GATE | GATE-CS-2005 | Question 40

Let P, Q and R be three atomic prepositional assertions. Let X denote (P v Q) → R and Y denote (P → R) v (Q → R). Which one of the following is a tautology?
(A) X ≡ Y
(B) X → Y
(C) Y → X
(D) ¬ Y → X


Answer: (B)

Explanation:

Answer : [ B ]
X = (P ⋁ Q) → R
  = ~(P ⋁ Q) ⋁ R

  = (~P ⋀ ~Q) ⋁ R

  = (~P ⋁ R) ⋀ (~Q ⋁ R)

  = (P → R) ⋀ (Q → R)

X → Y is true as (A ⋀ B) → (A ⋁ B) is always TRUE
but reverse implication is not always true. 


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