Bijection between sets of functions

[bedi]

For two sets X and Y let X^Y be the set of functions from Y to X.

Prove that there is a bijection between (X x Y)^Z and X^Z x Y^Z.

Attempt: I could not get any further from that "there must be a function S with S(f)=g and S(f')=g for any g, f' in X^Z x Y^Z, and where f is in (X x Y)^Z."

 

[hedipaldi]

see attachment.

 

[hedipaldi]

see attachment

 

[bedi]

Thank you, I think I'm done with proving that psi is an injection but I can't prove that it's also a surjection. Could you give me another hint?

 

[bedi]

This is what I tried for surjectivity so far: psi is clearly a surjection, as for every ordered pair (h,g) there is an f such that psi(f)=(h,g), because when we look at the values of f, h and g at z we see that psi(x,y)=(x,y). But this is the identity function and therefore a surjection. Is that correct?

 

[hedipaldi]

define f(z)=(h(z),g(z)).verify that this is a counter image of (h,g)