set concepts

Set TheorySet Theory

Professor OrrCPT120 ~ Quantitative

Analysis I

©1999 Indiana University Trustees

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


Key MathematiciansKey Mathematicians

These mathematicians influenced the development of set theory and logic:

Georg Cantor John Venn George Boole Augustus DeMorgan


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


……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


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)


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)


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


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


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


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


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}


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


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”


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


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


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


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


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


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}


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}


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


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


Venn DiagramsVenn Diagrams

Venn diagrams show relationships between sets and their elements

Universal Set

Sets A & B


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



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



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


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}



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}



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}

set concepts

Documents

set elements

developed set theory

laws of set theory

development of set theory

different set theory

set oris

theory cantor

universal set somesubsetsare