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
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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