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∅ </mi> <mo> ⊆ </mo> <mi> A </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \emptyset \subseteq A} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/101ea28c34ece6f8f6a828d5326e25cd334fc084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.004ex; height:2.509ex;" alt="{\displaystyle \emptyset \subseteq A}"> </noscript><span class="lazy-image-placeholder" style="width: 6.004ex;height: 2.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/101ea28c34ece6f8f6a828d5326e25cd334fc084" data-alt="{\displaystyle \emptyset \subseteq A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl></li> <li>Jede Menge bleibt bei <a href="https://de-m-wikipedia-org.translate.goog/wiki/Vereinigungsmenge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Vereinigungsmenge">Vereinigung</a> mit der leeren Menge unverändert: <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 \emptyset \cup A=A}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∅ </mi> <mo> ∪ </mo> <mi> A </mi> <mo> = </mo> <mi> A </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \emptyset \cup A=A} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24d4b4d0d830b299a647ec55727f8695ee36234" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.33ex; height:2.509ex;" alt="{\displaystyle \emptyset \cup A=A}"> </noscript><span class="lazy-image-placeholder" style="width: 10.33ex;height: 2.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24d4b4d0d830b299a647ec55727f8695ee36234" data-alt="{\displaystyle \emptyset \cup A=A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl></li> <li>Für jede Menge ist der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Schnittmenge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Schnittmenge">Durchschnitt</a> mit der leeren Menge die leere 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 \emptyset \cap A=\emptyset }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∅ </mi> <mo> ∩ </mo> <mi> A </mi> <mo> = </mo> <mi mathvariant="normal"> ∅ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \emptyset \cap A=\emptyset } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88a24bec2a593f460ac7a93d2e7efc9667ae641a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.749ex; height:2.509ex;" alt="{\displaystyle \emptyset \cap A=\emptyset }"> </noscript><span class="lazy-image-placeholder" style="width: 9.749ex;height: 2.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88a24bec2a593f460ac7a93d2e7efc9667ae641a" data-alt="{\displaystyle \emptyset \cap A=\emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl></li> <li>Für jede Menge ist das <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kartesisches_Produkt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kartesisches Produkt">kartesische Produkt</a> mit der leeren Menge die leere 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 \emptyset \times A=\emptyset }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∅ </mi> <mo> × </mo> <mi> A </mi> <mo> = </mo> <mi mathvariant="normal"> ∅ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \emptyset \times A=\emptyset } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4c10b1165eb08f0a2ff331b4ea98604f959af8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.007ex; height:2.509ex;" alt="{\displaystyle \emptyset \times A=\emptyset }"> </noscript><span class="lazy-image-placeholder" style="width: 10.007ex;height: 2.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4c10b1165eb08f0a2ff331b4ea98604f959af8" data-alt="{\displaystyle \emptyset \times A=\emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl></li> <li>Die einzige Teilmenge der leeren Menge ist die leere 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 A\subseteq \emptyset \Rightarrow A=\emptyset }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo> ⊆ </mo> <mi mathvariant="normal"> ∅ </mi> <mo stretchy="false"> ⇒ </mo> <mi> A </mi> <mo> = </mo> <mi mathvariant="normal"> ∅ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A\subseteq \emptyset \Rightarrow A=\emptyset } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cae8f03ab02f7b52a401db2174a38d957f38f816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.622ex; height:2.509ex;" alt="{\displaystyle A\subseteq \emptyset \Rightarrow A=\emptyset }"> </noscript><span class="lazy-image-placeholder" style="width: 15.622ex;height: 2.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cae8f03ab02f7b52a401db2174a38d957f38f816" data-alt="{\displaystyle A\subseteq \emptyset \Rightarrow A=\emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl></li> <li>Daraus folgt, dass die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Potenzmenge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Potenzmenge">Potenzmenge</a> der leeren Menge genau ein Element enthält, nämlich die leere Menge 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 {\mathcal {P}}(\emptyset )=\left\{\emptyset \right\}}"><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 mathvariant="normal"> ∅ </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow> <mo> { </mo> <mi mathvariant="normal"> ∅ </mi> <mo> } </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathcal {P}}(\emptyset )=\left\{\emptyset \right\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90bab1f6d2cf37822d9672b615415c3dbef96f68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.261ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(\emptyset )=\left\{\emptyset \right\}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.261ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90bab1f6d2cf37822d9672b615415c3dbef96f68" data-alt="{\displaystyle {\mathcal {P}}(\emptyset )=\left\{\emptyset \right\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl></li> <li>Für jede widersprüchliche Aussage