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GATE | GATE-CS-2016 (Set 2) | Question 37

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  • Difficulty Level : Hard
  • Last Updated : 11 Oct, 2021

Which one of the following well-formed formulae in predicate calculus is NOT valid?

(A) A
(B) B
(C) C
(D) D

Answer: (D)


Suppose if there are two statements P and Q,
   P=>Q = ~PvQ i.e.
   The only situation where implication fails is (=>) when P is true and Q is false. 
   i.e. A truth statement can't imply a false statement.
	So, for these type of questions it will be better to take option and 
	check for some arbitrary condition
   By looking options, we are pretty sure that A,B are correct
   Suppose X is any number and statement is P(x) = X is a prime number
                                            Q(x) = X is a non-prime number
   If we look at option D,
   Before Implication :- For all x, x is either prime or non-prime which is true
   After Implication  :-   For all x, x is prime or for all x, x is non-prime which is 
obviously false i.e. here, truth statement implies a false statement which is not valid.

   If we carefully look at option C,
   There exists a number x, which is both prime and non-prime which is false
   and a false statement can imply either true or false. So option (C) is correct
 So Answer is Option (D)

This explanation has been contributed by Anil Saikrishna.

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