chapter 10.1 and 10.2: boolean algebra

Chapter 10.1 and 10.2: Boolean Algebra

Based on Slides fromDiscrete Mathematical Structures: Theory and Applications

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Learning Objectives

Learn about Boolean expressions

Become aware of the basic properties of Boolean algebra

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Two-Element Boolean AlgebraLet B = {0, 1}.

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Two-Element Boolean Algebra

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Two-Element Boolean Algebra

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Two-Element Boolean Algebra

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Boolean Algebra

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Boolean Algebra

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Find a minterm that equals 1 ifx1 = x3 = 0 and x2 = x4 = x5 =1,and equals 0 otherwise.

x’1x2x’3x4x5

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

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Sum of products expression

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

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F(x,y,z) = (x + y) z’

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F(x,y,z) = (x + y) z’

F(x,y,z) = (x + y) z’= xyz’ + xy’z’ + x’yz’

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Functional Completeness

This means that the set of operators {. , +, '} is functionally complete.

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 ·,+,'

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Functional Completeness

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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

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{NAND} is functionally complete, since {. , '} is so andx' = x|xxy = (x|y)|(x|y)

NOR: