Question

Negation of the statement \((p \lor r) \implies (q \lor r)\) is :

• A. \(p \land q \land r\)
• B. \(\sim p \land q \land r\)
• C. \(p \land \sim q \land \sim r\)
• D. \(\sim p \land q \land \sim r\)

Hint:

A quick way to check logic is to use the principle: \((A \lor R) \land (\sim Q \land \sim R)\). Since we need both terms in the parenthesis to be true, and \(\sim R\) must be true, \(R\) must be false. If \(R\) is false, then \(A \lor R\) simplifies to just \(A\). Thus the statement becomes \(A \land \sim Q \land \sim R\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The negation of a conditional statement \(A \implies B\) is logically equivalent to \(A \land \sim B\).
Step 2: Key Formula or Approach:
1. \(\sim(A \implies B) \equiv A \land \sim B\)
2. De Morgan's Law: \(\sim(q \lor r) \equiv \sim q \land \sim r\)
Step 3: Detailed Explanation:
The given statement is \((p \lor r) \implies (q \lor r)\).
Using the property of negation of an implication: \[ \sim [(p \lor r) \implies (q \lor r)] \equiv (p \lor r) \land \sim(q \lor r) \] Apply De Morgan's Law to the second part: \[ \equiv (p \lor r) \land (\sim q \land \sim r) \] Using the associative and distributive laws: \[ \equiv ((p \lor r) \land \sim r) \land \sim q \] \[ \equiv ((p \land \sim r) \lor (r \land \sim r)) \land \sim q \] Since \(r \land \sim r\) is always False (\(F\)): \[ \equiv ((p \land \sim r) \lor F) \land \sim q \] \[ \equiv (p \land \sim r) \land \sim q \] \[ \equiv p \land \sim q \land \sim r \]
Step 4: Final Answer:
The negation is \(p \land \sim q \land \sim r\).