GATE CS 2021 Set 2 — Question 22
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Theory of Computation → Context-Free Languages → Closure Properties (CFL)
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Question
Let be a regular language and be a context-free language. Which of the following languages is/are context-free?
Correct answer
(B) L₁ ∪ L₂; (C) L₁ ∪ (L₂ ∪ L₂); (D) (L₁ ∩ L₂) ∪ (L₁ ∩ L₂)
Solution
Given: is regular and is context-free. Use the closure facts: regular languages are closed under complement; CFLs are closed under union and under intersection with a regular language, but not under complement.(A) — not necessarily context-free.
need not be a CFL. Taking gives , which can be non-context-free (e.g. the complement of is a CFL whose complement is not).(B) — context-free.
By De Morgan's law, , the intersection of a regular language with a CFL, which is always a CFL.(C) — context-free.
, so the language is , which is regular and hence context-free.(D) — context-free.
Factor out : , which is a CFL.Answer: (B), (C), (D)
need not be a CFL. Taking gives , which can be non-context-free (e.g. the complement of is a CFL whose complement is not).(B) — context-free.
By De Morgan's law, , the intersection of a regular language with a CFL, which is always a CFL.(C) — context-free.
, so the language is , which is regular and hence context-free.(D) — context-free.
Factor out : , which is a CFL.Answer: (B), (C), (D)
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