GATE Overflow | Mock GATE | Test 1 | Question: 46

Question

Which of the following statements is/are not correct?
(P) The class of all Turing Machines is countably infinite
(Q) The class of all DCFL's is countably infinite
(R) The class of all formal languages is uncountably infinite
(S) The set of all primes is countably infinite

• A. Only R
  
• B. Only R and S
  
• C. All are incorrect except P
  
• D. None of the above

Comments

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@Arjun 

Sir, I request to make more mock test of this level for GATE 2020

(Every question in this paper giving better clearity of concepts )

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(P) The class of all Turing Machines is countably infinite
(Q) The class of all DCFL's is countably infinite

is this 2 stmts meant that subset of RE/DCFL is countably infinite? like 2^RE , 2^CFL

Answer 1

All statements are correct.

D is the answer.

Comments

Set of prime numbers is a  subset of natural numbers and set of natural numbers is countably infinite, so S is also correct

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The rule you are talking about holds if the subset is finite.

An infinite subset of a countable infinite set maybe uncountable infinite.

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@gmrishikumar Any set is countably infinite if you can make one-to-one correspondence of its elements with natural numbers.

In case of prime numbers, you can say that -:

1st prime = 2

2nd prime = 3

3rd prime = 5.............and so on.

Thus, they are countably infinite.

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Updated the answer.

Answer 2

non recursively enumerable languages doesn't fall under formal language...

http://zasoby.open.agh.edu.pl/~11sustrojny/en/chomsky-classification/index.html

Chomsky created 4 classes we should obey Chomsky i guess

so class of all formal languages is countably infinite only..

and i didn't find anywhere its written as non r.e. is formal language... everywhere its 4 class upto r.e. only