Prove the semantic entailemnt I am suppose to prove the following: ∀x(P(x)$\vee$Q(x)) !$\models$ ∀xP(x)$\vee$∀xQ(x) Note that the !$\models$ means "does not entail". I have the solution to the problem, but there is no clear explanation, and therefore I do not understand it. I would have thought that the formula on the left **does** entail the one on the right... 1. It must be the case that M ⊨ι[x→a] P (x) ∨ Q(x) holds for all a ∈ A. (∀x case) 2. It must be the case that M⊨ι[x→a]P(x) or M⊨ι[x→a]Q(x) holds for all a∈ A. (∨ case) 3. Cannot conclude that M ⊨ι xP(x) or M ⊨ι xQ (x) holds. (∀x case) Can someone explain to me why we cannot conclude the following, it does not make sense to me.

Let $P(x)$ be "$x$ is even", and $Q(x)$ be "$x$ is odd." It's true that for all natural numbers $x$, $x$ is either odd or even. But it's not true that all $x$ are odd or all $x$ are even.

In general, It's easier for all x to either have one property or the other. It's harder for property $P$ to hold for all x or another $Q$ to hold all $x$.