The negation of the statement
(p v q) ∧ (q v(∼r)) is
(p v q) ∧ (q v(∼r)) is
- (p v r) ∧ (∼q)
- ((∼p v r)) ∧ (∼q)
- ((∼p ) v (∼q) v(∼r)
- ((∼p ) v (∼q) ∧(∼r)
The Correct Option is B
Solution and Explanation
Given the logical expression:
\[ (p \lor q) \land (q \lor (\sim r)) \]
Step 1: Taking the Negation
The negation of the expression is:
\[ \sim [(p \lor q) \land (q \lor (\sim r))] \]
Using De Morgan’s laws:
\[ \sim [(p \lor q) \land (q \lor (\sim r))] = \sim (p \lor q) \lor \sim (q \lor (\sim r)) \]
Step 2: Simplify Each Term
Simplify \( \sim (p \lor q) \):
\[ \sim (p \lor q) = \sim p \land \sim q \]
Simplify \( \sim (q \lor (\sim r)) \):
\[ \sim (q \lor (\sim r)) = \sim q \land r \]
Step 3: Combine Terms
Combine the simplified terms using the distributive property:
\[ \sim (p \lor q) \lor \sim (q \lor (\sim r)) = (\sim p \land \sim q) \lor (\sim q \land r) \]
Rewrite using distributive properties:
\[ (\sim p \lor r) \land (\sim q) \]
Final Answer:
The simplified negation of the given expression is:
\[ (\sim p \lor r) \land (\sim q) \]
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