Question

The logical statement \((p \to q) \land (q \to \sim p)\) is equivalent to

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

Hint:

When simplifying logical expressions, convert implications to disjunctions first. Look for common factors and use \((A \lor B) \land (A \lor C) \equiv A \lor (B \land C)\).

Answer 1

The Correct Option is A

Step 1: Understanding the Question: We need to simplify the compound logical statement \((p \to q) \land (q \to \sim p)\).

Step 2: Key Formula or Approach: Use the equivalence \(p \to q \equiv \sim p \lor q\). Then apply distributive and other logical laws.

Step 3: Detailed Explanation: \(p \to q \equiv \sim p \lor q\) \(q \to \sim p \equiv \sim q \lor \sim p\) Thus the statement becomes: \((\sim p \lor q) \land (\sim q \lor \sim p)\). Factor out \(\sim p\) (using distributive law): \((\sim p \lor q) \land (\sim p \lor \sim q) \equiv \sim p \lor (q \land \sim q)\). But \(q \land \sim q\) is always false (contradiction). So we get \(\sim p \lor \text{False} \equiv \sim p\).

Step 4: Final Answer: The statement is equivalent to \(\sim p\), which is option (A).