Boole's expansion theorem

Boole's expansion theorem expresses a Boolean function as the disjunction of its two restrictions on a variable, each conjoined with that variable or its complement.

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Shannon expansion: An identity that decomposes a Boolean function using its values when a chosen variable is fixed to 0 or 1. It is the modern name for the same two-case decomposition.
George Boole: An English mathematician and logician whose work founded modern symbolic logic and Boolean algebra. His algebraic treatment of logic gives the theorem its name.
Logic gate: An electronic circuit element that performs a Boolean operation on binary inputs. The expansion can be implemented using AND, OR, and NOT gates.
Algebraic normal form: A unique representation of a Boolean function as a polynomial over the two-element field. It represents functions with XOR and multiplication rather than disjunction and complemented literals.
Boolean derivative: An operator measuring how a Boolean function changes when one variable is flipped. It describes whether the two restricted forms differ, complementing the expansion's case split.
The Laws of Thought: George Boole's 1854 book presenting an algebraic account of the laws of human reasoning. It is the landmark work associated with Boole's logical algebra.
Multiplexer: A circuit that selects one of several input signals according to control inputs. A one-bit multiplexer directly realizes the theorem's choice between two restricted functions.
Zhegalkin polynomial: A polynomial representation of a Boolean function using addition and multiplication over the two-element field. Its polynomial expansion offers an alternative to Boole's disjunctive case split.
Cofactor (Boolean function): A Boolean function formed by fixing one or more variables in another Boolean function. The expansion's two terms use the cofactors obtained by setting the chosen variable to 0 and 1.
Claude Shannon: An American mathematician and engineer who established the use of Boolean algebra in digital circuit analysis. His switching-circuit work made the expansion a practical design tool.