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GATE CS 2017 Set 1 — Question 2

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MCQ+1 / -0.33MediumPredicate Logic & QuantifiersMathematical LogicEngineering Mathematics

Engineering Mathematics → Mathematical Logic → Predicate Logic & Quantifiers

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Question

Consider the first-order logic sentence F:∀x(∃yR(x,y))F: \forall x(\exists y R(x, y)). Assuming non-empty logical domains, which of the sentences below are implied by FF?
I. ∃y(∃xR(x,y))\exists y(\exists x R(x, y))
II. ∃y(∀xR(x,y))\exists y(\forall x R(x, y))
III. ∀y(∃xR(x,y))\forall y(\exists x R(x, y))
IV. ¬∃x(∀y¬R(x,y))\neg \exists x(\forall y \neg R(x, y))
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