Question

The statement $[p \land (q \lor r)] \lor [\sim r \land \sim q \land p]$ is equivalent to}

• A. $\sim r$
• B. $p$
• C. $\sim q$
• D. $q$

Hint:

When simplifying complex logical expressions, look for opportunities to apply De Morgan's laws and distributive laws. Identifying common sub-expressions or patterns like $A \land B \lor A \land \sim B \equiv A$ can greatly speed up the simplification process.

Answer 1

The Correct Option is A

Step 1: Simplify the given statement The given logical expression is: \[ S = [p \land (q \lor r)] \lor [\sim r \land \sim q \land p] \] Using De Morgan's Law: \[ \sim r \land \sim q = \sim (r \lor q) = \sim (q \lor r) \] So, the expression becomes: \[ S = [p \land (q \lor r)] \lor [p \land \sim (q \lor r)] \]
Step 2: Introduce substitution Let: \[ X = (q \lor r) \] Then: \[ S = (p \land X) \lor (p \land \sim X) \]
Step 3: Apply distributive law \[ S = p \land (X \lor \sim X) \]
Step 4: Use tautology law Since: \[ X \lor \sim X \equiv T \] We get: \[ S = p \land T \]
Step 5: Final simplification \[ S \equiv p \] Conclusion:
The given logical statement is equivalent to: \[ \boxed{p} \]