Equivalence Class of Element is Subset
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Theorem
Let $\RR$ be an equivalence relation on a set $S$.
The $\RR$-class of every element of $S$ is a subset of the set the element is in:
- $\forall x \in S: \eqclass x \RR \subseteq S$
Proof
\(\ds y\) | \(\in\) | \(\ds \eqclass x \RR\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tuple {x, y}\) | \(\in\) | \(\ds \RR\) | Definition of Equivalence Class | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds S \land y \in S\) | Definition of Relation | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \eqclass x \RR\) | \(\subseteq\) | \(\ds S\) | Definition of Subset |
$\blacksquare$