1. Q ⊃ ∼(R $ \lor $ S)
2. ∼R ⊃ T
3. U ⊃ S
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?
A) Q ⊃ (T • ∼U)
B) Q ⊃ (R • S)
C) Q ⊃ (S • ∼T)
D) Q ⊃ (∼T • ∼U)
E) Q ⊃ (U • T)

Question

Quizplus · Accepted Answer

Using Modus Tollens, we can derive ∼(R \lor S) ⊃ ∼Q from premise 1. Then, using contrapositive, we can derive Q ⊃ (R \lor S) from that. From premise 2, we can derive ∼T ⊃ ∼∼R or ∼T ⊃ R. Using Modus Ponens with premise 3, we can derive U ⊃ S. By combining these three derived propositions, we can use a series of steps and rules to arrive at Q ⊃ (T • ∼U).

1 Q ⊃ ∼(R ∨ \lor ∨ S) 2 ∼R ⊃ T 3

1 Q ⊃ ∼(R $\lor$ S)
2 ∼R ⊃ T
3