Definition:Renaming Mapping

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Definition

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

The renaming mapping $r: S / \mathcal R_f \to \operatorname {Im} \left({f}\right)$ is defined as:

$r: S / \mathcal R_f \to \operatorname {Im} \left({f}\right): r \left({\left[\!\left[{x}\right]\!\right]_{\mathcal R_f}}\right) = f \left({x}\right)$

where:

$\mathcal R_f$ is the equivalence induced by the mapping $f$;
$S / \mathcal R_f$ is the quotient set of $S$ determined by $\mathcal R_f$;
$\left[\!\left[{x}\right]\!\right]_{\mathcal R_f}$ is the equivalence class of $x$ under $\mathcal R_f$.