1) Given $L_1$ is a regular language and $L_2$ is a non-regular language, the intersection of $L_1$ and $L_2$ is a finite language, how to prove that the union of $L_1$ and $L_2$ is a non-regular language?
2) Given $L_1$ is a regular language and $L_2$ is a non-regular language, the intersection of $L_1$ and $L_2$ is an infinite language, how to prove that the union of $L_1$ and $L_2$ is a regular language.
I have tried my best to prove this, I tried pumping lemma and Demorgan's law and haven't worked it out. Asking for help with sincerity.
Best Answer
1) is equivalent to the following claim
This is easy to prove.
2) is false.