Proof related to Axioms of Probability

These problems and Proofs are adapted from the textbook: Probability and Random Process by Scott Miller 2ed Problem-1: Proof that for events A and B the following holds: \(Pr\left(A\cup B\right) = Pr\left(A\right)+Pr\left(B\right)-Pr\left(A\cap B\right)\) Solution: First, note that the sets \(A\cup B\) and \(B\) can be rewritten as \(A\cup B=\{A\cup B\}\cap\{\overline{B}\cup B\}=\{A\cap\overline{B}\}\cup B\) \(B=B\cap\{A\cup\overline{A}\}=\{A\cap B\}\cup\{\overline{A}\cap B\}.\) Hence, \(A\cup B\) can be expressed as the union of three mutually exclusive sets. \(A\cup B=\{\overline{B}\cap A\}\cup\{A\cap B\}\cup\{\overline{A}\cap B\}\) Next, rewrite \(\overline{B}\cap A\) as \( \overline{B} \cap A= \{\overline{B}\cap A\}\cup\{\overline{A}\cap A\}= A\cap\{\overline{A}\cup\overline{B}\} = A\cap\{\overline{A\cap B}\}\) Likewise, \(\overline{A}\cap B=B\cap\{\overline{A\cap B}\}\). Therefore \(A\cup B\) can be rewritten as the following...