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

To determine the negation of the given statement \( (p \Rightarrow q) \Rightarrow (q \Rightarrow p) \), we will first understand the implication and then apply the negation.

The implication \( p \Rightarrow q \) is logically equivalent to \( \sim p \lor q \). Similarly, the implication \( q \Rightarrow p \) is equivalent to \( \sim q \lor p \).

Let's express the given statement \( (p \Rightarrow q) \Rightarrow (q \Rightarrow p) \) in terms of logical operators:

• The inner statement \( p \Rightarrow q \) becomes \( \sim p \lor q \).
• The inner statement \( q \Rightarrow p \) becomes \( \sim q \lor p \).
• The overall implication \( (p \Rightarrow q) \Rightarrow (q \Rightarrow p) \) becomes \( \sim(\sim p \lor q) \lor (\sim q \lor p) \).

Next, we simplify using De Morgan's laws:

• The negation of \( (\sim p \lor q) \) becomes \( p \land \sim q \).
• The expression becomes \( (p \land \sim q) \lor (\sim q \lor p) \).

To find the negation of this entire implication statement, we use the equivalence for negation:

• Negation of \( A \Rightarrow B \) is \( A \land \sim B \).
• Here, \( A \) is \( \sim(\sim p \lor q) \), and its negation is \( p \land \sim q \).
• Thus, the negation becomes \( p \land \sim q \land \sim (\sim q \lor p) \).
• Expanding \( \sim (\sim q \lor p) \) gives \( q \land \sim p \).

Thus, the negation of the given statement is \( q \land \sim p \).

Therefore, the correct option is q ∧ (∼p).