Consensus Theorem ○◂ Entries

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Consensus Theorem ○◂｜Definition｜1st｜20251119205401-00-⌔

Consensus theorem - Wikipedia

Consensus theorem

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In Boolean algebra, the consensus theorem or rule of consensus¹ is the identity:

• ✤ $xy\vee \bar{x}z\vee yz=xy\vee \bar{x}z$

The consensus or resolvent of the terms $xy$ and $\bar{x}z$ is $yz$. 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 $y$ includes a term that is negated in $z$ (or vice versa), the consensus term $yz$ is false; in other words, there is no consensus term.

The conjunctive dual of this equation is:

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

Printed 2026-06-28.

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Footnotes

1. Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition 2003, p. 44 ↩

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