\begin{definition}[Definition:Equivalence Class]
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 $x$.
\end{definition}