Definition:Renaming Mapping

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 $\mathcal R_f$.

Also known as
This mapping can also be seen referred to as the mapping on $S / \mathcal R_f$ 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.

Also see

 * Existence of Renaming Mapping
 * Quotient Theorem for Sets
 * Renaming Mapping is Well-Defined
 * Renaming Mapping is Bijection