Lets say we have $L_1$, which is a regular language and $L_2$ which is not. Are $L_1 \cap L_2$, $L_1 \cup L_2$ , $L_1$ \ $L_2$ and $L_1 \cdot L_2$ are always non-regular languages?
We know that two regular languages always gives us a regular language under all of the above. I can't find anywhere any proof that combination of two languages, one regular and one non-regular results always in a regular or a non-regular language. It seems for me that it should be a non-regular in all above cases. Am I wrong?