Boolekalgebra

Boolekalgebra, or Boolean algebra, is an algebraic structure that captures the essentials of logical operations and set operations. It consists of a nonempty set B equipped with two binary operations: conjunction (∧) and disjunction (∨), a unary operation negation (¬), and distinguished elements 0 and 1. The structure satisfies the standard Boolean axioms: commutativity and associativity of ∧ and ∨; distributivity of ∧ over ∨ and ∨ over ∧; the identity laws a ∨ 0 = a and a ∧ 1 = a; the null laws a ∨ 1 = 1 and a ∧ 0 = 0; and the complement laws a ∨ ¬a = 1 and a ∧ ¬a = 0. De Morgan’s laws also hold: ¬(a ∨ b) = ¬a ∧ ¬b and ¬(a ∧ b) = ¬a ∨ ¬b. In many treatments, additional duals hold automatically. A Boolean algebra can also be viewed as a complemented distributive lattice with top element 1 and bottom element 0.

Boolean algebras model both propositional logic and set theory: elements can represent truth values, propositions, or subsets of a given set, with ∧ and ∨ corresponding to logical and set intersection, and ∨ to logical or and set union, while ¬ corresponds to logical negation and set complement. The power set of any set, under these operations, is a canonical example.

Historically, Boolean algebra was introduced by George Boole in the mid-19th century and later developed by others into a formal theory fundamental to logic, switching theory, and computer science. Applications span digital circuit design, database query languages, and formal verification.