1 / 25

Chapter 10.1 and 10.2: Boolean Algebra

Chapter 10.1 and 10.2: Boolean Algebra. Based on Slides from Discrete Mathematical Structures: Theory and Applications. Learning Objectives. Learn about Boolean expressions Become aware of the basic properties of Boolean algebra. Two-Element Boolean Algebra. Let B = {0, 1}.

kellya
Download Presentation

Chapter 10.1 and 10.2: Boolean Algebra

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


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

  16. Discrete Mathematical Structures: Theory and Applications

  17. Find a minterm that equals 1 if x1 = x3 = 0 and x2 = x4 = x5 =1, and equals 0 otherwise. x’1x2x’3x4x5 Discrete Mathematical Structures: Theory and Applications

  18. Therefore, the set of operators {. , +, ‘} is functionally complete. 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: • 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

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

  22. Discrete Mathematical Structures: Theory and Applications

  23. Discrete Mathematical Structures: Theory and Applications

  24. Functional Completeness Summery: A function f: Bn 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 ·,+,' This means that the set of operators {. , +, '} is functionally complete. 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 NOR: {NAND} is functionally complete, since {. , '} is so and x' = x|x xy = (x|y)|(x|y) Discrete Mathematical Structures: Theory and Applications

More Related