Question

The logically equivalent statement of $(\sim p \land q) \lor (\sim p \land \sim q) \lor (p \land \sim q)$ is

• A. $(\sim p) \land q$
• B. $(\sim p) \lor (\sim q)$
• C. $(\sim p) \land (\sim q)$
• D. $p \lor q$

Hint:

Look for terms that differ by only one variable (e.g., $X \land Y$ and $X \land \sim Y$). They combine to simply $X$.

Answer 1

The Correct Option is B

Step 1: Concept
Use the Distributive Law and the Law of Complements to simplify logical expressions.

Step 2: Meaning
The expression can be grouped to factor out common terms.

Step 3: Analysis
Group the first two terms: $(\sim p \land q) \lor (\sim p \land \sim q) = \sim p \land (q \lor \sim q)$. Since $q \lor \sim q$ is always True ($T$), this simplifies to $\sim p \land T = \sim p$. Now combine with the third term: $(\sim p) \lor (p \land \sim q)$. Use Distributive Law: $(\sim p \lor p) \land (\sim p \lor \sim q)$. Since $\sim p \lor p = T$, we get $T \land (\sim p \lor \sim q)$.

Step 4: Conclusion
The expression simplifies to $(\sim p \lor \sim q)$. Final Answer: (B)