If the universe of disclosure is set of integers, then which of the followings…

Correct answer: the three true statements are the ones asserting (1) for every integer n there exists an integer m with n^2 < m, (2) there exists an integer n such that n < m^2 for all integers m, and (3) there exists an integer n such that n·m = m for all integers m.

• Statement: For every integer n there exists an integer m with n^2 < m. True. Proof: take m = n^2 + 1, which is an integer greater than n^2.

• Statement: There exists an integer n such that for every integer m, n < m^2. True. Proof: choose n = -1. Since m^2 ≥ 0 for all integers m, we have -1 < m^2 for every m.

• Statement: There exists an integer n such that for every integer m, n·m = m. True. Proof: n = 1 satisfies 1·m = m for all integers m.

• Statement: There exist integers n and m with n^2 + m^2 = 6. False. Explanation: integer squares are 0,1,4,9,...; no pair of these sums to 6.

• Statement: There exist integers n and m with n + m = 4 and n - m = 1. False. Explanation: solving gives n = 5/2 and m = 3/2, which are not integers.

Therefore the correct choice is the one listing the first three statements (the inequalities and the multiplicative identity case); the other two statements are false.