Design a digital circuit that accepts an input $(3-$bit$)$ binary number $($A B C$)$ and generates an output $(2-$bit$)$ binary number $(\backslash (\backslash$mathbf$\{$X$\}\backslash$mathbf$\{$Y$\}\backslash ))$ where the output is the total number of $1\text{'}$s in the input.

a$.$ Preliminary work: Construct the truth table of the circuit.

b$.$ Preliminary work: Determine the sum$-$of$-$minterms $($SOM$)$ of the circuit.

c$.$ Preliminary work: Determine the product$-$of$-$maxterms $($POM$)$ of the circuit.

d$.$ Preliminary work: Using K$-$maps, simplify the circuit in terms of product$-$of$-$sums $($POS$).$

e$.$ Preliminary work: Draw the POS circuit by hand using elementary gates.