SUBTRACTOR $($NEW WORDING AND CLARIFICATION$)$

A$.$ Let $x_i$ and $y_i$ be column i bits. Let $b_{i-1}$ be the bit necessary for column $i-1$ taken from column $i$ we call this borrow out. Likewise let $b_{i+1}$ be the bit necessary for column i taken from column $i+1$ we call this borrow in$.$ Let $d_i$ be the result of subtraction for column $i.$ Then $d_i=x_i-y_i-b_{i-1}+2b_{i+1}$

First: Fill in the Truth Table for a Subtractor for $x-Y($with inputs X and Y $)$ given below. One row is already filled for your guidance.

Second: Generate $2$ K$-$maps for $b_{i+1}$ using $x_i,y_i,b_{i-1}$ as inputs. Write corresponding equations as Product of Sums. Then rearrange terms to get equations to allow you to construct a circuit for $b_{i+1}$ using inverter, XOR, and NOR gates.

$\backslash$table$[[$Truth Table for Full Subtractor,,,,,,,,,$],[$Inputs$,$outputs,Inputs,outputs,$],[b_{i-1},x_i,y_i,b_{i+1},d_i,b_{i-1},x_i,y_i,b_{i+1},d_i$