a) Let p(x),q(x) be open statements in the variable x, with a given universe. Prove that[T...

a) Let p(x),q(x) be open statements in the variable x, with a given universe. Prove that

[That is, prove that when the statement ∀x p(x) ˅ ∀x q(x) is true, then the statement ∀x [p(x) ˅ q(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.