Definition:Equivalence Class

Definition
Let $$\mathcal{R}$$ be an equivalence relation on $$S$$

Let $$x \in S$$.

Then 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 $$x$$.

This construction is justified by Relation Partitions a Set iff Equivalence.

Notation
The notation used to denote an equivalence class varies throughout the literature, but is usually some variant on the square bracket motif.

Other variants:


 * uses $$\overline x$$ for $$\left[\!\left[{x}\right]\!\right]_{\mathcal{R}}$$.


 * uses $$x / \mathcal{R}$$ for $$\left[\!\left[{x}\right]\!\right]_{\mathcal{R}}$$ (and compare the notation for quotient set).