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:
 * $\eqclass x {\mathcal R} = \set {y \in S: \tuple {x, y} \in \mathcal R}$

If $\mathcal R$ is an equivalence on $S$, then each $t \in S$ that satisfies $\tuple {x, t} \in \mathcal R$ (or $\tuple {t, x} \in \mathcal R$) 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:
 * $y \in \eqclass x {\mathcal R} \iff \paren {x, y} \in \mathcal R$


 * Relation Partitions Set iff Equivalence which justifies the construction.

Technical Note
On, the technique for rendering $\eqclass x {\mathcal R}$ in $\LaTeX$ is.

This is a custom construct which has been set up specifically for the convenience of the users of.

Note that there are two arguments to this operator: the part between the brackets, and the subscript.

If either part is a single symbol, then the braces can be omitted, for example: