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

To determine which option implies Geetha's conjecture, let's analyze each given statement in relation to her conjecture:

Geetha's conjecture is stated as:

\[ \forall x \big(P(x) \Rightarrow \exists y \, Q(x,y)\big) \]

This can be interpreted as: "For every integer \( x \), if \( P(x) \) is true, then there exists an integer \( y \) such that \( Q(x, y) \) is true."

Now, let's examine each option:

1. \(\exists x\big(P(x)\wedge \forall y\,Q(x,y)\big)\): This states that there exists an integer \( x \) such that \( P(x) \) is true and \( Q(x,y) \) is true for all \( y \). This does not imply Geetha's conjecture as it only guarantees the existence of one particular \( x \), not for all \( x \).
2. \(\forall x\,\forall y\,Q(x,y)\): This states that for all integers \( x \) and \( y \), \( Q(x, y) \) is true. This is a stronger statement than Geetha's conjecture because it implies that \( Q(x, y) \) is true irrespective of whether \( P(x) \) is true or false for all \( x \). Therefore, it indeed implies Geetha's conjecture.
3. \(\exists y\,\forall x\big(P(x)\Rightarrow Q(x,y)\big)\): This states there exists one \( y \) such that for all \( x \), if \( P(x) \) is true, then \( Q(x, y) \) is true. This is weaker than Geetha's conjecture because Geetha's conjecture allows \( y \) to depend on \( x \), while this option requires the same \( y \) for all \( x \).
4. \(\exists x\big(P(x)\wedge \exists y\,Q(x,y)\big)\): This simply states that there exists some integer \( x \) such that both \( P(x) \) and \( Q(x, y) \) are true. This does not imply Geetha's conjecture since it does not require \( Q(x, y) \) for all \( x \; P(x) \).

Based on the analysis above, the option \(\forall x\,\forall y\,Q(x,y)\) is the only one that implies Geetha's conjecture as it universally satisfies the condition without requiring any dependency on \( P(x) \).