Equivalence Class holds Equivalent Elements

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

Then:


 * $\tuple {x, y} \in \mathcal R \iff \eqclass x {\mathcal R} = \eqclass y {\mathcal R}$

Necessary Condition
First we prove that $\tuple {x, y} \in \mathcal R \implies \eqclass x {\mathcal R} = \eqclass y {\mathcal R}$.

Suppose:
 * $\tuple {x, y} \in \mathcal R: x, y \in S$

Then:

So:
 * $\eqclass x {\mathcal R} \subseteq \eqclass y {\mathcal R}$

Now:

... so we have shown that:
 * $\tuple {x, y} \in \mathcal R \implies \eqclass x {\mathcal R} = \eqclass y {\mathcal R}$.

Sufficient Condition
Next we prove that $\eqclass x {\mathcal R} = \eqclass y {\mathcal R} \implies \tuple {x, y} \in \mathcal R$.

By definition of set equality:
 * $\eqclass x {\mathcal R} = \eqclass y {\mathcal R}$

means:
 * $\paren {x \in \eqclass x {\mathcal R} \iff x \in \eqclass y {\mathcal R} }$

So by definition of equivalence class:
 * $\tuple {y, x} \in \mathcal R$

Hence by definition of equivalence relation: $\mathcal R$ is symmetric
 * $\tuple {x, y} \in \mathcal R$

So we have shown that
 * $\eqclass x {\mathcal R} = \eqclass y {\mathcal R} \implies \tuple {x, y} \in \mathcal R$

Thus, we have:

So by equivalence:
 * $\tuple {x, y} \in \mathcal R \iff \eqclass x {\mathcal R} = \eqclass y {\mathcal R}$