English Dictionary

Boolean algebra

名詞

(algebra) 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.)

日本語の意味

ブール代数：論理演算（論理和、論理積、補集合など）により定義される代数的構造。集合や論理回路、計算理論などで用いられ、0や1を含む定数や、分配法則、冪等性など特有の公理系に基づいた体系である。

In the seminar, we demonstrated that a Boolean algebra is an algebraic structure (Σ, ∨, ∧, ∼, 0, 1) in which the binary operations ∨ and ∧ are idempotent, the unary operator ∼ is an involution called complement, 0 and 1 are constants, (Σ, ∨, 0) and (Σ, ∧, 1) form commutative monoids, ∨ and ∧ distribute over each other, and combining two complementary elements with one binary operation yields the identity of the other.

ゼミでは、ブール代数が、二項演算 ∨ と ∧ が冪等で、単項演算 ∼ が補元と呼ばれる自己逆な演算であり、0 と 1 が定数で、(Σ, ∨, 0) と (Σ, ∧, 1) が可換モノイドをなすとともに、∨ と ∧ が互いに分配律を満たし、互いに補元である二つの元を一方の二項演算で結合すると他方の単位元になるような代数的構造であることを示しました。

意味（1）

(algebra) 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.

意味（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

意味（3）

(mathematics) The study of such algebras; Boolean logic, classical logic.

Boolean algebras （ plural ）