Exercise $2\mathrm{.}2\mathrm{.}8$ Fix $.$ Define the reversal of a string $w,w^R$ by induction

on $|w|$ as follows:

${}^R=.$

If $w=a$ for ain$,$ then $w^R=a(w^R).$

Prove the following.

If $w=uv$ then $w^R=v^R{w}^R.$

Prove that for all $j1,$ that $(w^R{)}^j=(w^j{)}^R.$

If $L=L()$ is regular, then so is $L({}^R).$