\begin{definition}[Definition:Quotient Mapping]
Let $\RR \subseteq S \times S$ be an equivalence on a set $S$.
Let $\eqclass s \RR$ be the $\RR$-equivalence class of $s$.
Let $S / \RR$ be the quotient set of $S$ determined by $\RR$.
Then $q_\RR: S \to S / \RR$ is the '''quotient mapping induced by $\RR$''', and is defined as:
:$q_\RR: S \to S / \RR: \map {q_\RR} s = \eqclass s \RR$
Effectively, we are defining a mapping on $S$ by assigning each element $s \in S$ to its equivalence class $\eqclass s \RR$.
If the equivalence $\RR$ is understood, $\map {q_\RR} s$ can be written $\map q s$.
\end{definition}