The statement $\sim (p\vee (\sim p\wedge q))$ is logically equivalent to which of the following?

Solution

1. Start with the given expression: $\sim (p\vee (\sim p\wedge q))$.
2. Apply De Morgan's law to distribute the negation: $\sim p\wedge \sim (\sim p\wedge q)$.
3. Apply De Morgan's law again to the second term: $\sim p\wedge (p\vee \sim q)$.
4. Distribute $\sim p$ over the parentheses: $(\sim p\wedge p)\vee (\sim p\wedge \sim q)$.
5. Since $(\sim p\wedge p)$ is a contradiction (False), the expression simplifies to $\sim p\wedge \sim q$.
6. Hence the answer is (B).