GATE | GATE CS 2010 | Question 65

Let L1 be a recursive language. Let L2 and L3 be languages that are recursively enumerable but not recursive. Which of the following statements is not necessarily true?
(A) L2 – L1 is recursively enumerable.
(B) L1 – L3 is recursively enumerable
(C) L2 ∩ L1 is recursively enumerable
(D) L2 ∪ L1 is recursively enumerable


Answer: (B)

Explanation:

A) Always True
(Recursively enumerable - Recursive ) is 
Recursively enumerable

B) Not always true
L1 - L3 = L1 intersection ( Complement L3 )
L1 is recursive , L3 is recursively enumerable 
but not recursive Recursively enumerable languages
are NOT closed under complement.

C) and D) Always true Recursively enumerable languages 
are closed under intersection and union. 

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