Welcome To

NAMASTE LECTURE SERIES

Welcome To

NAMASTE LECTURE SERIES

Set Unionand

Intersection

2009

Set Theory

Lecture # 1

Prof. Dr. Ram M. ShreshthaProf. Dr. Ram M. ShreshthaProf. Dr. Ram M. ShreshthaProf. Dr. Ram M. Shreshtha

At a GlanceAt a Glance

Presents

Set and Set Set and Set OperationsOperations

Set and Set Set and Set OperationsOperations

Learning ObjectivesLearning Objectives Learning ObjectivesLearning Objectives

After completing this lesson, students should After completing this lesson, students should be able:be able:

To describe a set using two standard To describe a set using two standard forms of notation. forms of notation.

To perform operations on sets. To perform operations on sets. To use Venn diagrams to illustrate and To use Venn diagrams to illustrate and

solve problems involving setssolve problems involving sets

In the In the universe universe we live, we have our we live, we have our Solar Solar

SystemSystem. Our Earth . Our Earth is inis in or or belongs tobelongs to the Solar System. the Solar System.

The living beings in the earth The living beings in the earth is dividedis divided into - into - Plant Plant

KingdomKingdom and and Animal KingdomAnimal Kingdom. The animal kingdom . The animal kingdom

containscontains two two FamiliesFamilies - vertebrate and Invertebrate. The - vertebrate and Invertebrate. The

huge human population huge human population is containedis contained in the vertebrate in the vertebrate

family. In particular, the population of a country family. In particular, the population of a country

generally consists of different generally consists of different Social ClassesSocial Classes. In every . In every

social class live people of different social class live people of different Income GroupsIncome Groups. A . A

SampleSample from every such income group may have some from every such income group may have some

bright students. A bright student often gets high scores bright students. A bright student often gets high scores

or or AggregateAggregate of marks in the examination. A student with of marks in the examination. A student with

high marks is often greeted with a high marks is often greeted with a BunchBunch of flowers. of flowers.

Words such as Aggregate, Bunch, Class, Family, Words such as Aggregate, Bunch, Class, Family,

Group, Kingdom, Population, Sample, System, etc. used Group, Kingdom, Population, Sample, System, etc. used

above in different context indicate that they all convey above in different context indicate that they all convey

something common. This idea or sense of commonness can something common. This idea or sense of commonness can

be found in phrases such as collection of books in a library or be found in phrases such as collection of books in a library or

deck of cards or ensemble of points also. deck of cards or ensemble of points also.

Mathematicians use the simple word SET to convey Mathematicians use the simple word SET to convey

the common idea contained in the words the common idea contained in the words aggregate, bunch, aggregate, bunch,

class, deck, ensemble, family, kingdom, population, sample, systemclass, deck, ensemble, family, kingdom, population, sample, system, ,

etc.. This word set is taken as a fundamental term (without etc.. This word set is taken as a fundamental term (without

making any attempt to define it) to define other making any attempt to define it) to define other

mathematical terms. mathematical terms.

This lesson is a brief introduction to the language and This lesson is a brief introduction to the language and

concepts involving sets, parts of sets, combination of sets, concepts involving sets, parts of sets, combination of sets,

etc. etc.

(Continued)

Part OnePart One

An Animated IllustrationAn Animated Illustration

Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

The Solar

System The Solar

System { }

Solar system

Earth Solar system Neptune Solar system

Basic ConceptsSet and Set-membershipSun is in or belongs to Solar system

Sun Earth is in or belongs to Solar system Neptune is in or belongs to Solar system

Sun

The Solar SystemThe Solar System

Sun

The Solar SystemThe Solar System

{Sun}

Sub-SystemsSub-Systems

{Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}

Star

Star, Planets

Planets

Singleton set

{Sun} Solar System

{Planets} Solar System

Inner planets

Outer planets

Superior planets

Sun

The Solar SystemThe Solar System

Mars

{Sun}

{Sun} is a subset of Solar System

Solar System

{Inner Planets} is a subset of Solar System

{Inner planets} Solar System

{Outer planets} is a subset of Solar System{Outer planets}

Solar System

Inner planets

Outer planets

Superior planets

Sun

The Solar SystemThe Solar System {Sun} Solar System

{Inner planets} Solar System

{Outer planets} Solar System

Superior planets Solar System

{Mars} Solar System

Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

The Solar System

(Sub-systems)

