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Consider the following types of languages: L1: Regular, L2: Context-free, L3: Recursive, L4: Recursively enumerable. Which of the following is/are TRUE? I. L̅3 ∪ L4 is recursively enumerable II. L̅2 ∪ L3 is recursive III. L1* ∩ L2 is context-free IV. L̅1 ∪ L̅2 is context-free

1. I only
2. I and III only
3. I and IV only
4. I, II and III only

Correct answer: I, II and III only

Solution

The correct options are true because: I states that the complement of a recursive language combined with a recursively enumerable language is recursively enumerable, which is accurate; II indicates that the union of the complement of a context-free language and a recursive language is recursive, which holds true; III asserts that the intersection of a regular language (which is closed under intersection) and a context-free language results in a context-free language, which is also correct.

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