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).

Also see

 * Residue class for the concept as it applies to congruence (number theory).