Definition:Equivalence Class

Definition
Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be an equivalence relation on $S$.

Let $x \in S$.

Then the equivalence class of $x$ under $\mathcal R$ is the set:
 * $\left[\!\left[{x}\right]\!\right]_\mathcal R = \left\{{y \in S: \left({x, y}\right) \in \mathcal R}\right\}$

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 $x$.

Also known as
The equivalence class of $x$ under $\mathcal R$ can be stated more tersely as the $\mathcal R$-equivalence class of $x$, or just the $\mathcal R$-class of $x$.

The term equivalence set can also occasionally be found for equivalence class.

Some sources, for example, use the term equivalence block.

Also see

 * Definition:Residue Class for the concept as it applies to Definition:Congruence Modulo Integer.


 * Condition for Membership of Equivalence Class:
 * $y \in \left[\!\left[{x}\right]\!\right]_\mathcal R \iff \left({x, y}\right) \in \mathcal R$


 * Relation Partitions Set iff Equivalence which justifies the construction.