Boolean algebra

Etymology

Definitions

1. 1

   An algebraic structure (Σ,∨,∧,∼,0,1) where ∨ and ∧ are idempotent binary operators, ∼ is…

   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.
2. 2

   Specifically, an algebra in which all elements can take only one of two values (typically…

   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. 3

   The study of such algebras

   The study of such algebras; Boolean logic, classical logic.

The neighborhood

• neighborBoolean lattice
• neighborBoolean ring
• neighbordistributive lattice
• neighborOckham algebra
• neighborDe Morgan algebra
• neighborKleene algebra
• neighborresiduated lattice
• neighborHeyting algebra
• neighborMV-algebra
• neighborcomplete Boolean algebra