1. SYEN 3330 Digital Systems
Chapter 2 Part 3
SYEN 3330 Digital Systems

2. Boolean Operator Precedence
SYEN 3330 Digital Systems

3. Review: Duality Principle
SYEN 3330 Digital Systems

4. Duality In Proofs
SYEN 3330 Digital Systems

5. Useful Theorems
SYEN 3330 Digital Systems

6. Proof of Simplification
SYEN 3330 Digital Systems

7. Proof of Concensus
SYEN 3330 Digital Systems

8. Proof of DeMorgan’s Law
SYEN 3330 Digital Systems

9. Boolean Function Evaluation
SYEN 3330 Digital Systems

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

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

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
SYEN 3330 Digital Systems

13. Minterms
SYEN 3330 Digital Systems

14. Maxterms
SYEN 3330 Digital Systems

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

16. Standard Order
SYEN 3330 Digital Systems

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.
SYEN 3330 Digital Systems

18. Index Example in Three Variables
SYEN 3330 Digital Systems

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

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

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

22. Function Tables for Both
Minterms of two variables
Maxterms of two variables
SYEN 3330 Digital Systems

23. Observations
SYEN 3330 Digital Systems

24. Minterm Function Example
SYEN 3330 Digital Systems

25. Minterm Function Example
F(A, B, C, D, E) = m2 + m9 + m17 + m23
SYEN 3330 Digital Systems

26. Maxterm Function Example
SYEN 3330 Digital Systems

27. Maxterm Function Example
SYEN 3330 Digital Systems

28. Cannonical Sum of Minterms
SYEN 3330 Digital Systems

29. Another SOM Example
SYEN 3330 Digital Systems

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

31. Canonical Product of Maxterms
SYEN 3330 Digital Systems

32. Product of Maxterm Example
SYEN 3330 Digital Systems

33. Function Complements
Then:
Or alternately:
SYEN 3330 Digital Systems

34. Conversion Between Forms
SYEN 3330 Digital Systems

35. Review of Canonical Forms
SYEN 3330 Digital Systems

36. Review: Indices
SYEN 3330 Digital Systems

37. Forms of Terms, Complements
SYEN 3330 Digital Systems

38. Review: Sum of Minterms Form
SYEN 3330 Digital Systems

39. Review: Product of Maxterms
SYEN 3330 Digital Systems

40. Review: Complements, Conversions
SYEN 3330 Digital Systems