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