1. Chapter 10.1 and 10.2: Boolean Algebra
Discrete Mathematical Structures:
Theory and Applications

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

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

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

5. Discrete Mathematical Structures: Theory and Applications

6. Discrete Mathematical Structures: Theory and Applications

7. Discrete Mathematical Structures: Theory and Applications

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

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

10. Discrete Mathematical Structures: Theory and Applications

11. Discrete Mathematical Structures: Theory and Applications

12. Discrete Mathematical Structures: Theory and Applications

13. Discrete Mathematical Structures: Theory and Applications

14. Boolean Algebra
Discrete Mathematical Structures: Theory and Applications

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

17. Find a midterm that equals 1 if x1 = x3 = 0 and x2 = x4 = x5 =1,
and equals 0 otherwise.
Discrete Mathematical Structures: Theory and Applications

18. Discrete Mathematical Structures: Theory and Applications

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

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

21. Discrete Mathematical Structures: Theory and Applications

22. Discrete Mathematical Structures: Theory and Applications

23. Functional Completness
The set of operators {. , +, ‘} is functionally complete.
Can we find a smaller set?
Yes, {. , ‘}, since x + y = (x’ + y’)’
{NAND}, {NOR} are functionally complete:
NAND: 1|1 = 0 and 1|0 = 0|1 = 0|0 = 1
NOR:
{NAND} is functionally complete, since {. , ‘} is so and
x’ = x|x
xy = (x|y)|(x|y)
Discrete Mathematical Structures: Theory and Applications