The consensus theorem removes a term that looks useful but never changes the output:
- =
- Dual: =
The term is called the consensus of and .
Why it works. Take any input where = 1, so Y = 1 and Z = 1. If X = 1, then = 1. If X = 0, then = 1. Either way the output is already 1, so adds nothing.
How to find a consensus term:
- Find two terms where one variable is plain in one and barred in the other.
- Drop that variable from both.
- AND what is left.
If that product (or a longer term that contains all of it) is already in the expression, it is a redundant term and you can delete it.
The pattern needs a complement pair. has none, so nothing can go. Consensus terms are also why a minimal SOP does not always use every prime implicant. In circuits, keeping a consensus term on purpose can remove a hazard.
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 |