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Then there exists a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math>. Given any element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, there is an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">{\sim}</annotation></semantics></math>. Every element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/axiom+of+quotients">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/Giraud%27s+axioms">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/coequaliser">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> <p>In <a class="existingWikiWord" href="/nlab/show/intensional+type+theory">intensional type theory</a> such as <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, <a class="existingWikiWord" href="/nlab/show/quotient+sets">quotient sets</a> could be constructed as a <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a>, and thus an equivalence class is an element of that higher inductive type.</p> <h3 id="as_subsets">As subsets</h3> <p>Again, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> be a set, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">{\sim}</annotation></semantics></math> be an equivalence relation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> be an element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. Then the <strong>equivalence class</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> consisting of those elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> that are equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/elementary+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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/foundations+of+mathematics">foundations of mathematics</a>, sets are not fundamental, but are defined as more basic <a class="existingWikiWord" href="/nlab/show/presets">presets</a> (sometimes called <a class="existingWikiWord" href="/nlab/show/types">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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> be a set, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">{\sim}</annotation></semantics></math> be an equivalence relation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. Then the quotient set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> equipped with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> (in place of the original equality on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>), and the equivalence class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/groupoids">groupoids</a> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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+groupoid">discrete</a>.</p> <p>So again, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> be a set, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">{\sim}</annotation></semantics></math> be an equivalence relation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. Then the quotient set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> and with a single morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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> </body></html> </div> <div class="revisedby"> <p> Last revised on May 20, 2022 at 13:31:52. 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