Consensus theorem

The Boolean identity that makes a consensus term redundant when two other terms are present: XY + X'Z + YZ = XY + X'Z.

Boolean algebra: An algebra of binary-valued variables and logical operations such as AND, OR, and NOT. The theorem is an identity within this algebra, using its operations and complement notation.
Literal (mathematical logic): A Boolean variable or its complement, such as X or X′. Each factor in the theorem’s product terms is a literal.
Static hazard: A temporary unwanted output change caused by unequal signal delays in a logic circuit. Removing a redundant consensus term can introduce a static hazard in a circuit.
Duality principle (Boolean algebra): The Boolean-algebra rule that exchanges AND with OR and 0 with 1 to produce a dual identity. Applying duality yields the corresponding product-of-sums consensus identity.
Boolean function: A function that maps one or more binary inputs to a binary output. The theorem preserves the output function while removing a redundant product term.
Boolean complement: The operation that reverses a Boolean value, mapping 0 to 1 and 1 to 0. The identity depends on the complementary pair X and X′.
Hazard-free logic: Logic-circuit design that prevents spurious output transitions caused by propagation delays. Keeping a consensus term can preserve hazard-free behavior despite its algebraic redundancy.
Consensus theorem (Boolean algebra): The dual Boolean identity X(Y + Z)(X′ + Y) = (Y + Z)(X′ + Y). This name distinguishes the product-of-sums form from the sum-of-products identity on this page.
Boolean minimization: The process of simplifying a Boolean function while preserving its input-output behavior. Recognizing the consensus term can reduce an expression without changing its function.
Implicant: A product term that implies a Boolean function is true whenever the term is true. The theorem’s three product terms are implicants, and one is covered by the other two.

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