# GATE | GATE CS 2019 | Question 44

Consider the first order predicate formula:

`∀x [( ∀z z|x ⇒ ((z = x) ∨ (z = 1))) ⇒ ∃w(w > x) ∧ (∀z z⏐w ⇒ ((w = z) ∨ (z = 1)))] `

Here ‘a⏐b’ denotes that ‘a divides b’, where a and b are integers. Consider the following sets:

• S1: {1, 2, 3, …, 100}
• S2: Set of all positive integers
• S3: Set of all integers

Which of the above sets satisfy φ ?
(A) S1 and S3
(B) S2 and S3
(C) S1, S2 and S3
(D) S1 and S2

Explanation: Given predicate φ:

`∀x [( ∀z z|x ⇒ ((z = x) ∨ (z = 1))) ⇒ ∃w(w > x) ∧ (∀z z⏐w ⇒ ((w = z) ∨ (z = 1)))] `

It simply says that if z is a prime number in the set then there exists another prime number is the set which is larger.

Therefore, it can not be satisfy in finite, like 97 ∈ S1 there exists no prime number in the set which is greater.

Only sets S2 and S3 satisfy φ.

Option (B) is correct.

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