1. Chapter 10.1 and 10.2: Boolean Algebra Based on Slides from Discrete Mathematical Structures: Theory and Applications

2. Discrete Mathematical Structures: Theory and Applications 2 Learning Objectives  Learn about Boolean expressions  Become aware of the basic properties of Boolean algebra

3. Discrete Mathematical Structures: Theory and Applications 3 Two-Element Boolean Algebra Let B = {0, 1}.

4. Discrete Mathematical Structures: Theory and Applications 4 Two-Element Boolean Algebra

5. Discrete Mathematical Structures: Theory and Applications 5

6. 6

7. 7

8. 8 Two-Element Boolean Algebra

9. Discrete Mathematical Structures: Theory and Applications 9 Two-Element Boolean Algebra

10. Discrete Mathematical Structures: Theory and Applications 10

11. Discrete Mathematical Structures: Theory and Applications 11

12. Discrete Mathematical Structures: Theory and Applications 12

13. Discrete Mathematical Structures: Theory and Applications 13

14. Discrete Mathematical Structures: Theory and Applications 14 Boolean Algebra

15. Discrete Mathematical Structures: Theory and Applications 15 Boolean Algebra

16. Discrete Mathematical Structures: Theory and Applications 16

17. Discrete Mathematical Structures: Theory and Applications 17 Find a minterm that equals 1 if x 1 = x 3 = 0 and x 2 = x 4 = x 5 =1, and equals 0 otherwise. x’ 1 x 2 x’ 3 x 4 x 5

18. Discrete Mathematical Structures: Theory and Applications 18 Therefore, the set of operators {., +, ‘} is functionally complete.

19. Discrete Mathematical Structures: Theory and Applications 19 Sum of products expression  Example 3, p. 710 Find the sum of products expansion of F(x,y,z) = (x + y) z’ Two approaches: 1)Use Boolean identifies 2)Use table of F values for all possible 1/0 assignments of variables x,y,z

20. Discrete Mathematical Structures: Theory and Applications 20 F(x,y,z) = (x + y) z’

21. Discrete Mathematical Structures: Theory and Applications 21 F(x,y,z) = (x + y) z’ F(x,y,z) = (x + y) z’= xyz’ + xy’z’ + x’yz’

22. Discrete Mathematical Structures: Theory and Applications 22

23. Discrete Mathematical Structures: Theory and Applications 23

24. Discrete Mathematical Structures: Theory and Applications 24 Functional Completeness This means that the set of operators {., +, '} is functionally complete. Summery: A function f: B n  B, where B={0,1}, is a Boolean function. For every Boolean function, there exists a Boolean expression with the same truth values, which can be expressed as Boolean sum of minterms. Each minterm is a product of Boolean variables or their complements. Thus, every Boolean function can be represented with Boolean operators ·,+,'

25. Discrete Mathematical Structures: Theory and Applications 25 Functional Completeness The question is: Can we find a smaller functionally complete set? Yes, {., '}, since x + y = (x'. y')' Can we find a set with just one operator? Yes, {NAND}, {NOR} are functionally complete: NAND: 1|1 = 0 and 1|0 = 0|1 = 0|0 = 1 {NAND} is functionally complete, since {., '} is so and x' = x|x xy = (x|y)|(x|y) NOR: