Let L1 be a recursive language. Let L2 and L3 be the languages that are recursively enumerable but not recursive. Which of the following statements is not necessarily true?a)L2 - L1 is recursively enumerable.b)L1 - L3 is recursively enumerable.c)L2 ∩L1 is recursively enumerable.d)L2∪ L1 is recursively enumerable.Correct answer is option 'B'. Can you explain this answer?

Question

Let L1 be a recursive language. Let L2 and L3 be the languages that are recursively enumerable but not recursive. Which of the following statements is not necessarily true?a)L2 - L1 is recursively enumerable.b)L1 - L3 is recursively enumerable.c)L2 &cap;L1 is recursively enumerable.d)L2&cup; L1 is recursively enumerable.Correct answer is option 'B'. Can you explain this answer?

Answer

L₁ → recursive

L₂,L₃ → recursively enumerable but not recursive.

So L₁ can be recursive enumerable.

RE - RE = RE

So L₁ → L₃ can be recursive enumerable.

Answer

Explanation:

L1 - L3 is not necessarily recursively enumerable
- L1 is a recursive language, meaning it is decidable and has a corresponding Turing machine that halts on all inputs.
- L3 is a recursively enumerable language, which means there exists a Turing machine that can enumerate all strings in the language.
- The set difference operation (L1 - L3) may result in a language that is not recursively enumerable, as the complement of a recursively enumerable language is not necessarily recursively enumerable.
- Therefore, L1 - L3 is not necessarily recursively enumerable and option (b) is the correct answer.

Other Options:
- L2 - L1 is recursively enumerable: The set difference of a recursively enumerable language (L2) and a recursive language (L1) is always recursively enumerable.
- L2 ∩ L1 is recursively enumerable: The intersection of a recursive language and a recursively enumerable language is always recursively enumerable.
- L2 ∪ L1 is recursively enumerable: The union of a recursive language and a recursively enumerable language is always recursively enumerable.
Therefore, the correct answer is option (b) as L1 - L3 is not necessarily recursively enumerable.

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