Given L1 is regular language and L1△L2 is regular is L2 regular?

Question

Let L1 and L2 be some languages under some alphabet Σ,
Given that L1 is regular and L1△L2 is regular
prove or disprove L2 must be also regular.

Im trying to figure out a counterexample yet couln't find one.

Answer 1

Words in $L2$ are either:

• in $L_1\triangle L_2$ but NOT in $L_1$ (then it originates from $L_2$) OR
• NOT in $L_1\triangle L_2$ but in $L_1$ (it is in both $L_1$ and $L_2$)

If both $L_1\triangle L_2$ and $L_1$ are regular, there are finite automata that accept them. For accepting $L_2$ you can run these in parallel and after reading the input word check the two conditions above. Based on that you accept or reject.

Thus $L_2$ is always regular.