An Adder is a digital logic circuit in electronics that performs the operation of additions of two number. Adders are classified into two types: half adder and full adder. The **full adder** (FA) circuit has three inputs: A, B and C_{in}, which add three input binary digits and generate two binary outputs i.e. carry and sum.

Let’s see an addition of single bits.

0+0+0 = 0

0+0+1 = 1

0+1+0 = 1

0+1+1 = 10

1+0+0 = 1

1+0+1 = 10

1+1+0 = 10

1+1+1 = 11

These are the least possible single-bit combinations. But the result for 1+1+1 is 11, the sum result must be re-written as a 2-bit output. Thus, the equations can be written as

0+0+0 = 0**0**

0+0+1 = 0**1**

0+1+0 = 0**1**

0+1+1 = 1**0**

1+0+0 = 0**1**

1+0+1 = 1**0**

1+1+0 = 1**0**

1+1+1 = 1**1**

The bold digit is the sum output. The output ‘1’of ‘1**0**’ is carry output. ‘SUM’ is the normal output and ‘CARRY’ is the carry output.

### Truth table of full adder

The truth table of the full Adder Circuit is shown in figure 2.

### Implementation of half adder

Logical expression for Sum,

Logical expression for Carry,

Carry = AB + BC_{in} + AC_{in}

The FA works by combining the operations of basic logic gates, with the simplest form using two **XOR**, one **OR** & two** AND** gate.

**Note:**

1. Minimum number of NAND Gate required implementing FA = 9

2. Minimum number of NOR Gate required implementing FA = 9

3. FA can be implemented by a combination of two half adder and one OR gate.

4. FA can be implemented by a combination of one 3×8 decoder and two OR gate.

**Excercise**

**Q.1 Which of the following circuits come under the class of combinational logic circuits?**

- Full adder
- Full subtracter
- Half adder
- J-K flip-flop
- Counter

Select the correct answer from the codes given below:

- 1 only
- 3 and 4
- 4 and 5
- 1, 2, and 3

**Ans.** option d is correct

**Q.2 A full adder can be made of**

- two half adders
- two half adders and a NOR gate
- two half adders and an OR gate
- two half adders and a AND gate

**Ans.** option c is correct