Condition for Membership of Equivalence Class

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

$\blacksquare$


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