# Renaming Mapping is Well-Defined

## Theorem

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

Let $r: S / \mathcal R_f \to \Img f$ be the renaming mapping, 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 $\mathcal R_f$.

The renaming mapping is always well-defined.

## Proof

By Relation Induced by Mapping is Equivalence Relation, we have that $\mathcal R_f$ is an equivalence relation.

To determine whether $r$ is well-defined, we have to determine whether $r: S / \mathcal R_f \to \Img f$ actually defines a mapping at all.

Consider a typical element $\eqclass x {\mathcal R_f}$ of $S / \mathcal R_f$.

Suppose we were to choose another name for the class $\eqclass x {\mathcal R_f}$.

Assume that $\eqclass x {\mathcal R_f}$ is not a singleton.

For example, let us choose $y \in \eqclass x {\mathcal R_f}, y \ne x$ such that:

- $\eqclass x {\mathcal R_f} = \eqclass y {\mathcal R_f}$

then:

- $\map r {\eqclass x {\mathcal R_f} } = \map r {\eqclass y {\mathcal R_f} }$

Hence from the definition, we have:

\(\displaystyle y\) | \(\in\) | \(\displaystyle \eqclass x {\mathcal R_f}\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \map f y\) | \(=\) | \(\displaystyle \map f x\) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map r {\eqclass x {\mathcal R_f} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map r {\eqclass y {\mathcal R_f} }\) |

Thus $r$ is well-defined.

$\blacksquare$

## 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: Theorem $10$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Theorem $6.5$ - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 7$: Relations: Exercise $2$