Chapter 10.1 and 10.2: Boolean Algebra

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Presentation transcript:

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 Discrete Mathematical Structures: Theory and Applications

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

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

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

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

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

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Boolean Algebra Discrete Mathematical Structures: Theory and Applications

Boolean Algebra Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

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

Discrete Mathematical Structures: Theory and Applications

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

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

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

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

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Functional Completeness The set of operators {. , +, ‘} is functionally complete. Can we find a smaller 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

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