Boolean algebra
nounEtymology Named after George Boole (1815–1864), an English mathematician, educator, philosopher and logician.
Algebra, Mathematics, Sciences An 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.)
- 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.
Algebra, Computing, Engineering, Human sciences, Logic, Mathematics, Natural sciences, Philosophy, Physical sciences, Sciences 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, Sciences The study of such algebras; Boolean logic, classical logic.
- Synonyms
- switching algebra
- Derived terms
- free Boolean algebra
- Related terms
- Boolean lattice, Boolean ring
- Hypernyms
- distributive lattice, Ockham algebra, De Morgan algebra, Kleene algebra, residuated lattice, Heyting algebra, MV-algebra
- Hyponyms
- complete Boolean algebra