Show that given four events A, B, C, D the probability of their union is given by P(A B C D) = P(A) + P(B) + P(C) + P(D) P(A B) P(A C) P(B C) P(A D) P(B D) P(C D) + P(A B C) + P(D B C) + P(A B D) + P(A D C) P(A B C D). Extend the inclusion and exclusion principle to an arbitrary finite union of k 2 subsets of the sample space, and give a proof of such extension.