 An adder is a digital logic circuit that is used to perform the addition of numbers. Adders are an integral part of computers and various electronic gadgets that require numerical computing. In electronic devices, an adder circuit is usually present inside the arithmetic logical unit. There are basically two types of adder circuits, namely half adder and full adder. Both half adder and full adder circuits fall under the category of combinational logic circuits.

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## Half Adder in Digital Logic

A half adder is a simple digital logic circuit that adds up two one-bit binary numbers. The inputs of the half adder are given as input 1 and input 2. These are typically referred to as A and B. The two outputs of the half adder are known as sum and carry. These are generally denoted by the English alphabets S and C. There are basically four combinations of the two inputs that can be fed to a half adder. ### Truth Table for a Half Adder

The truth table for a half adder that tends to establish a relationship between the inputs and outputs of the logic circuit by denoting the truth or falsity of the output in accordance with a particular combination of input is given as follows:

A EX-OR B

### Logical Expression for Carry

A AND B

A half adder can be implemented simply with the help of two logical gates, namely an AND gate and an exclusive-OR gate. The exclusive-OR gate takes A and B as inputs and outputs the sum, while the AND gate takes A and B as input and delivers carry as output.

## Full Adder in Digital Logic

A full adder is a digital logic circuit that obtains the sum of three one-bit binary numbers. The inputs of the full adder are given as input 1, input 2, and carry-in. These are typically referred to as A, B, and C-IN respectively. The two outputs of the full adder are known as sum and carry-out. These are generally denoted by S and C-OUT. There are eight possible combinations of the three inputs of a full adder. ### Truth Table for a Full Adder

The truth table for a full adder that tends to establish a relationship between the inputs and outputs of the logic circuit by denoting the truth or falsity of the output in accordance with a particular combination of input is given as follows: ### Logical Expression for SUM

= A’ B’ C-IN + A’ B C-IN’ + A B’ C-IN’ + A B C-IN

= C-IN (A’ B’ + A B) + C-IN’ (A’ B + A B’)

Therefore, Sum = C-IN XOR (A XOR B)

= (1,2,4,7)

### Logical Expression for C-OUT

= A’ B C-IN + A B’ C-IN + A B C-IN’ + A B C-IN

Therefore, C-OUT = A B + B C-IN + A C-IN

= (3,5,6,7)

A comparatively simpler method to evaluate the boolean expression for carry is given as follows:

= A B + A C-IN + B C-IN (A + A’)

= A B C-IN + A B + A C-IN + A’ B C-IN

= A B (1 +C-IN) + A C-IN + A’ B C-IN

= A B + A C-IN + A’ B C-IN

= A B + A C-IN (B + B’) + A’ B C-IN

= A B C-IN + A B + A B’ C-IN + A’ B C-IN

= A B (C-IN + 1) + A B’ C-IN + A’ B C-IN

= A B + A B’ C-IN + A’ B C-IN

= AB + C-IN (A’ B + A B’)

Therefore, C-OUT = AB + C-IN (A EX-OR B)