For $L_1, L_2 $ and $L_1 \in RE $ and $ L_1\notin R$ and $L_2 \in RE $ and $ L_2\notin R$
I was asked to prove/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 $L_1 \cup L_2 \in R$
For 1., I think any two disjoint langauges will suffice (because the empty set is decideable).
For 2., I think something along the lines of a language and its complement but I'm struggling to think of an example.
For 3,. it seems impossible but I have no idea how to prove it.
Any help/further insight would be welcomed!