Equivalence relation

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Definition

A relation $\sim$ in $X$ is an equivalence relation if it has the following properties:

• Reflexivity, $x\sim x$
• Symmetricity, $x\sim y$ implies $y\sim x$
• Transitivity, $x\sim y$ and $y\sim z$ implies $x\sim z$.

	Name	Definition
1	Reflexive	$\forall x\in X[(x,x) \in \sim]$. Often written $\forall x\in X[x\sim x]$.
2	Symmetric	$\forall x,y\in X[M$. Often written $\forall x,y \in X[x\sim y \implies y\sim x]$.
3	Transitive	$\forall x,y,z\in X[((x,y) \in \sim \wedge (y,z) \in \sim) \implies (x,z) \in \sim]$. Often written $\forall x,y,z \in X [(x\sim y \wedge y\sim z) \implies x\sim z]$.

Terminology

• Sometimes, letters and other designations are used with symbols to distinguish between different equivalence relations, such as $a \equiv_x b$.
  • For an $x\in X$, the equivalence class is written $[x]$ or $x_\sim$. That is, $\forall a\in X[a\in[x] \implies a\sim x]$.

See Also

• Relation
• Equivalence class
  • An equivalence class partitions a set.

Old Page

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