If L1 is Regular, and L1UL2 is regular, then L2 is?

Question

Consider L1, L2 ⊆ Ʃ* such that L1 and L1 ∪ L2 are regular.
(a) L2 is definitely regular

(b) L2 may not be regular

(c) L2 is context free

(d) None of above

Is it option B or C? How?

Answer 1

B.

let L1=(a+b)*

L1 contains all posible strings over A and B.

and L2= any language.

L1 union L2 =L1 =regular

Comments

Got the answer:

L1: a*b*c*-->reg
L2: a^nb^nc^n-->not cfl, and not regular

L1 U L2: a*b*c*-->reg

Hence option B.

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Yup, got it.. thanks...

Answer 2

option B is answer. L2 may not be regular, This can be see by following example-

let L1 = Σ^∗ and L2 = {aⁿbⁿ∣n>0} . Now,  (L1 ⋃ L2) is  Σ^∗ which is regular but L2 is not regular. 

Answer 3

may or may not be regular

Answer 4

B is the Answer

check this for detailed explanation https://gateoverflow.in/3876/given-that-a-language-la-l1-u-l2