GATE | CS | 2019 | Discrete Mathematics | Logic | 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 ?