Adding two numbers is one of the basic tasks performed by a computer. Addition operation of two binary digits can be represented as follows –

For all values of A and B except 11, the result is a single digit value but for AB = 11, the result is in two digits. The value at higher significant position, i.e., on this case is known as carry and has to be added to the pair of next higher significant bits as is done in real algebra also. Combinational Circuit that performs addition of two bits in known as HALF ADDER whereas the one in which we also take account of carry from lower significant bits is known as FULL ADDER.

In simple words, we can say that half adder adds two bits whereas a full adder adds three bits.

Half Adder

For designing a half adder, we will follow steps described follows –

  1. As we know that half adder is used to add two bits, from the statement of the problem it is clear that we have to add two bits in binary.
  2. As two bits have to be added, number of input variables will be 2. Now, the result when value for AB = 11 consists of two digits out of which higher significant bits is known as carry, we have to use two output variables, one of which is used for sum and another for carry.
  3. We give names A, B to input variables and c and s to output as carry and sum respectively.
  4. We know the required output value for given set of input values, so the truth table for half adder can be designed as follows –

Half Adder Truth Table

5. From above truth table, we can obtain Boolean function for both the output variables.

C = AB , S = A’ B + AB’ , =A+B

Logical Diagram of Half Adder

The main disadvantage of this circuit is that it can only add two inputs and if there is any carry it is neglected.

Full Adder

Full Adder is used to add three bits, i.e., when we also have a carry from the lower significant bit addition. Now, there will be three number of inputs and two outputs. Input variables are A, B which are the two bits that are to be added and CIN that is a carry from the addition of lower significant bit.

Full Adder Truth Table

The output S is an EX – OR between the input A and the half adder SUM output B. The COUT will be true only if any of the two inputs out of the three are HIGH or at logic 1.

Thus, a full adder circuit can be implemented with the help of two half adder circuits. The first half adder circuit will be used to add A and B to produce a partial sum. The second half adder logic can be used to add CIN to the sum produced by the first half adder circuit. Finally, the output S is obtained.

Logical Diagram of Full Adder

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