Let

${}_2=\{\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 1\end{array}\right],\left[\begin{array}{c}1\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 1\end{array}\right]\}.$

Here, ${}_2$ contains all columns of $0$ s and $1s$ of height two. A string of symbols in

${}_2$ gives two rows of $0$s and $1$s$.$ Consider each row to be a binary number and let

$C=\{win{}_2^{**}|$ the bottom row $ofwis$ three times the top row $\}.$

For example, $\left[\begin{array}{c}0\\ 0\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]\left[\begin{array}{c}1\\ 1\end{array}\right]\left[\begin{array}{c}0\\ 0\end{array}\right]$inC, but $\left[\begin{array}{c}0\\ 1\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]!$inC. Show that $C$ is regular.