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Boolean Algebra Simplification

Sep 9, 2025

Overview

This lecture covers examples of solving Boolean algebra expressions using key rules, De Morgan's Theorem, and basic simplification steps.

Example 1: Simplifying Boolean Expressions

  • Given the expression: ( a \cdot b + \overline{a \cdot b} + \overline{a} ).
  • Let ( x = a \cdot b ), so expression becomes ( x + \overline{x} + \overline{a} ).
  • ( x + \overline{x} = 1 ) as per Boolean rule.
  • ( 1 + \overline{a} = 1 ); the complement of 1 is 0, so final answer is 0.
  • Alternative method: Apply De Morgan's Theorem to rewrite and simplify further.
  • ( \overline{a \cdot b} ) becomes ( \overline{a} + \overline{b} ); substitute and simplify.
  • ( X \cdot \overline{X} = 0 ), confirming the result is 0.

Example 2: Applying De Morgan's Law and Simplification

  • Given a more complex Boolean expression involving complements and AND/OR operations.
  • Apply De Morgan’s Theorem: complements change AND to OR and vice versa.
  • For double complements, ( \overline{\overline{x}} = x ).
  • Multiply out terms after simplification and collect common factors.
  • Use Boolean rules: ( x \cdot \overline{x} = 0 ) and ( x + 1 = 1 ).
  • After simplification, only the necessary terms remain, leading to a final result.

Example 3: Solving for Variable Values

  • Given ( \overline{a} + a \cdot b = 0 ).
  • Apply the distributive law: ( \overline{a} + b = 0 ).
  • Both terms must be zero: ( \overline{a} = 0 ) (so ( a = 1 )), ( b = 0 ).
  • Solution: ( a = 1, b = 0 ).

Key Terms & Definitions

  • Boolean Algebra — A mathematical system for logic values using AND, OR, and NOT operations.
  • De Morgan's Theorem — Rules for negating AND/OR expressions: ( \overline{A \cdot B} = \overline{A} + \overline{B} ), ( \overline{A + B} = \overline{A} \cdot \overline{B} ).
  • Complement — The opposite value; for Boolean, ( \overline{1} = 0 ), ( \overline{0} = 1 ).
  • Distributive Law — In Boolean algebra, ( A + (B \cdot C) = (A + B) \cdot (A + C) ).

Action Items / Next Steps

  • Review and practice applying Boolean algebra rules and De Morgan's Theorem to simplify expressions.
  • Attempt similar Boolean simplification exercises as homework.