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:
 * $\left[\!\left[{x}\right]\!\right]_\mathcal R = \left\{{y \in S: \left({x, y}\right) \in \mathcal R}\right\}$

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 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 $\pi \left({x}\right)$ for $\left[\!\left[{x}\right]\!\right]_\mathcal R$.


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


 * uses $\bigsqcup_{\mathcal R} \mkern {-28 mu} {\raise 1pt x} \ \ $ for $\left[\!\left[{x}\right]\!\right]_\mathcal R$.


 * uses $p_{\mathcal R} \left({x}\right)$.


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

Also known as
The term equivalence set can also occasionally be found.

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

Also see

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


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