Let A be a nonempty set. What are all relations on A that are both equivalence relations and functions?

This is pretty simple.

To be an equivalence relation, it must be reflexive, so it must contain (x, x) for every x in A.

To be a function, it cannot contain both (x, y) and (x, z) where y ≠ z.

Since (x, x) is already there for every x, there cannot be any (x, y) in the relation where y ≠ x.

So the only relation on A that is both reflexive and a function is the identity relation {(x, x): x ∈ A}, or as a function f(x) = x. (It should be obvious that this is indeed an equivalence relation!)