oder nicht erfüllbare Eigenschaft <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 E(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> E </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle E(x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2411ec03b08babee0aa5cbf7a90ba80690a3a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.915ex; height:2.843ex;" alt="{\displaystyle E(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 4.915ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2411ec03b08babee0aa5cbf7a90ba80690a3a0c" data-alt="{\displaystyle E(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> 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 \emptyset =\left\{x\mid E(x)\right\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∅ </mi> <mo> = </mo> <mrow> <mo> { </mo> <mrow> <mi> x </mi> <mo> ∣ </mo> <mi> E </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mrow> <mo> } </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \emptyset =\left\{x\mid E(x)\right\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788a7653601f4b7f9242ab687587301327cdb05e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.767ex; height:2.843ex;" alt="{\displaystyle \emptyset =\left\{x\mid E(x)\right\}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.767ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788a7653601f4b7f9242ab687587301327cdb05e" data-alt="{\displaystyle \emptyset =\left\{x\mid E(x)\right\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, z. B. <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 \emptyset =\left\{x\mid x\neq x\right\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∅ </mi> <mo> = </mo> <mrow> <mo> { </mo> <mrow> <mi> x </mi> <mo> ∣ </mo> <mi> x </mi> <mo> ≠ </mo> <mi> x </mi> </mrow> <mo> } </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \emptyset =\left\{x\mid x\neq x\right\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a329b916247c8b449c51881c6b278bb80224955b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.611ex; height:2.843ex;" alt="{\displaystyle \emptyset =\left\{x\mid x\neq x\right\}}"> </noscript><span class="lazy-image-placeholder" style="width: 15.611ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a329b916247c8b449c51881c6b278bb80224955b" data-alt="{\displaystyle \emptyset =\left\{x\mid x\neq x\right\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> oder <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 \emptyset =\left\{x\in \mathbb {Z} \mid x+1=x+2\right\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∅ </mi> <mo> = </mo> <mrow> <mo> { </mo> <mrow> <mi> x </mi> <mo> ∈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> Z </mi> </mrow> <mo> ∣ </mo> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> <mo> = </mo> <mi> x </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> } </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \emptyset =\left\{x\in \mathbb {Z} \mid x+1=x+2\right\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3518735ec8b247cf2ac7f308d6363641d6b6dcbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.007ex; height:2.843ex;" alt="{\displaystyle \emptyset =\left\{x\in \mathbb {Z} \mid x+1=x+2\right\}}"> </noscript><span class="lazy-image-placeholder" style="width: 28.007ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3518735ec8b247cf2ac7f308d6363641d6b6dcbb" data-alt="{\displaystyle \emptyset =\left\{x\in \mathbb {Z} \mid x+1=x+2\right\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl></li> </ul> <dl> <dd> Damit ist die leere Menge insbesondere die <a href="https://de-m-wikipedia-org.translate.goog/wiki/L%C3%B6sungsmenge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lösungsmenge">Lösungsmenge</a> einer <a href="https://de-m-wikipedia-org.translate.goog/wiki/Gleichung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gleichung">Gleichung</a> oder <a href="https://de-m-wikipedia-org.translate.goog/wiki/Ungleichung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ungleichung">Ungleichung</a>, die keine Lösung besitzt. </dd> </dl> <ul> <li>Jede Existenzaussage über Elemente der leeren Menge, etwa <dl> <dd> <i>„Es existiert ein x aus <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 \emptyset }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∅ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \emptyset } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"> </noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 2.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" data-alt="{\displaystyle \emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, sodass gilt …“</i> </dd> </dl></li> </ul> <dl> <dd> ist falsch, denn es gibt kein Element, das die Bedingung erfüllen könnte. </dd> </dl> <ul> <li>Jede Allaussage über Elemente der leeren Menge, etwa <dl> <dd> <i>„Für alle Elemente 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 \emptyset }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∅ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \emptyset } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"> </noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 2.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" data-alt="{\displaystyle \emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> gilt …“</i> </dd> </dl></li> </ul> <dl> <dd> ist wahr, denn es gibt kein Element, für das die fragliche Forderung falsch sein könnte. </dd> </dl> <ul> <li>Sei <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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> eine Menge 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 f\colon A\to \emptyset }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo> : </mo> <mi> A </mi> <mo stretchy="false"> → </mo> <mi mathvariant="normal"> ∅ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f\colon A\to \emptyset } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ef87bb13270b55880f3d1979d7a14d3af3d398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.832ex; height:2.676ex;" alt="{\displaystyle f\colon A\to \emptyset }"> </noscript><span class="lazy-image-placeholder" style="width: 8.832ex;height: 2.676ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ef87bb13270b55880f3d1979d7a14d3af3d398" data-alt="{\displaystyle f\colon A\to \emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> eine Abbildung. 