I'm studying N. Jacobson's "Lectures in Advanced Algebra" (originally published in 1951, apparently a classic), and have encountered this puzzling paragraph:
:"This type of factorization of mappings ... is particularly useful when the set of inverse images coincides with ; for, in this case, the mapping is 1-1. Thus if , then and . Hence . Thus we obtain here a factorization where is 1-1 onto and is the natural mapping."
Note that in the above, Jacobson uses:
* for a general mapping from to ;
* for the Image of a representative element of under ;
* for the quotient set defined by the equivalence induced by ;
* for the quotient mapping from ;
* and for representative elements of ;
* for the renaming mapping .
The above, then, is a fairly terse proof of the Quotient Theorem for Surjections: the fact that a surjection can be factored into the canonical surjection and the renaming mapping, the latter of which is a bijection.
The puzzling statement is the one: "when the set of inverse images coincides with ".
Surely, as is a mapping, the quotient set of induced by always forms a partition of , and so the "set of inverse images MATH]\alpha^{-1} \left({a'}\right)[/tex] will always coincide with ?
What am I missing here?