Difference between revisions of "Equivalence relation"

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==Definition==
 
==Definition==
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A [[relation]] {{M|\sim}} in {{M|X}}<ref group="Notes">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}}:
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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}}:
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* Reflexivity, {{M|x\sim x}}
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* Symmetricity, {{M|x\sim y}} implies {{M|y\sim x}}
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* Transitivity, {{M|x\sim y}} and {{M|y\sim z}} implies {{M|x\sim z}}.
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| [[Relation#Types_of_relation|Reflexive]]
 
| [[Relation#Types_of_relation|Reflexive]]
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| {{M|\forall x\in X[(x,x) \in \sim]}}. Often written {{M|\forall x\in X[x\sim x]}}.
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| {{M|\forall x\in X[(x,x) \in \sim]}}. Which we write {{M|\forall x\in X[x\sim x]}}.
 
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| [[Relation#Types_of_relation|Symmetric]]  
 
| [[Relation#Types_of_relation|Symmetric]]  
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| {{M|\forall x,y\in X[M|(x,y) \in \sim \implies (y,x) \in \sim]}}. Often written {{M|\forall x,y \in X[x\sim y \implies y\sim x]}}.
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| {{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]}}.
 
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| [[Relation#Types_of_relation|Transitive]]
 
| [[Relation#Types_of_relation|Transitive]]
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| {{M|\forall x,y,z\in X[((x,y) \in \sim \wedge (y,z) \in \sim) \implies (x,z) \in \sim]}}. Often written {{M|\forall x,y,z \in X [(x\sim y \wedge y\sim z) \implies x\sim z]}}.
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| {{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]}}.
 
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==Terminology==
 
==Terminology==
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*Sometimes, letters and other designations are used with symbols to distinguish between different equivalence relations, such as {{M|a \equiv_x b}}.
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* An [[equivalence class]] is the name given to the set of all things which are equivalent under a given equivalence relation.
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**For an {{M|x\in X}}, the [[equivalence class]] is written {{M|[x]}} or {{M|x_\sim}}. That is, {{M|\forall a\in X[a\in[x] \implies a\sim x]}}.
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** Often denoted {{M|[a]}} for all the things equivalent to {{M|a}}
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*** This is not unique, if {{M|b\sim a}} then we could write {{M|[b]}} instead. ([[Equivalence classes are either equal or disjoint]])
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** Defined as {{M|1=[a]:=\{b\in X\ \vert\ b\sim a\} }}
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* If there are multiple equivalence relations at play, we often use different letters to distinguish them, eg {{M|\sim_\alpha}} and {{M|[\cdot]_\alpha}}
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* 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==
 
==See Also==
 
*[[Relation]]
 
*[[Relation]]
 
*[[Equivalence class]]
 
*[[Equivalence class]]
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**[[An equivalence class partitions a set]].
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**[[The equivalence classes of an equivalence relation partitions a set]].
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* [[Canonical projection of an equivalence relation]]
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* {{link|Passing to the quotient|function}} - things are often factored through the [[canonical projection of an equivalence relation]]
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* [[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
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** [[Factoring a function through the projection of an equivalence relation induced by that function yields an injection]]
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*** [[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]]
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==Notes==
 
==Notes==
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<references group="Notes"/>
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<references group="Note"/>
 
==References==  
 
==References==  
 
<references/>  
 
<references/>  

Latest revision as of 15:18, 12 February 2019

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Definition

A relation, [ilmath]\sim[/ilmath], in [ilmath]X[/ilmath][Note 1] is an equivalence relation if it has the following properties[1]:

Name Definition
1 Reflexive [ilmath]\forall x\in X[(x,x) \in \sim][/ilmath]. Which we write [ilmath]\forall x\in X[x\sim x][/ilmath].
2 Symmetric [ilmath]\forall x,y\in X[(x,y) \in \sim \implies (y,x) \in \sim][/ilmath]. Which we write [ilmath]\forall x,y \in X[x\sim y \implies y\sim x][/ilmath].
3 Transitive [ilmath]\forall x,y,z\in X[((x,y) \in \sim \wedge (y,z) \in \sim) \implies (x,z) \in \sim][/ilmath]. Which we write [ilmath]\forall x,y,z \in X [(x\sim y \wedge y\sim z) \implies x\sim z][/ilmath].

Terminology

  • An equivalence class is the name given to the set of all things which are equivalent under a given equivalence relation.
    • Often denoted [ilmath][a][/ilmath] for all the things equivalent to [ilmath]a[/ilmath]
    • Defined as [ilmath][a]:=\{b\in X\ \vert\ b\sim a\}[/ilmath]
  • If there are multiple equivalence relations at play, we often use different letters to distinguish them, eg [ilmath]\sim_\alpha[/ilmath] and [ilmath][\cdot]_\alpha[/ilmath]
  • Sometimes different symbols are employed, for example [ilmath]\cong[/ilmath] denotes a topological homeomorphism (which is an equivalence relation on topological spaces)

See Also

Notes

  1. ↑ This terminology means [ilmath]\sim \subseteq X\times X[/ilmath], as described on the relation page.

References

  1. ↑ Set Theory - Thomas Jech - Third millennium edition, revised and expanded

v • d • e
 
Common relations in mathematics
Overview of the common relations encountered almost everywhere
Generic
Equivalence relation, Function (AKA: mapping)
Orderings


Old Page

An equivalence relation is a special kind of relation

Required properties

Given a relation [ilmath]R[/ilmath] in [ilmath]A[/ilmath] we require the following properties to define a relation (these are restated for convenience from the relation page)

Reflexive

A relation [ilmath]R[/ilmath] if for all [ilmath]a\in A[/ilmath] we have [ilmath]aRa[/ilmath]

Symmetric

A relation [ilmath]R[/ilmath] is symmetric if for all [ilmath]a,b\in A[/ilmath] we have [ilmath]aRb\implies bRa[/ilmath]

Transitive

A relation [ilmath]R[/ilmath] is transitive if for all [ilmath]a,b,c\in A[/ilmath] we have [ilmath]aRb\text{ and }bRc\implies aRc[/ilmath]

Definition

A relation [ilmath]R[/ilmath] is an equivalence relation if it is:

  • reflexive
  • symmetric
  • transitive
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