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 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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> die leere Menge.</li> <li>Die leere Menge ist die einzige <a href="https://de-m-wikipedia-org.translate.goog/wiki/Basis_(Vektorraum)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Basis (Vektorraum)">Basis</a> des <a href="https://de-m-wikipedia-org.translate.goog/wiki/Nullvektorraum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nullvektorraum">Nullvektorraums</a>.</li> <li>Die leere Menge ist definitionsgemäß in jedem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Topologischer_Raum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Topologischer Raum">topologischen Raum</a> zugleich <a href="https://de-m-wikipedia-org.translate.goog/wiki/Abgeschlossene_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abgeschlossene Menge">abgeschlossen</a> und <a href="https://de-m-wikipedia-org.translate.goog/wiki/Offene_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Offene Menge">offen</a>.</li> <li>Jede endliche Teilüberdeckung enthält die leere Menge, also ist die leere Menge <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kompakter_Raum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kompakter Raum">kompakt</a>.</li> <li>Ebenfalls per definitionem ist die leere Menge in jedem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Ma%C3%9Fraum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Maßraum">Maßraum</a> eine <a href="https://de-m-wikipedia-org.translate.goog/wiki/Messbare_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Messbare Menge">messbare Menge</a> und besitzt das Maß 0.</li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Die_leere_Funktion">Die leere Funktion</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Leere_Menge&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Die leere Funktion" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <p>Die leere Menge ist insbesondere eine leere Menge geordneter Paare und damit eine <a href="https://de-m-wikipedia-org.translate.goog/wiki/Funktion_(Mathematik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Funktion (Mathematik)">Abbildung</a>. Daher gibt es für jede 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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> genau eine Abbildung</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 f\colon \emptyset \to A}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo> : </mo> <mi mathvariant="normal"> ∅ </mi> <mo stretchy="false"> → </mo> <mi> A </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f\colon \emptyset \to A} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ecdf5c87e0202c97c8dfdfb1df037806a28daef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.832ex; height:2.676ex;" alt="{\displaystyle f\colon \emptyset \to A}"> </noscript><span class="lazy-image-placeholder" style="width: 8.832ex;height: 2.676ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ecdf5c87e0202c97c8dfdfb1df037806a28daef" data-alt="{\displaystyle f\colon \emptyset \to A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, </dd> </dl> <p>nämlich <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 f=\emptyset }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo> = </mo> <mi mathvariant="normal"> ∅ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f=\emptyset } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965a558aeccec84531c0d2f3156b6823ad19d96f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.676ex;" alt="{\displaystyle f=\emptyset }"> </noscript><span class="lazy-image-placeholder" style="width: 5.54ex;height: 2.676ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965a558aeccec84531c0d2f3156b6823ad19d96f" data-alt="{\displaystyle f=\emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, die sogenannte <i>leere Abbildung</i> oder <i><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Funktion?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Leere Funktion">leere Funktion</a>.</i> Das kann man auch so formulieren:</p> <dl> <dd> Die leere Menge ist das <a href="https://de-m-wikipedia-org.translate.goog/wiki/Anfangsobjekt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Anfangsobjekt">Anfangsobjekt</a> in der Kategorie der Mengen. </dd> </dl> <p>Im Gegensatz dazu gibt es nur 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=\emptyset }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo> = </mo> <mi mathvariant="normal"> ∅ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A=\emptyset } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/882aebdcd58d66dc7aae7163265682303a9c1afa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.004ex; height:2.509ex;" alt="{\displaystyle A=\emptyset }"> </noscript><span class="lazy-image-placeholder" style="width: 6.004ex;height: 2.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/882aebdcd58d66dc7aae7163265682303a9c1afa" data-alt="{\displaystyle A=\emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> eine Funktion <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\to \emptyset }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo stretchy="false"> → </mo> <mi mathvariant="normal"> ∅ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A\to \emptyset } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8238deb489fc69482525253ef2c2a55b3fc024f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.52ex; height:2.509ex;" alt="{\displaystyle A\to \emptyset }"> </noscript><span class="lazy-image-placeholder" style="width: 6.52ex;height: 2.