$(2$ marks$)$ Recall that in class, we discussed the boolean half$-$adder. It takes two binary variables

$($bits$)x_1$ and $x_2$ as input, and outputs two bits $y_1$ and $y_2$ such that $y_2{y}_1=x_1+x_2.$ Recall that

the notation $y_2{y}_1$ refers to concatenation in this context. In lecture, we referred to $y_1$ as the sum

bit and $y_2$ as the carry bit, but we will just use generic labels for them here.

Suppose you want to build a double half$-$adder that operates on four input bits instead of two.

This adder takes four bits $x_1,x_2,x_3,$ and $x_4$ as input, and outputs three bits $y_1,y_2,$ and $y_3$ such

that $y_3{y}_2{y}_1=x_4{x}_3+x_2{x}_1.$ Give boolean formulas for each of the bits $y_1,y_2$ and $y_3.$ You may

use AND ${(}^{{?}^{?}}),$ OR $(vv),$NOT$(not)$ and XOR $(o+)$ clauses in your responses.

Show your work. Answers consisting of only formulas without justification will not be marked.