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

In the digital world, half adder and full adder are the combinational circuits which are designed to perform addition of input variables.adders are an important electronic device found in the computer. They are also found in many types of numeric data processing system. In this article, we will discuss both half adder and full adder theory with their truth tables and logic diagram.

The simple addition consists of only 4 possible operations which are  (0+0), (0+1), (1+0) and (1+1). So, all these four operations consist of 2 bits. The operations (0+0), (0+1), (1+0) produces a sum consisting 1 digit. But the remaining operation (1+1) produces a sum consists of 2 digits. Where the higher significant bit is called carry bit. So, we can say the half adder definition as a combinational circuit that performs the addition of 2 bits is called a half adder.

A half adder performs the addition of two inputs and it produces two outputs namely sum and carry. Below, in the half adder truth table, we termed A and B as inputs while the outputs sum and carry are named as S and C respectively.

### Half adder truth table :

From the above truth table, we have seen that the outputs of a half adder S and C are similar to XOR Gate and AND Gate respectively. So, we can construct a half adder circuit using these two gates as shown below.

### Half adder circuit diagram :

From the above half adder circuit diagram, we have seen that two input signals A and B are given to two different gates. We have given the same inputs to the XOR gate and also to the AND gate. So, when the voltage is applied to the inputs of XOR gate, the same input voltage will be received by the AND Gate also.

### Half adder boolean expression :

The half adder boolean expressions are :

• S = A’B + AB’
• C = AB

Full adder performs addition operation when the augend and addend number contains more than 2 digits. The carry obtained from the addition of 2 bits is added to the next higher pair of significant bits. So, here the addition operation involves 3 bits. They are augend bit, addend bit and carry bit respectively.

So, we can say the definition of full adder as the combinational circuit which is capable of performing addition operation of 3 bits is known as the full adder. A full adder adds 3 bits binary numbers and outputs 2 bits binary numbers. The outputs of the full adder are also referred as sum and carry

### Full adder circuit diagram :

From the above full adder circuit diagram, we can clearly notice that it is just a combination of two half adders which are joined by an OR gate.

Here, the first half adder is used to add the input signals A and B. The output produced by this half adder and the remaining input X is then fed to the inputs of the second half adder. This whole combination gives us the output S. If any one of the half adders produces a carry bit, the OR gate is implemented to handle this carry output.

### The full adder boolean expressions are :

• S = X’A’B + X’AB’ + XA’B’ + XAB
• C = AB + BX + AX