Half-Subtractor Circuit Overview

A half-subtractor is a combinational logic circuit that performs subtraction of two single-bit binary numbers. It takes two inputs, usually denoted as X (minuend) and Y (subtrahend), and produces two outputs: Difference (N) and Borrow (M).

The operation of a half-subtractor can be understood by looking at its truth table, which shows the outputs for all possible combinations of inputs.

Truth Table for Half-Subtractor

Let's construct the truth table for a half-subtractor with inputs X and Y, and outputs Difference (N) and Borrow (M).

Difference (N) Derivation

From the truth table, we can observe the pattern for the Difference output (N). The output N is '1' when the inputs X and Y are different (i.e., one is '0' and the other is '1'). This behavior is exactly what an Exclusive OR (XOR) gate performs.

• When X = 0 and Y = 1, N = 1.
• When X = 1 and Y = 0, N = 1.

Therefore, the Boolean expression for the Difference (N) is:

\(N = X \oplus Y\)

Borrow (M) Derivation

Now, let's analyze the Borrow output (M) from the truth table. The Borrow (M) output is '1' only in one specific case:

• When X = 0 and Y = 1, M = 1.

This means that a borrow is generated only when we try to subtract a '1' from a '0'. In Boolean algebra, this condition can be represented by the logical AND operation of X-bar (NOT X) and Y.

Therefore, the Boolean expression for the Borrow (M) is:

\(M = \overline{X}Y\)

Half-Subtractor Summary

Combining the derived expressions, for a half-subtractor circuit with X and Y as inputs:

• The Borrow (M) is given by \(M = \overline{X}Y\).
• The Difference (N = X - Y) is given by \(N = X \oplus Y\).

These expressions accurately describe the functionality of a half-subtractor. When comparing these with the provided options, we find that the option stating \(M = \overline{X}Y\) and \(N = X \oplus Y\) aligns perfectly with our derivations.