GATE | GATE-CS-2014-(Set-1) | Question 11
Consider the statement
“Not all that glitters is gold”
Predicate glitters(x) is true if x glitters and predicate gold(x) is true if x is gold. Which one of the following logical formulae represents the above statement?
The statement “Not all that glitters is gold” can be expressed as follows :
¬(∀x(glitters(x)⇒gold(x)) … (1)
Where ∀x(glitters(x)⇒gold(x) refers that all glitters is gold. Now ,
∃x¬(glitters(x)⇒gold(x)) … (2) , Since we know ¬∀x() = ∃x¬()
(Where ∀ refers to -> All and ∃x refers to -> There exists some).
As we know, A⇒B is true only in the case that either A is false or B is true. It can also defined in the other way :
A⇒B=¬A∨B (negationA or B ) … (3)
From equation (2) and (3) , we have
⇒∃x(glitters(x)∧¬gold(x)) … (4) , Negation cancellation ¬(¬) = () : and ¬(()∨()) = (¬()∧¬()) .
So Answer is (D) .
This solution is contributed by Nirmal Bharadwaj.
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