Let P(m,n) be the statement "m divides n" where the universe of discourse for…
2015
Let P(m,n) be the statement "m divides n" where the universe of discourse for both the variable is the set of positive integers. Determine the truth values of each of the following propositions:
(a) \(\forall m \forall n P(m, n)\) (b) \(\forall n P(1,n)\) (c) \(\exists m \forall n P(m,n)\)
Codes :
- A.
(a) False, (b) True, (c) True
- B.
(a) True, (b) False, (c) False
- C.
(a) False, (b) False, (c) False
- D.
(a) True, (b) True, (c) True
Attempted by 43 students.
Show answer & explanation
Correct answer: A
Answer: (a) False, (b) True, (c) True
For (a): False. The statement asserts that every positive integer m divides every positive integer n. This is not true: for example, take m = 2 and n = 3; 2 does not divide 3, so the universal claim fails.
For (b): True. The statement says that 1 divides every positive integer n. This holds because for any positive integer n we have n = 1 * n, so 1 divides n.
For (c): True. The statement asks whether there exists a positive integer m that divides every positive integer n. Choosing m = 1 works because 1 divides all positive integers, so the existential-universal statement is satisfied.
Therefore the correct truth values are: (a) false, (b) true, (c) true.
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