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ECE 15A
Fundamentals of Logic Design
Lecture 3
Malgorzata MarekSadowska
Electrical and Computer Engineering Department
UCSB
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Last time:
Fundamental laws of the algebra of sets
Comutative laws
(1a) XY = YX
(1b) X+Y = Y+X
Associative laws
(2a) X(YZ) = (XY)Z
(2b) X+(Y+Z) = (X+Y)+Z
Distributive laws
(3a) X(Y+Z) = XY+XZ
(3b) X+YZ= (X+Y)(X+Z)
Laws of Tautology
(4a) XX = X
(4b) X+X = X
Laws of Absorbtion
(5a) X(X+Y) = X
(5b) X+XY = X
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Last time:
Fundamental laws of the algebra of sets
Laws of Complementation
(6a) XX’=0
(6b) X+X’=1
Double Laws of Complementation
(7) (X’)’=X
Laws of De Morgan
(8a) (XY)’ = X’+Y’
(8b) (X+Y)’ = X’Y’
Operations with 0 and 1
(9a)
0X=0
(9b) 1+X=1
(10a) 1X=X
(10b)0+X=X
(11a)
0’=1
(11b) 1’=0
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Boolean Algebra
±
The 19th Century Mathematician, George
Boole, developed a math. system (algebra)
involving formal principles of reasoning,
Boolean Algebra.
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Later Claude Shannon (father of information
theory) showed (in his Master’s thesis!) how
to map Boolean Algebra to digital circuits.
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5
Today:
±
General treatment of Boolean algebras based
on definitions and axioms
²
Some duplication of the material introduced in
lecture #2
²
Earlier results were obtained based on intuitive
concepts
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Sets with algebraic structure
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Rules of combination defined between
elements of the set
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Examples:
²
Set of all integers
²
Set of all real numbers
¾
Rules of combination: addition, subtraction,
multiplication, division
²
Sets
¾
Intersection and union
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Binary operation
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A binary operation “&” on a set S is a rule which for
each ordered pair of elements (a,b) s.t. a,b belong to
S, assigns a unique
element c
S. We write c=a & b.
±
Examples
²
Subtraction is a binary operation on the set or rational
numbers
²
Subtraction is not a binary operation on the set of natural
numbers
²
Subtraction is a binary operation on the set of all integers.
±
Binary operation doesn’t need to have an intuitive
meaning (like +, , /, x).
∈
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Examples of abstract binary operations
±
Let S= { A, B, C }.
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We define # and & as follows:
#
A
B
C
A
A
C
B
B
C
B
A
C
B
A
C
&
A
B
C
A
A
B
C
B
A
B
C
C
A
B
C
A # B = C
B # C = A
A & B = B
B & A = A
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Definitions
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A binary operation @ on a set of elements S is
associative if and only if for every a,b,c
S
a @ (b @ c) = (a @ b) @ c
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A binary operation @ on a set of elements S is
commutative if and only if for every a,b S
a @ b = b
@ a
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If @ and % are two binary operations on the same set
S, @ is distributive over % if and only if for every
a,b,c
S
a @ (b % c) = (a @ b) % (a @ c)
∈
∈
∈
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Example
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Algebra of sets
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Intersection and union are commutative and
associative
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Each is distributive over the other
X(Y+Z) = XY+XZ
X+YZ= (X+Y)(X+Z)
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This note was uploaded on 05/29/2009 for the course ECE 15A taught by Professor M during the Winter '08 term at UCSB.
 Winter '08
 M

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