Equivalence relation

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Definition

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

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

Terminology

• An equivalence class is the name given to the set of all things which are equivalent under a given equivalence relation.
  • Often denoted $[a]$ for all the things equivalent to $a$
    • This is not unique, if $b\sim a$ then we could write $[b]$ instead. (Equivalence classes are either equal or disjoint)
  • Defined as $[a]:=\{b\in X\ \vert\ b\sim a\}$
• If there are multiple equivalence relations at play, we often use different letters to distinguish them, eg $\sim_\alpha$ and $[\cdot]_\alpha$
• Sometimes different symbols are employed, for example $\cong$ 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 $R$ 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