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Then there exists a set <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'>S/{\sim}</annotation></semantics></math>, the <strong>quotient set</strong> of <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> modulo <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo></mrow><annotation encoding='application/x-tex'>\sim</annotation></semantics></math>. Given any element <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_7' 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_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, there is an element <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>x</mi><msub><mo stretchy='false'>]</mo> <mo>∼</mo></msub></mrow><annotation encoding='application/x-tex'>[x]_{\sim}</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'>S/{\sim}</annotation></semantics></math>, the <strong>equivalence class</strong> of <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> modulo <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo></mrow><annotation encoding='application/x-tex'>{\sim}</annotation></semantics></math>. Every element of <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'>S/{\sim}</annotation></semantics></math> is of this form. Furthermore, <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>x</mi><msub><mo stretchy='false'>]</mo> <mo>∼</mo></msub></mrow><annotation encoding='application/x-tex'>[x]_{\sim}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>y</mi><msub><mo stretchy='false'>]</mo> <mo>∼</mo></msub></mrow><annotation encoding='application/x-tex'>[y]_{\sim}</annotation></semantics></math> are equal in <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'>S/{\sim}</annotation></semantics></math> if and only if <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∼</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x \sim y</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>.</p> <p>The <a class='existingWikiWord' href='/nlab/show/quotient+set'>axiom of quotients</a> is an axiom of <a class='existingWikiWord' href='/nlab/show/set+theory'>set theory</a> which states that the paragraph above is true. It corresponds to the clause in the definition of a <a class='existingWikiWord' href='/nlab/show/pretopos'>pretopos</a> (or in <a class='existingWikiWord' href='/nlab/show/Grothendieck+topos'>Giraud's axioms</a> for a <a class='existingWikiWord' href='/nlab/show/Grothendieck+topos'>Grothendieck topos</a>) that every <a class='existingWikiWord' href='/nlab/show/congruence'>congruence</a> has a <a class='existingWikiWord' href='/nlab/show/coequalizer'>coequaliser</a>. In most formulations of set theory, this axiom is not needed; instead, it is a theorem when equivalence classes are defined in one of the ways below.</p> <h3 id='as_subsets'>As subsets</h3> <p>Again, let <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> be a set, and let <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo></mrow><annotation encoding='application/x-tex'>{\sim}</annotation></semantics></math> be an equivalence relation on <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>. Let <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> be an element of <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>. Then the <strong>equivalence class</strong> of <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> modulo <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo></mrow><annotation encoding='application/x-tex'>{\sim}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/subset'>subset</a> of <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> consisting of those elements of <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> that are equivalent to <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>x</mi><msub><mo stretchy='false'>]</mo> <mo>∼</mo></msub><mo>≔</mo><mo stretchy='false'>{</mo><mi>y</mi><mo lspace='verythinmathspace'>:</mo><mi>S</mi><mspace width='thickmathspace'></mspace><mo stretchy='false'>|</mo><mspace width='thickmathspace'></mspace><mi>x</mi><mo>∼</mo><mi>y</mi><mo stretchy='false'>}</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> [x]_{\sim} \coloneqq \{ y\colon S \;|\; x \sim y \} .</annotation></semantics></math></div> <p>Then the <strong>quotient set</strong> <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'>S/{\sim}</annotation></semantics></math> is the collection of these equivalence classes.</p> <p>We may construct this collection using the <a class='existingWikiWord' href='/nlab/show/power+set'>power set</a> of <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>; therefore, this may be done in any <a class='existingWikiWord' href='/nlab/show/topos'>elementary topos</a> as well as in such diverse set theories as <a class='existingWikiWord' href='/nlab/show/ZFC'>ZFC</a>, <a class='existingWikiWord' href='/nlab/show/SEAR'>SEAR</a>, and <a class='existingWikiWord' href='/nlab/show/ETCS'>ETCS</a>. This definition of equivalence class is quite natural in <a class='existingWikiWord' href='/nlab/show/material+set+theory'>material set theory</a>, since it immediately produces a set (assuming that subsets are sets).