How to show that the following formula isn't valid in first order logic?

Question

I'm trying to understand how I can solve this task. I need to show that the following formula isn't valid in first order logic:

∀x(Px∨Qx) → ∀xPx∨∀xQx

I would appreciate if someone could show me how I can solve it. Thank you very much

Answer 1

I would appreciate if someone could show me how I can solve it.

The formula $\forall x~(Px\vee Qx)\to ((\forall x~Px)\vee (\forall x~Qx))$ may be declared invalid if there is some interpretation for the domain and predicates where where $\forall x~(Px\vee Qx)$ is true but neither $\forall x~Px$ nor $\forall x~Qx$ are true (ie: both are false).

Thus what you need to do is to show that this is so.   You should try constructing such an interpretation; a counterexample — a domain with predicates P and Q such that: everything is either P or Q, but something is not P and something is not Q.