Set theory.pdf

7/26/2019 Set theory.pdf

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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I

Set Theory

Jim Woodcock

January 2010


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Outline

1 Notation

2 Set Membership3 Finite and Infinite Sets

4 Equality of Sets

5 Empty Set

6 Subsets7 Power Set

8 Venn Diagrams

9 Union

10 Intersection11 Difference

12 Generalised Operations

13 Russells Paradox

14 Summary


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Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary


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Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary


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Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary

N i S M b hi Fi i d I fi i S E li f S E S S b P S V Di U i I


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Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary

N t ti S t M b hi Fi it d I fi it S t E lit f S t E t S t S b t P S t V Di U i I


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Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary



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Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary



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Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary



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Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary



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Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary



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Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary



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p q y p y g

Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary



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p q y p y g

Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary



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Outline

1 Notation


4 Equality of Sets

5 Empty Set


8 Venn Diagrams

9 Union



13 Russells Paradox

14 Summary



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Examples

The numbers 1, 3, 7, 4 and 10.

The solutions of the equationx2 3 x 2= 0.

The vowels of the alphabet: a, e, i, o,and u.

The people living on the earth.The studentsRobert, Catherine,andJonathan.

The students who are absent from the class.

The countriesEngland, France,andDenmark.

The capital cities of Europe.The numbers 2, 4, 6, 8, . . .

The rivers in England.



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Notation

set extension

A= {1, 3, 7, 10}

set comprehension

B={ x |x is even }

B={ x : N| x is even }

Ndenotes the set of counting numbers, 0, 1, 2, 3, . . .Convention:lower case (a,b,c) for elements, capitals (A,B,

Person,River) for set names



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Examples

A1 ={1, 3, 7, 10}

A2 ={ x : N| x2 3 x 2= 0 }

A3 ={a, e, i, o, u}

A4 ={ x :Person|x is living on the earth }A5 ={Robert, Catherine, Jonathan}

A6 ={ x :Student |x is absent from class }

A7 ={England, France, Denmark}

A8 ={ x :CapitalCity |x is in Europe }A9 ={2, 4, 6, 8, . . .}

A10={ x :River |x is in England }



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Set Membership

xA

x /A

Example

letA= {a, e, i, o, u}thena A, b /A, e A, f /A

letB={ x : N| x is even }then 3 /B, 6 B, 11 /B, 14 B



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Examples

The set of nonnegative integers less than 4.

The set of books in the Bodleian Library at the present

time.

The set consisting of people who spoke to Bertrand

Russell on the 7 June 1906.

The set of live dinosaurs in the Pitt-Rivers Museum.



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Examples

The set of integers greater than 3.

The set of all C++ programs that can be displayed on a

single screen.

The set of all C++ programs that would halt if run for a

sufficient time on a computer with unbounded storage.

The set of true propositions about the integers.

The set with two members, one of which is the set of even

integers, and other the set of odd integers.



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Finite and Infinite Sets

Examples

letMbe the set of the days of the week

thenMis finite

letN={2, 4, 6, 8, . . .}thenN is infinite

letP={ x :River |xis on Earth }although it may be difficult to count the number of rivers in

the world,Pis still a finite set



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Examples

suppose that statement labels in a (rather old-fashioned)

programming language must be either a single alphabetic

symbol or a single decimal digit

{A, B, C, . . . , Z}

{0, 1, 2, . . . , 9},

a variable name in the programming language BASICmust

be either an alphabetic symbol or an alphabetic symbol

followed by a single decimal digit



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Example

consider the four cubes puzzle

this involves four cubes each of whose faces is painted in

one of four different colours

the puzzle is to stack the cubes in such a way that each

vertical side of the stack contains squares of all four

colours

there are 3 123 =5, 184 different arrangements



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Equality of Sets

Example

{1, 2, 3, 4}={3, 1, 4, 2}{5, 6, 5, 7}={7, 5, 7, 6}

{ x : N| x2 3 x=2 }= {2, 1}={1, 2, 2, 1}



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Equality

The formal statement of when equality holds between sets is

A= B ( x :XxA xB)

ExampleSuppose that we have the following definitions:

Zeroes={ x : N| x2 3 x+ 2= 0 }SmallNums={1, 2}

Zeroes=SmallNums( x : Z

xZeroesxSmallNums)



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Empty Set

The empty set is denoted byExample

letAbe the set of people in the world who are older than

200 years

letB={ x : N| x2 =4 x is odd }



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Subsets

A B

Example

{1, 3, 5} {5, 4, 3, 2, 1}

{2, 4, 6} {6, 2, 4}

{ x : N| ( n: N x=2n) } { x : N| x is even }the set of women is a subset of the set of all humans

