Equivalence Relation

Definition: Equivalence Relation

An equivalence relation on the set $A$ is any relation $R\subseteq A\times A$ which is reflexive, transitive and symmetric.

NOTATION

Equivalence relations are usually denoted with $\sim$ instead of $R$.

INTUITION

The statement $x\sim y$ means that $x$ is equal to $y$ in the sense of $\sim$.

Definition: Equivalence Class

Let $A$ be a set with an equivalence relation $\sim$.

The equivalence class of an element $a\in A$ formed by $\sim$ is the set of all $x\in A$ which are equivalent to $a$.

$$\{x\in A\mid a\sim x\}$$

NOTATION

$$[a{]}_{\sim }$$