# Can intersection of non-context-free languages be context-free?

For two languages over the same alphabet that are not context-free, can their intersection be context-free, Or does at least one of them have to be context-free?

Let $$L$$ be an arbitrary language. Then $$\{ 0x : x \in L \} \cap \{ 1x : x \in L \} = \emptyset.$$
The intersection of two CFL may or may not be a CFL. We can show that both possibilities exist. For example if both $$L_1$$ and $$L_2$$ are regular then $$L_1∩ L_2$$ is regular and therefore CF.