Download - Welcome To NAMASTE LECTURE SERIES
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)
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}
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}
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.
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:
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
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.
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.
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:
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}
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
.
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.
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
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.
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).
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.”