Element in its own Equivalence Class

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Theorem

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

Then every element of $S$ is in its own $\mathcal R$-class:

$\forall x \in S: x \in \eqclass x {\mathcal R}$


Proof

\(\displaystyle \forall x \in S: \tuple {x, x}\) \(\in\) \(\displaystyle \mathcal R\) Definition of Equivalence Relation: $\mathcal R$ is Reflexive
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \eqclass x {\mathcal R}\) Definition of Equivalence Class

$\blacksquare$


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