GATE | GATE CS 2011 | Question 30

Which one of the following options is CORRECT given three positive integers x, y and z, and a predicate?

        
      P(x) = ¬(x=1)∧∀y(∃z(x=y*z)⇒(y=x)∨(y=1))

(A) P(x) being true means that x is a prime number
(B) P(x) being true means that x is a number other than 1
(C) P(x) is always true irrespective of the value of x
(D) P(x) being true means that x has exactly two factors other than 1 and x


Answer: (A)

Explanation:

 So the predicate is evaluated as
    P(x) = (¬(x=1))∧(∀y(∃z(x=y*z)⇒((y=x)∨(y=1))))
 P(x) being true means x ≠ 1 and
 For all y if there exists a z such that x = y*z then
 y must be x (i.e. z=1) or y must be 1 (i.e. z=x)
 
 It means that x have only two factors first is 1 
 and second is x itself.
 
This predicate defines the prime number.

Source: http://clweb.csa.iisc.ernet.in/rahulsharma/gate2011key.html

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