Boolean algebra

Boolean algebra (plural Boolean algebras)

  1. (algebra) An algebraic structure (\Sigma, \vee, \wedge, \sim, 0, 1) where \vee and \wedge are idempotent binary operators, \sim is a unary involutory operator (called "complement"), and 0 and 1 are nullary operators (i.e., constants), such that (\Sigma, \vee, 0) is a commutative monoid, (\Sigma, \wedge, 1) is a commutative monoid, \wedge and \vee distribute with respect to each other, and such that combining two complementary elements through one binary operator yields the identity of the other binary operator. (See Boolean algebra (structure)#Axiomatics.)
    The set of divisors of 30, with binary operators: g.c.d. and l.c.m., unary operator: division into 30, and identity elements: 1 and 30, forms a Boolean algebra.
    A Boolean algebra is a De Morgan algebra which also satisfies the law of excluded middle and the law of noncontradiction.
  2. (algebra, logic, computing) Specifically, an algebra in which all elements can take only one of two values (typically 0 and 1, or "true" and "false") and are subject to operations based on AND, OR and NOT
  3. (mathematics) The study of such algebras; Boolean logic, classical logic.
Synonyms Translations
  • French: algèbre de Boole, algèbre booléenne
  • German: boolesche Algebra
  • Italian: algebra booleana, reticolo booleano, algebra di Boole
  • Russian: бу́лева а́лгебра

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