Condition for Membership of Equivalence Class

Theorem
Let $\mathcal R$ be an equivalence relation on a set $S$.

Let $\eqclass x {\mathcal R}$ denote the $\mathcal R$-equivalence class of $x$.

Then:
 * $\forall y \in S: y \in \eqclass x {\mathcal R} \iff \tuple {x, y} \in \mathcal R$

Proof
From the definition of an equivalence class:


 * $\eqclass x {\mathcal R} = \set {y \in S: \tuple {x, y} \in \mathcal R}$

Let $y \in S$ such that $y \in \eqclass x {\mathcal R}$.

Then by definition $\tuple {x, y} \in \mathcal R$.

Similarly, let $\tuple {x, y} \in \mathcal R$.

Again by definition, $y \in \eqclass x {\mathcal R}$.