Chapter 10 1 and 10 2 Boolean Algebra

Chapter 10. 1 and 10. 2: Boolean Algebra Discrete Mathematical Structures: Theory and Applications

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

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

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

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

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

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Boolean Algebra Discrete Mathematical Structures: Theory and Applications 14

Boolean Algebra Discrete Mathematical Structures: Theory and Applications 15

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

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

Sum of products expression q 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 Discrete Mathematical Structures: Theory and Applications 20

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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 23