# Half Subtractor and Full Subtractor Theory with Diagram and Truth Table

## Half subtractor and full subtractor theory :

In digital electronics, half subtractor and full subtractor are one of the most important combinational circuit used. Half subtractor and full subtractor are basically electronic devices or we can say logical circuits which performs subtraction of two binary digits. In this article, we are going to discuss half subtractor and full subtractor theory and also discuss the terms like half subtractor and full subtractor boolean expression, half subtractor and full subtractor circuit diagram etc.

## Half subtractor :

As like addition operation of 2 binary digits, which produces SUM and CARRY, the subtraction of 2 binary digits also produces two outputs which are termed as difference and borrow. The simplest possible subtraction of 2-bit binary digits consists of four possible operations, they are 0-0, 0-1, 1-0 and 1-1. The operations 0-0, 1-0 and 1-1 produces a subtraction of 1-bit output whereas, the remaining operation 0-1  produces a 2-bit output. They are referred as difference and borrow bit respectively. This borrow bit is used for subtraction of the next higher pair bit.

So, we can define half subtractor as a combinational circuit which is capable of performing subtraction of 2-bit binary digits is known as a half subtractor. Here, the binary digit from which the other digit is subtracted is called minuend and the binary digit which is to be subtracted is known as the subtrahend.

## Half subtractor truth table and circuit diagram :

### Half subtractor truth table :

In the above truth table of half subtractor, the two input variables X and Y represents minuend and subtrahend respectively. The two output functions difference and borrow are termed as D and B respectively. Using the truth table of half subtractor, we can design the half subtractor circuit diagram as below.

### Half subtractor boolean expression :

The half subtractor boolean expressions are :

• D = (X’Y + XY’) = X ⊕ Y
• B = X’Y

## Full Subtractor :

When there is a situation where the minuend and subtrahend number contains more significant bit, then the borrow bit which is obtained from the subtraction of 2-bit binary digits is subtracted from the next higher order pair of bits. In such situation, the subtraction involves the operation of 3 bits. Such situation of subtraction can’t handle by a simple half subtractor. So, combining two half subtractor we can form another combinational circuit which can perform this type of operation. This circuit is known as the full subtractor.

So we can define full subtractor as a combinational circuit which takes three inputs and produces two outputs difference and borrow. Below is the truth table of the full subtractor, we have used three input variables X, Y and Z which refers to the term minuend, subtrahend and borrow bit respectively. The two outputs difference and borrow are named as D and B respectively.

### Full subtractor circuit diagram :

The construction of full subtractor circuit diagram involves two half subtractor joined by an OR gate as shown in the above circuit diagram of the full subtractor. The two borrow bits generated by two separate half subtractor are fed to the OR gate which produces the final borrow bit. The final difference bit is the combination of the difference output of the first half adder and the next higher order pair of bits.

### Full subtractor boolean expression :

The full subtractor boolean expressions are :

• (X’Y’Z + X’YZ’ + XY’Z’ + XYZ) = X ⊕ Y ⊕ Z
• (X’Y’Z + X’YZ’ + X’YZ + XYZ) = X'(Y ⊕ Z) + YZ