Add Sub Extra Credit Problem

Preliminaries

The Add$/$Sub circuit you studied in practice problem $10,$ week $3,$ is shown below. It adds two $4-$bit numbers $x$ and $Y.$ Now, let's assume that the user has two numbers $x$ and $Y$ that are both positive signed numbers, one of the following four scenarios might occur:

If the required is $x+Y,$ then the user inputs $x$ to the A input and $Y$ to the $B$ input and set the circuit to ADD $($i$.$e$.S=0).$

If the required is $x-Y,$ then the user inputs $x$ to the A input and $Y$ to the $B$ input and set the circuit to SUB $($i$.$e$.S=1).$

If the required is $-x+Y,$ then the user inputs the $2\text{'}$s complement of $x$ to the A input and $Y$ to the $B$ input and set the circuit to ADD $($i$.$e$.S=0).$

If the required is $-x-Y,$ then the user inputs the $2\text{'}$s complement of $x$ to the A input and the $2\text{'}$s complement of $Y$ to the $B$ input and set the circuit to ADD $($i$.$e$.S=0).$

Note the following two facts:

In the $3^{rd}$ and $4^{th}$ scenario, taking the $2^{\text{'}}\text{'}$s complement is done by hand first $($i$.$e$.$ the user has to take the $2\text{'}$s complement before feeding the number to the circuit$).$

In the third scenario, while the user could avoid taking the $2\text{'}$s complement $($by converting $-x+Y$ to $Y-x$ and thus applying the second scenario$),$ taking the two's complement is inevitable in the fourth scenario.