Boolean algebra

== English == === Etymology === Named after George Boole (1815–1864), an English mathematician, educator, philosopher and logician. === Noun === Boolean algebra (plural Boolean algebras) (algebra) An algebraic structure ( Σ , ∨ , ∧ , ∼ , 0 , 1 ) {\displaystyle (\Sigma ,\vee ,\wedge ,\sim ,0,1)} where ∨ {\displaystyle \vee } and ∧ {\displaystyle \wedge } are idempotent binary operators, ∼ {\displaystyle \sim } is a unary involutory operator (called "complement"), and 0 and 1 are nullary operators (i.e., constants), such that ( Σ , ∨ , 0 ) {\displaystyle (\Sigma ,\vee ,0)} is a commutative monoid, ( Σ , ∧ , 1 ) {\displaystyle (\Sigma ,\wedge ,1)} is a commutative monoid, ∧ {\displaystyle \wedge } and ∨ {\displaystyle \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.) (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 (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 ==== === See also === Boolean lattice Boolean ring