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<!DOCTYPE html><html> <head> <title>superset</title>  <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <link rel="stylesheet" href="LaTeXML.css" type="text/css"> <link rel="stylesheet" href="ltx-article.css" type="text/css"> <link rel="stylesheet" href="https://cdn.rawgit.com/holtzermann17/3f71ceeb3b055e1ddc3b6c11fb1f074c/raw/2bb23e3b173ff96840797fc0c3bcb8c54085df8e/LaTeXML.css" type="text/css"> <link rel="stylesheet" href="https://cdn.rawgit.com/holtzermann17/4bda0365b30858ac2fb83623185fe3ec/raw/cedd84ed3e3ad597c5d293f443ecfe4803741c6b/ltx-article.css" type="text/css"> <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML" type="text/javascript"></script> </head> <body> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">superset</h1> <div id="p1" class="ltx_para"> <br class="ltx_break"> </div> <div id="p2" class="ltx_para"> <p class="ltx_p">Given two sets <math id="p2.m1" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math> and <math id="p2.m2" class="ltx_Math" alttext="B" display="inline"><mi>B</mi></math>, <math id="p2.m3" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math> is a <em class="ltx_emph ltx_font_italic"><a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">superset</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Superset.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/superset"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup></em> of <math id="p2.m4" class="ltx_Math" alttext="B" display="inline"><mi>B</mi></math> if every <a class="nnexus_concept" href="http://mathworld.wolfram.com/Element.html">element</a> in <math id="p2.m5" class="ltx_Math" alttext="B" display="inline"><mi>B</mi></math> is also in <math id="p2.m6" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math>. We denote this <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">relation</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Relation.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/relationonobjects"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/relation"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> as <math id="p2.m7" class="ltx_Math" alttext="A\supseteq B" display="inline"><mrow><mi>A</mi><mo>⊇</mo><mi>B</mi></mrow></math>. This is <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">equivalent</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Equivalent.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/filterbasis"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/equivalenceofforcingnotions"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/equivalentmachines"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/equivalencerelation"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> to saying that <math id="p2.m8" class="ltx_Math" alttext="B" display="inline"><mi>B</mi></math> is a subset of <math id="p2.m9" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math>, that is <math id="p2.m10" class="ltx_Math" alttext="A\supseteq B\Leftrightarrow B\subseteq A" display="inline"><mrow><mrow><mi>A</mi><mo>⊇</mo><mi>B</mi></mrow><mo>⇔</mo><mrow><mi>B</mi><mo>⊆</mo><mi>A</mi></mrow></mrow></math>.</p> </div> <div id="p3" class="ltx_para"> <p class="ltx_p">Similar rules to those that hold for <math id="p3.m1" class="ltx_Math" alttext="\subseteq" display="inline"><mo>⊆</mo></math> also hold for <math id="p3.m2" class="ltx_Math" alttext="\supseteq" display="inline"><mo>⊇</mo></math>. If <math id="p3.m3" class="ltx_Math" alttext="X\supseteq Y" display="inline"><mrow><mi>X</mi><mo>⊇</mo><mi>Y</mi></mrow></math> and <math id="p3.m4" class="ltx_Math" alttext="Y\supseteq X" display="inline"><mrow><mi>Y</mi><mo>⊇</mo><mi>X</mi></mrow></math>, then <math id="p3.m5" class="ltx_Math" alttext="X=Y" display="inline"><mrow><mi>X</mi><mo>=</mo><mi>Y</mi></mrow></math>. Every set is a superset of itself, and every set is a superset of the <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">empty set</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/EmptySet.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/emptyset"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>.</p> </div> <div id="p4" class="ltx_para"> <p class="ltx_p">We say <math id="p4.m1" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math> is a <em class="ltx_emph ltx_font_italic"><a class="nnexus_concept" href="http://mathworld.wolfram.com/ProperSuperset.html">proper superset</a></em> of <math id="p4.m2" class="ltx_Math" alttext="B" display="inline"><mi>B</mi></math> if <math id="p4.m3" class="ltx_Math" alttext="A\supseteq B" display="inline"><mrow><mi>A</mi><mo>⊇</mo><mi>B</mi></mrow></math> and <math id="p4.m4" class="ltx_Math" alttext="A\neq B" display="inline"><mrow><mi>A</mi><mo>≠</mo><mi>B</mi></mrow></math>. This relation is sometimes denoted by <math id="p4.m5" class="ltx_Math" alttext="A\supset B" display="inline"><mrow><mi>A</mi><mo>⊃</mo><mi>B</mi></mrow></math>, but <math id="p4.m6" class="ltx_Math" alttext="A\supset B" display="inline"><mrow><mi>A</mi><mo>⊃</mo><mi>B</mi></mrow></math> is often used to mean the more general superset relation, so it should be made explicit when “proper superset” is intended, possibly by using <math id="p4.m7" class="ltx_Math" alttext="X\varsupsetneq Y" display="inline"><mrow><mi>X</mi><mo>⊋</mo><mi>Y</mi></mrow></math> or <math id="p4.m8" class="ltx_Math" alttext="X\supsetneqq Y" display="inline"><mrow><mi>X</mi><mo>⫌</mo><mi>Y</mi></mrow></math> (or <math id="p4.m9" class="ltx_Math" alttext="X\supsetneq Y" display="inline"><mrow><mi>X</mi><mo>⊋</mo><mi>Y</mi></mrow></math> or <math id="p4.m10" class="ltx_Math" alttext="X\varsupsetneqq Y" display="inline"><mrow><mi>X</mi><mo>⫌</mo><mi>Y</mi></mrow></math>).