Let p(x), q(x), and r(x) be the following open statements.
p(x): x2 - 7x + 10 = 0
q(x): x2 - 2x - 3 = 0
r(x): x < 0
(a) Determine the truth or falsity of the following statements, where the universe is all integers. If a statement is false, provide a counterexample or explanation.
(i) ∀x [p(x) → ¬r(x)]
(ii) ∀x [q(x) → r(x)]
(iii) ∃x[q(x) → r(x)]
(iv) ∃x [p(x) → r(x)]
(b) Find the answers to part (a) when the universe consists of all positive integers.
(c) Find the answers to part (a) when the universe contains only the integers 2 and 5.