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Boolean Simplification Rules

Oct 2, 2025

Overview

This lecture introduces key rules and laws for simplifying Boolean expressions, including official specification rules, general simplification rules, and the absorption rule.

Boolean Simplification Rules Overview

  • Boolean expressions can be simplified using rules instead of Karnaugh maps, especially when more than four variables are involved.
  • Five specification rules: De Morgan's Law, Distribution, Association, Commutation, Double Negation.
  • General AND and OR rules help quickly simplify basic expressions.
  • The absorption rule, while not required, assists in further simplification.

General AND & OR Rules

  • For AND: x AND 0 = 0; x AND 1 = x.
  • For OR: x OR 0 = x; x OR 1 = 1.
  • x OR x = x; x AND x = x.

De Morgan’s Laws

  • First Law: NOT (A AND B) = (NOT A) OR (NOT B).
  • Second Law: NOT (A OR B) = (NOT A) AND (NOT B).
  • De Morgan's law is applied one operator at a time.
  • Double negation can simplify nested NOT operations.

Simplification Example Using De Morgan’s Laws

  • Apply De Morgan’s law to invert operators and variables.
  • Remove double negations (NOT NOT A = A).
  • Use association to regroup or remove brackets.
  • Apply general rules (e.g., x OR x = x) for final simplification.

Other Key Rules

  • Double Negation: NOT NOT A = A.
  • Association: (A OR B) OR C = A OR (B OR C); (A AND B) AND C = A AND (B AND C).
  • Commutation: A OR B = B OR A; A AND B = B AND A.
  • Distribution: A AND (B OR C) = (A AND B) OR (A AND C); A OR (B AND C) = (A OR B) AND (A OR C).

Absorption Rule

  • A OR (A AND B) = A; A AND (A OR B) = A.
  • The operator outside the bracket must differ from the one inside.
  • The term outside must also appear inside the bracket.

Key Terms & Definitions

  • Boolean Expression — An equation using Boolean variables and logical operators (AND, OR, NOT).
  • De Morgan’s Law — Rules for converting between AND/OR using NOT operators.
  • Double Negation — Two NOTs in a row cancel each other.
  • Association — The grouping of variables does not affect the outcome.
  • Commutation — The order of variables does not affect the outcome.
  • Distribution — Allows factoring or expanding expressions.
  • Absorption — Eliminates redundant terms in expressions.

Action Items / Next Steps

  • Review the Boolean algebra rules and practice applying them to expressions.
  • Download and study the Boolean algebra cheat sheet from student.craigandave.org under A-Level revision.

Full transcript