Definition (Equivalence relation)

An equivalence relation on a set $X$ is a binary relation $\sim$ on $X$ satisfying

1. Reflexivity: $$a\sim a, \forall a\in X$$
2. Symmetry: $$a\sim b\implies b\sim a, \forall a,b\in X$$
3. Transitivity: $$a\sim b \ \& \ b\sim c\implies a\sim c,\forall a,b,c\in X$$

Definition (Equivalence class)

The equivalence class (or reduced residue class) of an element $a$ is defined as $$[a]=\{ x\in X:a\sim x \}$$