Question:
Let \( p, q \) and \( r \) be any three logical statements. Which one of the following is true?
Let \( p, q \) and \( r \) be any three logical statements. Which one of the following is true?
Show Hint
To negate a logic bracket: 1. Negate the first term. 2. Flip the sign (\( \wedge \) to \( \vee \), or \( \vee \) to \( \wedge \)). 3. Negate the second term.
Updated On: May 6, 2026
- \( \sim [p \wedge (\sim q)] \equiv (\sim p) \wedge q \)
- \( \sim (p \vee q) \wedge (\sim r) \equiv (\sim p) \vee (\sim q) \vee (\sim r) \)
- \( \sim [p \vee (\sim q)] \equiv (\sim p) \wedge q \)
- \( \sim [p \wedge (\sim q)] \equiv (\sim p) \wedge \sim q \)
- \( \sim [p \wedge (\sim q)] \equiv p \wedge q \)
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The Correct Option is C
Solution and Explanation
Concept:
The fundamental laws of logic used here are De Morgan's Laws. These laws describe how to distribute a negation (\( \sim \)) over a conjunction (\( \wedge \)) or a disjunction (\( \vee \)).
• \( \sim (p \vee q) \equiv (\sim p) \wedge (\sim q) \)
• \( \sim (p \wedge q) \equiv (\sim p) \vee (\sim q) \)
• Law of Double Negation: \( \sim (\sim p) \equiv p \)
Step 1: Evaluate Option (C) using De Morgan's Law.
The expression is \( \sim [p \vee (\sim q)] \). Applying De Morgan's Law, we distribute the negation and flip the disjunction to a conjunction: \[ \sim [p \vee (\sim q)] \equiv (\sim p) \wedge \sim (\sim q) \]
Step 2: Apply the Law of Double Negation.
The term \( \sim (\sim q) \) simplifies to \( q \). Therefore: \[ (\sim p) \wedge \sim (\sim q) \equiv (\sim p) \wedge q \] Since this matches the right-hand side of Option (C), it is the correct statement.
• \( \sim (p \vee q) \equiv (\sim p) \wedge (\sim q) \)
• \( \sim (p \wedge q) \equiv (\sim p) \vee (\sim q) \)
• Law of Double Negation: \( \sim (\sim p) \equiv p \)
Step 1: Evaluate Option (C) using De Morgan's Law.
The expression is \( \sim [p \vee (\sim q)] \). Applying De Morgan's Law, we distribute the negation and flip the disjunction to a conjunction: \[ \sim [p \vee (\sim q)] \equiv (\sim p) \wedge \sim (\sim q) \]
Step 2: Apply the Law of Double Negation.
The term \( \sim (\sim q) \) simplifies to \( q \). Therefore: \[ (\sim p) \wedge \sim (\sim q) \equiv (\sim p) \wedge q \] Since this matches the right-hand side of Option (C), it is the correct statement.
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