Boolean Algebra Laws: Complete Reference with Examples

Complete guide to Boolean algebra laws with proofs and examples. Learn Commutative, Associative, Distributive, De Morgan's, Absorption laws and how to apply them.

Fundamental Laws of Boolean Algebra

These laws allow you to simplify Boolean expressions and design more efficient digital circuits.

Identity Laws

  • A + 0 = A (OR with 0 gives original value)
  • A · 1 = A (AND with 1 gives original value)

Null/Domination Laws

  • A + 1 = 1 (OR with 1 always gives 1)
  • A · 0 = 0 (AND with 0 always gives 0)

Idempotent Laws

  • A + A = A
  • A · A = A

Complement Laws

  • A + A' = 1
  • A · A' = 0
  • (A')' = A (double negation)

Commutative Laws

  • A + B = B + A
  • A · B = B · A

Associative Laws

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

Distributive Laws

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

Absorption Laws

  • A + A·B = A
  • A · (A + B) = A

De Morgan's Laws

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

Memory trick: "Break the bar, change the sign" - when you distribute the NOT, change AND to OR (or vice versa).

Simplification Example

Simplify: A + A'B

  1. Factor: Cannot factor directly
  2. Apply: A + A'B = (A + A')(A + B) (Distributive)
  3. Simplify: = 1·(A + B) = A + B (Complement law)

Practice Tool

Use our Boolean Calculator to verify your simplifications step-by-step!

Frequently Asked Questions

What are De Morgan's laws?
De Morgan's laws state: (A+B)' = A'·B' and (A·B)' = A'+B'. They allow you to break the bar by changing AND to OR (or vice versa) and complementing each variable.
What is the absorption law?
The absorption law states: A + A·B = A and A·(A+B) = A. It allows you to remove redundant terms from Boolean expressions.
How do you apply Boolean laws to simplify expressions?
Start by factoring common terms, apply absorption to remove redundant parts, use De Morgan's to simplify negations, and check for complement pairs (A+A'=1, A·A'=0).