$10-$ Design a combinational circuit with three inputs $x,y,$ and z and three outputs F$1,$ F$2,$ and F$3$

with the minimum possible number of gates, based on the following specifications:

$(18$ points$)$

The output F$1$ is $1$ when the binary value of the inputs $($the $3$ bits together$)$ is equivalent to a

decimal number that is less than decimal $4,$ otherwise the output is $0.$

The output F$2$ is $1$ when the binary value of the inputs $($the $3$ bits together$)$ is equivalent to a

decimal number that is either $4$ or an odd decimal number, otherwise the output is $0.$

The output F$3$ is $1$ when the binary value of the inputs $($the $3$ bits together$)$ is equivalent to a

decimal number that is either $0$ or greater than decimal $4,$ otherwise the output is $0.$

a$)$ Your design steps must include: A truth table, K$-$maps, simplified Boolean expressions in

canonical and standard SOP forms, and implementations with logic gates.

Combine the logic diagrams into a single logic circuit having $3$ inputs and $3$ outputs.

In your implementations, assume the inputs and their complements are available

b$)$ Identify the minimum number of gates in your design. How many gate$-$delays from the

inputs to each of the three outputs? If the delays from the inputs to each output are not

equal, redraw the logic diagram of your circuit with equal delays from the inputs to each

output.