1. SYEN 3330 Digital SystemsJung H. Kim Chapter 2-3 1 SYEN 3330 Digital Systems Chapter 2 Part 3

2. SYEN 3330 Digital Systems Chapter 2-3 2 Boolean Operator Precedence

3. SYEN 3330 Digital Systems Chapter 2-3 3 Review: Duality Principle

4. SYEN 3330 Digital Systems Chapter 2-3 4 Duality In Proofs

5. SYEN 3330 Digital Systems Chapter 2-3 5 Useful Theorems

6. SYEN 3330 Digital Systems Chapter 2-3 6 Proof of Simplification

7. SYEN 3330 Digital Systems Chapter 2-3 7 Proof of Concensus

8. SYEN 3330 Digital Systems Chapter 2-3 8 Proof of DeMorgan’s Law

9. SYEN 3330 Digital Systems Chapter 2-3 9 Boolean Function Evaluation

10. SYEN 3330 Digital Systems Chapter 2-3 10 Expression Simplification Simplify to contain the smallest number of literals (complemented and uncomplemented variables):

11. SYEN 3330 Digital Systems Chapter 2-3 11 Complementing Functions This generate a lot of terms. You might want to simplify the expression first.

12. SYEN 3330 Digital Systems Chapter 2-3 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. SYEN 3330 Digital Systems Chapter 2-3 13 Minterms

14. SYEN 3330 Digital Systems Chapter 2-3 14 Maxterms

15. SYEN 3330 Digital Systems Chapter 2-3 15 Maxterms and Minterms The index above is important for describing which variables in the terms are true and which are complemented.

16. SYEN 3330 Digital Systems Chapter 2-3 16 Standard Order

17. SYEN 3330 Digital Systems Chapter 2-3 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. SYEN 3330 Digital Systems Chapter 2-3 18 Index Example in Three Variables

19. SYEN 3330 Digital Systems Chapter 2-3 19 Four Variables, Index 0-7

20. SYEN 3330 Digital Systems Chapter 2-3 20 Four Variables, Index 8-15

21. SYEN 3330 Digital Systems Chapter 2-3 21 Minterm and Maxterm Relationship Review: DeMorgan's Theorem (x  y) = (  x +  y) and (x + y) = (  x  y ) Note: For 2 variables: M 2 = (  x + y) and m 2 = (x  y) Thus M 2 is the complement of m 2 and vice-versa. Since DeMorgan's Theorem can be extended to n variables,this holds that for terms of n variables giving: MiMi and m i are complements.

22. SYEN 3330 Digital Systems Chapter 2-3 22 Function Tables for Both Minterms of two variables Maxterms of two variables

23. SYEN 3330 Digital Systems Chapter 2-3 23 Observations

24. SYEN 3330 Digital Systems Chapter 2-3 24 Minterm Function Example

25. SYEN 3330 Digital Systems Chapter 2-3 25 Minterm Function Example F(A, B, C, D, E) = m 2 + m 9 + m 17 + m 23

26. SYEN 3330 Digital Systems Chapter 2-3 26 Maxterm Function Example

27. SYEN 3330 Digital Systems Chapter 2-3 27 Maxterm Function Example

28. SYEN 3330 Digital Systems Chapter 2-3 28 Cannonical Sum of Minterms

29. SYEN 3330 Digital Systems Chapter 2-3 29 Another SOM Example

30. SYEN 3330 Digital Systems Chapter 2-3 30 Shorthand SOM Form Note that we explicitly show the standard variables in order and drop the “m” designators.

31. SYEN 3330 Digital Systems Chapter 2-3 31 Canonical Product of Maxterms

32. SYEN 3330 Digital Systems Chapter 2-3 32 Product of Maxterm Example

33. SYEN 3330 Digital Systems Chapter 2-3 33 Function Complements Or alternately: Then:

34. SYEN 3330 Digital Systems Chapter 2-3 34 Conversion Between Forms

35. SYEN 3330 Digital Systems Chapter 2-3 35 Review of Canonical Forms

36. SYEN 3330 Digital Systems Chapter 2-3 36 Review: Indices

37. SYEN 3330 Digital Systems Chapter 2-3 37 Forms of Terms, Complements

38. SYEN 3330 Digital Systems Chapter 2-3 38 Review: Sum of Minterms Form

39. SYEN 3330 Digital Systems Chapter 2-3 39 Review: Product of Maxterms

40. SYEN 3330 Digital Systems Chapter 2-3 40 Review: Complements, Conversions