For $L_1,L_2 \in RE - R $ , I want to prove or disprove if the following can occur:
- $L_1 \cap L_2 \in R$
- $L_1 \cup L_2 \in R$
- $L_1 \cap L_2 \in R$ and $L_1 \cup L_2 \in R$
What I did:
I think any two disjoint languages suffice, since the empty set is decidable.
I think something along the lines of a language and its complement but I'm struggling to think of an example.
- It seems impossible but I have no idea how to prove it.
Any help/further insight would be welcomed!