Bijectivity of sets of functions

Question

I am supposed to prove that $f: [X \to [Y \to Z]] \to [X * Y \to Z]$ is a bijection I get the definition and how to prove both injection and surjection, but I am quite confusd because the domain and range seem to be a set of functions

Any help on how should I start approaching this? Thank you.

Answer 1

It does seem that your domain and range are sets of functions. That complicates things a little, but remember that functions are the same if

• They have the same domain, and
• They have the same outputs for each element of that domain.

Going back to what you have to prove, you mentioned that you need to prove that $f$ is injective and surjective. What did those mean again?

• [Injective] If $f(a)=f(b)$ then $a=b$.
• [Surjective] If $y$ is in the range, then there exists $x$ in the domain so that $f(x)=y$.

Note that for both injectivity and surjectivity, you only test equality. As you noted, things are a little stranger than normal since the domain and range are both sets of functions, but testing equality is still very straightforward. You even get the first point for free since any two functions you might be comparing in your problem already have the same domain.