P$3.(20$ points$)$ Design a circuit that accepts a $3-$bit number $x=x_2{x}_1{x}_0$ as input and

generates a $6-$bit number $Y=y_5{y}_4{y}_3{y}_2{y}_1{y}_0$ as output, which is equal to the square of the

input number $($i$.$e$.,Y=x^2).$

a$)(10$ points$)$ Write down the truth table for the six output lines $y_5{y}_4{y}_3{y}_2{y}_1{y}_0$ that

jointly represent the number Y in binary.

b$)(10$ points$)$ Derive the simplest SOP expressions for each bit of the output. That is$,$

derive six expressions: one for $y_5,$ another for $y_4,$ and so on$.$