# Definition:Renaming Mapping

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## 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.

## Examples

### Projection of Plane onto $x$-axis

Let $P$ denote the Cartesian plane.

Let $X$ denote the $x$-axis of $P$.

Let $\pi_x: P \to X$ be the perpendicular projection of $P$ onto $X$.

Then:

- the equivalence relation $\RR_\pi$ induced by $\pi_x$ is:
- $p_1 \mathrel {\RR_\pi} p_2 \iff p_1$ and $p_2$ are on the same vertical line

- the quotient set $P / \RR_\pi$ of $P$ determined by $\RR_\pi$ is the set of points of the $x$-axis

- the equivalence class $\eqclass p {\RR_f}$ of $p$ under $\RR_f$ is the $x$-coordinate of $p$.

## Also see

- Condition for Mapping from Quotient Set to be Well-Defined
- Renaming Mapping is Well-Defined
- Renaming Mapping is Bijection

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 3$: Equivalence relations - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Factoring Functions