Full Adder

• When two (or more) bit binary numbers are added, the problem is somewhat more complicated.
• Example: Add binary representations of 3 (11) and 1 (01).

  	1	1	(3)
  +	0	1	(1)
  1	1		(carry)
  1	0	0	(result)

• The example illustrates that for all but the rightmost bit, not two but three bits must be added.
• The carry bit from the previous bit addition must be reflected.
• The truth table for the full adder (with carry input) is

  A1	B1	$\mbox{carry}_1$	$\mbox{carry}_2$	S1
  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

• The carry bit $\mbox{carry}_2$ is 1 if
  • both A₁ and B₁ are 1, or
  • exactly one of A₁ and B₁ is 1 and the input carry, $\mbox{carry}_1$, is 1.
• The sum bit S₁ is 1 if an odd number of the three inputs is on, i.e., S₁ is the XOR of the three inputs.
• Hence, the full adder can be realized as shown below.
• Notice that the full adder can be constructed from two half adders and an OR gate.

  

Figure 3.7: Full Adder