Textbook Problem

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.

1. Prove that for mutually disjoint events A and B,  $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right).$
2. Prove that for any events A and B, $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right).$