Full adder using 4:1 multiplexers

A full adder has inputs $A$, $B$, $C_{in}$ and outputs Sum $S$ and Carry $C_{out}$. Two 4:1 MUXes are used, one for each output, with $A$ and $B$ on the select lines ($S_1 = A$, $S_0 = B$) and $C_{in}$ on the data side.

Truth table

$$S = \Sigma m(1,2,4,7), \qquad C_{out} = \Sigma m(3,5,6,7)$$

Implementation tables

For data input $I_j$ ($AB = j$), the two rows are $C_{in} = 0$ (minterm $2j$) and $C_{in} = 1$ (minterm $2j+1$).

Sum:

Carry:

Circuit

               +-------+                  +-------+
  Cin -------->| I0    |      0 --------->| I0    |
  Cin --[NOT]->| I1 4:1|-> S  Cin ------->| I1 4:1|-> Cout
  Cin'-------->| I2    |      Cin ------->| I2    |
  Cin -------->| I3    |      1 --------->| I3    |
               +-------+                  +-------+
                 A  B                       A  B

Check

From the MUX equation:

$$C_{out} = A'B(C_{in}) + AB'(C_{in}) + AB(1) = (A \oplus B)C_{in} + AB$$

which is the standard carry expression. For $A = 1, B = 0, C_{in} = 1$: select $10$ gives $S = I_2 = C_{in}' = 0$ and $C_{out} = I_2 = C_{in} = 1$, i.e. $1 + 0 + 1 = 10_2$. All eight rows were verified.