$(20pts)$ In this problem, you will partially design a combinational circuit for a seven$-$segment display.

The circuit has four inputs $(x_1,x_2,x_3,x_4)$ and $7$ outputs $($A$,$ B$,$ C$,$ D$,$ E$,$ F$,$ G$).$ The $4-$bit input represents

a decimal number from $0$ to $9($e$.$g$.,x_1{x}_2{x}_3{x}_4$:$0011=3_{10}).$ The outputs are applied to a seven$-$segment

display to show the numbers using the format given below. Fill out the truth table only for A and $F,$

and then, write down the simplified equations for A and F$.$ Do not forget number $0,$ which is not

shown in the figure. Also, remember the "don't care" values for unused inputs.

$A=$

$F=$