Talk:Bijection iff Left and Right Inverse
The refactor template says:
- Because of the complexity of exactly what is being proved from which hypotheses, and the question of dependency on AoC etc., it is probably best to split the theorem up into two: the "if" and the "only if". This is what most texts on this subject do. There are yet more proofs for this theorem, all of which have merit. A suite of proofs for the "if" in one page, and an equivalent suite for the "only if", joined by an overall "iff" page, might be a profitable way to go.
To answer some of the questions:
Bijection iff left and right inverse is our statement.
The axiom of choice says (or, if you prefer, is equivalent to) that every surjective function has a right inverse.
Thus the statement left and right inverse if bijection depends on the axiom of choice. (injection iff left inverse is true regardless).
The statement: right inverse implies surjection is also independent, so, bijection if left and right inverse doesn't depend on the aoc.
Therefore, I do not think it is wholly necessary that this page is split, but if it is going to, one can use this.
The sentence: There are yet more proofs for this theorem, all of which have merit., I do not understand, so I cannot say anything about this. JSchoone 17:08, 2 March 2012 (EST)
- TL;DR - refactoring in progress. --prime mover 17:12, 2 March 2012 (EST)