Boolean algebra

plural Boolean algebras

1. algebraAn algebraic structure (Σ,∨,∧,∼,0,1) where ∨ and ∧ are idempotent binary operators, ∼ is a unary involutory operator (called "complement"), and 0 and 1 are nullary operators (i.e., constants), such that (Σ,∨,0) is a commutative monoid, (Σ,∧,1) is a commutative monoid, ∧ and ∨ 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.)

   A Boolean algebra is a De Morgan algebra which also satisfies the law of excluded middle and the law of noncontradiction.

   1 more example

   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.

2. algebraSpecifically, 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

   Synonyms: switching algebra

3. mathematicsThe study of such algebras; Boolean logic, classical logic.