Renaming Mapping is Well-Defined

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Theorem

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

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


The renaming mapping is always well-defined.


Proof 1

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

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

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

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

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

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

$\eqclass x {\RR_f} = \eqclass y {\RR_f}$

then:

$\map r {\eqclass x {\RR_f} } = \map r {\eqclass y {\RR_f} }$


Hence from the definition, we have:

\(\displaystyle y\) \(\in\) \(\displaystyle \eqclass x {\RR_f}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map f y\) \(=\) \(\displaystyle \map f x\)
\(\displaystyle \) \(=\) \(\displaystyle \map r {\eqclass x {\RR_f} }\)
\(\displaystyle \) \(=\) \(\displaystyle \map r {\eqclass y {\RR_f} }\)

Thus $r$ is well-defined.

$\blacksquare$


Proof 2

From Condition for Mapping from Quotient Set to be Well-Defined:

there exists a mapping $\phi: S / \RR \to T$ such that $\phi \circ q_\RR = f$

if and only if:

$\forall x, y \in S: \tuple {x, y} \in \RR \implies \map f x = \map f y$


But by definition of the equivalence induced by the mapping $f$:

$\forall x, y \in S: \tuple {x, y} \in \RR_f \implies \map f x = \map f y$

The result follows directly.

$\blacksquare$


Sources