# Definition:Renaming Mapping

## Definition

Let $f: S \to T$ be a mapping.

The renaming mapping $r: S / \RR_f \to \Img f$ is defined as:

$r: S / \RR_f \to \Img f: \map r {\eqclass x {\RR_f} } = \map f x$

where:

$\RR_f$ is the equivalence induced by the mapping $f$
$S / \RR_f$ is the quotient set of $S$ determined by $\RR_f$
$\eqclass x {\RR_f}$ is the equivalence class of $x$ under $\RR_f$.

## Also known as

This mapping can also be seen referred to as the mapping on $S / \RR_f$ induced by $f$.

However, the term induced mapping is used so often throughout this area of mathematics that it would make sense to use a less-overused term whenever possible.