Open In App

GATE | GATE CS 2013 | Question 47

Like Article
Like
Save
Share
Report

Which one of the following is NOT logically equivalent to ¬∃x(∀y(α)∧∀z(β))?

(A)

∀x(∃z(¬β)->∀y(α))

(B)

∀x(∀z(β)->∃y(¬α))

(C)

∀x(∀y(α)->∃z(¬β))

(D)

∀x(∃y(¬α)->∃z(¬β))



Answer: (A)

Explanation:

  Given statement is : ¬ ∃ x ( ∀y(α) ∧ ∀z(β) ) 

where Â¬ is a negation operator, âˆƒ is Existential Quantifier with the 
meaning of \"there Exists\", and âˆ€ is a Universal Quantifier 
with the meaning   \" for all \" , and Î±, Î² can be treated as predicates.

here we can apply some of the standard 
results of Propositional and 1st order logic on the given statement, 
which are as follows :

[ Result 1 : ¬(∀x P(x)) <=> ∃ x¬P(x), i.e. negation 
of \"for all\" gives \"there exists\" and negation also gets applied to scope of 
quantifier, which is P(x) here. And also negation of \"there exists\" gives \"for all\", 
and negation also gets applied to scope of quantifier  ]

[ Result 2 :  Â¬ ( A âˆ§ B ) = ( ¬A  âˆ¨ ¬B )  ]

[ Result 3 :  Â¬P  âˆ¨ Q <=> P -> Q ]

[ Result 4 : If P ->Q, then by Result of Contrapositive,  Â¬Q -> Â¬P  ]

Now we need to use these results as shown below:

 

¬ ∃ x ( ∀y(α) ∧ ∀z(β) )                 [ Given ]

=> âˆ€ x (¬∀y(α) ∨ Â¬âˆ€z(β) )          [ after applying Result 1 & Result 2 ]

=> âˆ€ x ( ∀y(α) -> ¬∀z(β) )     [after applying Result 3 ]

=> âˆ€ x ( ∀y(α) -> ∃z(¬β) )      [after applying Result 1]

which is same as the statement C. 

Hence the Given Statement is logically Equivalent
to the statement C.

Now, we can also prove that given statement is logically equivalent to the statement
 in option  B.

Let\'s see how !

The above derived statement is :

∀ x ( ∀y(α) -> ∃z(¬β) )

Now this statement can be written as (or equivalent to) :

=> ∀ x ( ∀z(β) -> ∃y(¬α) )     [after applying Result 4 ]

And this statement is same as statement B. 
Hence the Given statement is also logically equivalent 
to the statement B.

So, we can conclude that the Given statement is NOT logically equivalent to the 
statements A and D.

Hence, the correct answer is Option A and Option D. But in GATE 2013, marks were given to all for this question.


Quiz of this Question
Please comment below if you find anything wrong in the above post


Last Updated : 28 Jun, 2021
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads