Question

Negation of the Boolean expression \[ p \Leftrightarrow (q \Rightarrow p) \] is:

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

Hint:

Negation of a biconditional statement can be found using De Morgan’s laws.

Answer 1

The Correct Option is D

Step 1: Expanding the given expression.
The given statement \( p \Leftrightarrow (q \Rightarrow p) \) can be rewritten using logical equivalence: \[ p \Leftrightarrow (\sim q \vee p) \] which simplifies to: \[ (p \vee \sim q) \wedge (\sim p \vee (\sim q \vee p)) \] \[ = (p \vee \sim q) \wedge (p \vee \sim q \vee \sim p) \] Step 2: Finding the negation.
Negating both sides, \[ \sim ((p \vee \sim q) \wedge (p \vee \sim q \vee \sim p)) \] Applying De Morgan’s laws, \[ \sim p \wedge \sim q \] Thus, the correct answer is (D).