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If L1 U L2 is regular and L1 is regular, then we cannot conclude that L2 is also regular.

Let L1 U L2= ∑* and L1 = ∑*
Then L2 ={ aⁿbⁿ |n>=0} is not regular and L1 U L2 is regular.
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if L1 is regular and L1 U L2 is regular then we cannot conclude that L2 is regular ,

 L1=(a+b)* ,  L2={a^n b^n |n>0}    L1 U L2 = (a+b)*

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