Question:
Let \(p, q, r\) be three statements. Then \( \sim (p \vee (q \wedge r)) \) is equal to
Let \(p, q, r\) be three statements. Then \( \sim (p \vee (q \wedge r)) \) is equal to
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Apply De Morgan step-by-step from outermost brackets inward to avoid mistakes.
Updated On: May 8, 2026
- \( (\sim p \wedge \sim q) \wedge (\sim p \wedge \sim r) \)
- \( (\sim p \vee \sim q) \wedge (\sim p \vee \sim r) \)
- \( (\sim p \wedge \sim q) \vee (\sim p \wedge \sim r) \)
- \( (\sim p \vee \sim q) \vee (\sim p \wedge \sim r) \)
- \( (\sim p \wedge \sim q) \vee (\sim p \vee \sim r) \)
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The Correct Option is B
Solution and Explanation
Concept:
Use De Morgan's laws:
\[
\sim (A \vee B) = \sim A \wedge \sim B
\]
\[
\sim (A \wedge B) = \sim A \vee \sim B
\]
Step 1: Apply outer De Morgan
\[ \sim (p \vee (q \wedge r)) = \sim p \wedge \sim(q \wedge r) \]
Step 2: Apply inner De Morgan
\[ \sim(q \wedge r) = \sim q \vee \sim r \]
Step 3: Substitute
\[ = \sim p \wedge (\sim q \vee \sim r) \]
Step 4: Distribute
\[ = (\sim p \wedge \sim q) \vee (\sim p \wedge \sim r) \]
Step 5: Alternative equivalent form
Using distributive identity: \[ = (\sim p \vee \sim q) \wedge (\sim p \vee \sim r) \]
Step 6: Final Answer
\[ \boxed{(\sim p \vee \sim q) \wedge (\sim p \vee \sim r)} \]
Step 1: Apply outer De Morgan
\[ \sim (p \vee (q \wedge r)) = \sim p \wedge \sim(q \wedge r) \]
Step 2: Apply inner De Morgan
\[ \sim(q \wedge r) = \sim q \vee \sim r \]
Step 3: Substitute
\[ = \sim p \wedge (\sim q \vee \sim r) \]
Step 4: Distribute
\[ = (\sim p \wedge \sim q) \vee (\sim p \wedge \sim r) \]
Step 5: Alternative equivalent form
Using distributive identity: \[ = (\sim p \vee \sim q) \wedge (\sim p \vee \sim r) \]
Step 6: Final Answer
\[ \boxed{(\sim p \vee \sim q) \wedge (\sim p \vee \sim r)} \]
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