I have a math question. Prove that f : X�Y is surjective iff for each subset A c X, Y\f(A) c f(X\A).?

Question

I'm having a problem proving it

Answer 1

what year are you studying?

Answer 2

There is no money in math nerd

Answer 3

When a function is surjective, all the codomain have at least one x from the domain part. So if there is a set of domain, when we substract all unit in A, we will get the rest have to be the domain of the codomain whose domain is not in the A, but there is a chance that the domain without the A set has codomain in one of the f(q) and q is in the set A. So we have proven that f : XY is surjective if for each subset A c X, Y\f(A) c f(X\A). ,now we have to prove the other way around. Because Y\f(A) c f(X\A), we now know that there is some of the domain of x\a which has codomain in f(A). And because we know that Y\f(a) c f(x\a), we can say that Y\f(a) all have a domain, because if there is some of them that don't have domain, there have to be q E (X\A) where q doesn't have any codomain. And this is impossible because as we know each member of the domain in a function have to get a codomain.

And with these 2 facts, we can say that if each subset A c X, Y\f(A) c f(X\A), then f:X->Y is surjective. Because we can prove both ways, the problem is proven.

Sorry if it is confusing, I myself am not so sure too.