{1, 2, 3, 4, 5} { x : Z| 0< x


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Four Theorems

1 two setsAandBare equal if they each contain the other

A= B A BBA

2 the subset relationship isreflexive

for any setA, A A

3 since the empty set has no members, it is trivially a subset

of every other set

for any setA, A

4 the subset relation istransitive

A BBCA C



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Proper Subset

A B A BA=B

Examplethe set of natural numbers is a proper subset of the set of

integers

the set of even integers is a proper subset of the set of

integers



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Sets of Sets

it sometimes will happen that the objects of a set are sets

themselves

for example, the set of all subsets of A

the set{{2, 3}, {2}, {5, 6}}is a set of sets

its members are{2, 3}, {2}, and{5, 6}

it is not possible that a set has some members which are

sets themselves and some which arent

the type system will make sure that all the members of aset are the same kind of object



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Power Set

the family of all the subsets of any set Sis called the power

set ofS

PS

M={a, b}

PM={{a, b}, {a}, {b}, }

T ={4, 7, 8}

PT ={T, {4, 7}, {4, 8}, {7, 8}, {4}, {7}, {8}, }

A=

PA={}



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Examples

A= {1}

PA={, {1}}

A= {1, 2}

PA={, {1}, {2}, {1, 2}}

ifAis any (finite or infinite) set of natural numbers, then

APN

ifAis finite, then PAis also finite

otherwise, PAis infinite



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Disjoint Sets

AandBaredisjointiff they have no elements in common

Examples

A= {1, 3, 7, 8}andB={2, 4, 7, 9}the set of positive numbers and the set of negative

numbers

E={x, y, z}and F ={r, s, t}



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Venn Diagrams

Example

suppose thatA B

A

B


E l


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Example

suppose thatAandBare not comparable

A B



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letA={a, b, c, d}and B={c, d, e, f}

a

b

c

de

f


U i


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Union

we denote the union ofAwithBby

A B

Example

in the following Venn diagram, the union of the two sets A andBcontains every point marked

a

b

c

d

e

f


Examples


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Examples

letS={a, b, c, d}andT ={f, b, d, g}, then

S T ={a, b, c, d, f, g}

letPbe the set of positive real numbers and let Qbe theset of negative real numbers

thenP Qconsists of all the real numbers except zero


Intersection


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Intersection

the intersection of setAwith setBis the set of elements which

are common toAand toB

it is denoted by

A B


Examples


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Examples

in the following Venn diagram, the intersection of the two

setsA andBcontains only the pointscandd:

a

b

c

d

e

f



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letS={a, b, c, d}andT ={f, b, d, g}, then

S T ={b, d}

LetV ={2, 4, 6, . . .}, and letW ={3, 6, 9, . . .}, then

V W ={6, 12, 18, . . .}


Difference


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Difference

the difference between two setsAandBis the set of elements

which belong toAbut not toB; we denote it by

A \ B

which is read AminusB, or AwithoutB


Examples


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Examples

The difference between the two setsAandBcontains only

the pointsaandb:

LetS={a, b, c, d}andT ={f, b, d, g}, then

S\ T ={a, c}

a

b

c

d

e

f

letRbe the set of real numbers and let Qbe the set ofrational numbers

thenR\ Qconsists of the irrational numbers


Formal Definitions


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Formal Definitions

A B={ x :X |xA xB}

A B={ x :X |xA xB}

A \ B={ x :X |xA x /B}


Properties of Set Operations


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Properties of Set Operations

the operations of union and intersection are commutative and

associative

for arbitrary setsA,B, andC,

A B=B A

A B=B A

(A B) C=A (B C)

(A B) C=A (B C)


Theorems


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Theorems

the set operations of union and intersection distribute over each

other; that is, for arbitrary sets A, B,andC,A (B C) = (A B) (A C)

A (B C) = (A B) (A C)



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A A= AA A= A

A = AA = A \ BAA BCDA CB DA BCDA CB D

A A BA BAA BA B=BA BA B=BA \ =AA (B\ A) =A (B\ A) =A BA \ (B C) = (A \ B) (A \ C)A \ (B C) = (A \ B) (A \ C)


Generalised Operations


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p

A={ x :X |( S:A xS) }A={ x :X |( S:A xS) }


Russells Paradox


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the men of a certain town can be divided into two groups:

those who shave themselvesand those who do not

the barber shaves all those who do not shave themselves

so who shaves the barber?


Russells Paradox


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S={x :Person| (x shaves x)}= {x :Person|barber shaves x}

barberS

(barber shaves barber)barber /S

barber /S

barber shaves barber

barberS


Russells Paradox


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S={x :Person| x shaves x}

B={x :Person|barber shaves x}Person\ S=B

barber /S

(barber shaves barber)barber /B

barberS

barberSbarber shaves barber

barberB

barber /S


Summary


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sets, extension and display; finite and infinite; sets of sets

membership and non-membership /

equality: extensionality

empty set

subsets, powersets, proper subsets, disjoint sets

Venn diagrams

union, intersection, difference (minus)

generalised union and intersection

algebraic laws

axiomatic set theoryavoid the paradoxes

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