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If the truth value of the statement (P ∧ (~R)) → ((~R) ∧ Q) is F, then the truth value of which of the following is F? (1) P ∨ Q → ~R (2) R ∨ Q → ~P (3) ~(P ∨ Q) → ~R (4) ~(R ∨ Q) → ~P

1. P ∨ Q → ~R
2. R ∨ Q → ~P
3. ~(P ∨ Q) → ~R
4. ~(R ∨ Q) → ~P

Correct answer: ~(R ∨ Q) → ~P

Solution

The statement (P ∧ (~R)) → ((~R) ∧ Q) is false only when P is true and R is false, which leads to the conclusion that if R is false, then P must also be false for the implication to hold. Thus, the statement ~(R ∨ Q) → ~P must also be false, as it asserts that if neither R nor Q is true, then P must be false, contradicting the earlier condition.

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