# Would it be possible that the intersection of two non-context-free languages is context-free?

Assume two languages $L_1$ and $L_2$, both of which are non-context-free.

Let $L = L_1 \cap L_2$. Could $L$ be context-free?

• It's more interesting to ask whether the intersection of two context-free languages is context-free (though the answer is still negative). The union of two context-free languages is always context-free. So the only case remaining is the union of two non-context-free languages. What do you think, is it always non-context-free? – Yuval Filmus Oct 10 '13 at 2:42