Boolean Algebra

Definition:

  • Boolean algebra is a mathematical structure that deals with binary variables and logical operations. It is a set of elements together with two operations, typically denoted as  (logical AND),  (logical OR), and ¬ (logical NOT), satisfying specific axioms.

Basic Operations:

  1. Logical AND ():

    • Represents conjunction.
    • AB is true only when both A and B are true.
  2. Logical OR ():

    • Represents disjunction.
    • AB is true when at least one of A or B is true.
  3. Logical NOT (¬):

    • Represents negation.
    • ¬A is true when A is false, and vice versa.

Key Concepts:

  • Boolean Variables: Represented by 0 (false) and 1 (true).

  • Truth Tables: Tables that show the output of a logical operation for all possible input combinations.

  • Laws of Boolean Algebra:

    • Idempotent Law: AA=A, AA=A
    • Commutative Law: AB=BA, AB=BA
    • Associative Law: (AB)C=A(BC), (AB)C=A(BC)
    • Distributive Law: A(BC)=(AB)(AC), A(BC)=(AB)(AC)
    • Complement Law: A¬A=0, A¬A=1
    • Absorption Law: A(AB)=A, A(AB)=A
    • De Morgan's Law: ¬(AB)=¬A¬B, ¬(AB)=¬A¬B

Applications:

  • Logic Gates: Fundamental building blocks in digital circuits (AND, OR, NOT gates) are based on Boolean algebra.

  • Digital Electronics: Used in designing and analyzing digital circuits, computing systems, and hardware.

  • Computer Science: Boolean algebra forms the basis of Boolean logic used in programming and computer architecture.

Boolean Functions:

  • Boolean Expression: An expression formed using Boolean variables and operations.

  • Canonical Form: A form of Boolean expression where each term corresponds to a specific combination of variables that makes the function true.

  • Simplification: The process of reducing Boolean expressions to their simplest form using algebraic rules.