UGC NET CSE | December 2015 | Part 2 | Question: 7

Question

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:

1. $\forall m \forall n P(m, n)$,
2. $\forall n P(1,n)$
3. $\exists m \forall n P(m,n)$
   • A. a-True, b-True, c-False
   • B. a-True, b-False, c-False
   • C. a-False, b-False, c-False
   • D. a-True, b-True, c-True

Answer 1

∀m ∀n P(m, n) says that every number divides every other number and result should be a postive integer.

• Clearly it is a false proposition.
• Eg: if m=10 n=3 10 divides 3 does not follow the proposition
• a is  false

 ∀n P(1, n) says that any positive integer is divisible by 1 and result will be a +ve integer.

• That is correct
• b is true

∃m∀nP(m,n) says that there are some +ve integers which divides any other +ve integer.

• The proposition is correct
• example: 1
• c is true

Correct option must be

a-- false                b--true          c--true

None of the given options follows

Answer 2

Here order of propositions a and c are reversed .

in the original question proposition a    was ∃m∀nP(m,n)  and C was  ∀m∀nP(m,n),

now A is true as for for all numbers n there exists some m that divides n (even for prime numbers 1 and n satisfy this)

B is clearly true as 1 divides everything 

C is false as all n are not divisible by all m

hence ans is A