ECE2030 Introduction to Computer Engineering Lecture 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 Binary Boolean Operations All possible outcomes of a 2-input Boolean function ABF0F0 F1F1 F2F2 F3F3 F4F4 F5F5 F6F6 F7F7 F8F8 F9F9 F 10 F 11 F 12 F 13 F 14 F A·B ABAB A+B Identity A B Ā  B A+B ABAB A·B NULL

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

6 Function Evaluation ABCDE=00000 ABCDE=10000

7 Basic Identities of Boolean Algebra

8 Derivation of Simplification

9 Derivation of Consensus Theorem

10 Duality Principle dual of the expressionsA 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 Example in Lecture 4 A C B AC B Vdd F

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