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GATE CSE 2023 Q26: Statements that imply ∀x (P(x) ⇒ ∃y Q(x, y))

Question

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

∀x(P(x)  ⟹  ∃y Q(x,y)),\forall x \big(P(x) \implies \exists y\, Q(x, y)\big),

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

Try it: multiple select: one or more correct options

Select one or more options
Select all correct options. No negative marking, no partial marks.
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Answer: B, C (official GATE 2023 final answer key)

Concept used: An existential claim about one xx cannot imply a universal claim over all xx; ∃y∀x\exists y \forall x implies ∀x∃y\forall x \exists y.

First step: Discard options that only speak about some xx, then check that each remaining one gives a yy for every xx with P(x)P(x).

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Source: official GATE 2023 CS question paper and final answer key, organised by IIT Kanpur. Question paper · Answer key

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