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8238deb489fc69482525253ef2c2a55b3fc024f" data-alt="{\displaystyle A\to \emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Kardinalität_der_leeren_Menge"><span id="Kardinalit.C3.A4t_der_leeren_Menge"></span>Kardinalität der leeren Menge</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Leere_Menge&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Kardinalität der leeren Menge" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <p>Die leere Menge ist die einzige Menge mit der <a href="https://de-m-wikipedia-org.translate.goog/wiki/M%C3%A4chtigkeit_(Mathematik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mächtigkeit (Mathematik)">Kardinalität</a> (Mächtigkeit) <a href="https://de-m-wikipedia-org.translate.goog/wiki/Null?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Null">null</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 \left\vert \emptyset \right\vert =0.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> | </mo> <mi mathvariant="normal"> ∅ </mi> <mo> | </mo> </mrow> <mo> = </mo> <mn> 0. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left\vert \emptyset \right\vert =0.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/791fa070e7d52ed5b4109f4bd718796ef38192ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.364ex; height:2.843ex;" alt="{\displaystyle \left\vert \emptyset \right\vert =0.}"> </noscript><span class="lazy-image-placeholder" style="width: 7.364ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/791fa070e7d52ed5b4109f4bd718796ef38192ed" data-alt="{\displaystyle \left\vert \emptyset \right\vert =0.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Sie ist daher auch der einzige Repräsentant der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kardinalzahl_(Mathematik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kardinalzahl (Mathematik)">Kardinalzahl</a> 0 und der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Ordinalzahl?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ordinalzahl">Ordinalzahl</a> 0. Insbesondere ist sie eine <a href="https://de-m-wikipedia-org.translate.goog/wiki/Endliche_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Endliche Menge">endliche Menge</a>.</p> <p>Die leere Menge ist auch die einzige Menge, die durch ihre Kardinalität bereits eindeutig bestimmt ist. (Für jede andere Kardinalzahl ist die Klasse der Mengen dieser Kardinalität sogar <a href="https://de-m-wikipedia-org.translate.goog/wiki/Echte_Klasse?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Echte Klasse">echt</a>.)</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Literatur">Literatur</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Leere_Menge&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Literatur" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul> <li>Oliver Deiser: <cite style="font-style:italic">Einführung in die Mengenlehre</cite>. Die Mengenlehre Georg Cantors und ihre Axiomatisierung durch Ernst Zermelo. 3. Auflage. Springer Verlag, Berlin, Heidelberg 2010, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/9783642014444?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 978-3-642-01444-4</a>, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Digital_Object_Identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007/978-3-642-01445-1">10.1007/978-3-642-01445-1</a></span>.<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:Leere+Menge&rft.au=Oliver+Deiser&rft.btitle=Einf%C3%BChrung+in+die+Mengenlehre&rft.date=2010&rft.doi=10.1007%2F978-3-642-01445-1&rft.edition=3.&rft.genre=book&rft.isbn=9783642014444&rft.place=Berlin%2C+Heidelberg&rft.pub=Springer+Verlag" style="display:none"> </span></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Leere_Menge&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Weblinks" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> <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"> <noscript> <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" data-file-width="300" data-file-height="300"> </noscript><span class="lazy-image-placeholder" style="width: 16px;height: 16px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" data-alt="" data-width="16" data-height="16" data-mw-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-class="mw-file-element"> </span></span></span> </div><b><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikibooks.org/wiki/Mathe_f%25C3%25BCr_Nicht-Freaks:_Leere_Menge_und_Allklasse" class="extiw" title="b:Mathe für Nicht-Freaks: Leere Menge und Allklasse">Wikibooks: Mathe für Nicht-Freaks: Leere Menge und Allklasse</a></b> – Lern- und Lehrmaterialien </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Leere_Menge&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Einzelnachweise" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>Bearbeiten</span> </a> </span> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-1">↑</a></span> <span class="reference-text">Deiser, S. 31.</span></li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-2">↑</a></span> <span class="reference-text"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Willard_van_Orman_Quine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Willard van Orman Quine">Willard van Orman Quine</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Set Theory And Its Logic</cite>. Belknap Press of Harvard University Press, Cambridge, USA 1963, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/0674802071?