Let $p$ and $q$ be two propositions. The compound statement $(p\vee q)\wedge (\neg p\vee q)$ is logically equivalent to which of the following?

Solution

1. The given expression is $(p\vee q)\wedge (\neg p\vee q)$.
2. By the distributive law, this is equivalent to $(p\wedge \neg p)\vee q$.
3. Since $p\wedge \neg p$ is a contradiction (False), the expression becomes $F\vee q$.
4. The disjunction of a false statement with $q$ is $q$.
5. Hence the answer is (B).