GATE CS 2026 Set 1 — Question 51
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Theory of Computation → Context-Free Languages → Closure Properties (CFL)
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Question
Let and be two languages over a finite alphabet, such that and are regular languages.Which of the following statements is/are always true?
Correct answer
(C) L₂ is context-free
Solution
Given that is a regular language. The class of regular languages is closed under the complement operation. Therefore, is also a regular language. Since every regular language is a context-free language (Regular CFL), is necessarily context-free. Thus, statement (C) is always true.To see why the other statements are not necessarily true, consider the following counter-example:
Let , which is a regular language.
Let , which is a known non-context-free language (and hence non-regular).
Then:
Let , which is a regular language.
Let , which is a known non-context-free language (and hence non-regular).
Then:
1., which is a regular language. Both conditions of the question are satisfied.
2.(A) is regular: False, as is not regular in this case.
3.(B) is regular: False, as , which is not regular.
4.(D) is context-free: False, as is not context-free.
Therefore, only statement (C) is always true.Continue learning with Success Tracker
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