Definition:Renaming Mapping

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

• Quotient Theorem for Sets

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