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In addition to the axioms, Boolean algebra also has various theorems that are derived from the axioms and provide additional rules for manipulating Boolean expressions. Here are some common theorems of Boolean algebra:

Theorems of Boolean Algebra:

  1. Double Negation Theorem:
  2. Complement Theorem:
    • OR (NOT ) = 1
    • AND (NOT A) = 0
  3. Consensus Theorem:
  4. Absorption Laws:
    • OR () = A AND A
    • OR ) =A AND (A)
  5. Reduction Theorem:
    • OR ( AND ) =
    • AND ( OR ) =
  6. Involution Law:
  7. Complementarity:
    • OR (NOT  A) = 1
    •  A AND (NOT) = 0

These theorems, along with the axioms, provide a comprehensive set of rules for manipulating and simplifying Boolean expressions.

Boolean Functions:

Boolean functions are mathematical functions that operate on one or more Boolean inputs and produce a single Boolean output. They are represented using Boolean expressions and can be implemented using logic gates in digital circuits.

Boolean functions can be classified based on their properties:

  1. Monotonic Functions: These functions either increase or remain the same as each input variable increases from false (0) to true (1).
  2. Self-Dual Functions: Functions that produce the same output when all input variables are complemented (i.e., replaced by their negation).
  3. Linear Functions: Functions where the output is a linear combination (XOR) of the input variables.
  4. Symmetric Functions: Functions whose value remains the same when the order of input variables is changed.
  5. Threshold Functions: Functions that produce a true output only if a certain number of input variables are true.

Boolean functions play a crucial role in digital logic design, as they represent the behavior of digital systems and circuits. They are used in various applications such as digital signal processing, cryptography, and computer algorithms.