1. V ≡ (W $ \lor $ ∼X)
2. ∼Y ≡ ∼V
3. (W ⊃ Y) ⊃ Z
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?
A) V
B) Y
C) Z
D) W ⊃ Z
E) V ⊃ W

Question

Quizplus · Accepted Answer

From premise 2, we have ∼V ≡ ∼Y. Substituting ∼V for ∼Y in premise 3 gives (W ⊃ ∼V) ⊃ Z. We can use Material Implication rule to rewrite this as ∼(W ⊃ ∼V) ∨ Z. Now, using De Morgan's law, we get (W ∧ V) ∨ Z. Finally, from premise 1, we have V ≡ (W $ \lor $ ∼X). Substituting V for (W $ \lor $ ∼X) in the previous equation gives (W ∧ (W $ \lor $ ∼X)) ∨ Z. By Distribution rule, we get (W ∨ Z) ∧ ((W $ \lor $ ∼X) ∨ Z). Now, by Commutation rule, we can rewrite this as (W $ \lor $ ∼X) ∨ Z. Therefore, the proposition Z is derivable from the given premises.

1 V ≡ (W ∨ \lor ∨ ∼X) 2 ∼Y ≡ ∼V 3

1 V ≡ (W $\lor$ ∼X)
2 ∼Y ≡ ∼V
3