Proving existence of right inverse to a surjection

[fuzuli]

I am stuck at a proof and do not even have an idea where to start and how to start:

Let X and Y be sets, and let f : X → Y be a surjection. Prove that there is an injection g : Y → X such that f (g(y)) = y for every y ∈ Y.

Could you please show me a way?

 

[jbunniii]

Hint: you will need to use the axiom of choice to define g. For each y in Y, you need to choose an x such that f(x) = y.