Definition:Equivalence Class

If $$\mathcal{R}$$ is an equivalence on $$S$$, and $$x \in S$$, the equivalence class of $$x$$ under $$\mathcal{R}$$, or the $$\mathcal{R}$$-equivalence class of $$x$$, or just the $$\mathcal{R}$$-class of $$x$$, is the set:

$$\left[\left[{x}\right]\right]_{\mathcal{R}} = \left\{{y \in S: \left({x, y}\right) \in \mathcal{R}}\right\}$$

Thus:

$$y \in \left[\left[{x}\right]\right]_{\mathcal{R}} \iff \left({x, y}\right) \in \mathcal{R}$$

If $$\mathcal{R}$$ is an equivalence on $$S$$, then each $$t \in S$$ that satisfies $$\left({x, t}\right) \in \mathcal{R}$$ (or $$\left({t, x}\right) \in \mathcal{R}$$) is called a $$\mathcal{R}$$-relative of $$x$$.

That is, the equivalence class of $$x$$ under $$\mathcal{R}$$ is the set of all $$\mathcal{R}$$-relatives of $$x$$.

Results
An element is in its own $$\mathcal{R}$$-class.

$$\forall x \in S: x \in \left[\left[{x}\right]\right]_{\mathcal{R}}$$

Proof:

$$\forall x \in S: \left({x, x}\right) \in \mathcal{R}$$ Equivalence relation is reflexive

$$\Longrightarrow x \in \left[\left[{x}\right]\right]_{\mathcal{R}}$$

The $$\mathcal{R}$$-class of an element is a subset of the set the element is in:

$$\forall x \in S: \left[\left[{x}\right]\right]_{\mathcal{R}} \subseteq S$$

Proof:

$$y \in \left[\left[{x}\right]\right]_{\mathcal{R}}$$

$$\Longrightarrow \left({x, y}\right) \in \mathcal{R}$$

$$\Longrightarrow x \in S \land y \in S$$ Definition of Relation

$$\Longrightarrow \left[\left[{x}\right]\right]_{\mathcal{R}} \subseteq S$$ Definition of a subset

No $$\mathcal{R}$$-class is empty.

Proof:

$$\forall \left[\left[{x}\right]\right]_{\mathcal{R}} \subseteq S: \left[\left[{x}\right]\right]_{\mathcal{R}} \ne \varnothing$$

$$\Longrightarrow \forall \left[\left[{x}\right]\right]_{\mathcal{R}} \subseteq S: \exists x \in S: x \in \left[\left[{x}\right]\right]_{\mathcal{R}}$$

$$\Longrightarrow \left[\left[{x}\right]\right]_{\mathcal{R}} \ne \varnothing$$ Definition of the Empty Set