Full Adder

A full adder accepts two input bits and an input carry and generates a sum and a carry, in its output. The logic symbol of a full adder is,

full adder symbol diagram

Truth table of a full adder is given below,

A B CinCout
00000
00110
01010
01101
10010
10101
11001
11111

Expression for Sum and Carry can be derived using the following K-Map.

full adder kmap

A full adder should add Cout with other input bits A and B. We have used XOR operation to perform addition of two bits A and B. To add input carry, Cin, with the input bits, it must be XOR-ed with A⊕B. Logic expression for sum and carry can be derived from the truth table.

\Sigma=(A \oplus B) \oplus C_{in}\\ C_{out}=AB+AC_{in}+BC_{in}

The expression for Cout can be simplified into

{C_{out}=AB+(A \oplus B)C_{in}}

This means two XOR gates are used to implement full adder sum function. The first will generate A⊕B and the second will add this output with an input carry Cin. The complete logic circuit of a full adder will be,

full adder circuit diagram

Implementation of Full Adder using Half Adders

A full adder can also be implemented using two half adder as follows.

full adder using 2 half adders

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