Full Adder

Let . The addition of these two bits along with a carry-in bit will result in a sum bit and possibly a carry out bit as shown below:

We can see that and (see XOR).

Constructing a Full Adder using Half Adders

See Half Adder

Suppose the first half adder does this:
and the second half adder does this: .

If , then , and if , . Thus, both and are not at the same time, so we can write the carry out of the full adder as the logical or of and .

From the diagram, we have and , where and .

Adder and Subtractor Unit

See two's complement.