Difference between revisions of "Equivalence relation"

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(Refactoring this page to be more in line with other pages on relations.)
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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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{| class="wikitable" border="1"
 
{| class="wikitable" border="1"
 
|-
 
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! 1
 
! 1
 
| [[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]}}.
 
|-
 
|-
 
! 2
 
! 2
 
| [[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[M|(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
 
! 3
 
| [[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 writen {{M|\forall x,y,z \in X [(x\sim y \wedge y\sim z) \implies x\sim z]}}.
 
|}
 
|}
 
==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]]
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**[[An equivalence class partitions a set]].
 
**[[An equivalence class partitions a set]].
 
==Notes==
 
==Notes==
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<references group="Notes"/>
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<references group="Note"/>
 
==References==  
 
==References==  
 
<references/>  
 
<references/>  

Revision as of 17:18, 18 March 2016

This page is currently being refactored (along with many others)
Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.

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[M[/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 writen [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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