Half Subtractor in Digital Logic

Last Updated on October 17, 2023 by Ankit Kochar

In the realm of digital electronics, understanding the fundamentals of subtracting binary numbers is pivotal. One of the fundamental building blocks in this domain is the Half Subtractor circuit. This circuit is responsible for performing the subtraction of two binary digits and producing both the difference and borrow outputs. In this article, we’ll delve into the inner workings of the Half Subtractor circuit, explore its truth table, and discuss its significance in more complex digital circuits.

What is Half Subtractor?

A half subtractor is a digital logic circuit that performs the subtraction of two binary bits. It has two inputs, A and B, and two outputs, Difference (d) and Borrow (b). The output d is the difference between the two inputs, A and B, and output b represents a borrow, indicating that the subtraction of B from A requires borrowing from the next digit in a larger binary number.

A half subtractor can be used in combination with other half subtractors to build a full subtractor circuit, which can perform binary subtraction of larger binary numbers.

The block diagram for the half subtractor is shown below:

Let us make the truth table according to the behavior of a Half Subtractor on giving various different inputs.

Half Subtractor Truth Table

The truth table for Half Subtractor is given as

From the truth table, we can see that the difference (d) output is equal to 1 if either A is 1 or B is 0. This means that the half subtractor will produce a 1 in the difference output if there is no borrow from the next lower order bit. If both A and B inputs are 1, then the half subtractor produces a 0 in the difference output and a 1 in the borrow out (b) output. This indicates that there is a borrow from the next lower order bit.

K-Map for Half Subtractor

After making the Truth Table for the Half Subtractor, let us now derive the Boolean Expression for both the outputs of Half Subtractor i.e., “d” and “b”.

K-Map :
K-Map is the official way for deriving the boolean expressions using the truth table for a particular digital circuit. Let us make the K-Map for the Half Subtractor.

• For Difference (d):
  

• For Borrow (b):
  

So by using the K-Map we get the Logical Boolean Expressions for the output as:

1. d = A’B + AB”, which can be written as
   d = A ⊕ B
2. b = A’B

Construction of Half Subtractor Circuit

Since, we get the Boolean Expression, now we are constructing the Circuit Diagram for the Half Subtractor.

The Half Subtractor can be constructed using the following basic Logic Gates:

• 1 AND Gate
• 1 Ex-OR Gate
• 1 NOT Gate

Before making Circuit Diagram, it is good for us to learn the above-mentioned logic gates in brief:

1. AND Gate:
   An AND gate is a digital logic gate that outputs a 1 (high voltage) if and only if all its inputs are 1. Its Symbol is

The truth table for AND Gate is given below:

2. Ex-OR Gate:
   An XOR (Exclusive OR) gate is a digital logic gate that outputs a 1 if and only if exactly one of its inputs is 1. Its Symbol is

The truth table for Ex-OR Gate is given below:

3. NOT Gate:
   A NOT gate is a digital logic gate that outputs the opposite logic level of its single input, effectively inverting it. Its Symbol is

The truth table for NOT Gate is given below:

Circuit Diagram for Half Subtractor using Basic Gates:

The circuit diagram according to the above mentioned boolean expression is given below:

Half Subtractor using only NAND Gates:

We can also construct a Half Subtractor using only NAND Gates as shown in the circuit diagram given below:

Here, X and Y are inputs and D represents “difference” between bits and B denotes “borrow”. Lets’ see the application of Half Subtractor.

Applications of Half Subtractor

A half subtractor is a combinational circuit that is used to perform the subtraction of two binary bits. There are several application of Half Subtractor, including

• Digital subtraction – The half subtractor can be used to build larger digital circuits for binary subtraction.
• BCD subtraction – BCD (Binary-Coded Decimal) subtraction can be performed using half subtractors.
• Error detection – In digital communication systems, half subtractors can be used to detect errors in the transmission of binary data.
• Arithmetic logic unit (ALU) – The half subtractor can be used in the design of an ALU (Arithmetic Logic Unit) in a computer’s central processing unit (CPU).
• Other digital circuits – The half subtractor can be used in other digital circuits such as multiplexers, decoders, and comparators.

Limitations of Half Subtractor Circuit

A half subtractor has some limitations, which are listed below:

• Only handles subtraction of two single-bit binary numbers.
• Cannot perform addition or handle multi-bit numbers.
• Only gives a difference and a borrow output, not the sum and carry.
• Cannot handle negative numbers or perform signed subtraction.

Conclusion
The Half Subtractor circuit, a cornerstone of digital electronics, is a simple yet crucial component for performing binary subtraction. Its operation, governed by Boolean logic gates, facilitates the subtraction of binary numbers and produces outputs that help manage borrow operations. Understanding the Half Subtractor is a fundamental step towards comprehending more complex digital circuits, paving the way for the development of powerful computational systems.

Frequently Asked Questions (FAQs) Related to Half Subtractor Circuit

Some Frequently Asked Questions on Half Subtractor Circuit are

1. What is the difference between a Half Subtractor and a Full Subtractor?
A Half Subtractor subtracts two binary numbers and produces both the difference and borrow outputs. A Full Subtractor, on the other hand, is more advanced and can handle three binary inputs, providing a more comprehensive solution for subtraction with borrow.

2. How is the borrow output used in digital circuits?
The borrow output from a Half Subtractor or Full Subtractor is typically fed into the next higher-order subtraction stage, ensuring that borrow operations are correctly managed when subtracting multi-digit binary numbers.

3. Can Half Subtractor circuits be used for decimal subtraction?
While Half Subtractors are designed for binary subtraction, similar principles can be applied to decimal subtraction, but with different components and logic operations to handle the base-10 system.

4. Are there integrated circuits (ICs) available for Half Subtractors?
Yes, some ICs include Half Subtractor functionality, making it easier to implement in digital circuit designs. These ICs often contain multiple logic gates for more complex operations as well.