Question

The negative of \( (p \land (\sim q)) \lor (\sim p) \) is equivalent to:

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

Hint:

Use De Morgan’s laws to simplify the negation of logical statements: \[ \sim (A \lor B) = (\sim A \land \sim B) \quad \text{and} \quad \sim (A \land B) = (\sim A \lor \sim B). \]

Answer 1

The Correct Option is A

To determine the negation of \( (p \land (\sim q)) \lor (\sim p) \): \[ \sim \big[(p \land (\sim q)) \lor (\sim p)\big] \] Applying De Morgan's laws yields: \[ \big[\sim (p \land (\sim q)) \land \sim (\sim p)\big] \] Further simplification results in: \[ \big[(\sim p \lor q) \land p\big] \] Combining terms, we get: \[ p \land q \] Conclusion: The negation of \( (p \land (\sim q)) \lor (\sim p) \) is \( p \land q \).