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

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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 Algebra
Let 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 if x1 x3 0 and
x2 x4 x5 1, and equals 0 otherwise.
x1x2x3x4x5
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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
• Use Boolean identifies
• 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 xyz xyz
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Functional Completeness
Summery A function f Bn ? B, where B0,1, 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.
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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 11 0 and 10 01 00 1
NOR
NAND is functionally complete, since . , ' is
so and x' xx xy (xy)(xy)