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# Full Subtractor in Details

Last Updated on April 13, 2023 by Prepbytes Full Subtractor is a basic combinational circuit used in a modern digital computer for performing the subtraction of two binary numbers. Full Subtractor has various applications in the domain of Digital Electronics. In this article, we will study Full Subtractor, its operations along with its circuit. So, let us start with the definition of a Full Subtractor.

## What is Full Subtractor?

A full subtractor is a digital logic circuit that performs the subtraction of two binary numbers. It has three inputs: A, B, and Borrow In (bin), and two outputs: Difference (b) and Borrow Out (b). The full subtractor circuit can perform the subtraction of two bits, taking into account borrowing from a previous subtraction if necessary.

The operation of a full subtractor is straightforward. Given two inputs, A and B, and borrow in the signal, bin, the full subtractor calculates the difference between A and b and produces a binary output, d, representing the difference. It also produces a carry-out signal, b, which indicates whether a borrow was necessary to perform the subtraction.

The block diagram for Full Subtractor is given below: In other words, a full subtractor performs the binary subtraction operation A – B – bin, and the output Difference (d) represents the difference between the two inputs, and the output Borrow Out (bout) represents the borrow signal.

Let us make Full Subtractor Truth Table as per its behavior to different inputs.

### Full Subtractor Truth Table

The below table summarizes the Full Subtractor Truth Table: In the above Full Subtractor truth table, the inputs are A and B, and the Borrow In (bin) signal. The outputs are the Difference (d) and Borrow Out (b) signals. The truth table shows the relationship between the inputs and outputs of the full subtractor circuit.

### K-Map for Full Subtractor

After making the Full Subtractor Truth Table, let us now derive the Boolean Expression for both the outputs of Full 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 Full Subtractor.

• For Difference (d): • For Borrow (b): So by using the K-Map we get the Logical Boolean Expressions for the output as:
d = A’B’bin + A’Bbin’ + AB’bin’ + ABbin
b = A’bin + A’B + Bbin

### Construction of Full Subtractor using Logic Gates

A full subtractor can be constructed using various types of digital logic gates, such as AND, OR, NOT, and XOR gates. Here’s how a full subtractor can be constructed using logic gates:

1. Difference (d): To find the difference between two bits A and B, an XOR gate is used. The output of the XOR gate is the difference between the two inputs.
2. Borrow Out (b): To find the borrow out signal, an AND gate is used to find the AND operation between the inputs A and B. The output of the AND gate is then passed to a NOT gate to obtain the inverted signal, which is the Borrow Out signal.
3. Outputs: The outputs of the full subtractor are the difference (d) and the borrow out (b) signals. The difference (d) is obtained from the XOR gate, and the borrow out (b) is obtained from the NOT gate.

The circuit diagram for the above is shown below: ### Construction of Full Subtractor using Half Subtractor

A full subtractor can be constructed using two half subtractors. The full subtractor performs the binary subtraction of two bits and considers the borrowed signal from a previous subtraction. Here’s how a full subtractor can be constructed using two half subtractors:

1. First Half Subtractor: The first half subtractor performs the subtraction of the two bits A and B.
2. Second Half Subtractor: The second half subtractor performs the subtraction of the borrow in (Bin) signal and the difference obtained from the first half subtractor.
3. Outputs: The outputs of the full subtractor are the difference (D) and the borrow out (Bout) signals. The difference (D) is obtained from the second half subtractor, and the borrow out (Bout) is obtained from the first half subtractor.

Here is the circuit diagram for the Full Subtractor using Half Subtractors. ### Applications of Full Subtractor

The applications of a full subtractor are many and varied, including:

• Arithmetic circuits: Full subtractors are used in arithmetic circuits for performing binary subtraction.
• ALU (Arithmetic Logic Unit): The full subtractor is an essential component of an ALU, which performs arithmetic and logical operations in a computer’s central processing unit (CPU).
• Binary number representation: Full subtractors are used in the representation of negative binary numbers.
• Error Correction: Full subtractors are used in error correction codes for detecting and correcting errors in digital signals.
• Digital filters: Full subtractors are used in digital filters to subtract one signal from another to produce a filtered output.
• Digital clocks: Full subtractors are used in digital clocks to subtract one number from another to obtain the elapsed time.
• Microcontrollers: Full subtractors are used in microcontrollers to perform subtraction operations as part of their instruction set.
• Digital signal processing: Full subtractors are used in digital signal processing to perform subtraction operations on signals.

So, the full subtractor is a versatile combinational logic circuit that is widely used in various digital systems and applications. It is used in arithmetic circuits, ALUs, error correction, digital filters, digital clocks, microcontrollers, and digital signal processing, among others.

Conclusion
In conclusion, the full subtractor is a useful digital logic circuit that performs binary subtraction of two numbers while taking into account borrowing from a previous subtraction. It has three inputs and two outputs. The full subtractor circuit is widely used in various digital systems and applications, such as arithmetic circuits, ALUs, error correction codes, digital filters, digital clocks, microcontrollers, and digital signal processing, among others.

The full subtractor is a simple, yet versatile digital logic circuit that plays a crucial role in many digital systems and applications. It is an essential component in digital arithmetic and has the ability to perform binary subtraction with borrowing, making it an important component in digital systems that require binary arithmetic operations.