To complete the truth table for the circuit with 4 inputs (x, y, z, w) that activates output f = 1 when the number of 1s in the inputs is odd, we need to consider all possible combinations of inputs.
The possible input combinations for four binary variables (x, y, z, w) range from 0000 to 1111, totaling 16 combinations. For each combination, we will count the number of 1s:
| x | y | z | w | Number of 1s | f (Output) |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 1 | 1 | 2 | 0 |
| 0 | 1 | 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 | 2 | 0 |
| 0 | 1 | 1 | 0 | 2 | 0 |
| 0 | 1 | 1 | 1 | 3 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 2 | 0 |
| 1 | 0 | 1 | 0 | 2 | 0 |
| 1 | 0 | 1 | 1 | 3 | 1 |
| 1 | 1 | 0 | 0 | 2 | 0 |
| 1 | 1 | 0 | 1 | 3 | 1 |
| 1 | 1 | 1 | 0 | 3 | 1 |
| 1 | 1 | 1 | 1 | 4 | 0 |
In this truth table, the output f is 1 when the count of 1s (in the Number of 1s column) is odd. For the combinations where the output f = 1, we have:
- 0001
- 0010
- 0100
- 0111
- 1000
- 1011
- 1101
- 1110
This methodology can be applied to analyze and design circuits that depend on counting binary values effectively.