Geetha has a conjecture about integers, which is of the form $$\forall x\Big(P(x) \implies \exists y Q(x,y)\Big),$$ 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?

Geetha's conjecture is given by:

We need to determine which statement $S$ logically implies $G$ (i.e., $S \implies G$ is valid):

• Option B ($\forall x \forall y Q(x, y)$): If for all $x$ and all $y$, $Q(x, y)$ is true, then for every $x$, there definitely exists at least one $y$ such that $Q(x, y)$ is true (since the domain of integers is non-empty). Thus, the consequent $\exists y Q(x, y)$ is always true for every $x$, making the conditional $P(x) \implies \exists y Q(x, y)$ true for all $x$. Therefore, $\forall x \forall y Q(x, y) \implies G$.

• Option C ($\exists y \forall x (P(x) \implies Q(x, y))$): Suppose there exists an integer $y_0$ such that for all $x$, $P(x) \implies Q(x, y_0)$. Then for any arbitrary $x$, if $P(x)$ holds, $Q(x, y_0)$ must hold. Since $y_0$ is a witness, $\exists y Q(x, y)$ holds. Thus, for every $x$, $P(x) \implies \exists y Q(x, y)$. Therefore, $\exists y \forall x (P(x) \implies Q(x, y)) \implies G$.

• Options A and D ($\exists x (\dots)$): Both state existential properties for only *some* integer $x$. An existential statement about a single element cannot guarantee that a condition holds universally for *all* integers $x$. Hence, neither implies $G$.

Thus, the correct options are B and C.