1. ECE/CS 352 Digital System Fundamentals© T. Kaminski & C. Kime 1 ECE/CS 352 Digital Systems Fundamentals Spring 2001 Chapter 2 Part 3 Tom Kaminski & Charles R. Kime

2. ECE/CS 352 Digital System Fundamentals Chapter 2 2 Boolean Operator Precedence

3. ECE/CS 352 Digital System Fundamentals Chapter 2 3 Review: Duality Principle

4. ECE/CS 352 Digital System Fundamentals Chapter 2 4 Duality In Proofs

5. ECE/CS 352 Digital System Fundamentals Chapter 2 5 Useful Theorems

6. ECE/CS 352 Digital System Fundamentals Chapter 2 6 Proof of Simplification

7. ECE/CS 352 Digital System Fundamentals Chapter 2 7 Proof of Concensus

8. ECE/CS 352 Digital System Fundamentals Chapter 2 8 Proof of DeMorgan’s Law

9. ECE/CS 352 Digital System Fundamentals Chapter 2 9 Boolean Function Evaluation

10. ECE/CS 352 Digital System Fundamentals Chapter 2 10 Expression Simplification Simplify to contain the smallest number of literals (complemented and uncomplemented variables):

11. ECE/CS 352 Digital System Fundamentals Chapter 2 11 Complementing Functions This generate a lot of terms. You might want to simplify the expression first.

12. ECE/CS 352 Digital System Fundamentals Chapter 2 12 Canonical Forms It is useful to specify Boolean functions of n variables in a manner that is easy to compare. Two such Canonical Forms are in common usage:  Sum of Minterms  Product of Maxterms

13. ECE/CS 352 Digital System Fundamentals Chapter 2 13 Minterms

14. ECE/CS 352 Digital System Fundamentals Chapter 2 14 Maxterms

15. ECE/CS 352 Digital System Fundamentals Chapter 2 15 Maxterms and Minterms The index above is important for describing which variables in the terms are true and which are complemented.

16. ECE/CS 352 Digital System Fundamentals Chapter 2 16 Standard Order

17. ECE/CS 352 Digital System Fundamentals Chapter 2 17 Purpose of the Index The index for the minterm or maxterm, expressed as a binary number, is used to determine whether the variable is shown in the true form or complemented form.

18. ECE/CS 352 Digital System Fundamentals Chapter 2 18 Index Example in Three Variables

19. ECE/CS 352 Digital System Fundamentals Chapter 2 19 Four Variables, Index 0-7

20. ECE/CS 352 Digital System Fundamentals Chapter 2 20 Four Variables, Index 8-15

21. ECE/CS 352 Digital System Fundamentals Chapter 2 21 Minterm and Maxterm Relationship

22. ECE/CS 352 Digital System Fundamentals Chapter 2 22 Function Tables for Both Minterms of two variables Maxterms of two variables

23. ECE/CS 352 Digital System Fundamentals Chapter 2 23 Observations

24. ECE/CS 352 Digital System Fundamentals Chapter 2 24 Minterm Function Example

25. ECE/CS 352 Digital System Fundamentals Chapter 2 25 Minterm Function Example F(A, B, C, D, E) = m 2 + m 9 + m 17 + m 23

26. ECE/CS 352 Digital System Fundamentals Chapter 2 26 Maxterm Function Example

27. ECE/CS 352 Digital System Fundamentals Chapter 2 27 Maxterm Function Example

28. ECE/CS 352 Digital System Fundamentals Chapter 2 28 Cannonical Sum of Minterms

29. ECE/CS 352 Digital System Fundamentals Chapter 2 29 Another SOM Example

30. ECE/CS 352 Digital System Fundamentals Chapter 2 30 Shorthand SOM Form Note that we explicitly show the standard variables in order and drop the “m” designators.

31. ECE/CS 352 Digital System Fundamentals Chapter 2 31 Canonical Product of Maxterms

32. ECE/CS 352 Digital System Fundamentals Chapter 2 32 Product of Maxterm Example

33. ECE/CS 352 Digital System Fundamentals Chapter 2 33 Function Complements Or alternately: Then:

34. ECE/CS 352 Digital System Fundamentals Chapter 2 34 Conversion Between Forms

35. ECE/CS 352 Digital System Fundamentals Chapter 2 35 Review of Canonical Forms

36. ECE/CS 352 Digital System Fundamentals Chapter 2 36 Review: Indices

37. ECE/CS 352 Digital System Fundamentals Chapter 2 37 Forms of Terms, Complements

38. ECE/CS 352 Digital System Fundamentals Chapter 2 38 Review: Sum of Minterms Form

39. ECE/CS 352 Digital System Fundamentals Chapter 2 39 Review: Product of Maxterms

40. ECE/CS 352 Digital System Fundamentals Chapter 2 40 Review: Complements, Conversions