Equivalence class

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, when the elements of some <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> 
 
 
 
 S
 
 
 {\displaystyle S}
 
 have a notion of equivalence (formalized as an <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a>), then one may naturally split the set 
 
 
 
 S
 
 
 {\displaystyle S}
 
 into equivalence classes. These equivalence classes are constructed so that elements 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 b
 
 
 {\displaystyle b}
 
 belong to the same equivalence class <a href="/facts/If%2c_and_only_if/bYSxGJ66">if, and only if</a>, they are equivalent.
Formally, given a set 
 
 
 
 S
 
 
 {\displaystyle S}
 
 and an equivalence relation 
 
 
 
 ∼
 
 
 {\displaystyle \sim }
 
 on 
 
 
 
 S
 ,
 
 
 {\displaystyle S,}
 
 the equivalence class of an element 
 
 
 
 a
 
 
 {\displaystyle a}
 
 in 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is denoted 
 
 
 
 [
 a
 ]
 
 
 {\displaystyle [a]}
 
 or, equivalently, 
 
 
 
 [
 a
 
 ]
 
 ∼
 
 
 
 
 {\displaystyle [a]_{\sim }}
 
 to emphasize its equivalence relation 
 
 
 
 ∼
 
 
 {\displaystyle \sim }
 
, and is defined as the set of all elements in 
 
 
 
 S
 
 
 {\displaystyle S}
 
 with which 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is 
 
 
 
 ∼
 
 
 {\displaystyle \sim }
 
-related. The definition of equivalence relations implies that the equivalence classes form a <a href="/facts/Partition_of_a_set/VeP9g8CA">partition</a> of 
 
 
 
 S
 ,
 
 
 {\displaystyle S,}
 
 meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 by 
 
 
 
 ∼
 ,
 
 
 {\displaystyle \sim ,}
 
 and is denoted by 
 
 
 
 S
 
 /
 
 
 ∼
 
 .
 
 
 {\displaystyle S/{\sim }.}

When the set 
 
 
 
 S
 
 
 {\displaystyle S}
 
 has some structure (such as a <a href="/facts/Group_(mathematics)/IQTYqINt">group operation</a> or a <a href="/facts/Topological_space/v2ut7CbK">topology</a>) and the equivalence relation 
 
 
 
 ∼
 ,
 
 
 {\displaystyle \sim ,}
 
 is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include <a href="/facts/Quotient_space_(linear_algebra)/WmQ3lqdI">quotient spaces in linear algebra</a>, <a href="/facts/Quotient_space_(topology)/muE0qQVC">quotient spaces in topology</a>, <a href="/facts/Quotient_group/HlwpHtjb">quotient groups</a>, <a href="/facts/Homogeneous_space/YGebqcJA">homogeneous spaces</a>, <a href="/facts/Quotient_ring/X2WkXyEw">quotient rings</a>, <a href="/facts/Quotient_monoid/rgSdk2kt">quotient monoids</a>, and <a href="/facts/Quotient_category/y3CATQTt">quotient categories</a>.

Equivalence class open-in-new

Equivalence class