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 0-674-80207-1</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>359 (HC) / 380 (PB)</span> (englisch).<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:Leere+Menge&rft.au=Willard+van+Orman+Quine&rft.btitle=Set+Theory+And+Its+Logic&rft.date=1963&rft.genre=book&rft.isbn=0674802071&rft.pages=359+%28HC%29+%2F+380+%28PB%29&rft.place=Cambridge%2C+USA&rft.pub=Belknap+Press+of+Harvard+University+Press" style="display:none"> </span> – Hier: Seite 19.<br><a href="https://de-m-wikipedia-org.translate.goog/wiki/Willard_van_Orman_Quine?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Willard van Orman Quine">Willard van Orman Quine</a>: <cite style="font-style:italic">Mengenlehre und ihre Logik</cite> (= <cite style="font-style:italic">Logik und Grundlagen der Mathematik (deutsche Übersetzung)</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>10</span>). Vieweg+Teubner Verlag, 1973, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/3528082941?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 3-528-08294-1</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>264</span>.<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:Leere+Menge&rft.au=Willard+van+Orman+Quine&rft.btitle=Mengenlehre+und+ihre+Logik&rft.date=1973&rft.genre=book&rft.isbn=3528082941&rft.pages=264&rft.pub=Vieweg%2BTeubner+Verlag&rft.series=Logik+und+Grundlagen+der+Mathematik+%28deutsche+%C3%9Cbersetzung%29" style="display:none"> </span> – Hier: <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.de/books?id%3DT3GeBgAAQBAJ%26pg%3DPA47%26lpg%3DPA47%26dq%3DQuine%2Bmengenlehre%2Bund%2Bihre%2Blogik%26ots%3DgiTUVyatvF%26sig%3DMsLkICsoRFy5y3OxIVYUTgA7JTE%26hl%3Dde%26ved%3D0ahUKEwjU4KW66-7YAhUGchQKHeCgBEkQ6AEILjAA%23v%3Dsnippet%26q%3D%2522Wenn%2520nach%2520Kapitel%25204%2522%26f%3Dfalse">Seite 14.</a></span></li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-3">↑</a></span> <span class="reference-text">Akihiro Kanamori: <i><style data-mw-deduplicate="TemplateStyles:r246413598">.mw-parser-output .webarchiv-memento{color:var(--color-base,#202122)!important}</style><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20180201075321/https://pdfs.semanticscholar.org/79f4/eb47076249937c2057d53226c6f853298826.pdf">The Empty Set, The Singleton, And The Ordered Pair.</a> (<a href="https://de-m-wikipedia-org.translate.goog/wiki/Web-Archivierung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Begrifflichkeiten" title="Web-Archivierung"><span class="webarchiv-memento">Memento</span></a> vom 1. Februar 2018 im </i><a href="https://de-m-wikipedia-org.translate.goog/wiki/Internet_Archive?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Internet Archive">Internet Archive</a><i>)</i> In: <i>The Bulletin of Symbolic Logic.</i> Bd. 9, Nr. 3, Sept. 2003, Seite 289 (<a href="https://de-m-wikipedia-org.translate.goog/wiki/Norbert_Wiener?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Norbert Wiener">Norbert Wiener</a> zitierend).</span></li> </ol> <style data-mw-deduplicate="TemplateStyles:r248673343">.mw-parser-output div.NavFrame{border-width:1px;border-style:solid;border-left-color:var(--dewiki-rahmenfarbe1);border-right-color:var(--dewiki-rahmenfarbe1);border-top-color:var(--dewiki-rahmenfarbe1);border-bottom-color:var(--dewiki-rahmenfarbe1);clear:both;font-size:95%;margin-top:1.5em;min-height:0;padding:2px;text-align:center}.mw-parser-output div.NavPic{float:left;padding:2px}.mw-parser-output div.NavHead{background-color:var(--dewiki-hintergrundfarbe5);font-weight:bold}.mw-parser-output div.NavFrame:after{clear:both;content:"";display:block}.mw-parser-output div.NavFrame+div.NavFrame,.mw-parser-output div.NavFrame+link+div.NavFrame,.mw-parser-output div.NavFrame+style+div.NavFrame{margin-top:-1px}.mw-parser-output .NavToggle{float:right;font-size:x-small}@media screen{html.skin-theme-clientpref-night .mw-parser-output .NavPic span[typeof="mw:File"] img{background-color:#c8ccd1}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .NavPic span[typeof="mw:File"] img{background-color:#c8ccd1}}</style> <div class="NavFrame navigation-not-searchable" role="navigation"> <div class="NavHead"> Axiome und Axiomenschemata der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Zermelo-Fraenkel-Mengenlehre?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Zermelo-Fraenkel-Mengenlehre">Zermelo-Fraenkel-Mengenlehre</a> </div> <div class="NavContent"> <p><b>Axiome:</b> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Extensionalit%C3%A4tsaxiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Extensionalitätsaxiom">Extensionalitätsaxiom</a> | <a href="https://de-m-wikipedia-org.translate.goog/wiki/Fundierungsaxiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fundierungsaxiom">Fundierungsaxiom</a> | <a class="mw-selflink-fragment" href="https://de-m-wikipedia-org.translate.goog/wiki/Leere_Menge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Leermengenaxiom">Leermengenaxiom</a> | <a href="https://de-m-wikipedia-org.translate.goog/wiki/Paarmenge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Paarmenge">Paarmengenaxiom</a> | <a href="https://de-m-wikipedia-org.translate.goog/wiki/Vereinigungsaxiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Vereinigungsaxiom">Vereinigungsaxiom</a> | <a href="https://de-m-wikipedia-org.translate.goog/wiki/Potenzmengenaxiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Potenzmengenaxiom">Potenzmengenaxiom</a> | <a href="https://de-m-wikipedia-org.translate.goog/wiki/Unendlichkeitsaxiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Unendlichkeitsaxiom">Unendlichkeitsaxiom</a> | <a href="https://de-m-wikipedia-org.translate.goog/wiki/Auswahlaxiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Auswahlaxiom">Auswahlaxiom</a></p> <p><b>Axiomenschemata:</b> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Aussonderungsaxiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Aussonderungsaxiom">Aussonderungsaxiom</a> | <a