Question

Construct a circuit for a full subtractor using AND gates, OR gates, and inverters. A full subtractor has two bits and a borrow as input, and produces as output a difference bit and a borrow.

Short answer

The output for difference is \(b\overline x y + bx\overline y + \overline b x\overline y \) and output for borrow is\(b\overline {xy} + bxy\).

Step-by-step solution

Defining of gates

There are three types of gates.

It is also called NOT gate.

Output result for difference

The input of the circuit is two bits x and y and a borrow b.

First we find the difference \(\left( {x - y} \right)\). it is 1 when exactly one of x and y is equal to 1.The borrow also needs t be decreased by 1 when x is 1 and y is 0.

Then the output for difference is \(b\overline x y + bx\overline y + \overline b x\overline y \).

Construct a circuit

Find the output for borrow

When decreased the borrow by 1 while x is 1 and y=0., then the it changed to its complement.

However, b can only 1 when b was originally equal to 1, while x=1 or x=0 with y is 0.

Then the output of borrow is \(b\overline {xy} + bxy\).

The circuit is

Therefore,the output for difference is \(b\overline x y + bx\overline y + \overline b x\overline y \) and output for borrow is \(b\overline {xy} + bxy\).