Consider 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))
(A) IV only
(B) I and IV only
(C) II only
(D) II and III only


Answer: (B)

Explanation: Given, first order logic sentence
F: ∀x (∃y R(x, y)) with following sentences:
(i) ∃y (∃x R(x, y)) is true. Because we have ∀x (∃y R(x, y)) → ∃x (∃y R(x, y)) → ∃y (∃x R(x, y)).
(ii) ∃y (∀x R(x, y)) is false. Because we have ∀x (∃y R(x, y)) ← ∃y (∀x R(x, y)).
(iii) ∀y (∃x R(x, y)) is false. Because for ∃y can not imply ∀y.
(iv) ∼∃x (∀y ∼R(x, y)) is true. Because ∼∃x (∀y ∼R(x, y)) = ∀x (∼ ∃y ∼R(x, y)) = ∀x (∃y ∼∼R(x, y)) = ∀x (∃y R(x, y)).

This explanation is contributed by Mithlesh Upadhyay.

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