$1\mathrm{.}31$ For any string $w=w_1{w}_{2}cdots{w}_n,$ the reverse of $w,$ written $w^R,$ is the string $w$ in reverse order, $w_{n}cdots{w}_2{w}_1.$ For any language $A,$ let $A^R=\{w^R|winA\}.$ Show that if $A$ is regular, so is $A^R.$

$1\mathrm{.}32$ Let