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Chapter 10.1 and 10.2: Boolean Algebra

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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}.

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Chapter 10.1 and 10.2: Boolean Algebra
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  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