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Boolean Operations

Boolean Operations

Fundamental logical operations that combine or manipulate truth values, forming the basis of digital logic and computational reasoning.

Boolean Operations

Boolean operations are the fundamental logical functions that operate on boolean values (true/false, 1/0) and form the cornerstone of digital logic and modern computing. Named after mathematician George Boole, these operations provide the mathematical framework for processing binary information.

Core Operations

AND Operation

  • Outputs true only when all inputs are true
  • Represented by symbols: ∧, &, •
  • Example: (true AND true) = true, (true AND false) = false
  • Used in circuit design for serial conditions

OR Operation

  • Outputs true if at least one input is true
  • Represented by symbols: ∨, |, +
  • Example: (true OR false) = true, (false OR false) = false
  • Essential for parallel processing scenarios

NOT Operation

  • Inverts the input value
  • Represented by symbols: ¬, !, ~
  • Example: NOT(true) = false
  • Fundamental for creating logical negation

Compound Operations

Building on the core operations, several compound operations emerge:

  1. XOR (Exclusive OR)

  2. NAND (NOT AND)

  3. NOR (NOT OR)

Applications

Boolean operations find widespread use in:

Properties

Important mathematical properties include:

  • Commutativity: A OR B = B OR A
  • Associativity: (A AND B) AND C = A AND (B AND C)
  • Distributivity: A AND (B OR C) = (A AND B) OR (A AND C)
  • De Morgan's Laws for operation transformation

Historical Impact

The development of boolean operations has profoundly influenced:

Boolean operations continue to evolve with new applications in quantum computing and machine learning, while remaining fundamental to classical computing and logic design.

See Also