Adder plays an important role in digital devices. The addition process of the digital circuits is performed by an adder.

By using adder all the mathematical procedures of computers are done. In much computational processing, adders are used in the ALU(Arithmetic Logic Unit ).

Know what are the subtractors in Digital Electronics.

Adders are mainly two types. They are,

It adds two binary bits. It has no carry addition ability that's why it is called half adder.

It has the ability to add two bits and also it can add carry. That's why it is called a full adder.

### For two single bits here we will get a sum and a carry as output for each input combination.

 A B Sum Carry 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1

A truth table for half adder

### In this circuit, there is a system that can add two bits and also two carries. It is called the Full adder. Let, 3 input sign of a full adder is A, B, Ci(Ci is the carry input) and two outputs are S(Addition) and Co(Carry output).

Here is a truth table for a full adder circuit.

 Input Output A B Ci S Co 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1

Output equation

S = ĀB̄Cᵢ +ĀBC̄ᵢ + AB̄C̄ᵢ+ABCᵢ

C₀ = ĀBCᵢ+AB̄Cᵢ+ABC̄ᵢ+ABC

And also the equation of output carry can be modified like this -

From modified equations of the addition of full adder (S) and carry output (Cₒ) the circuit diagram using X-OR gate will be like this-

Full adder representation by using the basic logical gates

A full adder can be represented by using the basic logic gates only. The circuit diagram is given below-

The full adder reduces the complexity of a logical circuit.

We have to use two half adders and also we need an extra OR gate for adding the carries for designing a full adder using a half adder.

Let, the input signal for full adder is X and Y, carry Cᵢ and outputs are S (Addition) Co(Carry).

In the figure, if we see for the first half adder-

Sum,
S₁ = X⊕Y ----(1)

Carry

C₁ = XY------(2)

Sum,

S₂ = S₁⊕Cᵢ
= X⊕Y⊕Cᵢ [From (1)]
= S

Now,

C₂ = S₁Cᵢ = (X⊕Y)Cᵢ-----(3)

Now total carry(C₀) will be the addition of first half adder carry(C₁) and second half adder carry(C₂)

So,

C₀ = C₁+C₂ = XY+(X⊕Y)Cᵢ [From (2) and (3) ]

This is the full adder carry.
From here we get,
Sum of full adder output, S = X⊕Y⊕Cᵢ
Total carry of the full adder, C₀ = XY+(X⊕Y)Cᵢ

Let the 3 input signals of full adder are X, Y, Cᵢ ( Cᵢ is the carry input) and outputs are S(Sum) Cₒ (Carry output). The truth table for the full adder is given below.

 Input Output X Y Cᵢ S Cₒ 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1

So, considering  all S = 1  we get,

Again for all Cₒ = 1, we get,

Here is the diagram of the half adder to the full adder circuit.