href="https://de-m-wikipedia-org.translate.goog/wiki/Ersetzungsaxiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ersetzungsaxiom">Ersetzungsaxiom</a></p> </div> </div> </section> </div> <noscript> <img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=mobile&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> Abgerufen von „<a dir="ltr" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/w/index.php?title%3DLeere_Menge%26oldid%3D252333812">https://de.wikipedia.org/w/index.php?title=Leere_Menge&oldid=252333812</a>“ </div> </div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"><a class="last-modified-bar" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Leere_Menge&action=history&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"> <div class="post-content last-modified-bar__content"><span class="minerva-icon minerva-icon-size-medium minerva-icon--modified-history"></span> <span class="last-modified-bar__text modified-enhancement" data-user-name="Hutch" data-user-gender="male" data-timestamp="1737099136"> <span>Zuletzt bearbeitet am 17. 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href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ast.wikipedia.org/wiki/Conxuntu_vac%25C3%25ADu" title="Conxuntu vacíu – Asturisch" lang="ast" hreflang="ast" data-title="Conxuntu vacíu" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target"><span>Asturianu</span></a></li> <li class="interlanguage-link interwiki-az mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://az.wikipedia.org/wiki/Bo%25C5%259F_%25C3%25A7oxluq" title="Boş çoxluq – Aserbaidschanisch" lang="az" hreflang="az" data-title="Boş çoxluq" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li> <li class="interlanguage-link interwiki-ba mw-list-item"><a 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data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target"><span>Български</span></a></li> <li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bs.wikipedia.org/wiki/Prazan_skup" title="Prazan skup – Bosnisch" lang="bs" hreflang="bs" data-title="Prazan skup" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Conjunt_buit" title="Conjunt buit – Katalanisch" lang="ca" hreflang="ca" data-title="Conjunt buit" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ckb.wikipedia.org/wiki/%25DA%25A9%25DB%2586%25D9%2585%25DB%2595%25DA%25B5%25DB%2595%25DB%258C_%25D8%25A8%25DB%2595%25D8%25AA%25D8%25A7%25DA%25B5" title="کۆمەڵەی بەتاڵ – Zentralkurdisch" lang="ckb" hreflang="ckb" data-title="کۆمەڵەی بەتاڵ" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/wiki/Pr%25C3%25A1zdn%25C3%25A1_mno%25C5%25BEina" title="Prázdná množina – Tschechisch" lang="cs" hreflang="cs" data-title="Prázdná množina" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cv.wikipedia.org/wiki/%25D0%259F%25D1%2583%25D1%2588%25C4%2583_%25D0%25B9%25D1%258B%25D1%2588" title="Пушă йыш – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Пушă йыш" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target"><span>Чӑвашла</span></a></li> <li class="interlanguage-link interwiki-da mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://da.wikipedia.org/wiki/Tomme_m%25C3%25A6ngde" title="Tomme mængde – Dänisch" lang="da" hreflang="da" data-title="Tomme mængde" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://el.wikipedia.org/wiki/%25CE%259A%25CE%25B5%25CE%25BD%25CF%258C_%25CF%2583%25CF%258D%25CE%25BD%25CE%25BF%25CE%25BB%25CE%25BF" title="Κενό σύνολο – Griechisch" lang="el" hreflang="el" data-title="Κενό σύνολο" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Empty_set" title="Empty set – Englisch" lang="en" hreflang="en" data-title="Empty set" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eo.wikipedia.org/wiki/Malplena_aro" title="Malplena aro – Esperanto" lang="eo" hreflang="eo" data-title="Malplena aro" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Conjunto_vac%25C3%25ADo" title="Conjunto vacío – Spanisch" lang="es" hreflang="es" data-title="Conjunto vacío" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/wiki/T%25C3%25BChi_hulk" title="Tühi hulk – Estnisch" lang="et" hreflang="et" data-title="Tühi hulk" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Multzo_huts" title="Multzo huts – Baskisch" lang="eu" hreflang="eu" data-title="Multzo huts" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25D9%2585%25D8%25AC%25D9%2585%25D9%2588%25D8%25B9%25D9%2587_%25D8%25AA%25D9%2587%25DB%258C" title="مجموعه تهی – Persisch" lang="fa" hreflang="fa" data-title="مجموعه تهی" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Tyhj%25C3%25A4_joukko" title="Tyhjä joukko – Finnisch" lang="fi" hreflang="fi" data-title="Tyhjä joukko" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fiu-vro.wikipedia.org/wiki/T%25C3%25BChi_hulk" title="Tühi hulk – Võro" lang="vro" hreflang="vro" data-title="Tühi hulk" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li> <li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/Ensemble_vide" title="Ensemble vide – Französisch" lang="fr" hreflang="fr" data-title="Ensemble vide" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gl.wikipedia.org/wiki/Conxunto_baleiro" title="Conxunto baleiro – Galicisch" lang="gl" hreflang="gl" data-title="Conxunto baleiro" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%2594%25D7%25A7%25D7%2591%25D7%2595%25D7%25A6%25D7%2594_%25D7%2594%25D7%25A8%25D7%2599%25D7%25A7%25D7%2594" title="הקבוצה הריקה – Hebräisch" lang="he" hreflang="he" data-title="הקבוצה הריקה" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hi.wikipedia.org/wiki/%25E0%25A4%25B0%25E0%25A4%25BF%25E0%25A4%2595%25E0%25A5%258D%25E0%25A4%25A4_%25E0%25A4%25B8%25E0%25A4%25AE%25E0%25A5%2581%25E0%25A4%259A%25E0%25A5%258D%25E0%25A4%259A%25E0%25A4%25AF" title="रिक्त