Given ∑ = {a, b}, which of the following sets is not countable ?
(A) Set of all strings over ∑
(B) Set of all languages over ∑
(C) Set of all regular languages over ∑
(D) Set of all languages over ∑ accepted by Turing machines
Answer: (B)
Explanation:
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(A) The set ∑ ={a, b} is countable because each element of this set can be mapped with natural number and also generated in the following order:
Given ∑ ={a, b}. So, order will be a,b,aa,ab,ba,bb,aaa,aab … Therefore it will mapped with natural number. Hence it is countable. - (B) Here, we see that set of languages over z is the power set of strings over ∑ which is an infinite set and As we know that power set of an infinite set is uncountable. Hence the set of languages becomes an uncountable set a, so we can prove this using Cantor’s diagonalisation method.
- (C) The set of all regular languages is a subset of the set of all recursively enumerable languages. And we know that a subset of a countable set is always countable.
- (D) Set of all language over ∑ accepted by Turing machine is countable.
Option (B) is correct.
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