Gate2017 cs Q28

28Consider the first-order logic sentence
F: ∀ x (∃ y R(x,y)).
Assuming non-empty logical domains, which of the sentences below are implied by F?
I. ∃y (∃x R(x,y)) II. ∃y (∀x R(x,y)) III. ∀y (∃x R(x,y)) IV. ∼∃x (∀y ~R(x,y))

Option : B
Explanation :
∀x(∃yR(x,y)) ⇒ ∃y∃xR(x,y)
∃y∀xR(x,y) ⇒ ∀x∃yR(x,y)
∀x∃yR(x,y) ⇏ ∃y∀xR(x,y)
-∃x(∀y - R(x,y)) ⇔ ∀x∃yR(x,y)

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