Design of $2-$bit unsigned adder:

Consider the addition of two $2-$bit unsigned integers X $($x$1$ x$0)$ and Y $($y$1$ y$0)$ producing a $3$ bit sum Z $($z$2$ z$1$ z$0).$

$($a$)$ Draw the truth table showing the $3-$bit output $($Z$)$ for all possible $2-$bit inputs $($X and Y$)$ and from the truth table, write the $($un$-$simplified$)$ Boolean functions for each of the three outputs $($z$2,$ z$1,$ z$0)$ in terms of the four input bits $($x$1,$ x$0,$ y$1,$ y$0).$

$($c$)$ Using the Boolean identities, simplify each of the output functions into the corresponding minimal Boolean function. Show all the steps in your simplification indicating the identity used in each step.

$($e$)$ Draw the logic circuits implementing your minimal Boolean functions to generate the output bits from the input.