GATE | GATE-CS-2016 (Set 2) | Question 28

Consider the following types of languages:

L1 Regular,
L2: Context-free,
L3: Recursive,
L4: Recursively enumerable. 

Which of the following is/are TRUE?

I.   L3' U L4 is recursively enumerable
II.  L2  U L3 is recursive
III. L1* U L2 is context-free
IV.  L1 U L2' is context-free 

(A) I only

(B) I and III only
(C) I and IV only
(D) I, II and III only

Answer: (D)

Explanation: St 1: As L3 is Recursive and recursive languages are closed under complementation, L3’ will also be recursive and hence RE. L3’ U L4 is also recursive enumerable as recursive enumerable languages are closed under union.
St 2: As L2 is Context- Free, it will be recursive as well. L2 U L3 is recursive because as recursive languages are closed under union.
St 3: L1* is regular because regular languages are closed under kleene –closure. L1* U L2 is context free as union of regular and context free is context free.
St 4: L2’ may or may not be context free because CFL are not closed under complementation. So it is not true.

So I, II and III are correct.

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