a) Let p(x, y) denote the open statement “x divides y,”where the universe for each of the...

a) Let p(x, y) denote the open statement “x divides y,”where the universe for each of the variables a, 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)

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ii) p(3, 27)

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iii) ∀y p(1, y)

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iv) ∀x p(x 0)

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v) ∀x p(x,x)

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vi) ∀y  p(x, y)

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vii) ∃y∀x p(x,y)

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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)

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ii) ∀y ∃x p(x, y)

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iii) ∃x ∀x p(x, y)

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iv) ∃y ∀x p(x, y)