Boolean algebra

Boolean algebra is a sub-discipline of mathematics and computer science that deals with operations on logical values. Named after George Boole, a 19th-century mathematician and logician, Boolean algebra is fundamental to the design and operation of computer hardware and software.

History

Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854). Boole's system was based on binary variables with two possible values, 0 and 1, and on the three basic operations, AND, OR, and NOT.

Basic operations

Boolean algebra includes three basic operations:

• AND: The AND operation, also known as conjunction, takes two Boolean inputs and returns 1 if both inputs are 1, and 0 otherwise.
• OR: The OR operation, also known as disjunction, takes two Boolean inputs and returns 1 if at least one input is 1, and 0 otherwise.
• NOT: The NOT operation, also known as negation, takes a single Boolean input and returns the opposite value.

Properties

Boolean algebra has many properties that are similar to other algebraic structures, such as commutativity, associativity, and distributivity. It also has additional properties that are unique to Boolean algebra, such as idempotency, involution, and the laws of De Morgan.

Applications

Boolean algebra is used extensively in computer science and electrical engineering. It is fundamental to the design of digital circuits, including logic gates, flip-flops, and microprocessors. In software, Boolean algebra is used in the design of algorithms, in programming languages, and in database query languages.

See also

• Binary number
• Logic gate
• Truth table
• De Morgan's laws

• Venn Diagram of Boolean Operations
  Venn Diagram of Boolean Operations
• Logic Gates
  Logic Gates
• De Morgan's Theorems
  De Morgan's Theorems