समुच्चय – Hindi" lang="hi" hreflang="hi" data-title="रिक्त समुच्चय" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hr.wikipedia.org/wiki/Prazni_skup" title="Prazni skup – Kroatisch" lang="hr" hreflang="hr" data-title="Prazni skup" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hu.wikipedia.org/wiki/%25C3%259Cres_halmaz" title="Üres halmaz – Ungarisch" lang="hu" hreflang="hu" data-title="Üres halmaz" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hy.wikipedia.org/wiki/%25D4%25B4%25D5%25A1%25D5%25BF%25D5%25A1%25D6%2580%25D5%25AF_%25D5%25A2%25D5%25A1%25D5%25A6%25D5%25B4%25D5%25B8%25D6%2582%25D5%25A9%25D5%25B5%25D5%25B8%25D6%2582%25D5%25B6" title="Դատարկ բազմություն – Armenisch" lang="hy" hreflang="hy" data-title="Դատարկ բազմություն" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target"><span>Հայերեն</span></a></li> <li class="interlanguage-link interwiki-id mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://id.wikipedia.org/wiki/Himpunan_kosong" title="Himpunan kosong – Indonesisch" lang="id" hreflang="id" data-title="Himpunan kosong" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://io.wikipedia.org/wiki/Ensemblo_vakua" title="Ensemblo vakua – Ido" lang="io" hreflang="io" data-title="Ensemblo vakua" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://is.wikipedia.org/wiki/T%25C3%25B3mamengi" title="Tómamengi – Isländisch" lang="is" hreflang="is" data-title="Tómamengi" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Insieme_vuoto" title="Insieme vuoto – Italienisch" lang="it" hreflang="it" data-title="Insieme vuoto" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E7%25A9%25BA%25E9%259B%2586%25E5%2590%2588" title="空集合 – Japanisch" lang="ja" hreflang="ja" data-title="空集合" 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-ka mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ka.wikipedia.org/wiki/%25E1%2583%25AA%25E1%2583%2590%25E1%2583%25A0%25E1%2583%2598%25E1%2583%2594%25E1%2583%259A%25E1%2583%2598_%25E1%2583%25A1%25E1%2583%2598%25E1%2583%259B%25E1%2583%25A0%25E1%2583%2590%25E1%2583%2595%25E1%2583%259A%25E1%2583%2594" title="ცარიელი სიმრავლე – Georgisch" lang="ka" hreflang="ka" data-title="ცარიელი სიმრავლე" data-language-autonym="ქართული" data-language-local-name="Georgisch" class="interlanguage-link-target"><span>ქართული</span></a></li> <li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://kk.wikipedia.org/wiki/%25D0%2591%25D0%25BE%25D1%2581_%25D0%25B6%25D0%25B8%25D1%258B%25D0%25BD" title="Бос жиын – Kasachisch" lang="kk" hreflang="kk" data-title="Бос жиын" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target"><span>Қазақша</span></a></li> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ko.wikipedia.org/wiki/%25EA%25B3%25B5%25EC%25A7%2591%25ED%2595%25A9" title="공집합 – Koreanisch" lang="ko" hreflang="ko" data-title="공집합" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ku.wikipedia.org/wiki/Koma_vala" title="Koma vala – Kurdisch" lang="ku" hreflang="ku" data-title="Koma vala" data-language-autonym="Kurdî" data-language-local-name="Kurdisch" class="interlanguage-link-target"><span>Kurdî</span></a></li> <li class="interlanguage-link interwiki-la mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://la.wikipedia.org/wiki/Copia_vacua" title="Copia vacua – Latein" lang="la" hreflang="la" data-title="Copia vacua" data-language-autonym="Latina" data-language-local-name="Latein" class="interlanguage-link-target"><span>Latina</span></a></li> <li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lmo.wikipedia.org/wiki/Insema_voeud" title="Insema voeud – Lombardisch" lang="lmo" hreflang="lmo" data-title="Insema voeud" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lt.wikipedia.org/wiki/Tu%25C5%25A1%25C4%258Dia_aib%25C4%2597" title="Tuščia aibė – Litauisch" lang="lt" hreflang="lt" data-title="Tuščia aibė" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target"><span>Lietuvių</span></a></li> <li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lv.wikipedia.org/wiki/Tuk%25C5%25A1a_kopa" title="Tukša kopa – Lettisch" lang="lv" hreflang="lv" data-title="Tukša kopa" data-language-autonym="Latviešu" data-language-local-name="Lettisch" class="interlanguage-link-target"><span>Latviešu</span></a></li> <li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mk.wikipedia.org/wiki/%25D0%259F%25D1%2580%25D0%25B0%25D0%25B7%25D0%25BD%25D0%25BE_%25D0%25BC%25D0%25BD%25D0%25BE%25D0%25B6%25D0%25B5%25D1%2581%25D1%2582%25D0%25B2%25D0%25BE" title="Празно множество – Mazedonisch" lang="mk" hreflang="mk" data-title="Празно множество" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target"><span>Македонски</span></a></li> <li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ml.wikipedia.org/wiki/%25E0%25B4%25B6%25E0%25B5%2582%25E0%25B4%25A8%25E0%25B5%258D%25E0%25B4%25AF%25E0%25B4%2597%25E0%25B4%25A3%25E0%25B4%2582" title="ശൂന്യഗണം – Malayalam" lang="ml" hreflang="ml" data-title="ശൂന്യഗണം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li> <li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mn.wikipedia.org/wiki/%25D0%25A5%25D0%25BE%25D0%25BE%25D1%2581%25D0%25BE%25D0%25BD_%25D0%25BE%25D0%25BB%25D0%25BE%25D0%25BD%25D0%25BB%25D0%25BE%25D0%25B3" title="Хоосон олонлог – Mongolisch" lang="mn" hreflang="mn" data-title="Хоосон 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ms.wikipedia.org/wiki/Set_kosong" title="Set kosong – Malaiisch" lang="ms" hreflang="ms" data-title="Set kosong" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Lege_verzameling" title="Lege verzameling – Niederländisch" lang="nl" hreflang="nl" data-title="Lege verzameling" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nn.wikipedia.org/wiki/Den_tomme_mengda" title="Den tomme mengda – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Den tomme