Question

Geetha has a conjecture about integers, of the form \[ \forall x \big(P(x) \Rightarrow \exists y \, Q(x,y)\big) \] where \(P\) is a statement about integers and \(Q\) is a statement about pairs of integers. Which of the following option(s) would imply Geetha's conjecture?

• A. \(\exists x\big(P(x)\wedge \forall y\,Q(x,y)\big)\)
• B. \(\forall x\,\forall y\,Q(x,y)\)
• C. \(\exists y\,\forall x\big(P(x)\Rightarrow Q(x,y)\big)\)
• D. \(\exists x\big(P(x)\wedge \exists y\,Q(x,y)\big)\)

Hint:

To test whether a stronger statement implies \(\forall x\,(P(x)\Rightarrow \exists y\,Q(x,y))\), ask: for each \(x\) with \(P(x)\), can I point to at least one \(y\) (possibly depending on \(x\)) making \(Q(x,y)\) true?
Statements that provide a global} witness \(y\) for all such \(x\) (as in (C)) or make \(Q\) true for all \(x,y\) (as in (B)) will imply the target; statements about only some} \(x\) do not.

Answer 1

The Correct Option is B

Target statement (Geetha's conjecture): \(\forall x\big(P(x)\Rightarrow \exists y\,Q(x,y)\big)\).
This reads: for every \(x\), if \(P(x)\) holds, then there exists (possibly \(x\)-dependent) \(y\) such that \(Q(x,y)\) holds.
Check (A): \(\exists x\big(P(x)\wedge \forall y\,Q(x,y)\big)\).
This asserts that there is some single \(x_0\) for which \(P(x_0)\) holds and \(Q(x_0,y)\) holds for all \(y\).
It provides no information about other values of \(x\) where \(P(x)\) might be true.
Therefore (A) does not imply the target \(\forall x\big(P(x)\Rightarrow \exists y\,Q(x,y)\big)\). \(\Rightarrow\) False.
Check (B): \(\forall x\,\forall y\,Q(x,y)\).
If \(Q(x,y)\) is true for all \(x\) and all \(y\), then in particular for any \(x\) with \(P(x)\) true, we can choose any \(y\) and \(Q(x,y)\) will hold.
Hence \(\forall x\big(P(x)\Rightarrow \exists y\,Q(x,y)\big)\) follows immediately. \(\Rightarrow\) True.
Check (C): \(\exists y\,\forall x\big(P(x)\Rightarrow Q(x,y)\big)\).
This says there exists a single fixed \(y_0\) such that for all \(x\), if \(P(x)\) then \(Q(x,y_0)\).
Then for each \(x\) with \(P(x)\), we can witness the existential in the target by choosing that same \(y_0\).
Thus \(\forall x\big(P(x)\Rightarrow \exists y\,Q(x,y)\big)\) holds. \(\Rightarrow\) True.
Check (D): \(\exists x\big(P(x)\wedge \exists y\,Q(x,y)\big)\).
This only guarantees one particular \(x_0\) with \(P(x_0)\) and some \(y_0\) s.t.\ \(Q(x_0,y_0)\).
It gives no guarantee for other \(x\) where \(P(x)\) might hold.
Therefore (D) does not imply the target universal conditional. \(\Rightarrow\) False.
\[ \boxed{\text{Options that imply Geetha's conjecture: (B) and (C).}} \]