Mindboggling set of set 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^Y)^Z and X^(Y x Z).

Attempt: These all are so confusing that I don't even know how to start.

 

[haruspex]

Start by considering what an element of (X^Y)^Z would be like. What would it do?

 

[bedi]

It's elements are the ordered pairs (z,(y,x)) aren't they? Where z€Z etc. and if this is true then the elements of X^(Y x Z) are similarly the ordered pairs ((y,z),x). Am I correct?

 

[haruspex]

Its elements are the ordered pairs (z,(y,x)) aren't they?

No. It would be a function from Z to X^Y, right? So for each zεZ it would pick out a function from Y to X. And if you supply that function with an element of Y it will give you an element of X. So in total, giving that function from Z to X^Y both an element of Z and an element of Y you find an element of X. Do you see it now?