mengda" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://no.wikipedia.org/wiki/Den_tomme_mengden" title="Den tomme mengden – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Den tomme mengden" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Zbi%25C3%25B3r_pusty" title="Zbiór pusty – Polnisch" lang="pl" hreflang="pl" data-title="Zbiór pusty" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Conjunto_vazio" title="Conjunto vazio – Portugiesisch" lang="pt" hreflang="pt" data-title="Conjunto vazio" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Mul%25C8%259Bime_vid%25C4%2583" title="Mulțime vidă – Rumänisch" lang="ro" hreflang="ro" data-title="Mulțime vidă" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%259F%25D1%2583%25D1%2581%25D1%2582%25D0%25BE%25D0%25B5_%25D0%25BC%25D0%25BD%25D0%25BE%25D0%25B6%25D0%25B5%25D1%2581%25D1%2582%25D0%25B2%25D0%25BE" title="Пустое множество – Russisch" lang="ru" hreflang="ru" data-title="Пустое множество" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://scn.wikipedia.org/wiki/Nzemi_vacanti" title="Nzemi vacanti – Sizilianisch" lang="scn" hreflang="scn" data-title="Nzemi vacanti" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sh.wikipedia.org/wiki/Prazan_skup" title="Prazan skup – Serbokroatisch" lang="sh" hreflang="sh" data-title="Prazan skup" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Empty_set" title="Empty set – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Empty set" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sk.wikipedia.org/wiki/Pr%25C3%25A1zdna_mno%25C5%25BEina" title="Prázdna množina – Slowakisch" lang="sk" hreflang="sk" data-title="Prázdna množina" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Prazna_mno%25C5%25BEica" title="Prazna množica – Slowenisch" lang="sl" hreflang="sl" data-title="Prazna množica" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li> <li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sq.wikipedia.org/wiki/Bashk%25C3%25ABsia_boshe" title="Bashkësia boshe – Albanisch" lang="sq" hreflang="sq" data-title="Bashkësia boshe" data-language-autonym="Shqip" data-language-local-name="Albanisch" class="interlanguage-link-target"><span>Shqip</span></a></li> <li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sr.wikipedia.org/wiki/%25D0%259F%25D1%2580%25D0%25B0%25D0%25B7%25D0%25B0%25D0%25BD_%25D1%2581%25D0%25BA%25D1%2583%25D0%25BF" title="Празан скуп – Serbisch" lang="sr" hreflang="sr" data-title="Празан скуп" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Tomma_m%25C3%25A4ngden" title="Tomma mängden – Schwedisch" lang="sv" hreflang="sv" data-title="Tomma mängden" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li> <li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://szl.wikipedia.org/wiki/Pusto_mynga" title="Pusto mynga – Schlesisch (Wasserpolnisch)" lang="szl" hreflang="szl" data-title="Pusto mynga" data-language-autonym="Ślůnski" data-language-local-name="Schlesisch (Wasserpolnisch)" class="interlanguage-link-target"><span>Ślůnski</span></a></li> <li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ta.wikipedia.org/wiki/%25E0%25AE%25B5%25E0%25AF%2586%25E0%25AE%25B1%25E0%25AF%258D%25E0%25AE%25B1%25E0%25AF%2581%25E0%25AE%2595%25E0%25AF%258D_%25E0%25AE%2595%25E0%25AE%25A3%25E0%25AE%25AE%25E0%25AF%258D" title="வெற்றுக் கணம் – Tamil" lang="ta" hreflang="ta" data-title="வெற்றுக் கணம்" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://th.wikipedia.org/wiki/%25E0%25B9%2580%25E0%25B8%258B%25E0%25B8%2595%25E0%25B8%25A7%25E0%25B9%2588%25E0%25B8%25B2%25E0%25B8%2587" title="เซตว่าง – Thailändisch" lang="th" hreflang="th" data-title="เซตว่าง" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target"><span>ไทย</span></a></li> <li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/wiki/Bo%25C5%259F_k%25C3%25BCme" title="Boş küme – Türkisch" lang="tr" hreflang="tr" data-title="Boş küme" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target"><span>Türkçe</span></a></li> <li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tt.wikipedia.org/wiki/%25D0%2591%25D1%2583%25D1%2588_%25D0%25BA%25D2%25AF%25D0%25BF%25D0%25BB%25D0%25B5%25D0%25BA" title="Буш күплек – Tatarisch" lang="tt" hreflang="tt" data-title="Буш күплек" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatarisch" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D1%2580%25D0%25BE%25D0%25B6%25D0%25BD%25D1%258F_%25D0%25BC%25D0%25BD%25D0%25BE%25D0%25B6%25D0%25B8%25D0%25BD%25D0%25B0" title="Порожня множина – Ukrainisch" lang="uk" hreflang="uk" data-title="Порожня множина" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vi.wikipedia.org/wiki/T%25E1%25BA%25ADp_h%25E1%25BB%25A3p_r%25E1%25BB%2597ng" title="Tập hợp rỗng – Vietnamesisch" lang="vi" hreflang="vi" data-title="Tập hợp rỗng" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vls.wikipedia.org/wiki/Lege_verzoamelienge" title="Lege verzoamelienge – Westflämisch" lang="vls" hreflang="vls" data-title="Lege verzoamelienge" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://wuu.wikipedia.org/wiki/%25E7%25A9%25BA%25E9%259B%2586" title="空集 – Wu" lang="wuu" hreflang="wuu" data-title="空集" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh.wikipedia.org/wiki/%25E7%25A9%25BA%25E9%259B%2586" title="空集 – Chinesisch" lang="zh" hreflang="zh" data-title="空集" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-classical.wikipedia.org/wiki/%25E7%25A9%25BA%25E9%259B%2586" title="空集 – Klassisches Chinesisch" lang="lzh" hreflang="lzh" data-title="空集" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-yue.wikipedia.org/wiki/%25E7%25A9%25BA%25E9%259B%2586" title="空集 – Kantonesisch" lang="yue" hreflang="yue" data-title="空集" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod">Diese Seite wurde zuletzt am 17. 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Leere Menge – Wikipedia