Before revision:  The input to this problem is a set system (X, S ), where X is a set of elements and S is a collection of subsets of X, and an integer k <= \le |X|.
After revision:  The input to this problem is a set system (X, S ), where X is a set of elements and \mathcal{S is a collection of subsets of X, and an integer k <= \le |X|.