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width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/9019/#Item_5" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #16 to #17: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <h1 id='subsets'>Subsets</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definitions'>Definitions</a><ul><li><a href='#in_material_set_theory'>In material set theory</a></li><li><a href='#in_structural_set_theory'>In structural set theory</a></li><li><a href='#internal_subsets'>Internal subsets</a></li></ul></li><li><a href='#remarks'>Remarks</a></li><li><a href='#related_concepts'>Related concepts</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>A subset of a given <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is a set <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> that may be viewed as contained within <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p> <h2 id='definitions'>Definitions</h2> <h3 id='in_material_set_theory'>In material set theory</h3> <p>In <a class='existingWikiWord' href='/nlab/show/diff/material+set+theory'>material set theory</a>, a <strong>subset</strong> of a <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is a set <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> with the <em>property</em> that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>B</mi><mspace width='thickmathspace' /><mo>⇒</mo><mspace width='thickmathspace' /><mi>x</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'> x \in B \;\Rightarrow\; x \in A </annotation></semantics></math></div> <p>for any object <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> whatsoever. One writes <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>⊆</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>B \subseteq A</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>B \subset A</annotation></semantics></math> (depending on the author) if <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> has this property. Set theory's <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+extensionality'>axiom of extensionality</a> says that <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>A = B</annotation></semantics></math> if (and only if) <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊆</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>A \subseteq B</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>⊆</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>B \subseteq A</annotation></semantics></math> (although this is only strong enough for <span class='newWikiWord'><span><ins class='diffins'> </ins> well-founded<del class='diffmod'> set</del><ins class='diffmod'> sets</ins></span><a href='/nlab/new/well-founded+set'>?</a></span><span><del class='diffmod'> s).</del><ins class='diffmod'> ).</ins></span></p> <h3 id='in_structural_set_theory'>In structural set theory</h3> <p>In <a class='existingWikiWord' href='/nlab/show/diff/structural+set+theory'>structural set theory</a>, this definition doesn't make sense, because there is no global membership predicate <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∈</mo></mrow><annotation encoding='application/x-tex'>\in</annotation></semantics></math> (and there may not be a global <a class='existingWikiWord' href='/nlab/show/diff/equality'>equality predicate</a> either). Accordingly, we define a <strong>subset</strong> of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> to be a set <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> with the <em>structure</em> of an <a class='existingWikiWord' href='/nlab/show/diff/injection'>injection</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>B</mi><mo>↪</mo><mi>A</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> i\colon B \hookrightarrow A .</annotation></semantics></math></div> <p>We can still define a <em>local</em> membership predicate <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∈</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>\in_A</annotation></semantics></math> as follows: Given an element <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and a subset <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> (technically, <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>B</mi><mo>,</mo><mi>i</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(B,i)</annotation></semantics></math>) of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>,</p> <div class='maruku-equation' id='eq:indef'><span class='maruku-eq-number'>(1)</span><math class='maruku-mathml' display='block' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><msub><mo>∈</mo> <mi>A</mi></msub><mi>B</mi><mspace width='thickmathspace' /><mo>⇔</mo><mspace width='thickmathspace' /><mo>∃</mo><mo stretchy='false'>(</mo><mi>y</mi><mo lspace='verythinmathspace'>:</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thickmathspace' /><mi>x</mi><mo>=</mo><mi>i</mi><mo stretchy='false'>(</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> x \in_A B \;\Leftrightarrow\; \exists(y\colon B),\; x = i(y) .</annotation></semantics></math></div> <p>Similarly, we can define a local containment predicate <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>⊆</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>\subseteq_A</annotation></semantics></math> as follows: Given subsets <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>,</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><msub><mo>⊆</mo> <mi>A</mi></msub><mi>C</mi><mspace width='thickmathspace' /><mo>⇔</mo><mspace width='thickmathspace' /><mo>∀</mo><mo stretchy='false'>(</mo><mi>x</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thickmathspace' /><mi>x</mi><mo>∈</mo><mi>B</mi><mspace width='thickmathspace' /><mo>⇒</mo><mspace width='thickmathspace' /><mi>x</mi><mo>∈</mo><mi>C</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> B \subseteq_A C \;\Leftrightarrow\; \forall(x\colon A),\; x \in B \;\Rightarrow\; x \in C .</annotation></semantics></math></div> <p>We can also define a local equality predicate <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>=</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>=_A</annotation></semantics></math> on subsets of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><msub><mo>=</mo> <mi>A</mi></msub><mi>C</mi><mspace width='thickmathspace' /><mo>⇔</mo><mspace width='thickmathspace' /><mi>B</mi><mo>⊆</mo><mi>C</mi><mspace width='thickmathspace' /><mo>∧</mo><mspace width='thickmathspace' /><mi>C</mi><mo>⊆</mo><mi>B</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> B =_A C \;\Leftrightarrow\; B \subseteq C \;\wedge\; C \subseteq B .