Question:
Negation of the Boolean expression
\[
p \Leftrightarrow (q \Rightarrow p)
\]
is:
Negation of the Boolean expression
\[
p \Leftrightarrow (q \Rightarrow p)
\]
is:
Show Hint
Negation of a biconditional statement can be found using De Morgan’s laws.
Updated On: Apr 2, 2025
- \( \sim p \wedge q \)
- \( p \wedge \sim q \)
- \( \sim p \vee \sim q \)
- \( \sim p \wedge \sim q \)
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The Correct Option is D
Solution and Explanation
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).
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).
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