Renaming Mapping is Bijection/Different approaches

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Different approaches to Renaming Mapping is Bijection

Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts considers the case where $r$ is an injection, but does not stress its bijective aspects from this particular perspective:

This type of factorization of mappings ... is particularly useful when the set of inverse images $\map {\alpha^{-1} } {a'}$ coincides with $\overline S$; for, in this case, the mapping $\overline a$ is 1-1.
Thus if $\overline a \overline \alpha = \overline b \overline \alpha$, then $a \alpha = b \alpha$ and $a \sim b$. Hence $\overline a = \overline b$. Thus we obtain here a factorization $\alpha = \nu \overline \alpha$ where $\overline \alpha$ is 1-1 onto $T$ and $\nu$ is the natural mapping.


Note that in the above, Jacobson uses:

$\alpha$ for $f$
$a'$ for the image of a representative element $a$ of $S$ under $\alpha$
$\overline S$ for $S / \RR_f$
$\nu$ for the quotient mapping $q_{\RR_f}: S \to S / \RR_f$
$\overline a$ and $\overline b$ for representative elements of $\overline S$
$\overline \alpha$ for the renaming mapping $r$.


T.S. Blyth: Set Theory and Abstract Algebra takes the approach of deducing the existence of the mapping $r$, and then determining under which conditions it is either injective or surjective.

From there, the surjective restriction of $r$ is taken, and $\RR$ is then identified with the equivalence induced by $f$.

Hence the bijective nature of $r$ is deduced rather than constructed.


Sources

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