# GATE | Sudo GATE 2020 Mock I (27 December 2019) | Question 65

Consider the following statements:

I. If L1 and L2 are languages such that L2, L1L2, and L2L1 are all regular, then L1 must be regular.
II. If R and S are two regular expressions, then a solution of x = Rx + S is x = RS*.
III. If L is a regular language, then {wwR | w in L} must be a regular language. (Here, wR denotes the reverse of string w.)

Which of the following option is correct?
(A) Statements I and II are false. Statement III is true.
(B) All statements I, II, and III are false.
(C) Only statement I and III are false. Statement II is true.
(D) None of these.

Explanation: I. False, consider L1 = { 02i | i >0 }, and L1 = {0}*.
II. False, it is Arden’s Theorem, stats that if R and S are two regular expressions, then a solution of x = Rx + S is x = R*S not RS*.
III. False, a common mistake is confusing the language above with LLR.

Option (B) is correct.

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