Propositional logic-Predicates

Question

I have this problem in my Discrete structures course
Show why : ∀x P(x) ∨∀x Q(x) is not logically equivalent to ∀x(P(x)∨Q(x)) . Please help solve this

Answer 1

Let $P(x)$ denote the proposition that $x$ is even ($x \in \mathbb{Z}$) and $Q(x)$ denote the proposition that $x$ is odd. If the domain of discourse is $\mathbb{Z}$, then $$\forall x (P(x) \vee Q(x))$$ is true, but $\forall x P(x) \vee \forall x Q(x)$ is false.

Answer 2

A true statement:

For each integer $n$, $n$ is even or $n$ is odd.

Is the following statement true?

Either every integer is even, or every integer is odd.

The first is $\forall n\big(P(n)\lor Q(n)\big)$, where $P(n)$ is n is even and $Q(n)$ is n is odd, and the universe of discourse is the set of integers; the second is $\forall n\big(P(n)\big)\lor\forall n\big(Q(n)\big)$.