Question

Let \(p, q, r\) be three statements. Then \( \sim (p \vee (q \wedge r)) \) is equal to

• A. \( (\sim p \wedge \sim q) \wedge (\sim p \wedge \sim r) \)
• B. \( (\sim p \vee \sim q) \wedge (\sim p \vee \sim r) \)
• C. \( (\sim p \wedge \sim q) \vee (\sim p \wedge \sim r) \)
• D. \( (\sim p \vee \sim q) \vee (\sim p \wedge \sim r) \)
• E. \( (\sim p \wedge \sim q) \vee (\sim p \vee \sim r) \)

Hint:

Apply De Morgan step-by-step from outermost brackets inward to avoid mistakes.

Answer 1

The Correct Option is B

Understanding the 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
\[ \boxed{(\sim p \vee \sim q) \wedge (\sim p \vee \sim r)} \]