Booleans

Booleans describe a simple data type or concept that represents truth values. In logic, a boolean can take one of two values: true or false. In computing, booleans are used to control program flow and to represent the result of comparisons or conditions.

The primary Boolean operations are AND, OR, and NOT. These can be combined to form more complex expressions, including XOR, NAND, and NOR. Truth tables list the output for every possible combination of inputs and are used to reason about logic independent of language or hardware.

Boolean algebra provides a set of algebraic rules for combining and simplifying Boolean expressions. Laws such as De Morgan's rules and distributive, associative, and commutative properties enable the reduction of logical formulas and the design of efficient circuits.

Representations of booleans vary by context. In hardware, booleans correspond to binary signals (1 and 0). In software, many languages define a boolean type with literals true and false; some languages treat other values as truthy or falsy.

Historically, George Boole introduced Boolean algebra in the 19th century, providing a mathematical framework for logic. The theory formed the basis for digital circuit design, later expanded by Claude Shannon in the 20th century to apply to electrical engineering and computer hardware.

Booleans underpin modern computing and information processing, including conditional branching in software, filtering and search operations, and the design of digital circuits and processors.