</annotation></semantics></math></div> <p>In foundations that already have a global equality predicate on sets (and functions between equal sets), this local equality predicate must be interpreted as an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a>; then a <strong>subset</strong> of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is technically an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+class'>equivalence class</a> of injections to <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> rather than simply an injection to <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p> <p>In any case, if <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is a subset of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> is a <strong>superset</strong> of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p> <h3 id='internal_subsets'>Internal subsets</h3> <p>One could define the subset relation as a <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> using the <a class='existingWikiWord' href='/nlab/show/diff/internal+logic'>internal logic</a> of a <a class='existingWikiWord' href='/nlab/show/diff/set+theory'>set theory</a>. The inclusion <a class='existingWikiWord' href='/nlab/show/diff/relation'>relation</a> between two sets <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> is defined as the <a class='existingWikiWord' href='/nlab/show/diff/support+of+a+set'>support</a> of the <a class='existingWikiWord' href='/nlab/show/diff/injection+set'>injection set</a> between <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊆</mo><mi>B</mi><mo>≔</mo><mrow><mo>[</mo><mi mathvariant='normal'>Inj</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>]</mo></mrow></mrow><annotation encoding='application/x-tex'>A \subseteq B \coloneqq \left[\mathrm{Inj}(A, B)\right]</annotation></semantics></math></div> <p>A <strong>internal subset</strong> of a <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> is a set <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> with an element <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>∈</mo><mi>A</mi><mo>⊆</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p \in A \subseteq B</annotation></semantics></math>. A <strong>internal superset</strong> of a <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is a set <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> with an element <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>∈</mo><mi>A</mi><mo>⊆</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p \in A \subseteq B</annotation></semantics></math>.</p> <h2 id='remarks'>Remarks</h2> <p>As you can see from the right-hand sides of the above sequence of definitions, one usually suppresses the subscript <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> from the notation. Even the right-hand side of <a class='maruku-eqref' href='#eq:indef'>(1)</a> may use a local equality relation on elements of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>. It may be necessary to distinguish <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo lspace='verythinmathspace'>:</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'>x\colon E</annotation></semantics></math> (the <em><a class='existingWikiWord' href='/nlab/show/diff/type'>typing declaration</a></em> introducing a variable <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> for an element of a given set <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>) from <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><msub><mo>∈</mo> <mi>A</mi></msub><mi>E</mi></mrow><annotation encoding='application/x-tex'>x \in_A E</annotation></semantics></math> (the <em><a class='existingWikiWord' href='/nlab/show/diff/proposition'>proposition</a></em> that a given element <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> of a given set <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is a member of a given subset <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>). Some authors may use <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>x \in A</annotation></semantics></math> for either or both of these, trusting on context (particularly whether <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> has been introduced before) to distinguish them. Another notational convenience is to suppress the injection <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>, writing <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> instead of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo stretchy='false'>(</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>i(y)</annotation></semantics></math>.</p> <p>The concept of subset as it appears here generalises to <a class='existingWikiWord' href='/nlab/show/diff/subobject'>subobject</a> in category theory. To be precise, a subset of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is exactly a subobject of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> when <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is thought of as an object of the category <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>. The concept of superset then generalises to a notion of <a class='existingWikiWord' href='/nlab/show/diff/extension'>extension</a> analogous to that of <a class='existingWikiWord' href='/nlab/show/diff/field+extension'>field extension</a>.</p> <p>When the abstract set <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is fixed, a subset <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_611b694eea030f0c42d064c5352ea79aca9956ae_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> may be thought of as a <strong><a class='existingWikiWord' href='/nlab/show/diff/concrete+structure'>concrete</a> set</strong>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subtype'>subtype</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cofinite+subset'>cofinite subset</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/saturated+subset'>saturated subset</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/causally+closed+subset'>causally closed subset</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/space+of+finite+subsets'>space of finite subsets</a></p> </li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on November 15, 2023 at 04:23:43. See the <a href="/nlab/history/subset" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/subset" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/9019/#Item_5">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/subset/16" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/subset" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/subset" accesskey="S" class="navlink" id="history" rel="nofollow">History (16 revisions)</a> <a href="/nlab/show/subset/cite" style="color: black">Cite</a> <a href="/nlab/print/subset" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/subset" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>