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 often some variant on the square bracket motif.

Other variants:


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


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


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