Given that $\forall xQ(x)$ is true, is $\exists xQ(x)$ also true?

Question

Given that $\forall xQ(x)$ is true, is $\exists xQ(x)$ also true?

Thanks in advance.

Answer 1

Yes—as long as the set that the universal quantifier $\forall$ is ranging over is nonempty. For exmaple, $\forall x\in[0,1]\ Q(x)$ does imply $\exists x\in[0,1]\ Q(x)$, but $\forall x\in\emptyset\ Q(x)$ does not imply $\exists x\in\emptyset\ Q(x)$.