Complement function: how to prove surjective? Given some set A and a complement function C(K) = A - K from the power set of A onto the power set of A, how can I formally prove that it is surjective?
I think I get it, but can't get it on paper. (For context, I have already proven that it is injective.) Every element of the power set of A will be both in the domain and range because the power set of A minus one set yields the rest of its power set, and
P(A) - empty set = P(A)
and
P(A) - P(A) = empty set
... but that's not equivalent to saying there's a function f(x) = P(x) - {x} , right?

To prove that a function is surjective, it's enough to prove that it's bijective; and so it's enough to prove that it has an inverse.
What is the inverse of $C: K \mapsto A \setminus K$?
