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- In matematica, informatica ed elettronica, l'algebra di Boole, anche detta algebra booleana o reticolo booleano, è un ramo dell'algebra astratta che comprende tutte le algebre che operano con i soli valori di verità 0 o 1, detti variabili booleane o logiche. La struttura algebrica studiata dall'algebra booleana è finalizzata all'elaborazione di espressioni nell'ambito del calcolo proposizionale. Essendo un reticolo dotato di particolari proprietà, l'algebra booleana risulta criptomorfa, cioè associata biunivocamente e in modo da risultare logicamente equivalente, ad un insieme parzialmente ordinato reticolato. Ogni algebra booleana risulta criptomorfa ad un particolare tipo di anello, chiamato anello booleano. Tale algebra permette di definire gli operatori logici AND, OR e NOT, la cui combinazione permette di sviluppare qualsiasi funzione] logica e consente di trattare in termini esclusivamente algebrici le operazioni insiemistiche dell'intersezione, dell'unione e della complementazione, oltre a questioni riguardanti singoli bit 0 e 1, sequenze binarie, matrici binarie e diverse altre funzioni binarie. L'algebra di Boole, sviluppata nel 1854 da George Boole, un matematico inglese dell'University College di Cork, assume un ruolo importante in vari ambiti, in particolare nella logica matematica e nell'elettronica digitale, dove nella progettazione dei circuiti elettronici riveste grande importanza il teorema di Shannon, introdotto da Claude Shannon intorno al 1940 e utilizzato per scomporre una funzione booleana complessa in funzioni più semplici, o per ottenere un'espressione canonica da una tabella della verità o da un'espressione non canonica.
- ブール代数（ブールだいすう）あるいはブール環（ブールかん）とは、ジョージ・ブールが19世紀中頃に考案した論理数学の代表的な概念。ブール束（束論#ブール束）ともいう。ブール代数の研究は代数的構造としての束の理論が築かれるひとつの契機ともなった。 論理回路の設計には必須の知識である。組み合わせ回路（論理回路#組み合わせ回路）はブール代数の式で表現できる。
- Boolean algebra, as developed in 1854 by George Boole in his book An Investigation of the Laws of Thought, is a variant of ordinary elementary algebra differing in its values, operations, and laws. Instead of the usual algebra of numbers, Boolean algebra is the algebra of truth values 0 and 1, or equivalently of subsets of a given set. The operations are usually taken to be conjunction ∧, disjunction ∨, and negation ¬, with constants 0 and 1. And the laws are definable as those equations that hold for all values of their variables, for example x∨(y∧x) = x. Applications include mathematical logic, digital logic, computer programming, set theory, and statistics. Boole's algebra predated the modern developments in abstract algebra and mathematical logic; it is however seen as connected to the origins of both fields. In an abstract setting, Boolean algebra was perfected in the late 19th century by Jevons, Schröder, Huntington, and others until it reached the modern conception of an (abstract) mathematical structure. For example, the empirical observation that one can manipulate expressions in the algebra of sets by translating them into expressions in Boole's algebra is explained in modern terms by saying that the algebra of sets is a Boolean algebra. In fact, M. H. Stone proved in 1936 that every Boolean algebra is isomorphic to a field of sets. In the 1930s, while studying switching circuits, Claude Shannon observed that one could also apply the rules of Boole's algebra in this setting, and he introduced switching algebra as a way to analyze and design circuits by algebraic means in terms of logic gates. Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as the two-element Boolean algebra. In circuit engineering settings today, there is little need to consider other Boolean algebras, thus "switching algebra" and "Boolean algebra" are often used interchangeably. Efficient implementation of Boolean functions is a fundamental problem in the design of combinatorial logic circuits. Modern electronic design automation tools for VLSI circuits often rely on an efficient representation of Boolean functions known as (reduced ordered) binary decision diagrams (BDD) for logic synthesis and formal verification. Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra. Thus, Boolean logic is sometimes used to denote propositional calculus performed in this way. Boolean algebra is not sufficient to capture logic formulas using quantifiers, like those from first order logic. Although the development of mathematical logic did not follow Boole's program, the connection between his algebra and logic was later put on firm ground in the setting of algebraic logic, which also studies the algebraic systems of many other logics. The problem of determining whether the variables of a given Boolean (propositional) formula can be assigned in such a way as to make the formula evaluate to true is called the Boolean satisfiability problem (SAT), and is of importance to theoretical computer science, being the first problem shown to be NP-complete. The closely related model of computation known as a Boolean circuit relates time complexity to circuit complexity.
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- ブール代数（ブールだいすう）あるいはブール環（ブールかん）とは、ジョージ・ブールが19世紀中頃に考案した論理数学の代表的な概念。ブール束（束論#ブール束）ともいう。ブール代数の研究は代数的構造としての束の理論が築かれるひとつの契機ともなった。 論理回路の設計には必須の知識である。組み合わせ回路（論理回路#組み合わせ回路）はブール代数の式で表現できる。
- In matematica, informatica ed elettronica, l'algebra di Boole, anche detta algebra booleana o reticolo booleano, è un ramo dell'algebra astratta che comprende tutte le algebre che operano con i soli valori di verità 0 o 1, detti variabili booleane o logiche. La struttura algebrica studiata dall'algebra booleana è finalizzata all'elaborazione di espressioni nell'ambito del calcolo proposizionale.
- Boolean algebra, as developed in 1854 by George Boole in his book An Investigation of the Laws of Thought, is a variant of ordinary elementary algebra differing in its values, operations, and laws. Instead of the usual algebra of numbers, Boolean algebra is the algebra of truth values 0 and 1, or equivalently of subsets of a given set. The operations are usually taken to be conjunction ∧, disjunction ∨, and negation ¬, with constants 0 and 1.
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