The problem revolves around the validity of the logical sentence:

$(\exists x P(x) \land \exists x Q(x)) \rightarrow \exists x (P(x) \land Q(x))$

We need to determine under which conditions, if any, this sentence could be invalid. Let's break down each option:

1. Understanding the statement:
   • The left side of the implication, $\exists x P(x) \land \exists x Q(x)$, means there exists at least one $x$ where $P(x)$ is true and at least one (possibly different) $x$ where $Q(x)$ is true.
   • The right side of the implication, $\exists x (P(x) \land Q(x))$, means there exists an $x$ for which both $P(x)$ and $Q(x)$ are simultaneously true.

For this implication to be true, if we can find any interpretation where $\exists x P(x)$ and $\exists x Q(x)$ are true but no single $x$ makes both $P(x)$ and $Q(x)$ true, then the sentence would be false.

Let's evaluate each option with this logic in mind:

• Option A claims the sentence is valid (always true). This is incorrect because there are cases where $P(x)$ and $Q(x)$ hold for different values of $x$, making it impossible for a single $x$ to satisfy both $P(x) \land Q(x)$.

• Option B proposes that the sentence is false under the interpretation where $P(x)$ is $x < 3$ and $Q(x)$ is $x > 6$. In this case:

  • There exist values of $x$ such that $P(x)$ is true (e.g., $x = 1, 2$) and values such that $Q(x)$ is true (e.g., $x = 7, 8$), but there is no single $x$ that satisfies both $x < 3$ and $x > 6$.
  • Therefore, $\exists x P(x)$ and $\exists x Q(x)$ are both true, but $\exists x (P(x) \land Q(x))$ is false, making the statement invalid under this interpretation. This option is correct.

• Option C suggests $P(x)$ is $x > 3$ and $Q(x)$ is $x > 6$. However, for values $x > 6$, both $P(x)$ and $Q(x)$ are true for any such $x$. This would make the sentence true, so Option C is incorrect.

• Option D uses the conditions $P(x)$ as $x < -3$ and $Q(x)$ as $x > 6$. Similar to Option B, there would be no overlap between values of $x$ satisfying $P(x)$ and $Q(x)$, making $\exists x (P(x) \land Q(x))$ false. This interpretation would also make the sentence invalid, but it does not match the specific counterexample given in Option B. Option D is also incorrect.

Conclusion: The correct answer is B.

Would you like further details or clarification?

Here are five related questions to deepen understanding:

1. What would change if we replaced $\exists$ with $\forall$ in this sentence?
2. How does the interpretation of quantifiers affect the truth of logical sentences?
3. Can we construct other pairs of predicates $P(x)$ and $Q(x)$ that invalidate the sentence?
4. What is the impact of the domain (positive integers, real numbers) on logical statements?
5. How can Venn diagrams help visualize the truth of such logical statements?

Tip: When analyzing logical implications, always consider if a counterexample exists, as finding just one invalidating case disproves the general validity.