GATE CS 2021 Set 1 — Question 11
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Theory of Computation → Context-Free Languages → Closure Properties (CFL)
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Question
Suppose that is a regular language and is a context-free language. Which one of the following languages is NOT necessarily context-free?
Correct answer
(C) L₁ - L₂
Solution
Let's analyze the closure properties for each option:
- (A) : The intersection of a regular language and a context-free language is always a context-free language.
- (B) : The concatenation of a regular language and a context-free language is always a context-free language.
- (D) : The union of a regular language and a context-free language is always a context-free language.
- (C) : This can be written as . Since context-free languages are not closed under complementation, is not necessarily context-free. Therefore, the intersection of a regular language and a non-context-free language is not necessarily context-free. For example, if and is a context-free language whose complement is not context-free, then is not context-free.
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