Consider the first-order logic sentence \(F:\forall x(\exists yR(x,y))\).…

GATE · 2017 · CS · Set 1 · Computer Science & IT

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

I.    ∃y(∃xR(x,y))\exists y(\exists xR(x,y))

II    ∃y(∀xR(x,y))\exists y(\forall xR(x,y))

III.    ∀y(∃xR(x,y))\forall y(\exists xR(x,y))

IV.    ¬∃x(∀y¬R(x,y))¬\exists x(\forall y¬R(x,y))

  1. A.

    IV only

  2. B.

    I and IV only

  3. C.

    II only

  4. D.

    II and III only

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