Let X be a subset of {1,2,3,...,n} such that |X| is even. Define f(X) to

be the set obtained from X by adding the element n when n is not in set X, and deleting the element n when n is in set X. (That is, f(X) = X union {n} if n is not in X, and f(X) = X \ {n} if n is in X.) Prove that f is a 1-1 correspondence between the subsets of {1, 2,....,n} with an even number of elements and the ones with an odd number of elements.