Let L1 be the 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−L3is recursively enumerable.
c.)L2∩L3is recursively enumerable.
d.)L2∪L3is recursively enumerable.
My approach:- I am able to falsify a,c and d option. Now for b option i took one example where my L1 is universal language so when i do L1-L3,i get the $L3^c$(L3 complement).So complement of RE but Not REC is NOT RE always.So it is not required that L1-L3 will be RE. But on the other hand if i solve b option as follow:-
= L1 ∩ $(L3)^c$
= REC ∩ $(RE but Not REC)^c$
=REC ∩ NOT RE
So the final expression will give me empty language and it is regular.Is it possible for the final expression to have non empty set above?