Full adder from half adders

A full adder adds three bits $A$, $B$, $C_{in}$ and gives sum $S$ and carry $C_{out}$. A half adder adds two bits: $S = X \oplus Y$, $C = XY$.

Full adder expressions

$$S = A \oplus B \oplus C_{in}$$ $$\begin{aligned} C_{out} &= A'BC_{in} + AB'C_{in} + ABC_{in}' + ABC_{in} \\ &= AB + C_{in}(A'B + AB') = AB + C_{in}(A \oplus B) \end{aligned}$$

Construction with two half adders and an OR gate

• HA1 ($A$, $B$): $S_1 = A \oplus B$, $C_1 = AB$
• HA2 ($S_1$, $C_{in}$): $S = S_1 \oplus C_{in} = A \oplus B \oplus C_{in}$, $C_2 = C_{in}(A \oplus B)$
• OR: $C_{out} = C_1 + C_2 = AB + C_{in}(A \oplus B)$

       +-----+  S1   +-----+
A ---->| HA1 |------>| HA2 |-----------> S
B ---->|     | Cin ->|     |
       +-----+       +-----+
          | C1          | C2
          +---->[OR]<---+
                  +--------------------> Cout

Check: $A=1, B=0, C_{in}=1$: $S_1 = 1, C_1 = 0$; $S = 0, C_2 = 1$; $C_{out} = 1$. So $1+0+1 = 10_2$. Correct.