Let $\sum_{3}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \\0 \end{bmatrix},\begin{bmatrix} 0\\0 \\1 \end{bmatrix},\begin{bmatrix} 0\\1 \\0 \end{bmatrix},…….,\begin{bmatrix} 1\\1 \\1 \end{bmatrix}\end{Bmatrix}.$
$\sum_{3}$ contains all size $3$ columns of $0’s$ and $1’s.$ A string of symbols in $\sum_{3}$ gives three rows of $0’s$ and $1’s.$ Consider each row to be a binary number and let
$B=\{w\in\sum_{3}^{*}|\text{the bottom row of $w$ is the sum of the top two rows}\}.$
For example$,\begin{bmatrix} 0\\0 \\1 \end{bmatrix}$ $\begin{bmatrix} 1\\0 \\0 \end{bmatrix}$ $\begin{bmatrix} 1\\1 \\0 \end{bmatrix}\in B,$ but $\begin{bmatrix} 0\\0 \\1 \end{bmatrix}$ $\begin{bmatrix} 1\\0 \\1 \end{bmatrix}\notin B$
Show that $B$ is regular. $\text{(Hint$:$Working with $B^{R}$ is easier. You may assume the result claimed in question $31$})$
