# Identifying class of language

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


= 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?

The intersection of a recursive language and a non-RE language isn't necessarily empty. Take the recursive langauge to be $\Sigma^*$, as you've already done once.