Boolean algebra

English

Etymology

Named after George Boole (1815–1864), an English mathematician, educator, philosopher and logician.

Noun

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

• (Specifically ...): switching algebra

Hypernyms

• Kleene algebra
  • De Morgan algebra
    • Ockham algebra
      • distributive lattice
• Heyting algebra
  • residuated lattice
• MV-algebra

Hyponyms

• complete Boolean algebra

Derived terms

• free Boolean algebra

Translations

algebraic structure

• Chinese:

  Mandarin: 布爾代數／布尔代数 (zh) (Bù'ěr dàishù)

• Czech: Booleova algebra f, booleovská algebra f
• French: algèbre de Boole (fr) f, algèbre booléenne f
• German: boolesche Algebra f
• Hungarian: Boole-algebra (hu)
• Italian: algebra booleana f, reticolo booleano m, algebra di Boole (it) f
• Japanese: ブール代数 (Būru-daisū)
• Korean: 불 대수 (Bul daesu), 부울 대수 (Buul daesu)
• Macedonian: Булова алге́бра f (Bulova algébra)
• Romanian: algebră booleană f
• Russian: бу́лева а́лгебра f (búleva álgebra)
• Serbo-Croatian: Booleova algebra f
• Swedish: Boolesk algebra

See also

• Boolean lattice
• Boolean ring