$1\mathrm{.}47$ Let $=\{1,$#$\}$ and let

$Y=\{\begin{array}{cc}w|w=x_1\text{\#}{x}_2\text{\#}\text{c}\text{d}\text{o}\text{t}\text{s}& \text{\#}{x}_k\end{array}$ for $k0,$ each $x_{i}in{1}^{**},$ and $x_i{x}_j$ for $\{$:$ij\}.$

Prove that $Y$ is not regular.

IN EITHER $1\mathrm{.}47($above$)$ or $1\mathrm{.}53($below$),$ AT YOUR CHOOSING, YOU MUST USE THE MYHILL$-$NERODE THEOREM

TO PROVE NON$-$REGULARITY, WHILE IN THE OTHER YOU MUST USE THE PUMPING LEMMA.