# Equivalence Class holds Equivalent Elements

## Theorem

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

Then:

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

## Proof

### 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:

 $\displaystyle z$ $\in$ $\displaystyle \eqclass x {\mathcal R}$ $\displaystyle \leadsto \ \$ $\displaystyle \tuple {x, z}$ $\in$ $\displaystyle \mathcal R$ Definition of Equivalence Class $\displaystyle \leadsto \ \$ $\displaystyle \tuple {z, x}$ $\in$ $\displaystyle \mathcal R$ Definition of Equivalence Relation: $\mathcal R$ is symmetric $\displaystyle \leadsto \ \$ $\displaystyle \tuple {z, y}$ $\in$ $\displaystyle \mathcal R$ Definition of Equivalence Relation: $\mathcal R$ is transitive $\displaystyle \leadsto \ \$ $\displaystyle \tuple {y, z}$ $\in$ $\displaystyle \mathcal R$ Definition of Equivalence Relation: $\mathcal R$ is symmetric $\displaystyle \leadsto \ \$ $\displaystyle z$ $\in$ $\displaystyle \eqclass y {\mathcal R}$ Definition of Equivalence Class

So:

$\eqclass x {\mathcal R} \subseteq \eqclass y {\mathcal R}$

Now:

 $\displaystyle \tuple {x, y} \in \mathcal R$ $\implies$ $\displaystyle \eqclass x {\mathcal R} \subseteq \eqclass y {\mathcal R}$ (see above) $\displaystyle \tuple {x, y} \in \mathcal R$ $\implies$ $\displaystyle \tuple {y, x} \in \mathcal R$ Definition of Equivalence Relation: $\mathcal R$ is symmetric $\displaystyle$ $\leadsto$ $\displaystyle \eqclass y {\mathcal R} \subseteq \eqclass x {\mathcal R}$ from above $\displaystyle$ $\leadsto$ $\displaystyle \eqclass y {\mathcal R} = \eqclass x {\mathcal R}$ Definition of Set Equality

... so we have shown that:

$\tuple {x, y} \in \mathcal R \implies \eqclass x {\mathcal R} = \eqclass y {\mathcal R}$.

$\Box$

### 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:

 $\displaystyle \tuple {x, y} \in \mathcal R$ $\implies$ $\displaystyle \eqclass x {\mathcal R} = \eqclass y {\mathcal R}$ $\displaystyle \eqclass x {\mathcal R} = \eqclass y {\mathcal R}$ $\implies$ $\displaystyle \tuple {x, y} \in \mathcal R$

$\Box$

So by equivalence:

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

$\blacksquare$