Definition:Equivalence Class

Definition
Let $S$ be a set.

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

Let $x \in S$.

Then the equivalence class of $x$ under $\RR$ is the set:
 * $\eqclass x \RR = \set {y \in S: \tuple {x, y} \in \RR}$

If $\RR$ is an equivalence on $S$, then each $t \in S$ that satisfies $\tuple {x, t} \in \RR$ (or $\tuple {t, x} \in \RR$) is called a $\RR$-relative of $x$.

That is, the equivalence class of $x$ under $\RR$ is the set of all $\RR$-relatives of $x$.

Also known as
The equivalence class of $x$ under $\RR$ can be stated more tersely as the $\RR$-equivalence class of $x$, or just the $\RR$-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:
 * $y \in \eqclass x \RR \iff \paren {x, y} \in \RR$


 * Relation Partitions Set iff Equivalence which justifies the construction.