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 $6.3 \ (1)$
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