ECE2030 Introduction to Computer Engineering Lecture 5: Boolean Algebra

1. ECE2030 Introduction to Computer EngineeringLecture 5: Boolean Algebra Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering Georgia Tech

2. What is Boolean Algebra • An algebra dealing with • Binary variables by alphabetical letters • Logic operations: OR, AND, XOR, etc • Consider the following Boolean equation • A Boolean function can be represented by a truth table which list all combinations of 1’s and 0’s for each binary value

3. Fundamental Operators • NOT • Unary operator • Complements a Boolean variable represented as A’, ~A, or Ā • OR • Binary operator • A “OR”-ed with B is represented as A + B • AND • Binary operator • A “AND”-ed with B is represented as AB or A·B • Can perform logical multiplication

4. B A·B A AB A+B  Ā A·B NULL B A+B AB Identity Binary Boolean Operations • All possible outcomes of a 2-input Boolean function

5.         Precedence of Operators • Precedence of Operator Evaluation (Similar to decimal arithmetic) • () : Parentheses • NOT • AND • OR

6. ABCDE=00000 ABCDE=10000 Function Evaluation

7. Basic Identities of Boolean Algebra

8. Derivation of Simplification

9. Derivation of Consensus Theorem

10. Duality Principle • A Boolean equation remains valid if we take the dual of the expressions on both sides of the equals sign • Dual of expressions • Interchange 1’s and 0’s • Interchange AND () and OR (+)

11. Duality Principle

12. Simplification Examples

13. DeMorgan’s Law  

14. Vdd A C B F B A C Example in Lecture 4

15. Vdd A C B F B A C Another Way to Draw It