Equivalence relation

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Definition

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

	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]
    • This is not unique, if [ilmath]b\sim a[/ilmath] then we could write [ilmath][b][/ilmath] instead. (Equivalence classes are either equal or disjoint)
  • 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

• Relation
• Equivalence class
  • The equivalence classes of an equivalence relation partitions a set.
• Canonical projection of an equivalence relation
• Passing to the quotient - 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

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