Editing Equivalence class (section)

{{Short description|Mathematical concept}}
{{About|equivalency in mathematics|equivalency in music|equivalence class (music)}}
{{Redirect|Quotient map|Quotient map in topology|Quotient map (topology)}}

[[File:Congruent non-congruent triangles.svg|thumb|370px|[[Congruence (geometry)|Congruence]] is an example of an equivalence relation. The leftmost two triangles are congruent, while the third and fourth triangles are not congruent to any other triangle shown here. Thus, the first two triangles are in the same equivalence class, while the third and fourth triangles are each in their own equivalence class.]]

In [[mathematics]], when the elements of some [[Set (mathematics)|set]] <math>S</math> have a notion of equivalence (formalized as an [[equivalence relation]]), then one may naturally split the set <math>S</math> into '''equivalence classes'''. These equivalence classes are constructed so that elements <math>a</math> and <math>b</math> belong to the same '''equivalence class''' [[if, and only if]], they are equivalent.

Formally, given a set <math>S</math> and an equivalence relation <math>\sim</math> on <math>S,</math> the {{em|equivalence class}} of an element <math>a</math> in <math>S</math> is denoted <math>[a]</math> or, equivalently, <math>[a]_{\sim}</math> to emphasize its equivalence relation <math>\sim</math>, and is defined as the set of all elements in <math>S</math> with which <math>a</math> is <math>\sim</math>-related. The definition of equivalence relations implies that the equivalence classes form a [[Partition of a set|partition]] of <math>S,
</math>  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 <math>S</math> by <math>\sim,</math> and is denoted by <math>S /{\sim}.</math>

When the set <math>S</math> has some structure (such as a [[group (mathematics)|group operation]] or a [[topological space|topology]]) and the equivalence relation <math>\sim,</math> is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include [[quotient space (linear algebra)|quotient spaces in linear algebra]], [[quotient space (topology)|quotient spaces in topology]], [[quotient group]]s, [[homogeneous space]]s, [[quotient ring]]s, [[quotient monoid]]s, and [[quotient category|quotient categories]].