Question

If the truth value of the statement
(P∧(∼R))→((∼R)∧Q)
is F, then the truth value of which of the following is F?

• A. P∨Q→∼R
• B. R∨Q→∼P
• C. ∼(P∨Q)→∼R
• D. ∼(R∨Q)→∼P

Answer 1

The Correct Option is D

To solve this problem, we need to determine under what condition the truth value of the statement \((P \land (\neg R)) \rightarrow ((\neg R) \land Q)\) is False. Then we compare this with each of the provided options to find the one that also evaluates to False.

1. Recall that the implication \(A \rightarrow B\) (read as "A implies B") is False only when \(A\) is True and \(B\) is False.
2. For the given statement \((P \land (\neg R)) \rightarrow ((\neg R) \land Q)\) to be False:
   • \(P \land \neg R\) must be True.
   • \((\neg R) \land Q\) must be False.
3. Break down these conditions:
   • \(P \land \neg R\) being True implies \(P = \text{True}\) and \(\neg R = \text{True}\) (hence, \(R = \text{False}\)).
   • \((\neg R) \land Q = \text{False}\) implies \(\neg R = \text{True}\) (hence, \(R = \text{False}\)) and \(Q = \text{False}\).
4. Therefore, the condition for False evaluation is \(P = \text{True}\), \(R = \text{False}\), and \(Q = \text{False}\).
5. Evaluate each option under these conditions:
   • Option 1: \(P \lor Q \rightarrow \neg R\)
     • Substitute values: \(\text{True} \lor \text{False} \rightarrow \text{True}\) simplifies to \(\text{True} \rightarrow \text{True}\) (True).
   • Option 2: \(R \lor Q \rightarrow \neg P\)
     • Substitute values: \(\text{False} \lor \text{False} \rightarrow \text{False}\) simplifies to \(\text{False} \rightarrow \text{False}\) (True).
   • Option 3: \(\neg(P \lor Q) \rightarrow \neg R\)
     • Substitute values: \(\neg(\text{True} \lor \text{False}) \rightarrow \text{True}\) simplifies to \(\text{False} \rightarrow \text{True}\) (True).
   • Option 4: \(\neg(R \lor Q) \rightarrow \neg P\)
     • Substitute values: \(\neg(\text{False} \lor \text{False}) \rightarrow \text{False}\) simplifies to \(\text{True} \rightarrow \text{False}\) (False).
   • This is the correct answer as it matches the truth value condition.

Therefore, the correct answer is \(\neg(R \lor Q) \rightarrow \neg P\).