Let p(n), q(n) represent the open statementsp(n):n is odd q(n): n2 is oddfor the universe...

Let p(n), q(n) represent the open statements

p(n):n is odd q(n): n2 is odd

for the universe of all integers. Which of the following statements are logically equivalent to each other?

a) If the square of an integer is odd, then the integer is odd.

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b) ∀n [p(n) is necessary for q(n]

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c) The square of an odd integer is odd.

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d) There are some integers whose squares are odd.

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e) Given an integer whose square is odd, that integer is likewise odd.

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f) ∀n [¬p(x)→ ¬q(n)]

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g) ∀n [p(n) is sufficient for q(n)]