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 see

 * Existence of Renaming Mapping
 * Renaming Mapping is Well-Defined
 * Renaming Mapping is a Bijection