Full adder using a single 4:1 MUX

A full adder adds $A$, $B$ and $C_{in}$ and gives

$$S = A \oplus B \oplus C_{in}, \qquad C_{out} = AB + C_{in}(A \oplus B)$$

Use $A$ and $B$ as the select lines ($S_1 = A$, $S_0 = B$) of the 4:1 MUX and express each output in terms of $C_{in}$.

Truth table grouped by A, B

Realization

• Sum from the single 4:1 MUX: $I_0 = C_{in}$, $I_1 = C_{in}'$, $I_2 = C_{in}'$, $I_3 = C_{in}$ (one NOT gate gives $C_{in}'$).
• Carry from gates: $C_{out} = AB + C_{in}(A\oplus B)$, using one XOR, two AND and one OR gate (or the simpler $C_{out} = AB + BC_{in} + AC_{in}$).

Cin ------+----------> I0 +-------+
          +-[NOT]--+-> I1 | 4:1   |
                   +-> I2 | MUX   |--> S
Cin -----------------> I3 |       |
                          +-------+
                          S1=A S0=B

A --+--[AND]---------------+
B --+--[   ]               +--[OR]--> Cout
A,B --[XOR]--[AND]---------+
Cin ---------[   ]

Check: $A=1, B=0, C_{in}=1$ selects $I_2 = C_{in}' = 0$ so $S = 0$, and $C_{out} = 0 + 1\cdot1 = 1$. This matches $1+0+1 = 10_2$.

(If a second 4:1 MUX were allowed, $C_{out}$ would use $I_0=0, I_1=C_{in}, I_2=C_{in}, I_3=1$.)