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 byalwaysforms apartitionof , and so the "set of inverse images MATH]\alpha^{-1} \left({a'}\right)[/tex] willalwayscoincide with ?

What am I missing here?