boolealgebran

Boolealgebran, also known as Boolean algebra, is a branch of algebra that studies variables that take only two values, typically true and false. It provides an algebraic framework for logic, enabling the manipulation of logical expressions through a small set of operations and laws.

Historically, the system originated with George Boole in the mid-19th century, who introduced an algebraic approach to logic. It was further developed in the 20th century into a formal mathematical theory that underpins digital logic and computer science. The term boolealgebran is used in some languages as the name for this theory.

Core operations are conjunction (and), disjunction (or), and negation (not). In mathematical terms, these are meet, join, and complement in a distributive lattice. Typical laws include associativity and commutativity of and/or, distributivity of and over or, De Morgan's laws, and complement laws: x ∨ ¬x = 1 and x ∧ ¬x = 0, along with identities x ∨ 0 = x and x ∧ 1 = x. Double negation and De Morgan's laws are common forms: ¬¬x = x; ¬(x ∧ y) = ¬x ∨ ¬y; ¬(x ∨ y) = ¬x ∧ ¬y.

Boolean algebra can be modeled by sets, with ∨ as union, ∧ as intersection, and ¬ as complement with respect to a universal set. It can also be represented by truth values in a two-element algebra, or by bitwise operations in computer hardware. Boolean functions map binary inputs to a binary output and are often simplified using algebraic or graphical methods such as Karnaugh maps.

Applications include the design and analysis of digital circuits, computer algorithms, search and database queries, and the formal semantics of programming languages. Boolean algebra provides a compact language for reasoning about logical statements and binary data.