{Inner planets} I {Outer planets } O{Sun}

Inner Planets Solar system

Outer Planets Solar system

{Sun} Solar system

Basic Concepts : Set and Subset

Inner planets

Outer planets

Superior planets

Sun

The Solar SystemThe Solar System

{Sun}

{Outer planets} {Superior planets}

Solar System

{Mars} =

{Mars} {Inner planets} {Inner planets}=

{Outer planets} {Inner planets} =

Basic Concept : Union of Sets

{Outer Planets}

Mars

JupiterSaturnUranus

NeptunePluto

Set Union

{Mars} {Outer Planets} = {Superior Planets}

{Mars}

{Outer Planets}

MercuryVenusEarth

Mars

JupiterSaturnUranus

NeptunePluto

I O = P

{Inner Planets}

Set Union

Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

The Solar System

(Sub-systems)

{Inner planets} I {Outer planets} O

{Sun} {Inner Planets} {Outer Planets} = Solar System

{Sun}

{Inner Planets} Solar system

{Outer Planets} Solar system

{Sun} Solar system

Basic Concepts : Set , Subset and Union

{Mars} {Outer Planets} = { Superior Planets }

Inner planets

Outer planets

Superior planets

Sun

The Solar SystemThe Solar System

{Sun}

{Outer planets} { } {Mars} =

{Mars} {Inner planets}{Mars}=

{Outer planets} {Inner planets} =

=

Ø

Ø

{Inner planets} {Superior planets} = {Mars}

(Intersecting sets or Sets with common element)

(Disjoint sets or sets with no common element) (Intersecting sets or Sets with common element)

{Inner Planets}

{Superior Planets}

MercuryVenusEarthMars Mars

JupiterSaturnUranus

NeptunePluto

SET

INTERSECT ION

I S = {Mars}

Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

The Solar System(Sub-systems)

The Solar System(Sub-systems)

{Inner planets} I {Outer planets} O

{Superior planets} S

{Sun}

{Inner Planets} {Outer Planets} =

{ Sun} {nner Planets} =

Basic Concepts (Empty set, intersecting and disjoint

sets}

{Inner Planets} {Superior Planets} = { Mars}

Sun

The Solar SystemThe Solar System

Complement of { Sun } = {Planets}

{ Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune }

Complement of { Planets } = {Sun}

Inner planets

The Solar SystemThe Solar System

Outer planets

Superior planets

Sun

{ Sun} Inner Planets Outer Planets = Solar System

Outer Planets Superior Planets

Inner Planets Outer Planets =

Superior Planets Inner Planets = {MarsMars}

Part Two

A Bit of History

Modern mathematics begins with the

notion of a set. It lies at the foundation of almost all branches of mathematics. Georg Cantor (1845 – 1915) is known as the father of the theory of sets. George Boole and John Venn were the two other 19th century mathematicians who made valuable contributions to the theory of

sets.

Pioneers of Set Theory

Set

The word set is synonymous with words such as:

"aggregate", " bunch", “collection”, "dump", "ensemble", “family", " group" etc.

Examples:The aggregate of marks;

A bunch of flowers; A collection of stamps; A dump of waste materials;

An ensemble of points; A family of plants;A group of students

Set notation

A set is usually denoted by capital letters such as

A, B, C, …, X, Y, Z;and the elements or members of a set by

the small letters such asa, b, c, …, x, y, z.

Set Membership

Set membership is indicated by the symbol "" and non-membership is denoted by "".

We writex X to mean “ x is an element or member of X”

x A to mean “ x does not belong to A”

Set Specification

a) By description/extension:It is the explicit listing of the objects or members or elements in between two braces or curly brackets { }.Examples:

1. A = {a, b, …, y, z}, 2. N = {1, 2, …, 11, 12 …}

Set Specification(Continued)

b) By comprehension/ intension:It is the specification by a membership condition or rule for inclusion in the set.

Examples:A = {x : x satisfies a property P(x)}

= {x : x P(x)} = {x : P(x)}.This is the set-builder form. Here,

the braces {x} is read as “The set of all x’s” the colon “:” as “ such that”.

V={The set of vowels in the English alphabet}

Some Special Characteristics

A set is basically unordered.It consists of distinct (or unequal) objects;

Multiple listing of elements is not done.A set may consists of sets.

A set may have nothing else or may be empty.A set with one element is called a singleton set.

A set may have a fixed (or finite) number of elements.

Successor Element

In a non-empty non-singleton set, every element (except the last, if there is one such

element) may be followed by a unique element, called the successor of the former.

Examples:In the set of counting numbers

2 is the successor of 13 is the successor of 2

and so on.

Ordered Pair

In a pair of numbers (i.e., a set with two elements),

one number may be followed by another or

one may be the successor of the other. Such a set is said to be ordered.

An ordered pair in which b is a successor of a is denoted by (a, b).

Two ordered pairs (a, b) and (c, d) are said to be equal

if and only if a = c and b = d.

The Two Speical Cases

A set without any element is known as an empty set.