</p> </div> <div id="p5" class="ltx_para"> <p class="ltx_p">One will occasionally see a <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">collection</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Collection.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/collection"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> <math id="p5.m1" class="ltx_Math" alttext="C" display="inline"><mi>C</mi></math> of subsets of some set <math id="p5.m2" class="ltx_Math" alttext="X" display="inline"><mi>X</mi></math> made into a <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">partial order</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/PartialOrder.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/partialorder"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> “by containment”. Depending on <a class="nnexus_concept" href="http://planetmath.org/conceptlattice">context</a> this can mean defining a partial order where <math id="p5.m3" class="ltx_Math" alttext="Y\leq Z" display="inline"><mrow><mi>Y</mi><mo>≤</mo><mi>Z</mi></mrow></math> means <math id="p5.m4" class="ltx_Math" alttext="Y\subseteq Z" display="inline"><mrow><mi>Y</mi><mo>⊆</mo><mi>Z</mi></mrow></math>, or it can mean defining the opposite partial order: <math id="p5.m5" class="ltx_Math" alttext="Y\leq Z" display="inline"><mrow><mi>Y</mi><mo>≤</mo><mi>Z</mi></mrow></math> means <math id="p5.m6" class="ltx_Math" alttext="Y\supseteq Z" display="inline"><mrow><mi>Y</mi><mo>⊇</mo><mi>Z</mi></mrow></math>. This is frequently used when applying Zorn’s lemma.</p> </div> <div id="p6" class="ltx_para"> <p class="ltx_p">One will also occasionally see a collection <math id="p6.m1" class="ltx_Math" alttext="C" display="inline"><mi>C</mi></math> of subsets of some set <math id="p6.m2" class="ltx_Math" alttext="X" display="inline"><mi>X</mi></math> made into a <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">category</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Category.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/category"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>, usually by defining a single abstract <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">morphism</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Morphism.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/structurehomomorphism"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> <math id="p6.m3" class="ltx_Math" alttext="Y\to Z" display="inline"><mrow><mi>Y</mi><mo>→</mo><mi>Z</mi></mrow></math> whenever <math id="p6.m4" class="ltx_Math" alttext="Y\subseteq Z" display="inline"><mrow><mi>Y</mi><mo>⊆</mo><mi>Z</mi></mrow></math> (this being a special case of the general method of treating <a class="nnexus_concept" href="http://planetmath.org/preorder">pre-orders</a> as categories). This allows a concise definition of <a class="nnexus_concept" href="http://planetmath.org/sheaf1">presheaves</a> and sheaves, and it is generalized when defining a site. </p> <table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle"> <tbody class="ltx_tbody"> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">Title</th> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">superset</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Canonical name</th> <td class="ltx_td ltx_align_left ltx_border_r">Superset</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Date of creation</th> <td class="ltx_td ltx_align_left ltx_border_r">2013-05-24 14:35:12</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Last modified on</th> <td class="ltx_td ltx_align_left ltx_border_r">2013-05-24 14:35:12</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Owner</th> <td class="ltx_td ltx_align_left ltx_border_r">yark (2760)</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Last modified by</th> <td class="ltx_td ltx_align_left ltx_border_r">unlord (1)</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Numerical id</th> <td class="ltx_td ltx_align_left ltx_border_r">13</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Author</th> <td class="ltx_td ltx_align_left ltx_border_r">yark (1)</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Entry type</th> <td class="ltx_td ltx_align_left ltx_border_r">Definition</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Classification</th> <td class="ltx_td ltx_align_left ltx_border_r">msc 03E99</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Related topic</th> <td class="ltx_td ltx_align_left ltx_border_r">Subset</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Related topic</th> <td class="ltx_td ltx_align_left ltx_border_r">SetTheory</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Defines</th> <td class="ltx_td ltx_align_left ltx_border_r">proper superset</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Defines</th> <td class="ltx_td ltx_align_left ltx_border_r">contains</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">Defines</th> <td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">contained</td> </tr> </tbody> </table> </div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Thu Feb 8 20:22:24 2018 by <a href="http://dlmf.nist.gov/LaTeXML/">LaTeXML <img src="data:image/png;base64,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" alt="[LOGO]"></a> </div></footer> </div> </body> </html>