4. For any set V the power set P(V) of V is the set of all subsets of V. If V is a vector space, let S(V) denote the set (a subsetof P(V)) of all subspaces of V. For a finite-dimensional vector space V define o: P(V) S(V): S span S. (a) Prove that o(o(S)) = o(S) for every S in P(V). (b) Prove that (S C S) 0(S) Co(S) for all S and S in P(V). (c) Prove that o(S US) = o(S) + 0(S) for all S and S in P(V).