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
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 17.6$: Equivalence classes