</p> <p>Any element <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_32' 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_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is a <strong>representative</strong> of its equivalence class <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>x</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[x]</annotation></semantics></math>. Every equivalence class has at least one representative, and its representatives are all equivalent. The set of representatives <em>is</em> the equivalence class in the material set-theoretic sense.</p> <p>One usually defines properties of equivalence classes and functions on quotient sets by defining them for an arbitrary representative, then proving that the result is independent of the representative chosen. This does <em>not</em> require the <a class='existingWikiWord' href='/nlab/show/axiom+of+choice'>axiom of choice</a>.</p> <h3 id='redefined_equality'>Redefined equality</h3> <p>In some <a class='existingWikiWord' href='/nlab/show/foundation+of+mathematics'>foundations of mathematics</a>, sets are not fundamental, but are defined as more basic <a class='existingWikiWord' href='/nlab/show/preset'>presets</a> (sometimes called <a class='existingWikiWord' href='/nlab/show/type'>types</a> or, confusingly, sets). By definition, a set (sometimes called a <a class='existingWikiWord' href='/nlab/show/setoid'>setoid</a>) is a preset equipped with an equivalence (pre)relation.</p> <p>Once more, let <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> be a set, and let <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo></mrow><annotation encoding='application/x-tex'>{\sim}</annotation></semantics></math> be an equivalence relation on <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>. Then the quotient set <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'>S/{\sim}</annotation></semantics></math> is the the underlying preset of <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> equipped with <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo></mrow><annotation encoding='application/x-tex'>\sim</annotation></semantics></math> (in place of the original equality on <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>), and the equivalence class <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>x</mi><msub><mo stretchy='false'>]</mo> <mo>∼</mo></msub></mrow><annotation encoding='application/x-tex'>[x]_{\sim}</annotation></semantics></math> is simply <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>.</p> <h3 id='as_objects_of_a_groupoid'>As objects of a groupoid</h3> <p>One can also consider sets as <a class='existingWikiWord' href='/nlab/show/groupoid'>groupoids</a> (or <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/infinity-groupoid'>groupoids</a>) with the property of being <a class='existingWikiWord' href='/nlab/show/discrete+category'>discrete</a>.</p> <p>So again, let <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> be a set, and let <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo></mrow><annotation encoding='application/x-tex'>{\sim}</annotation></semantics></math> be an equivalence relation on <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>. Then the quotient set <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'>S/{\sim}</annotation></semantics></math> is the (higher) groupoid whose objects are the same as those of <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> and with a single morphism from <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> iff <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∼</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x \sim y</annotation></semantics></math> (and none otherwise); <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>x</mi><msub><mo stretchy='false'>]</mo> <mo>∼</mo></msub></mrow><annotation encoding='application/x-tex'>[x]_{\sim}</annotation></semantics></math> is simply <math class='maruku-mathml' display='inline' id='mathml_2eddda15c5875a54087cd9b1fa42ef4b65b72fac_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> again.</p> <h2 id='examples'>Examples</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/isomorphism+class'>isomorphism class</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy+class'>homotopy class</a></p> </li> </ul> <p> </p> </div>  <div class="revisedby"> <p> Revision on May 20, 2022 at 08:43:22 by <a href="/nlab/author/Urs+Schreiber" style="color: #005c19">Urs Schreiber</a> See the <a href="/nlab/history/equivalence+class" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/14283/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/show/equivalence+class" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (to current)</span><span class="backintime"><a href="/nlab/revision/equivalence+class/2" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (2 more)</span><a href="/nlab/show/equivalence+class" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/equivalence+class/3" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/equivalence+class" accesskey="S" class="navlink" id="history" rel="nofollow">History (3 revisions)</a><a href="/nlab/rollback/equivalence+class?rev=3" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/equivalence+class/3/cite" style="color: black">Cite</a> <a href="/nlab/source/equivalence+class/3" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>