Equivalence Class is Unique

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Theorem

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


For each $x \in S$, the one and only one $\RR$-class to which $x$ belongs is $\eqclass x \RR$.


Proof

This follows directly from the Fundamental Theorem on Equivalence Relations: the set of $\RR$-classes forms a partition of $S$.

$\blacksquare$


Sources