View source for Equivalence relation

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==Definition==
A [[relation]], {{M|\sim}}, in {{M|X}}<ref group="Note">This terminology means {{M|\sim \subseteq X\times X}}, as described on the [[relation]] page.</ref> is an ''equivalence relation'' if it has the following properties{{rSTTJ}}:
{| class="wikitable" border="1"
|-
!
! Name
! Definition
|-
! 1
| [[Relation#Types_of_relation|Reflexive]]
| {{M|\forall x\in X[(x,x) \in \sim]}}. Which we write {{M|\forall x\in X[x\sim x]}}.
|-
! 2
| [[Relation#Types_of_relation|Symmetric]] 
| {{M|\forall x,y\in X[(x,y) \in \sim \implies (y,x) \in \sim]}}. Which we write {{M|\forall x,y \in X[x\sim y \implies y\sim x]}}.
|-
! 3
| [[Relation#Types_of_relation|Transitive]]
| {{M|\forall x,y,z\in X[((x,y) \in \sim \wedge (y,z) \in \sim) \implies (x,z) \in \sim]}}. Which we write {{M|\forall x,y,z \in X [(x\sim y \wedge y\sim z) \implies x\sim z]}}.
|}

==Terminology==
* An [[equivalence class]] is the name given to the set of all things which are equivalent under a given equivalence relation.
** Often denoted {{M|[a]}} for all the things equivalent to {{M|a}}
*** This is not unique, if {{M|b\sim a}} then we could write {{M|[b]}} instead. ([[Equivalence classes are either equal or disjoint]])
** Defined as {{M|1=[a]:=\{b\in X\ \vert\ b\sim a\} }}
* If there are multiple equivalence relations at play, we often use different letters to distinguish them, eg {{M|\sim_\alpha}} and {{M|[\cdot]_\alpha}}
* Sometimes different symbols are employed, for example {{M|\cong}} denotes a [[topology (subject)|topological]] ''[[homeomorphism]]'' (which is an equivalence relation on [[topological space|topological spaces]])
==See Also==
*[[Relation]]
*[[Equivalence class]]
**[[The equivalence classes of an equivalence relation partitions a set]].
* [[Canonical projection of an equivalence relation]]
* {{link|Passing to the quotient|function}} - things are often factored through the [[canonical projection of an equivalence relation]]
* [[Equivalence relation induced by a map]] - while many things induce equivalence relations, the "purity" of any [[function]] doing so means this ought to be here
** [[Factoring a function through the projection of an equivalence relation induced by that function yields an injection]]
*** [[If a surjective function is factored through the canonical projection of the equivalence relation induced by that function then the yielded function is a bijection]]

==Notes==
<references group="Note"/>
==References== 
<references/> 
{{Relations navbox|plain}} 
{{Definition|Set Theory|Abstract Algebra}} 



=Old Page=

An equivalence relation is a special kind of [[Relation|relation]]

==Required properties==
Given a relation {{M|R}} in {{M|A}} we require the following properties to define a relation (these are restated for convenience from the [[Relation|relation]] page)
===Reflexive===
A relation {{M|R}} if for all {{M|a\in A}} we have {{M|aRa}}
===Symmetric===
A relation {{M|R}} is symmetric if for all {{M|a,b\in A}} we have {{M|aRb\implies bRa}}
===Transitive===
A relation {{M|R}} is transitive if for all {{M|a,b,c\in A}} we have {{M|aRb\text{ and }bRc\implies aRc}}

==Definition==
A relation {{M|R}} is an equivalence relation if it is:
* reflexive
* symmetric
* transitive


{{Definition|Set Theory}}