Question

C3PO has run into the Falcon’s computer again. This time he destroys the subtraction circuit, so it must be rebuilt. Fortunately, C3PO has R2D2 to help him out. R2 suggests that they first build a circuit that can subtract one 1-bit number from another. In particular, it will realize the function x – y, where x and y are both 1-bit wide numbers. The output of the circuit is 2-bits wide. The outputs include “d” for difference and “b_out” for borrow.

a) Draw the truth table for the 1-bit subtraction circuit.

b) Give an expression for the output “d” in canonical SOP.

c) Give an expression for the output “b_out” in canonical POS.

Unfortunately, a 1-bit subtractor isn’t sufficient. C3PO knows that the Falcon needs to be able to handle more bits. R2 tells him that he needs to create a circuit with three inputs: b_in, x, and y. The circuit acts like a column in the type of subtraction you are already familiar with from grade school. The b_in input is coming from the b_out output of the column to the right. Once again, y is being subtracted from x. The outputs of this three-input circuit are “d” for difference and “b_out” for borrow.

d) Draw the truth table for the 1-bit subtractor circuit with b_in, x, and y as inputs.

e) Give an expression for the output “d” in canonical SOP.

f) Give an expression for the output “b_out” in canonical POS.

g) If you could implement “d” using any set of standard gates, what is the most efficient implementation you could make?

Answer 1

Hello,
Please find the answer to the first 4 questions attached as under. Please give a thumbs up rating if you find the answer useful! Have a rocking day ahead!

a)

x	y	d	b_out
0	0	0	0
1	0	1	0
0	1	1	1
1	1	0	0

b.    d = xy' + x'y

c. b_out = (x'+y')(x+y')(x+y)

d.

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