(a) Let p(x, y) denote the open statement "x divides y," where the universe for each of the variables x, y comprises all integers. (In this context "divides" means "exactly divides" or "divides evenly.") Determine the truth value of each of the following statements; if a quantified statement is false, provide an explanation or a counterexample.
(i) p(3, 7)
(ii) p(3, 27)
(iii) ∀y p(1, y)
(iv) ∀x p(x, 0)
(v) ∀x p(x, x)
(vi) ∀y ∃x p(x, y)
(vii) ∃y ∀x p(x, y)
(viii) ∀x ∀y [(p(x, y) ∧ p(y, x)) → (x = y)]
(b) Determine which of the eight statements in part (a) will change in truth value if the universe for each of the variables x, y were restricted to just the positive integers.
(c) Determine the truth value of each of the following statements. If the statement is false, provide an explanation or a counterexample. [The universe for each of x, y is as in part (b).]
(i) ∀x ∃y p(x, y)
(ii) ∀y ∃x p(x, y)
(iii) ∃x ∀y p(x, y)
(iv) ∃y ∀x p(x, y)