Problem $2.(10$ points$)$ And now we move to adding three$-$bit numbers: x will consist of three

bits x$2$ x$1$ x$0$ and y will consist of three bits y$2$ y$1$ y$0.$ If we add them, the result will be z$,$

consisting of four bits z$3$ z$2$ z$1$ z$0.$

w$3$ w$2$ w$1$

x$2$ x$1$ x$0$

$+$ y$2$ y$1$ y$0$

z$3$ z$2$ z$1$ z$0$

Where w$3,$ w$2$ and w$1$ are called third, second and first carry bits respectively.

a$)$ Write a boolean formula for z$2.$ Submit a file named p$2$a$.$py$.$

b$)$ Write a boolean formula for w$3.$ Submit a file named p$2$b$.$py$.$

Note: Your Boolean formulas for z$2$ and w$3$ should only consist of x$0,$ x$1,$ x$2,$ y$0,$ y$1$ and y$2.$