Question

Negation of (p⇒q)⇒(q⇒p) is

• A. p∨(∼q)
• B. (∼p)∨q
• C. q∧(∼p)
• D. (∼q)∧p

Hint:

Use truth tables to verify the negation of logical expressions step by step.

Answer 1

The Correct Option is C

Step 1: Understand the implication.
The given expression is \((p \to q) \to (q \to p)\). We want to find the negation of this expression.
Step 2: Rewrite the expression using logical equivalences.
Recall that an implication \(A \to B\) is logically equivalent to \(\sim A \lor B\). So, we can rewrite the expression as: \[ (p \to q) \to (q \to p) \equiv \sim(p \to q) \lor (q \to p). \] Now, apply the equivalence \(p \to q \equiv \sim p \lor q\) and \(q \to p \equiv \sim q \lor p\) to obtain: \[ (\sim p \lor q) \to (\sim q \lor p). \] Step 3: Apply negation to the entire expression.
Next, we apply negation to the entire expression. The negation of an implication \(\sim (A \to B)\) is \(A \land \sim B\). So, we have: \[ \sim \left( \sim(p \to q) \lor (q \to p) \right). \] Using De Morgan’s law, this becomes: \[ \sim(\sim p \lor q) \land \sim(\sim q \lor p). \] Step 4: Simplify the expression.
Simplifying the negations inside: \[ \sim(\sim p \lor q) = p \land \sim q, \quad \sim(\sim q \lor p) = q \land \sim p. \] Thus, the negation becomes: \[ (p \land \sim q) \land (q \land \sim p). \] This simplifies to: \[ q \land \sim p. \] Final Answer: \(q \land \sim p\).