Download Opening Page: Set Theory

Set Theory
Professor Orr
CPT120 ~ Quantitative Analysis I
Why Study Set Theory?
Understanding set theory helps people
to …
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see things in terms of systems
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organize things into groups
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begin to understand logic
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Key Mathematicians
These mathematicians influenced the
development of set theory and logic:
Georg Cantor
 John Venn
 George Boole
 Augustus DeMorgan
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Georg Cantor
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developed set
theory
set theory was not
initially accepted
because it was
radically different
set theory today is
widely accepted
and is used in
many areas of
mathematics
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Trustees
1845 -1918
…Cantor
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the concept of infinity was expanded by
Cantor’s set theory
Cantor proved there are “levels of infinity”
an infinitude of integers initially ending
with  or 0
an infinitude of real numbers exist
between 1 and 2;
there are more real numbers than there are
integers…
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John Venn

studied and taught
logic and probability
theory
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articulated Boole’s
algebra of logic
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devised a simple way
to diagram set
operations (Venn
Diagrams)
1834-1923
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George Boole
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self-taught
mathematician with an
interest in logic
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developed an algebra of
logic (Boolean Algebra)

featured the operators
–
–
–
–
and
or
not
nor (exclusive or)
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1815-1864
Augustus De Morgan
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developed two laws of
negation
interested, like other
mathematicians, in
using mathematics to
demonstrate logic
furthered Boole’s work
of incorporating logic
and mathematics
formally stated the laws
of set theory
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1806-1871
Basic Set Theory Definitions
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A set is a collection of elements
An element is an object contained in a set
If every element of Set A is also contained
in Set B, then Set A is a subset of Set B
– A is a proper subset of B if B has more
elements than A does

The universal set contains all of the
elements relevant to a given discussion
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Simple Set Example
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the universal set is
a deck of ordinary
playing cards
each card is an
element in the
universal set
some subsets are:
–
–
–
–
face cards
numbered cards
suits
poker hands
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Set Theory Notation
Symbol
Meaning
Upper case
Lower case
{ }
 or 




| or :
| |
designates set name
designates set elements
enclose elements in set
is (or is not) an element of
is a subset of (includes equal sets)
is a proper subset of
is not a subset of
is a superset of
such that (if a condition is true)
the cardinality of a set
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Set Notation: Defining Sets

a set is a collection of objects

sets can be defined two ways:
– by listing each element
– by defining the rules for membership

Examples:
– A = {2,4,6,8,10}
– A = {x|x is a positive even integer <12}
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Set Notation Elements


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an element is a member of a set
notation:
 means “is an element of”
 means “is not an element of”
Examples:
– A = {1, 2, 3, 4}
1A
6A
2A
zA
– B = {x | x is an even number  10}
2B
9B
4B
zB
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Subsets
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a subset exists when a set’s members are
also contained in another set
notation:
 means “is a subset of”
 means “is a proper subset of”
 means “is not a subset of”
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Subset Relationships
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A = {x | x is a positive integer  8}
set A contains: 1, 2, 3, 4, 5, 6, 7, 8
B = {x | x is a positive even integer  10}
set B contains: 2, 4, 6, 8
C = {2, 4, 6, 8, 10}
set C contains: 2, 4, 6, 8, 10
Subset Relationships
AA
AB
AC
BA
BB
BC
CA
CB
CC
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Set Equality
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Two sets are equal if and only if they contain
precisely the same elements.
The order in which the elements are listed is
unimportant.
Elements may be repeated in set definitions
without increasing the size of the sets.
Examples:
A = {1, 2, 3, 4} B = {1, 4, 2, 3}
A  B and B  A; therefore, A = B and B = A
A = {1, 2, 2, 3, 4, 1, 2} B = {1, 2, 3, 4}
A  B and B  A; therefore, A = B and B = A
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Cardinality of Sets
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Cardinality refers to the number of
elements in a set
A finite set has a countable number of
elements
An infinite set has at least as many
elements as the set of natural
numbers
notation: |A| represents the cardinality of
Set A
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Finite Set Cardinality
Set Definition
Cardinality
A = {x | x is a lower case letter}
|A| = 26
B = {2, 3, 4, 5, 6, 7}
|B| = 6
C = {x | x is an even number  10}
|C|= 4
D = {x | x is an even number  10}
|D| = 5
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Infinite Set Cardinality
Set Definition
Cardinality
A = {1, 2, 3, …}
|A| = 
B = {x | x is a point on a line}
|B| = 
C = {x| x is a point in a plane}
|C| = 1
0
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0
Universal Sets
The universal set is the set of all things
pertinent to to a given discussion
and is designated by the symbol U
Example:
U = {all students at IUPUI}
Some Subsets:
A = {all Computer Technology students}
B = {freshmen students}
C = {sophomore students}
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The Empty Set
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Any set that contains no elements is
called the empty set
the empty set is a subset of every set
including itself
notation: { } or 
Examples ~ both A and B are empty
A = {x | x is a Chevrolet Mustang}
B = {x | x is a positive number  0}
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The Power Set ( P )
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The power set is the set of all subsets
that can be created from a given set
The cardinality of the power set is 2 to
the power of the given set’s cardinality
notation: P (set name)
Example:
A = {a, b, c}
where |A| = 3
P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, }
and |P (A)| = 8
In general, if |A| = n, then |P (A) | = 2n
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Special Sets
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Z represents the set of integers
– Z+ is the set of positive integers and
– Z- is the set of negative integers
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N represents the set of natural numbers
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ℝ represents the set of real numbers
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Q represents the set of rational numbers
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Venn Diagrams
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Venn diagrams show relationships between
sets and their elements
Sets A & B
Universal Set
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Venn Diagram Example 1
Set Definition
Elements
A = {x | x  Z+ and x  8}
12345678
B = {x | x  Z+; x is even and  10} 2 4 6 8 10
AB
BA
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Venn Diagram Example 2
Set Definition
Elements
A = {x | x  Z+ and x  9}
123456789
B = {x | x  Z+ ; x is even and  8} 2 4 6 8
AB
BA
AB
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Venn Diagram Example 3
Set Definition
Elements
A = {x | x  Z+ ; x is even and  10} 2 4 6 8 10
B = x  Z+ ; x is odd and x  10 }
13579
AB
BA
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Venn Diagram Example 4
Set Definition
U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 6, 7}
B = {2, 3, 4, 7}
C = {4, 5, 6, 7}
A = {1, 2, 6, 7}
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Venn Diagram Example 5
Set Definition
U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 6, 7}
B = {2, 3, 4, 7}
C = {4, 5, 6, 7}
B = {2, 3, 4, 7}
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Venn Diagram Example 6
Set Definition
U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 6, 7}
B = {2, 3, 4, 7}
C = {4, 5, 6, 7}
C = {4, 5, 6, 7}
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Trustees