Let P(m, n) be the statement “m divides n” where the universe of discourse for…

2013

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)

  1. A.

    Both I and II are true

  2. B.

    Both I and II are false

  3. C.

    I – false & II – true

  4. D.

    I – true & II – false

Attempted by 117 students.

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Correct answer: C

Answer: I is false; II is true.

  1. I: The statement ∀m ∀n P(m,n) means every positive integer m divides every positive integer n. This is false. Counterexample: take m = 2 and n = 3; 2 does not divide 3.

  2. II: The statement ∃m ∀n P(m,n) asks whether there exists a positive integer m that divides every positive integer n. This is true because m = 1 divides every positive integer.

Therefore, the correct truth values are: first statement false, second statement true.

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