GATE CS 2017 Set 2 — Question 4
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Theory of Computation → Context-Free Languages → Closure Properties (CFL)
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Question
Let be any two context-free languages and be any regular language. Then which of the following is/are CORRECT?I. is context-free.
II. is context-free.
III. is context-free.
IV. is context-free.
II. is context-free.
III. is context-free.
IV. is context-free.
Correct answer
(B) I and III only
Solution
I. Context-free languages (CFLs) are closed under union. So, is context-free. (Correct)
II. CFLs are NOT closed under complementation. So, may not be context-free. (Incorrect)
III. . Since is regular, its complement is also regular. The intersection of a CFL and a regular language is always a CFL. So, is context-free. (Correct)
IV. CFLs are NOT closed under intersection. So, may not be context-free. (Incorrect)
Therefore, only I and III are correct.
II. CFLs are NOT closed under complementation. So, may not be context-free. (Incorrect)
III. . Since is regular, its complement is also regular. The intersection of a CFL and a regular language is always a CFL. So, is context-free. (Correct)
IV. CFLs are NOT closed under intersection. So, may not be context-free. (Incorrect)
Therefore, only I and III are correct.
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