Discussion
Let P(m, n) be the statement "m divides n" where the universe of discourse for both the variables is the set of positive integers. Determine the truth values of each of the following propositions :
I. ∀m ∀n P(m, n),
II. ∃m ∀n P(m, n)
Answer: C
Statement I says that every m divides every n, which is false, since for instance 2 does not divide 3. Statement II says that there is an m that divides every n. Take m = 1, which divides every positive integer, so II is true.
So I is false and II is true. The options calling both true or both false are wrong, and the one that says I true and II false has them the wrong way round.
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