What does '[n]' stand for? Do you just mean the natural number n? (I.e., the von Neumann finite ordinal n?). If you do, then I would write this way:

Let f be the function from the powerset of n into the powerset of n defined by:

f(x) = {k in n | k not in x} for x subset of n.

First, just as an aside, though it is not used in the following proof, it is easy to show that, if n is odd, then, for all x subset of n, if card(x) is even then card(f(x)) is odd. (Easy since n is odd and therefore card(n)=n is odd, and an even number plus an even number is even.)

Show that f is bijection from the powerset of n onto the powerset of n:

Suppose x and y are different subsets of n. Without loss of generality, let k be in x but not in y. Then k in f(y) but not f(x), so f is 1-1. So f is an injection from powerset of n into the powerset of n.

Suppose x is a subset of n. So x = f({k in n | k not in x}). So f is onto the powerset of n.

So f is a bijection from the powerset of n onto the powerset of n.