Set TheorySet Theory
Professor OrrCPT120 ~ Quantitative
Analysis I
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Why Study Set Theory?Why Study Set Theory?
Understanding set theory helps people to …
see things in terms of systems
organize things into groups
begin to understand logic
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Key MathematiciansKey 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 Georg Cantor 1845 -19181845 -1918
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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……CantorCantor 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 an infinitude of real numbers exist
between 1 and 2; there are more real numbers than
there are integers…
0
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John Venn John Venn 1834-19231834-1923
studied and taught logic and probability theory
articulated Boole’s algebra of logic
devised a simple way to diagram set operations (Venn Diagrams)
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George Boole George Boole 1815-18641815-1864
self‑taught mathematician with an interest in logic
developed an algebra of logic (Boolean Algebra)
featured the operators– and– or– not– nor (exclusive or)
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Augustus De Morgan Augustus De Morgan 1806-18711806-1871
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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Basic Set Theory DefinitionsBasic Set Theory Definitions
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 ExampleSimple Set Example 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 NotationSet Theory NotationSymbol Meaning
Upper case designates set name
Lower case designates set elements
{ } enclose elements in set
or 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
| or : such that (if a condition is true)
| | the cardinality of a set
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Set Notation: Defining SetsSet 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 ElementsSet Notation Elements 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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SubsetsSubsets
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 RelationshipsSubset Relationships 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 CB A B B B CC A C B C C
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Set EqualitySet Equality 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 SetsCardinality of Sets
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 CardinalityFinite 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 CardinalityInfinite 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| =
0
0
1
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Universal SetsUniversal Sets
The universal set is the set of all things pertinent to to a given discussionand 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 SetThe Empty Set
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 emptyA = {x | x is a Chevrolet Mustang}B = {x | x is a positive number 0}
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The Power Set ( The Power Set ( P P )) 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| = 3P (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 SetsSpecial Sets
Z represents the set of integers – Z+ is the set of positive integers and– Z- is the set of negative integers
N represents the set of natural numbers
ℝ represents the set of real numbers
Q represents the set of rational numbers
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Venn DiagramsVenn Diagrams
Venn diagrams show relationships between sets and their elements
Universal Set
Sets A & B
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Venn Diagram Example 1Venn Diagram Example 1
Set Definition ElementsA = {x | x Z+ and x 8} 1 2 3 4 5 6
7 8B = {x | x Z+; x is even and 10} 2 4
6 8 10A BB A
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Venn Diagram Example 2Venn Diagram Example 2
Set Definition ElementsA = {x | x Z+ and x 9} 1 2 3 4 5
6 7 8 9B = {x | x Z+ ; x is even and 8} 2 4
6 8
A BB AA B
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Venn Diagram Example 3Venn Diagram Example 3
Set Definition ElementsA = {x | x Z+ ; x is even and 10} 2 4
6 8 10B = x Z+ ; x is odd and x 10 } 1 3
5 7 9
A BB A
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Venn Diagram Example 4Venn Diagram Example 4Set 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 5Venn 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 6Venn 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}