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Set Theory

Professor Orr

CPT120 ~ Quantitative Analysis I


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1999 Indiana UniversityTrustees

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 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 1845 -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

manyareas ofmathematics


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Cantor

the concept of infinity was expanded byCantors set theory

Cantor proved there are levels of infinity

an infinitude of integers initially endingwith 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 1834-1923

studied and taught

logic and probability

theory

articulated Booles

algebra of logic

devised a simple way

to diagram setoperations (Venn

Diagrams)


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George Boole 1815-1864

self-taughtmathematician with an

interest in logic

developed an algebra oflogic (Boolean Algebra)

featured the operators and

or not

nor (exclusive or)


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Augustus De Morgan 1806-1871

developed two laws of

negation

interested, like other

mathematicians, in

using mathematics to

demonstrate logic

furthered Booles work

of incorporating logic

and mathematics

formally stated the laws

of set theory


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Basic Set Theory Definitions

A set is a collection of elements

An element is an object contained in a set

If every element of SetA is also contained

in Set B, then SetA is a subset of Set B A is aproper subsetof B if B has more

elements than A does

The universal set contains all of theelements relevant to a given discussion


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Simple Set Example

the universal set isa deck of ordinary

playing cards

each card is an

element in theuniversal set

some subsets are:

face cards

numbered cards suits

poker hands


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Set Theory Notation

Symbol 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 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|xis a positive even integer


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Set Notation Elements

an element is a member of a set

notation: means is an element ofmeans is not an element of Examples:

A ={1, 2, 3, 4}

1 A 6 A2 A z A

B ={x | x is an even number 10}2 B 9 B4 B z B


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Subsets

a subsetexists when a sets members are

also contained in another set

notation:

means is a subset ofmeans is a proper subset ofmeans is not a subset of


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Subset 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 Equality

Two sets are equalif and only if they containprecisely the same elements.

The order in which the elements are listed is

unimportant.

Elements may be repeated in set definitionswithout 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 = AA = {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

Cardinality refers to the number of

elements in a set

A f in i teset has a countable number of

elements An inf in i teset has at least as many

elements as the set ofnatural

numbers

notation: |A| represents the cardinal i ty ofSet A


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Finite Set Cardinality

Set Defini t io n Cardinal i ty

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 CardinalitySet Defini t ion Cardinal i ty

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 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

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 ( P ) The power set is the set of all subsets

that can be created from a given set

The cardinal i tyof the power set is 2 to

the power of the given sets 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

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 ofnatural num bers

represents the set of real numbers Q represents the set of rational numbers


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Venn Diagrams

Venn diagrams show relationships between

sets and their elements

Universal Set

Sets A & B


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Venn Diagram Example 1

Set Definition Elements

A = {x | x Z+ and x 8} 1 2 3 4 5 6 7 8

B = {x | x Z+; x is even and 10} 2 4 6 8 10

A BB A


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A = {x | x Z+ and x 9} 1 2 3 4 5 6 7 8 9

B = {x | x Z+ ; x is even and 8} 2 4 6 8

A BB AA B


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A = {x | x Z+ ; x is even and 10} 2 4 6 8 10

B = x Z+ ; x is odd and x 10 } 1 3 5 7 9

A BB A


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Set DefinitionU= {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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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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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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