Full Adder

combinatorial-logic

Definition

Full Adder

A full adder is a 1-bit adder and an extension of the half adder that takes in three inputs $A,B,C_{in}$ and produces an output carry bit $C_{out}$ and the result $S$. The output carry bit $C_{out}$ is the overflow bit and the result bit $S$ is the bit addition. The input carry bit $C_{in}$ is the output carry bit of a previous addition.

$A$	$B$	$C_{in}$	$C_{out}$	$S$
0	0	0	0	0
0	0	1	0	1
0	1	0	0	1
0	1	1	1	0
1	0	0	0	1
1	0	1	1	0
1	1	0	1	0
1	1	1	1	1

We can observe that $C_{out}$ is the XOR of $A,B,C_{in}$ and $C_{out}$ is given by a disjunction of conjunctions:

$$\begin{array}{rl}{C}_{out}& =A\oplus B\oplus C_{in}\\ S& =(A\wedge B)\vee (A\wedge C_{in})\vee (B\wedge C_{in})\end{array}$$