Consensus theorem

In Boolean algebra, the consensus theorem or rule of consensus is the identity:

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.

The conjunctive dual of this equation is:

The consensus or consensus term of two conjunctive terms of a disjunction is defined when one term contains the literal ${\displaystyle a}$ and the other the literal ${\displaystyle {\bar {a}}}$, an opposition. The consensus is the conjunction of the two terms, omitting both ${\displaystyle a}$ and ${\displaystyle {\bar {a}}}$, and repeated literals; the consensus is undefined if there is more than one opposition. For example, the consensus of ${\displaystyle {\bar {x}}yz}$ and ${\displaystyle w{\bar {y}}z}$ is ${\displaystyle w{\bar {x}}z}$.

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