Equivalence relation

From Maths
Revision as of 22:20, 8 October 2016 by Alec (Talk | contribs) (See Also: Linking out to new pages)

Jump to: navigation, search
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[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 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


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