Proving Reversal of a Language in Recursive Way

Question

We define the $reverse$ of a string as follows:

$(x_1x_2...x_n)^R=x_nx_{n-1}...x_1$

where $x_1,x_2,...,x_n \in \Sigma$. We can also define the reverse of a language by

$L^R= \lbrace s' | \exists s \in L : s' = s^R \rbrace$.

Prove that if $L$ is a regular language, $L^R$ is as well.

$Hint:$ Define the $reverse$ operation for regular expressions, recursively.

Although I have the hint I can't find the answer. Can you help me?

Answer 1

Hint

Try to solve the following questions $$ (K \cup L)^R = {?} \qquad (KL)^R = {?} \qquad (L^*)^R = {?} $$