# Definition:Renaming Mapping

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## Contents

## Definition

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

The **renaming mapping** $r: S / \mathcal R_f \to \Img f$ is defined as:

- $r: S / \mathcal R_f \to \Img f: \map r {\eqclass x {\mathcal R_f} } = \map f x$

where:

- $\mathcal R_f$ is the equivalence induced by the mapping $f$
- $S / \mathcal R_f$ is the quotient set of $S$ determined by $\mathcal R_f$
- $\eqclass x {\mathcal R_f}$ 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

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: I. Basic Concepts*... (previous) ... (next): Introduction $\S 3$: Equivalence relations - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{I}$: Factoring Functions - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Theorem $6.5$