Lecture 5: Boolean Algebra and Simplification Techniques

Introduction

• Continuation of Boolean algebra discussion.
• Focus on converting between different forms: Sum of Products (SOP) and Product of Sums (POS).

Problem 2: Converting SOP to POS

• Problem 2.6a in the textbook.
• Expression: ( AB + C'D' )
• Convert from Sum of Products (SOP) to Product of Sums (POS).

Key Concepts

• Double Complement Rule: ( X'' = X )
  • Verified using truth table.
• De Morgan's Theorems:
  1. Complement of a sum is the product of the complements.
  2. Complement of a product is the sum of the complements.

Solution Steps

1. Start with ( AB + C'D' ).
2. Apply double complement: ( (AB + C'D')'' ).
3. Use De Morgan's Theorem to rewrite: ( (AB)'(C'D')' ).
4. Apply De Morgan again: ( (A' + B')(C'' + D'') ).
5. Simplify double complements: ( (A' + B')(C + D) ).
6. Results in Product of Sums: ( (A + C')(A + D')(B + C')(B + D') )

Practice Exercise

• Convert POS back to SOP.
• Verify results through simplification properties.

Simplification Techniques

• Use Boolean simplification theorems.
• Focus on expressions like ( X + X'Y ) and their simplifications.

Example Problem 21A

• Expression: ( A' + B' + C ) AND ( A' + B' + C' )
  • Simplifies to zero using the fact that a term ANDed with its complement is zero.

Example Problem 21C

• Expression: ( AB + C'D ) AND ( AB' )
  • Use theorem ( XY + X'Y = Y ) for simplification.
  • Leads to understanding simplification through truth tables.

Circuit Simplification

Problem 213A

• Circuit Simplification
  • Given a complex circuit, find a simpler equivalent.
  • Example of using simplification properties to reduce circuit complexity.

Example

• Circuit using inverters and OR gates; simplifies to a single wire.

Homework Problems

• 5.1: Identify expressions equal to zero for all conditions except one specific condition.
• 5.2: Analyze a given circuit to find the correct output.

Conclusion

• Practice converting between SOP and POS forms.
• Familiarize with De Morgan’s laws and simplification theorems.
• Apply Boolean simplification techniques for circuit analysis.