Equivalence Class

Definition

$[a]=E_a:=\{s\in S:x\sim a\}$ where $S$ is a set, $\sim$ is a Equivalence Relation on $S$, and $a\in S$

A set of elements that are related to each other by an Equivalence Relation

Facts

Let $X\ne \varnothing$ and $E$ be Equivalence Relation on $X$

• $E_x\ne \varnothing$
• $E_x=E_y\iff xEy$
• $E_x\cap E_y\ne \varnothing \iff xEy$