Assume that A, B, and C are three events in a sample space . Show that the

Assume that A, B, and C are three events in a sample space Ω. Show that the following relations hold:

(i) P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A′BC) − P(AB′C)

−P(ABC′) − 2P(ABC);

(ii) P(ABC) ≥ P(A) + P(B) + P(C) − 2;

(iii) P(A′BC) + P(AB′C) + P(ABC′) ≤ 2P(A′ ∪ B′ ∪ C′);

(iv) P(AB′C′) + P(A′BC′) + P(A′B′C) ≤ 2P(A ∪ B ∪ C).