In other words, the multiplexer works like the input selector of a home music system. Only one input is selected at a time, and the selected input is transmitted to the single output. While on the music system, the selection of the input is made manually, the multiplexer chooses its input based on a binary number, the address input.
The truth table for a multiplexer is huge for all but the smallest values of n. We therefore use an abbreviated version of the truth table in which some inputs are replaced by `-' to indicate that the input value does not matter.
Here is such an abbreviated truth table for n = 3. The full truth table would have 2(3 + 23) = 2048 rows.
| a2 | a1 | a0 | d7 | d6 | d5 | d4 | d3 | d2 | d1 | d0 | | | x |
| - | - | - | - | - | - | - | - | - | - | - | --- | - |
| 0 | 0 | 0 | - | - | - | - | - | - | - | 0 | | | 0 |
| 0 | 0 | 0 | - | - | - | - | - | - | - | 1 | | | 1 |
| 0 | 0 | 1 | - | - | - | - | - | - | 0 | - | | | 0 |
| 0 | 0 | 1 | - | - | - | - | - | - | 1 | - | | | 1 |
| 0 | 1 | 0 | - | - | - | - | - | 0 | - | - | | | 0 |
| 0 | 1 | 0 | - | - | - | - | - | 1 | - | - | | | 1 |
| 0 | 1 | 1 | - | - | - | - | 0 | - | - | - | | | 0 |
| 0 | 1 | 1 | - | - | - | - | 1 | - | - | - | | | 1 |
| 1 | 0 | 0 | - | - | - | 0 | - | - | - | - | | | 0 |
| 1 | 0 | 0 | - | - | - | 1 | - | - | - | - | | | 1 |
| 1 | 0 | 1 | - | - | 0 | - | - | - | - | - | | | 0 |
| 1 | 0 | 1 | - | - | 1 | - | - | - | - | - | | | 1 |
| 1 | 1 | 0 | - | 0 | - | - | - | - | - | - | | | 0 |
| 1 | 1 | 0 | - | 1 | - | - | - | - | - | - | | | 1 |
| 1 | 1 | 1 | 0 | - | - | - | - | - | - | - | | | 0 |
| 1 | 1 | 1 | 1 | - | - | - | - | - | - | - | | | 1 |
We can abbreviate this table even more by using a letter to indicate the value of the selected input, like this:
| a2 | a1 | a0 | d7 | d6 | d5 | d4 | d3 | d2 | d1 | d0 | | | x |
| - | - | - | - | - | - | - | - | - | - | - | --- | - |
| 0 | 0 | 0 | - | - | - | - | - | - | - | c | | | c |
| 0 | 0 | 1 | - | - | - | - | - | - | c | - | | | c |
| 0 | 1 | 0 | - | - | - | - | - | c | - | - | | | c |
| 0 | 1 | 1 | - | - | - | - | c | - | - | - | | | c |
| 1 | 0 | 0 | - | - | - | c | - | - | - | - | | | c |
| 1 | 0 | 1 | - | - | c | - | - | - | - | - | | | c |
| 1 | 1 | 0 | - | c | - | - | - | - | - | - | | | c |
| 1 | 1 | 1 | c | - | - | - | - | - | - | - | | | c |
The same way we can simplify the truth table for the multiplexer, we can also simplify the corresponding circuit. Indeed, our simple design method would yield a very large circuit. The simplified circuit looks like this: