Consensus theorem
 |
| Variable inputs | Function values | |||
| X | Y | Z | xy + x'z + yz | xy + x'z |
| 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 |
In Boolean algebra, the consensus theorem is a simplification of the following terms:
- xy + x'z + yz = xy + x'z
Proof for this theorem is:
LHS = xy + x'z + (x + x')yz
= xy + x'z + xyz + x'yz
= xy + xyz + x'z + x'yz
= xy(1 + z) + x'z(1 + y)
= xy + x'z
= RHS
The dual of this equation is:
- (x + y)(x' + z)(y + z) = (x + y)(x' + z)
The consensus term refers to the redundant term.
In digital logic, including the consensus term can eliminate race hazards.
References
- Roth, Charles H. Jr. and Kinney, Larry L. (2004, 2010). "Fundamentals of Logic Design", 6th Ed., p. 66ff.