# $L_1 ∩ L_2$ is not regular while $L_1$ is regular and $L_2$ is not regular language

Could you give me an example of languages $$L_1$$ (regular) and $$L_2$$ (not regular) where $$L_1 \cap L_2$$ is not regular?

• What did you try to find one yourself? Nov 18, 2019 at 6:32
• Seriously, what have you tried? What properties would L1 have so it doesn’t turn a non-regular language into a regular one? There is an obvious L1 that will work with any non-regular language. Nov 18, 2019 at 9:08

Take $$L_2$$ to be any nonregular language, say $$\{0^n 1^n: n \ge 1\}$$ and let $$L_1 = (0+1)^*$$ be the set of all strings over the alphabet $$\{0,1\}$$. Then $$L_2$$ is a subset of $$L_1$$, and their intersection is $$L_2$$, which is nonregular.
Suppose $$\Sigma = \{a\}$$. A straighforward example is $$L_1= a^*$$, $$L_2 = \{a^p| \text{p is prime}\}$$. So, $$L_2 \cap L_1 = L_2$$ and it is not regular.