jacobrhcp
Oct23-08, 01:25 PM
1. The problem statement, all variables and given/known data
if A is not empty and s:A->B is injective, then there is a surjective function f:B->A such that f(s(a))=a for all a in A. Do not use the Axiom of Choice
3. The attempt at a solution
for all b', b'' in B s(b')=s(b'') means b'=b''. So s^-1 (c) is unique in A. Because any point in A is sent to at most one point in B, we can just let f send every point in B of the form s(c) to c.
now we only need to send all the other points somewhere. Here I need to 'pick' some point once again. Why do I not need the AC here?
if A is not empty and s:A->B is injective, then there is a surjective function f:B->A such that f(s(a))=a for all a in A. Do not use the Axiom of Choice
3. The attempt at a solution
for all b', b'' in B s(b')=s(b'') means b'=b''. So s^-1 (c) is unique in A. Because any point in A is sent to at most one point in B, we can just let f send every point in B of the form s(c) to c.
now we only need to send all the other points somewhere. Here I need to 'pick' some point once again. Why do I not need the AC here?