Given the following statements: (i) Recursive enumerable sets are closed under complementation. (ii) Recursive sets are closed under complements. Which is/are the correct statements?a)I onlyb)II onlyc)Both I and IId)Neither I nor IICorrect answer is option 'B'. Can you explain this answer?

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Answer

Explanation:

Recursive sets:
- Recursive sets are sets that can be decided by a Turing machine that always halts.
- They are also known as decidable sets because there exists an algorithm that can determine membership in these sets.

Recursive enumerable sets:
- Recursive enumerable sets are sets for which there exists a Turing machine that will enumerate their elements.
- These sets may not halt for elements that are not in the set, but will eventually list all elements that are in the set.

Statement (i): Recursive enumerable sets are closed under complementation.
- This statement is true because if a set is recursively enumerable, then its complement is also recursively enumerable.
- The complement of a recursively enumerable set can be generated by running the Turing machine for the set and outputting elements not in the set.

Statement (ii): Recursive sets are closed under complements.
- This statement is false because recursive sets are not necessarily closed under complementation.
- A recursive set may not have a recursive complement, meaning there may not be a Turing machine that can decide membership in the complement.

Therefore, the correct statement is II only because recursive sets are not always closed under complements, while recursive enumerable sets are closed under complementation.

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