# half subtractor and full subtractor

Half subtractor and full subtractor is a part of combinational circuits. Combinational circuits are the circuits whose output characteristics depends upon their input signals provided at a particular time. Examples of combinational circuits are – multiplexer ( MUX ), demultiplexer ( DEMUX ), half subtractor and full subtractor, encoder, decoder, half adder and full adder. The combinational circuits does not include any memory, so the previous state of output and input signals does not influence the current or next state.

## What is Subtractor ?

A digital logic circuit that is widely used for subtraction of numbers is known as subtractor. Just like adders, subtractors are also used in processors that calculates addresses and similar activities.

Subtractors are classified into two types –

1. Half Subtractors
2. Full Subtractors

## Half Subtractor

A half subtractor is a combinational logic circuit. It has two inputs ( one bit each ) termed as A and B that generates difference ( D ) and borrow ( Br ) as output. Half subtractor is designed using three logic gates that is AND gate, NOT gate and XOR gate. The output of difference is obtained from XOR gate and the output of borrow is obtained from AND gate.

### Truth Table for Half Subtractor

The truth table of half subtractor is written by the concept of binary subtraction. The binary subtraction is done as shown below –

0 – 0 = 0

0 – 1 = 1, borrow 1

1 – 0 = 1

1 – 1 = 0

The expression is written only when logic signal 1 is obtained in output. Expression will be written in SOP form where A = 1 and A’ = 0 also B = 1 and B’ = 0.

 A B D Br D Br 0 0 0 0 – – 0 1 1 1 A’B A’B 1 0 1 0 AB’ – 1 1 0 0 – –

### Expression for Half Subtractor

The logic expression for half subtractor is given as

Difference ( D ) = A’B + AB’ = A ⊕ B

Borrow ( Br ) = A’B

## Full Subtractor

A full subtractor is a combinational logic circuit. It has three inputs ( one bit each ) termed as A, B and C that generates difference ( D ) and borrow ( Br ) in output. Full subtractor is designed using two XOR gate, two NOT gate, two AND gate, and one OR gate. The output of Difference ( D ) is obtained from XOR gate and the output of borrow ( Br ) is obtained from OR gate. As shown in circuit below.

### Truth Table for Full Subtractor

The truth table of full subtractor is also written by the concept of binary subtraction.

0 – 0 = 0

0 – 1 = 1, borrow 1

1 – 0 = 1

1 – 1 = 0

Also, similar to half subtractor expression is written only when logic 1 is obtained in output. The expression is written in SOP form where A = 1 and A’ = 0 also B = 1 and B’ = 0. Similarly, C = 1 and C’ = 0.

 A B C D Br D Br 0 0 0 0 0 – – 0 0 1 1 1 A’B’C A’B’C 0 1 0 1 1 A’BC’ A’BC’ 0 1 1 0 1 – A’BC 1 0 0 1 0 AB’C’ – 1 0 1 0 0 – – 1 1 0 0 0 – – 1 1 1 1 1 ABC ABC

### Expression for Full Subtractor

The logic expression for full subtractor is given as –

Difference ( D ) = A’B’C + A’BC’ + AB’C’ + ABC

Borrow ( Br ) = A’B’C + A’BC’ + A’BC + ABC

Final expression –

Difference ( D ) = A ⊕ B ⊕ C

Borrow = A’B + BC + A’C

### Difference Between Half Subtractor and Full Subtractor

 Half Subtractor Full Subtractor It is a type of combinational logic circuit that subtracts two binary digit of one bit each and provides output in the form of difference and borrow. It is a type of combinational logic circuit that subtracts three binary digits of one bit each and provides output on the form of difference and borrow. It does not add the borrow obtained from previous subtraction to the next subtraction. It subtracts the borrow obtained from A, B to C. A half subtractor circuit is designed using only one AND gate, NOT gate and XOR gate. A full subtractor circuit is designed using two XOR gate, two NOT gate, two AND gate and one OR gate. Half subtractor has two inputs Full subtractor has 3 inputs Half subtractor is used in digital measuring devices and calculators, etc. Full subtractor is used in digital processors and subtraction of multiple bits, etc. The logic expression for half subtractor is given asDifference ( D ) = A’B + AB’ = A ⊕ BBorrow ( Br ) = A’B The logic expression for full subtractor is given asDifference ( D ) = A ⊕ B ⊕ CCarry = A’B + BC + A’C

Author

Akash Sharma