Condition for Membership of Equivalence Class

Theorem

Let $\RR$ be an equivalence relation on a set $S$.

Let $\eqclass x \RR$ denote the $\RR$-equivalence class of $x$.

Then:

$\forall y \in S: y \in \eqclass x \RR \iff \tuple {x, y} \in \RR$

Proof

From the definition of an equivalence class:

$\eqclass x \RR = \set {y \in S: \tuple {x, y} \in \RR}$

Let $y \in S$ such that $y \in \eqclass x \RR$.

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

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

Again by definition, $y \in \eqclass x \RR$.

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Theorem $6.3 \ (1)$