Notes on Boolean Functions: Sum of Product and Product of Sum Representation

Introduction

• Video by ALL ABOUT ELECTRONICS
• Focus on:
  • Representing Boolean functions in Sum of Product (SOP) and Product of Sum (POS) forms.
  • Understanding minterms and maxterms.

Understanding Boolean Functions

• Logic circuit output is a function of digital inputs.
• Can be represented using:
  • Truth tables
  • Boolean expressions

Sum of Product (SOP) Form

Definition

• Represents a Boolean function as a sum of product terms.
• Each product term is formed using logical AND operation on input variables (true or complemented).

Example Product Terms

• a.b: Both variables are true.
• a.c.b': a and c are true; b is complemented.

SOP Expression Structure

• Logical OR operation among product terms.
• Represented by:
  • Product terms (AND) combined with a plus sign (OR).

Types of SOP Forms

1. Canonical SOP Form
   • Each product term contains all variables.
2. Non-Canonical SOP Form
   • Product terms may not include all variables.
   • Example: a variable missing in terms.

Example Expressions

• f1 and f2: Both expressions can be identified as SOP based on criteria above.

Product of Sum (POS) Form

Definition

• Represents a Boolean function as a product of sum terms.
• Each sum term is formed using logical OR operation.

Example Sum Terms

• Logical OR operation among different variables.

POS Expression Structure

• Logical AND operation among sum terms.
• Represented by:
  • Sum terms (OR) combined with a dot (AND).

Types of POS Forms

1. Canonical POS Form
   • Each sum term contains all variables.
2. Non-Canonical POS Form
   • Sum terms may not include all variables.

Example Expressions

• f1 and f2: Identified as POS based on the same criteria.

Minterms and Maxterms

Minterms

• Product terms in canonical SOP form.
• Consists of all variables in true or complemented form.
• For n variables, there are 2^n minterms.

Writing Minterms from a Truth Table

• Minterm representation example for two variables a and b:
  • 00 -> a'b' (m0)
  • 01 -> a'b (m1)
  • 10 -> ab' (m2)
  • 11 -> ab (m3)

Maxterms

• Sum terms in canonical POS form.
• Each maxterm includes all variables, represented in true or complemented form based on values.
• For n variables, 2^n maxterms.

Writing Maxterms from a Truth Table

• Maxterm representation for two variables a and b:
  • 00 -> a + b (M0)
  • 01 -> a + b' (M1)
  • 10 -> a' + b (M2)
  • 11 -> a' + b' (M3)

Converting Between Forms

• Minterms and maxterms are complements of each other.
• Example of converting SOP to POS:
  • Identify minterms for which the function is 0.
  • Write maxterms accordingly.

Conclusion

• Understanding how to represent Boolean expressions in SOP and POS forms is crucial in digital electronics.
• Key concepts of minterms and maxterms facilitate the conversion and understanding of logic functions.