Project ForgeGeetha 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?
(A) ∃x(P(x) ∧ ∀y Q(x,y)) (B) ∀x∀y Q(x,y) (C) ∃y∀x(P(x) ⟹ Q(x,y)) (D) ∃x(P(x) ∧ ∃y Q(x,y))
- A∃x(P(x) ∧ ∀y Q(x,y))
- ∀x∀y Q(x,y)
- ∃y∀x(P(x) ⟹ Q(x,y))
- D∃x(P(x) ∧ ∃y Q(x,y))
Explanation
(A) and (D) both only assert something about ONE specific x (existentially), saying nothing about every other x — they can't imply a universally-quantified claim about ALL x. (B): if Q(x,y) holds unconditionally for every x and y, then in particular for any x with P(x) true, Q(x,y) holds for every y, so ∃y Q(x,y) trivially follows — this implies the conjecture. (C): if a single fixed y makes P(x)⟹Q(x,y) true for every x, then whenever P(x) holds for some x, Q(x,y) holds for that fixed y — so ∃y' Q(x,y') holds (witnessed by y'=y) — this also implies the conjecture. So (B) and (C) work, (A) and (D) don't.
Source: GATE CS 2023 (2023)
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