(a) Let p (x), q (x) be open statements in the variable x, with a given universe. Prove that
∀x p(x) ∨ ∀x q(x) ⇒ ∀x [p(x) ∨ q(x)].
[That is, prove that when the statement ∀x p(x) ∨ ∀x q(x) is true, then the statement ∀x [p(x) ∨ g(x)] is true.]
(b) Find a counterexample for the converse in part (a). That is, find open statements p(x), q (x) and a universe such that ∀x [p(x) ∨ q(x)] is true, while ∀x p(x) ∨ ∀x q(x) is false.