In symbols, = { x | x ≠ x }.Examples:

1.The set of male-students in a girls’ school.2. The set of numbers in a fish pond.

An important convention is to acceptthe empty set or null set or void set as

a finite set

a) Empty set.

The set of all elements under consideration. It is usually denoted by the letter U.

The set of numbers is the universal set for ordinary arithmetical operations.

b) Universal set.

Venn Diagrams

A Venn-diagram is the representation of the elements of a set by points inside a simple closed curve such as a circle or ellipse.

The universal set U is usually denoted by a rectangle and its subset by a simple closed curve within the rectangle as follows:

Venn-Euler Diagram

a . e . i . o . . u

b. c. d . . . . . V . x . y . z

U

Subset

A set A is a subset of a set B

if every element of A is an element. In symbols, we write

A B if x A then x B.

Examples: The set of boy-students in a class is

a subset of the set of all students

2. a)

0 1 2 3 4 0 1 2

T

T

0 1 2

2. b)

0 1 2

S

S = T

S

Cases such as 2(a) and 2(b), when taken together, are denoted by the single notation:

2. The set of point in the circumference of a circle is a subset of the set of points of the circular region including the boundary

Here T is called the superset of S.

Equal set

Two sets are said to be equal if every element is also and element of the other.

In other words, A set A is said to be equal to a set B if A is a subset of B and B is a subset of A if every element of A is an element of B.

In symbols, we write A = B iff A B and B A.

p p

2. A

= A B

B

1. The set {1, 2, 3} = {3, 2, 1};

but {123}≠ {321}

Examples:

Proper Subset

A subset A of a set B is a proper subset if the sets are not equal.

A B = { x | if x A then x B } and A ≠ B.

Examples:1. The set {1, 5, 9} is a proper subset of

the set {1, 2, 3, 4, 5, 6, 7, 8, 9}.2. The set of points on a line

is a proper subset of the set of points in a plane.

Proper subset

c

B A

B

A

c

Diagram1923

Joh n

Venn

Venn1834

Power Set

The power set of a set X is the set of all subsets of the set X.

The power set contains both the empty set and the set X.

In symbols, we write(X)= = 2℘ X = { A | if x A then x X }.

Example:The power set of S = {1, 2, 3} is the set

(S)=2℘ S ={,{1},{2},{3},{1,2},{1,3},{2,3},{1,2},{1,2,3}}

Equivalent Sets

Two sets A and B are said to be equivalent if to each element of A there

corresponds one element of B and to each element of B there corresponds one

element of A ( i.e., if there is a one-to-one correspondence between the two sets).

In symbols, we writeA ≈ B

to mean A is equivalent to B.

The set N of countries

{Nepal, China, India}

and the set C of capitals

{Kathmandu, Beijing, Delhi}

are equivalent as shown below:

Nepal China India

↕ ↕ ↕

Kathmandu Beijing Delhi

Examples:

One-to-one correspondence

1, 2, 3, … , 101, 102,…

2, 4, 6, … , 202, 204,…

Cardinality

Two sets are said to have the same cardinality or cardinal number

if they are equivalent.

In case, a set A is finite, the cardinality of the set A is the number of elements in A.

It is denoted by |A| or # (A).Examples:

1. The cardinality of the empty set is 0.

Cardinality(Continued)

2. For the setsN = {Nepal, China, India}

and C = {Kathmandu, Beijing, Delhi},

a) The cardinality of the set N,

|N| or # (N) = 3,b) The cardinality of the set C,

|C| or # (C) = 3.

Also, # (N) = 3 = # (C).

Set Operations

a.Set UnionThe union of two sets is the set

consisting of the elements of both sets. In symbols, the union of two sets A and B

is denoted byA B = { x | x A or x B }.

Examples:1. {a, b} {1, 2, 3} = {a, b,1, 2, 3}

2. {1, 2} {1, 2, 3} = {1, 2, 3}

Elements in at least one of the two sets:

A B

U

AB

Venn Diagram

3. {{2, 3, 5} {3, 5, 7}} = {2, 3, 5, 3, 5,7}= {2, 3, 5, 7}

The intersection of two sets is the set of the elements common to both sets.

In symbols, the intersection of the two sets A and B is denoted by

A ∩ B = { x | x A and x B} Examples:

1. The intersection of the sets A = {1, 2} and B = {2, 3} is the singleton set {2}.

2. Parallel lines do not intersect.

Set Intersection

Elements in exactly one of the two sets:

A B

U

AB

Venn Diagram

AB = { x | x A , x B }

Intersection

3. {a,b,c}{2,3} =

4. {2,4,6}{3,4,5} = {4}

.

