Prove that for any events A and B P(A ∪ B) = P(A) + P(B) - P(A ∩ B).?

Just as the difference rule gives rise to a formula for the probability of the complement of an event, so the addition and inclusion/exclusion rules give rise to formulas for the probability of the union of mutually disjoint events and for a general union of (not necessarily mutually exclusive) events.