Boolean Algebra

Boolean algebra is the branch of algebra in which variables take only two values — typically denoted 0 and 1, or false and true — and the operations are defined by logical rather than arithmetic rules. It was developed by George Boole in the mid-nineteenth century as a formalization of logical inference, and was later recognized by Claude Shannon as directly applicable to the design of electrical switching circuits.

The basic operations are AND (conjunction, analogous to multiplication), OR (disjunction, analogous to addition with the rule 1+1=1), and NOT (negation, or complement). These three operations are functionally complete: any logical function, no matter how complex, can be expressed using only combinations of AND, OR, and NOT.

Boolean algebra is the foundational formalism of digital logic design, computer programming, database query languages, and formal verification. Its power lies in the structural isomorphism between logical propositions and switching circuits: every Boolean expression corresponds to a circuit, and every circuit to an expression.