Question:
\( \sim [(\sim p) \wedge q] \) is logically equivalent to:
\( \sim [(\sim p) \wedge q] \) is logically equivalent to:
Show Hint
Double negation (\( \sim \sim \)) always cancels out, just like a negative times a negative in math.
Updated On: May 6, 2026
- \( \sim (p \vee q) \)
- \( \sim [p \wedge (\sim q)] \)
- \( p \wedge (\sim q) \)
- \( p \vee (\sim q) \)
- \( (\sim p) \vee (\sim q) \)
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The Correct Option is D
Solution and Explanation
Concept:
We use De Morgan's Law: \( \sim (A \wedge B) \equiv (\sim A) \vee (\sim B) \).
Step 1: Distribute the negation.
Apply negation to the terms inside the square brackets and flip the conjunction to a disjunction: \[ \sim [(\sim p) \wedge q] \equiv \sim (\sim p) \vee (\sim q) \]
Step 2: Simplify.
Using the Law of Double Negation, \( \sim (\sim p) \) becomes \( p \). So the expression becomes: \[ p \vee (\sim q) \] This matches Option (D).
Step 1: Distribute the negation.
Apply negation to the terms inside the square brackets and flip the conjunction to a disjunction: \[ \sim [(\sim p) \wedge q] \equiv \sim (\sim p) \vee (\sim q) \]
Step 2: Simplify.
Using the Law of Double Negation, \( \sim (\sim p) \) becomes \( p \). So the expression becomes: \[ p \vee (\sim q) \] This matches Option (D).
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