vacuously true for empty set

Question

I don't understand why if $D$ is an empty set, then $P(x)\rightarrow Q(x) $ is true. My understanding is, if $D$ is an empty set, then there is no $x$ to choose, so we can't tell truth values of $P(x)$ or $Q(x)$, because $x$ is empty

Answer 1

The statement $\forall x\in\varnothing, P(x)\rightarrow Q(x)$ isn't false. Indeed, if it were false, then its contrary $$\exists x\in\varnothing, P(x)\wedge \neg Q(x)$$ would be true. Since $\varnothing$ is empty, such an $x$ can't exist, hence the contrary is false and the first statement is (vacuously) true.