1. ∼R ≡ S
2. T ⊃ R
3. ∼(R $ \lor $ T) ⊃ U
-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) S ⊃ R
B) S ⊃ T
C) S ⊃ (R • T)
D) S ⊃ U
E) S ⊃ (R • U)

Question

Quizplus · Accepted Answer

From premise 1, we have ∼R ≡ S.

Using Material Implication, we can convert premise 2 to ∼T ∨ R.

By applying DeMorgan's Law on premise 3, we get R • T ⊃ U.

Combining everything together, we can derive:

∼T ∨ (∼R ≡ S) (premise 1)
∼T ∨ (∼S ≡ R) (Material Equivalence on premise 1)
(∼T ∨ ∼S) ≡ R (Commutativity and Associativity of ≡)
R ≡ (∼T ∨ ∼S) (Equivalent Rearrangement)
(R ≡ ∼T) ∨ (R ≡ ∼S) (Material Equivalence)
∼(R • T) ⊃ ∼U (Contrapositive of premise 3)
(R • T) ∨ ∼U (Material Implication)
(R ≡ ∼T) ∨ U (Distributivity of ≡ over ∨)
∼S ∨ U (Substitution of ∼T for R in previous step)
S ⊃ U (Material Implication)

Therefore, choice D is the correct answer.

1 ∼R ≡ S 2 ∨ \lor ∨ T) ⊃ U -Which of the Following Propositions Is Derivable

1 ∼R ≡ S
2 $\lor$ T) ⊃ U
-Which of the Following Propositions Is Derivable