Equivalence relation

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An equivalence relation is a special kind of relation

Required properties

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

Reflexive

A relation $R$ if for all $a\in A$ we have $aRa$

Symmetric

A relation $R$ is symmetric if for all $a,b\in A$ we have $aRb\implies bRa$

Transitive

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

Definition

A relation $R$ is an equivalence relation if it is:

• reflexive
• symmetric
• transitive