(Solution) - a Let p x y denote the open statement x divides -(2025 Original AI-Free Solution)

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(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)