Theorem: SC $\in$ NPC

Proof:

1.

Given C, check that all elements of X are members of some set in C and that .

2.

L' = VC

3.

Given $\langle$G, k$\rangle$ $\in$ VC, define F such that each element of F is a subset for a vertex v in G containing v and all vertices reachable by an edge from v.
Let X = V. Then $\langle$X,F, k$\rangle$ $\in$ SC.

4.

If C is the vertex cover of $\langle$G, k$\rangle$ $\in$ VC, then every vertex u in G is incident from an edge (u,v) where either u $\in$ C or v $\in$ C. Thus all vertices will appear in some set in F, and the sets in F corresponding to the vertices in C make up the set covering of $\langle$X, F, k$\rangle$ $\in$ SC.