Question $1$:

Design a circuit that counts the number of $1$s present in $3$ inputs A$,$ B and C$.$ Its output is a two$-$bit number X$1$X$0,$ representing that count in binary.

a$.$ Construct the truth table for designing this circuit.

b$.$ Use the summation format to represent the Boolean expression$($s$)$ for the desired circuit.

c$.$ Use K$-$Maps to produce Boolean expression$($s$)$ for the simplest possible circuit.$]$

d$.$ Use basic gates to draw the logic diagram for the expression$($s$)$ derived from c$.$

e$.$ Show how DeMorgan$$s law may be used to convert the simplified circuit to use NAND gates.

f$.$ Use the expressions derived from e to draw the logic diagram.