$($a$)$ For any string w $=$ w$1$w$2$

$...$ wn

$,$ the reverse of w$,$ written w

R

$,$ is the string w in reverse

order, wn

$...$ w$2$w$1$

$.$ For any language L$,$ let L

R $=\{$w

R

$|$ w in L$\}.$ Show that if L is

regular, so is L

R

$.$

$1$

ECE $374$ B Homework $2$ Fall $2024$

$($b$)$ Let

$\backslash$Sigma $3=$

$$

$$

$$

$$

$$

$0$

$0$

$0$

$$

$,$

$$

$$

$0$

$0$

$1$

$$

$,$

$$

$$

$0$

$1$

$0$

$$

$,...,$

$$

$$

$1$

$1$

$1$

$$

$$

$$

$$

$$

$.$

$\backslash$Sigma $3$

contains all size $3$ columns of $0$s and $1$s$.$ A string of symbols in $\backslash$Sigma $3$ gives three rows

of $0$s and $1$s$.$ Consider each row to be a binary number and let