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ECE 15A
Fundamentals of Logic Design
Lecture 3
Malgorzata MarekSadowska
Electrical and Computer Engineering Department
UCSB
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Boolean Algebra
The 19th Century Mathematician, George
Boole, developed a math. system (algebra)
involving formal principles of reasoning,
Boolean Algebra.
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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Today:
General treatment of Boolean algebras based
on definitions and axioms
Some duplication of the material introduced in
lecture #2
Earlier results were obtained based on intuitive
concepts
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Sets with algebraic structure
Rules of combination defined between
elements of the set
Examples:
Set of all integers
Set of all real numbers
Rules of combination: addition, subtraction,
multiplication, division
Sets
Intersection and union
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Binary operation
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
Examples
Subtraction is a binary operation on the set or rational numbers
Subtraction is not a binary operation on the set of natural
numbers
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 }.
#
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
B & A = A
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Definitions
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
A binary operation @ on a set of elements S is
commutative if and only if for every a,b
S
a @ b = b
@ a
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
Let S= { A, B, C }.
#
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
Is # associative?
(A#B)#C = C#C= C
A #(B#C) = A#A = A
Is & associative?
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Example
Algebra of sets
Intersection and union are commutative and
associative
Each is distributive over the other
X(Y+Z) = XY+XZ
X+YZ= (X+Y)(X+Z)
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Example
Natural numbers
Addition and multiplication are commutative and
associative
Multiplication is distributive over addition
Addition is not distributive over multiplication
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 Winter '08
 M

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