GATE | GATE CS 2020 | Question 49

Which one of the following predicate formulae is NOT logically valid ?

Note that W is a predicate formula without any free occurrence of x.
(A) ∀x(p(x) ∨ W) ≡ ∀x(px) ∨ W
(B) ∃x(p(x) ∧ W) ≡ ∃xp(x) ∧ W
(C) ∀x(p(x) → W) ≡ ∀xp(x) → W
(D) ∃x(p(x) → W) ≡ ∀xp(x) → W


Answer: (C)

Explanation: ∀x (p(x) → W) ≡ ∀x p(x) → W is wrong.

Since ∀x [p(x) → W]

≡ ∀x [¬p(x) ∨ W]
≡ ∀x (¬p(x) ∨ W
≡ ¬(∃x p(x)) ∨ W
≡ ∃x p(x) → W 

Option (C) is correct.

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