GATE | GATE CS 2008 | Question 48

Which of the following statements is false?
(A) Every NFA can be converted to an equivalent DFA
(B) Every non-deterministic Turing machine can be converted to an equivalent deterministic Turing machine
(C) Every regular language is also a context-free language
(D) Every subset of a recursively enumerable set is recursive


Answer: (D)

Explanation:
A language is recursively enumerable if there exists a Turing machine that accepts every string of the language, and does not accept strings that are not in the language.
Strings that are not in the language may be rejected or may cause the Turing machine to go into an infinite loop.

A recursive language can’t go into an infinite loop, it has to clearly reject the string, but a recursively enumerable language can go into an infinite loop.

So, every recursive language is also recursively enumerable.
Thus, the statement ‘Every subset of a recursively enumerable set is recursive’ is false.

 
Thus, option (D) is the answer.

 
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