4:1 mux implementing a 3-variable function — multiplexers and decoders
coreShannon expansion in hardware
Answer
Y column 00011011
Why this example is worth doing
A function of three variables fits a four-input mux if the third variable is allowed to appear on the data inputs, each of which becomes 0, 1, C or C̄. That is Shannon expansion — splitting a function on its variables and handling the cofactors separately — and it halves the part count. The page derives the residues from the truth table and names the theorem, since it is the same decomposition that underlies binary decision diagrams.
Try your own input in the Multiplexers & decoders. Multiplexers from 2:1 to 16:1 and decoders from 2:4 to 4:16, with the selection logic derived.
How the answer is reached
Output table
| A | B | C | Y |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |