Peter Linz Edition 4 Exercise 4.3 Question 14 (Page No. 123)

Question

Prove or disprove the following statement: If $L_1$ and $L_2$ are non regular languages, then $L_1 ∪ L_2$ is
also non regular.

Answer 1

The statement " If L1 and L2 are non regular languages then L1 U L2 is also non regular" is false.
Because,

Let L1={ aⁿbⁿ | n>=0 } which is non regular language (L1 is a deterministic context free language)
L2 = complement of L1 , which is also a non regular language. L2 is also deterministic context free language as dcfl is closed under complement operation.

Here L1 U L2 = ∑*, which is regular language.

But if L2 is not the complement language of L1 then L1 U L2 will not be regular language.