Disjoint setsTwo sets are disjoint

if they have no elements in common. In symbols, if A and B are disjoint, then

A ∩ B = .Examples:

1. The set of vowels and the set of consonants of the English alphabet are

disjoint. 2. The set of points of intersection of two

parallel lines is empty and

The two sets of points are disjoint.

A B

Disjoint sets

No common point or members

The difference of two set A and B (A followed by B)

is the set of elements belonging to the first set but not to the second set.

In symbols, the difference of two sets A and B is denoted by

A ~ B = A - B = A \ B= { x | x A and x B∉ }

Difference of two sets

Venn Diagram

A - B = { x | x A , x B }

Elements in first set but not in the second:

A

B

U

A-B

A B A - B 1.

Examples:

Venn Diagram

2. {1, 2 , 3, 4 , 5, 6 } {2, 3, 5, 7, 9, 11}

3.The difference of the set of points bounded by a circle and the set of points on the circumference of the circle is the interior of the circle or the set of points inside the circle.

= {1,4,6}

Complement of a Set The complement of

a set A is a set consisting of those elements not belonging to the set A.

In symbols, the complement of a set A with respect to a given universal set is

denoted by

= A' = ~A = { x | x A } = U \ A Ac

1. In the set of English alphabet, the complement of the set of consonants is the set of vowels.

2. In the solar system, the complement of the set consisting of the sun is the set of all planets.

U

U - A

A

Complement of

A

Symmetric difference:

The union of the differences A – B and B – A between two sets A and B is called the symmetric difference of A and B.

In symbols, A ∆ B = (A – B) (B – A).

Symmetric Difference

A-B B-A

UA ∆ B

Elements in exactly one of the two sets:

(A – B) (B – A)

Examples:

The symmetric difference of the sets A = {1, 2, 3} and B = {2, 3, 4}

is A ∆ B

= (A – B) (B – A) = ({1,2,3}–{2,3,4})({2,3,4}–

{1,2,3})= {1} {4} = {1,4}

Some standard sets of numbers N = {1, 2, 3, … 11, 12, … }

Z+ = {0, 1, 2, 3, … 11, 12, … }

Z = {…, -2, -1, 0, 1, 2, …}

Q = {p/q | p, q in Z and q ≠ 0}

Q* = {The set of irrational numbers}

R = {The set of real numbers }

R R = The Cartesian plane.

Paradoxes:

give rise to what is known as paradox.

Popular questions such as

1. Does the set of all sets contain ltself? 2. Is there is a bibliography that lists all bibliographies that don't list themselves? 3. In a village, there is a barber (a man) who shaves all those men who do not shave themselves. Who shaves the barber? "

Paradox

Paradoxes are set-theoretic constructions Paradoxes are set-theoretic constructions leading to contradiction.leading to contradiction.

For instance,For instance,

The set of all sets that do not contain themselvesThe set of all sets that do not contain themselves..

In particular, the set of students of set theory who In particular, the set of students of set theory who themselves do not do set theory.themselves do not do set theory.

Barber’s ParadoxBarber’s Paradox

In a small village, a barber claims In a small village, a barber claims that “I shave anyone who does that “I shave anyone who does not shave himself, and none not shave himself, and none else.”else.”

Let A = the set of men who do not Let A = the set of men who do not shave themselves.shave themselves.

Let B = the set of men that the Let B = the set of men that the barber shaves.barber shaves. A = BA = B

Russell’s Paradox (1901)Russell’s Paradox (1901)

Let M be the set of all sets Let M be the set of all sets that do not contain that do not contain themselves as members. themselves as members. That is, That is,

M = { A | A ∉ M = { A | A ∉ A }.A }.Question: Does M Question: Does M contains itself ?contains itself ?

Let A ={x | xx }. Is AA? Both the assumption that A is a member of A and A is not a member of A lead to a contradiction (If R ={x | xx} then RR iff RR.).

Bertrand Russell 1872-1970

Cantor’s Paradox (189?)Cantor’s Paradox (189?)

The cardinality of a set is smaller The cardinality of a set is smaller than the cardinality of its power set.than the cardinality of its power set.

The set of all sets is its own power The set of all sets is its own power set.set.Therefore, the cardinality of the set Therefore, the cardinality of the set of all sets is smaller than itselfof all sets is smaller than itself

Crisis Paradoxes threaten set theory and the Paradoxes threaten set theory and the

mathematical analysis based on it.mathematical analysis based on it.

Intuitionists like Brouwer demanded a Intuitionists like Brouwer demanded a complete rejection of Cantor’s set complete rejection of Cantor’s set

theory.theory.

Mathematician D. Hilbert announcedMathematician D. Hilbert announced

““No one can drive us from the heaven No one can drive us from the heaven which Cantor created for us.”which Cantor created for us.”