roblem $8.[10$ points$]$ Let $=\{0,1,+,=\}$ and ADD $=\{$x$=$y$+$z $|$ x$,$ y$,$ z are binary integers, and x is the sum of y and z$\}.$ Show that ADD is not regular. Problem $7.[20$ points$]$ Let $=\{0,1\}.($a$)$ Let A $=\{0$ku$0$k $|$ k $1$ and u in $\}.$ Show that A is regular. $($b$)$ Let B $=\{0$k$1$u$0$k $|$ k $1$ and u in $\}.$ Show that B is not regular. Problem $8.[10$ points$]$ Find a mathematical invention that was initially created without a clear real$-$world application but was later proven to be useful. $($a$)[2$ points$]$ What search query did you use? $($b$)[2$ points$]$ When was it invented? $($c$)[2$ points$]$ Who invented it$?($d$)[2$ points$]$ Who discovered its broader usefulness? $($e$)[2$ points$]$ Is it still widely used today?