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 $$x$$.