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Geetha has a conjecture about integers, which is of the form ∀x (P(x) ⇒ ∃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. ∃x (P(x) ∧ ∀y Q(x,y))
2. ∀x∀y Q(x,y)
3. ∃y∀x (P(x) ⇒ Q(x,y))
4. ∃x (P(x) ∧ ∃y Q(x,y))

Correct answer: ∀x∀y Q(x,y)

Solution

The option ∀x∀y Q(x,y) implies that for every integer x, the statement Q holds for all integers y, which directly satisfies the condition of Geetha's conjecture that for each x where P(x) is true, there exists at least one y such that Q(x,y) is also true.

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