Consensus theorem

Variable inputs			Function values
x	y	z	${\displaystyle xy\vee {\bar {x}}z\vee yz}$	${\displaystyle xy\vee {\bar {x}}z}$
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 or rule of consensus^[1] is the identity:

${\displaystyle xy\vee {\bar {x}}z\vee yz=xy\vee {\bar {x}}z}$

The consensus or resolvent of the terms ${\displaystyle xy}$ and ${\displaystyle {\bar {x}}z}$ is ${\displaystyle yz}$. It is the conjunction of all the unique literals of the terms, excluding the literal that appears unnegated in one term and negated in the other. If ${\displaystyle y}$ includes a term that is negated in ${\displaystyle z}$ (or vice versa), the consensus term ${\displaystyle yz}$ is false; in other words, there is no consensus term.

The conjunctive dual of this equation is:

${\displaystyle (x\vee y)({\bar {x}}\vee z)(y\vee z)=(x\vee y)({\bar {x}}\vee z)}$