# Element in its own Equivalence Class

## 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

$\blacksquare$