Boolean Algebra Laws

Comprehensive reference of Boolean algebra laws and identities with descriptions, examples, and applications in digital logic design.

Showing 19 of 19 laws

Identity Law

Basic Laws

Formula

A + 0 = A, A · 1 = A

Description

OR with 0 or AND with 1 leaves the variable unchanged

Example

X + 0 = X, Y · 1 = Y

Null Law

Basic Laws

Formula

A + 1 = 1, A · 0 = 0

Description

OR with 1 always gives 1, AND with 0 always gives 0

Example

X + 1 = 1, Y · 0 = 0

Idempotent Law

Basic Laws

Formula

A + A = A, A · A = A

Description

OR or AND of a variable with itself equals the variable

Example

X + X = X, Y · Y = Y

Complement Law

Basic Laws

Formula

A + A' = 1, A · A' = 0

Description

OR of a variable and its complement is 1, AND is 0

Example

X + X' = 1, Y · Y' = 0

Involution Law

Basic Laws

Formula

(A')' = A

Description

Double complement of a variable equals the variable itself

Example

(X')' = X

Commutative Law (OR)

Commutative Laws

Formula

A + B = B + A

Description

Order of variables in OR operation can be changed

Example

X + Y = Y + X

Commutative Law (AND)

Commutative Laws

Formula

A · B = B · A

Description

Order of variables in AND operation can be changed

Example

X · Y = Y · X

Associative Law (OR)

Associative Laws

Formula

(A + B) + C = A + (B + C)

Description

Grouping of variables in OR operation can be changed

Example

(X + Y) + Z = X + (Y + Z)

Associative Law (AND)

Associative Laws

Formula

(A · B) · C = A · (B · C)

Description

Grouping of variables in AND operation can be changed

Example

(X · Y) · Z = X · (Y · Z)

Distributive Law (AND over OR)

Distributive Laws

Formula

A · (B + C) = A · B + A · C

Description

AND distributes over OR

Example

X · (Y + Z) = X · Y + X · Z

Distributive Law (OR over AND)

Distributive Laws

Formula

A + (B · C) = (A + B) · (A + C)

Description

OR distributes over AND

Example

X + (Y · Z) = (X + Y) · (X + Z)

De Morgan's Law 1

De Morgan's Laws

Formula

(A + B)' = A' · B'

Description

Complement of OR equals AND of complements

Example

(X + Y)' = X' · Y'

De Morgan's Law 2

De Morgan's Laws

Formula

(A · B)' = A' + B'

Description

Complement of AND equals OR of complements

Example

(X · Y)' = X' + Y'

Absorption Law 1

Absorption Laws

Formula

A + (A · B) = A

Description

Variable absorbs its AND with another variable

Example

X + (X · Y) = X

Absorption Law 2

Absorption Laws

Formula

A · (A + B) = A

Description

Variable absorbs its OR with another variable

Example

X · (X + Y) = X

Absorption Law 3

Absorption Laws

Formula

A + (A' · B) = A + B

Description

Simplified absorption with complement

Example

X + (X' · Y) = X + Y

Absorption Law 4

Absorption Laws

Formula

A · (A' + B) = A · B

Description

Simplified absorption with complement

Example

X · (X' + Y) = X · Y

Consensus Law 1

Consensus Laws

Formula

(A · B) + (A' · C) + (B · C) = (A · B) + (A' · C)

Description

Third term is redundant in consensus

Example

(X · Y) + (X' · Z) + (Y · Z) = (X · Y) + (X' · Z)

Consensus Law 2

Consensus Laws

Formula

(A + B) · (A' + C) · (B + C) = (A + B) · (A' + C)

Description

Third term is redundant in dual consensus

Example

(X + Y) · (X' + Z) · (Y + Z) = (X + Y) · (X' + Z)

About Boolean Algebra

Boolean algebra is a branch of mathematics that deals with operations on logical values. Named after George Boole, it forms the theoretical foundation of digital logic design and computer science. Boolean algebra uses binary variables (0 and 1, or FALSE and TRUE) and logical operations.

Fundamental Concepts

Basic Operations

  • AND (·): Logical multiplication
  • OR (+): Logical addition
  • NOT ('): Logical complement

Boolean Values

  • 0: FALSE, Low, Off
  • 1: TRUE, High, On

Law Categories

Basic Laws

Fundamental identities including Identity, Null, Idempotent, Complement, and Involution laws that form the foundation of Boolean algebra.

Commutative & Associative Laws

Laws that allow changing the order and grouping of variables in Boolean expressions without affecting the result.

Distributive Laws

Laws that describe how AND and OR operations distribute over each other, essential for expression manipulation.

De Morgan's Laws

Critical laws for converting between AND and OR operations through complementation, fundamental in logic design.

Absorption Laws

Laws that allow simplification of Boolean expressions by eliminating redundant terms.

Consensus Laws

Advanced laws for eliminating redundant terms in complex Boolean expressions.

Applications in Digital Logic

  • Logic Circuit Simplification: Reducing the number of gates and complexity
  • Karnaugh Map Construction: Systematic approach to Boolean minimization
  • Digital System Design: Optimizing hardware implementations
  • Computer Architecture: Processor design and instruction set optimization
  • Software Development: Conditional logic and algorithm optimization

Notation Conventions

Common Symbols

  • AND: A · B, A ∧ B, AB
  • OR: A + B, A ∨ B
  • NOT: A', Ā, ¬A, ~A
  • XOR: A ⊕ B, A ⊻ B
  • NAND: (A · B)', A ↑ B
  • NOR: (A + B)', A ↓ B

Proof Techniques

Boolean laws can be proven using several methods:

  • Truth Tables: Exhaustive verification for all input combinations
  • Algebraic Manipulation: Using known laws to derive new ones
  • Venn Diagrams: Visual representation of set operations
  • Perfect Induction: Systematic verification of all cases

Historical Context

Boolean algebra was developed by George Boole in the mid-19th century as "An Investigation of the Laws of Thought" (1854). It remained largely theoretical until Claude Shannon's 1938 thesis demonstrated its application to electrical circuits, laying the foundation for digital computing.