Geetha has a conjecture about integers, which is of the form \(\forall x \,…

GATE · 2023 · CS · Computer Science & IT

Geetha has a conjecture about integers, which is of the form

∀x (P(x)⇒∃y Q(x,y))\forall x \, (P(x) \Rightarrow \exists y \, Q(x, y))

where P is a statement about integers, and Q is a statement about pairs of integers. Which of the following (one or more) option(s) would imply Geetha’s conjecture?

  1. A.

    ∃x(P(x)∧∀yQ(x,y))∃x(P(x) ∧ ∀yQ(x, y))

  2. B.

    ∀x∀y Q(x,y)\forall x \forall y \, Q(x, y)

  3. C.

    ∃y∀x (P(x)⇒Q(x,y))\exists y \forall x \, (P(x) \Rightarrow Q(x, y))

  4. D.

    ∃x (P(x)∧∃y Q(x,y))\exists x \, (P(x) \land \exists y \, Q(x, y))

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