NP-completeness

4. NP-completeness. Let X be a finite set and F a family of subsets of X such that every element of X appears in at least one subset in F. We say that a subset C of F is a set cover for X if X -SEc S (that is, the union of the sets in C is X). The cardinality of a set cover C is the number of elements in C. (Note that an element of C is a subset of X.) The Set Cover problem is: Given an in- stance (X,F), and a nonnegative integer k, decide if X has a set cover C in F of cardinality k. For example, if X = { 1 , 2, 3, 4, 5), F= { {1),(2),(3),(25), (1 ,3,5).( 1,2,5),(24.5),{ 1.45)), and k = 2, then the answer is Yes because X has a set cover C in F of cardinality 2, namely C = {(1 , 3, 5), {2, 4, 5)). Show that the Set Cover problem is NP-complete. (Hint. Reduce from Vertex Cover.)