# Equivalence Relation

DEFINITION1: Let $X$ be a set and $R\subset{X\times{X}}$. If the relation $R$ is reflexive, symmetric and transitive, then the relation $R$ is called an "equivalence relation" and denoted by $R=\sim$ in general.

DEFINITION2: Let $\sim$ be an equivalence relation over a set $X$ and $a\in{X}$. The set defined as $\{x\in{X}\:|\:a\sim{x}\}\subset{X}$ is called the “equivalence class” of $a$ under $\sim$ and denoted by $\overline{a}$, $[a]$ or $[a]_{\sim}$. Since $a\sim{a}$ for every $a\in{X}$, then $a\in{\overline{a}}$. So the equivalence class $\overline{a}$ is non-empty for every $a\in{X}$. The family of all the equivalence classes of the relation $\sim$ is called the “quotient set” of $X$ by $\sim$ and denoted by ${^X}/{_\sim}$

I.e.,

${^X}/{_\sim}=\{\overline{a}\:|\:a\in{X}\}\subset{\mathbf{P}(X)}$.

DEFINITION3: Let $\sim$ be an equivalence relation over a set $X$ and $a,b\in{X}$. If $b\in{\overline{a}}$, then $b$ is called a “representative class” of the equivalence class $\overline{a}$.