Question

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

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

Hint:

To negate a biconditional expression, use the fact that \( p \Leftrightarrow q \) is true if and only if both \( p \) and \( q \) have the same truth value. The negation is the case where the truth values differ.

Answer 1

The Correct Option is D

Step 1: We are given the expression \( p \Leftrightarrow (q \Rightarrow p) \), and we need to find its negation. Recall that the biconditional \( p \Leftrightarrow q \) is true if both \( p \) and \( q \) have the same truth value. This can be rewritten as: \[ p \Leftrightarrow (q \Rightarrow p) \equiv (p \Rightarrow (q \Rightarrow p)) \land ((q \Rightarrow p) \Rightarrow p). \] However, to simplify: - The expression \( q \Rightarrow p \) is equivalent to \( \sim q \lor p \). Thus, the expression becomes: \[ p \Leftrightarrow (\sim q \lor p). \] Next, we negate the biconditional. The negation of \( p \Leftrightarrow (\sim q \lor p) \) is: \[ \sim (p \Leftrightarrow (\sim q \lor p)) = (\sim p) \land (\sim (\sim q \lor p)). \] Now, simplify: \[ \sim (\sim q \lor p) = q \land \sim p. \] So, the negation of the given expression is: \[ (\sim p) \land (\sim q). \]