Difference between revisions of "Equivalence relation"

From Maths
Jump to: navigation, search
(Created page with "An equivalence relation is a special kind of relation ==Required properties== Given a relation {{M|R}} in {{M|A}} we require the following properties to define a...")
 
m (Fixing mathematical notation typo)
 
(9 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
{{Refactor notice|review=yes}}
 +
 +
__TOC__
 +
 +
==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]]
 
An equivalence relation is a special kind of [[Relation|relation]]
  

Latest revision as of 15:18, 12 February 2019

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.
This page is waiting for a final review, then this notice will be removed.

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
Retrieved from "https://www.maths.kisogo.com/index.php?title=Equivalence_relation&oldid=9412"