In this lecture, we will learn about the Full Subtractor, its circuit diagram, full subtractor circuit, full subtractor truth table, full subtractor using half subtractor, full subtractor expression, K-map of the full subtractor and many more in a very detailed analysis.

Subtracting the two binary digits is the most basic operation performed by digital computers. For this subtraction, there are binary adders used.

There are two types of binary Subtractors:

In this section, we will learn digital circuits which are used to subtract two binary numbers called Full f subtractors. We already learn the Half subtractor in the previous lecture, you can learn from the above link.

**Note: **Before getting the knowledge of the subtractors, you can also learn the digital circuits that add the two binary numbers.

There are two types of binary adders:

**What is a Full Subtractor**

As we have seen in the Half subtractor lecture, there is a disadvantage of half subtractor which is overcome by the Full subtractor.

The Full subtractor is a combinational circuit with three inputs A, B, and B_{in}, and two outputs namely D and B.

A is the minuend, B is the subtrahend, B_{in} is the borrow produced by the previous stage, D is the difference output and B is the horror output.

**Full Subtractor Block Diagram**

The block diagram of a full subtractor is shown in the Figure below:

**Full Subtractor Truth Table**

The Full Subtractor Truth Table is one that gives the relationship between the input and output of a logic circuit. The following is the truth table of the full-subtractor:

**Full Subtractor K Map**

The equations for the difference bit (D) and the output borrow bit (B) can be found using the K-Map (or Karnaugh Map), a technique for simplifying Boolean algebra.

**Full Subtractor Expression**

From the above K-map of full subtractor, we can write the boolean expression for sull subtractor as per below:

**Simplification of Difference Output (D):**

`\mathbf{ D = \bar A \bar B B_{in} + \bar A B \bar B_{in} + A \bar B \bar B_{in} + AB B_{in}}`

`\mathbf{ D = B_{in}(\bar A \bar B + AB) + \bar B_{in} (\bar A B + A \bar B)}`

`\mathbf{ \therefore D = B_{in}(\overline{A \oplus B}) + \bar B_{in} (A \oplus B)}`

Let \mathbf{A \oplus B = C}, then

`\mathbf{ \therefore D = B_{in}\bar C + \bar B_{in} C}`

`\mathbf{ \therefore D = B_{in} \oplus A \oplus B}`

**Simplification of Borrow Output (B):**

From the above k-map, we can write the Borrow output as,

`\mathbf{ B = \bar A B_{in} + \bar A B + BB_{in}}`

**Full Subtractor Circuit Diagram**

we can realize the full subtractor using two XOR gates, two NOT gates, two AND gates, and one OR gate. The circuit diagram of the full subtractor is shown in the below figure:

We can write the expression for the Borrow output as

`\mathbf{B = (\overline {A \oplus B} ) B_{in} + \bar A B}`

We can realize the above equation same as the Borrow output as we have seen above.

**Proof:**

**Full Subtractor Using Half Subtractor**

The below figure shows the implementation of a full subtractor using a half subtractor. To implement the Full subtractor using the half subtractor we have to use 2 half subtractors and one OR Gate.

**Full Subtractor Using NAND Gates**

We can realize the full subtractor circuit using NAND gates only as shown in the figure below:

From the logic circuit of the full subtractor using NAND Gates, we can see that** 9 NAND Gates** are required to realize the full subtractor using NAND Gates.

**Applications of Full Subtractor**

The full subtractor has the following notable applications:

- In the ALU (Arithmetic Logic Unit) of computer CPUs, full subtraction is employed.
- For arithmetic operations like subtraction, full subtractors are frequently employed in electronic calculators and many other digital devices.
- Different microcontrollers employ full subtractors for arithmetic subtraction.
- They are utilized in PC program counts and timers.
- In processors, full subtractors are also used to compute addresses, tables, etc.
- Systems based on networking and DSP (Digital Signal Processing) also use full subtractors.

**FAQs on Full Subtractor**

**What is a full subtractor?**

A full subtractor is a combinational circuit that performs subtraction involving three bits, namely A (minuend), B (subtrahend), and Bin (borrow-in).

**What is a full subtractor and its application?**

Full subtractors are used in ALU (Arithmetic Logic Unit) in computer CPUs. Full subtractors are extensively used to perform arithmetical operations like subtraction in electronic calculators and many other digital devices.

**What are the advantages of a full subtractor?**

We can cascade single-bit full Subtractor circuits and could subtract two multiple-bit binary numbers.

**How many OR gates are required in a full subtractor?**

The full subtractor can be implemented using two half subtractors and an OR gate.

**What is the difference between a half subtractor and a full subtractor?**

The difference between a half subtractor and a full subtractor is the output of the half subtractor is the Ex-OR of two inputs. However, the difference output of the full subtractor is the XOR of three inputs.