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 A comprehensive overview of Boolean algebra, its laws, and methods for converting and simplifying Boolean expressions. ​

Objectives

  • Introduction to Boolean algebra
  • Explanation of Boolean laws with examples
  • Concepts of minterms and maxterms
  • Conversion to sum of minterms and product of maxterms forms
  • Algebraic conversions
  • Standard forms

Introduction

  • Boolean algebra, invented by George Boole in 1854, is used to represent the activity of electrical circuits.
  • It involves a set ( B = {0, 1} ) with binary operators (+, ·) and a unary operator (-).

Boolean Laws

  • Identity Element: ( x \cdot 1 = x ), ( x + 0 = x )
  • Idempotence: ( x \cdot x = x ), ( x + x = x )
  • Complement: ( x + \overline{x} = 1 ), ( x \cdot \overline{x} = 0 )
  • Involution: ( \overline{\overline{x}} = x )
  • Commutative Property: ( x \cdot y = y \cdot x ), ( x + y = y + x )
  • Associative Property: ( (x \cdot y) \cdot z = x \cdot (y \cdot z) ), ( (x + y) + z = x + (y + z) )
  • Distributive Property: ( x \cdot (y + z) = (x \cdot y) + (x \cdot z) ), ( x + (y \cdot z) = (x + y) \cdot (x + z) )
  • De Morgan’s Theorem: ( \overline{x \cdot y} = \overline{x} + \overline{y} ), ( \overline{x + y} = \overline{x} \cdot \overline{y} )

Minterms and Maxterms

  • Minterms: AND terms with every variable in true or complemented form. There are ( 2^n ) minterms for ( n ) variables.
  • Maxterms: OR terms with every variable in true or complemented form. There are ( 2^n ) maxterms for ( n ) variables.

Sum of Minterms (SOM) and Product of Maxterms (POM)

  • SOM: Canonical form representing the sum of minterms that evaluate to ‘1’.
  • POM: Canonical form representing the product of maxterms that evaluate to ‘0’.

Converting Forms

  • To SOM: Use a truth table to list all minterms that evaluate to ‘1’.
  • To POM: Use a truth table to list all maxterms that evaluate to ‘0’.

Algebraic Conversions

  • To SOM: Expand terms to list all minterms explicitly.
  • To POM: Apply distributive laws to list all maxterms explicitly.

Standard Forms

  • Standard Sum-of-Products (SOP): Equations written as an OR of AND terms.
  • Standard Product-of-Sums (POS): Equations written as an AND of OR terms.

Observations

  • Any function can be implemented by ORing minterms or ANDing maxterms.
  • The same Boolean function can be expressed in both SOM and POM forms.
  • The form with fewer literals depends on the number of ‘1’ or ‘0’ entries in the function table.

Boolean Laws – Lesson 6

Engineering Institute of Technology