Set Theory

Preparatory Quantitative Methods

Sirish Kumar Gouda and Vinu CT

May 26, 2014

Agenda

I Some basic Notations

I Basics of Set theory

I De�nitionI Set RepresentationI Set OperationsI Venn diagrams

Notations

Some notations that you will come across in the next few weeks

I N - Set of natural numbers

I N0- Set of natural numbers inclusive of zero (Whole numbers)

I Z - Set of integer numbers

I Q - Set of rational numbers

I R+- Set of non-negative real numbers

I C - Set of complex numbers

Notations

I∑n

i=1xi - Sum of the numbers xi : x1 + x2 + · · ·+ xn

I∏n

i=1xi - Product of the numbers xi : x1 ∗ x2 ∗ · · · ∗ xn

I n! - n factorial, i.e., n.(n − 1).(n − 2)....1

I min{a, b} - Minimum of the numbers a and b

I max{a, b} - Maximum of the numbers a and b

I |x | - Absolute value of the real number x

I [x ]- Greatest integer y such that y ≤ x

Notations

I [a, b]- Closed interval such that a ≤ x ≤ b

I (a, b)- Open interval such that a < x < b

I ∀ - For allI ∃ - there exists

I´- Inde�nite Integral and

´ ba - De�nite Integral

I ddx - Di�erential and ∂

∂x - Partial di�erential

I (nk) ornCk - Binomial coe�cient or Number of combinations

of k from n

Set theory

�No one shall expel us from the paradise that Cantor has createdfor us� - David Hilbert

De�nition

I A set is collection of objects(elements) de�ned in a preciseway, so that any given object is either

I in the setI or not in the set

I Elements of a set are known as its members

I All elements are picked from a Universal set usuallyrepresented by U

I Elements of a set are unique i.e there are no repeats

I Order of the elements is irrelevant

Representation

I Extension - via explicit enumeration

I A = {2, 4, 6, 8, ...}I B = {1, 4, 9, 16}I C = {1, 2, 6, 24, 120, ...}

I Intension - via a de�ning property

I A = {2k | k is a positive integer }I B = ?I C = ?

Sets of numbers

Figure : Relationship between numbers

Source: http://www.mathsisfun.com/sets/number-types.html

Is this a set?

I A is the collection of all the scores in IPL 7

I B is the collection of percentiles of all students of IIMB PGP2014-16 Batch

I Scams = { A Raja, 2G, CWG, Sonia G, Rahul G, Manmohanji}

I F = {k!| kεN0}

Is this a set?

I A is the collection of all the scores in IPL 7

I B is the collection of percentiles of all students of IIMB PGP2014-16 Batch

I Scams = { A Raja, 2G, CWG, Sonia G, Rahul G, Manmohanji}

I F = {k!| kεN0}

Is this a set?

I A is the collection of all the scores in IPL 7

I B is the collection of percentiles of all students of IIMB PGP2014-16 Batch

I Scams = { A Raja, 2G, CWG, Sonia G, Rahul G, Manmohanji}

I F = {k!| kεN0}

Is this a set?

I A is the collection of all the scores in IPL 7

I B is the collection of percentiles of all students of IIMB PGP2014-16 Batch

I Scams = { A Raja, 2G, CWG, Sonia G, Rahul G, Manmohanji}

I F = {k!| kεN0}

Set Notations and Operations

I A is a subset of B (written A⊆B) if every element of A is anelement of B

I Example: A = {Sachin, Dravid, Ganguly} and B is a collectionof all cricketers who scored 10,000 or more runs in ODIs

I Sets A and B are disjoint if no element of A is an element of Band vice-versa

I Sets A and B are equal if every element of A is an element ofB and vice-versa

I Exercise: A = {2k | k is a positive integer } and B is setcontaining the pair wise sum of all prime numbers (________ conjecture !!!)

Set Notations and Operations

I A is a subset of B (written A⊆B) if every element of A is anelement of B

I Example: A = {Sachin, Dravid, Ganguly} and B is a collectionof all cricketers who scored 10,000 or more runs in ODIs

I Sets A and B are disjoint if no element of A is an element of Band vice-versa

I Sets A and B are equal if every element of A is an element ofB and vice-versa

I Exercise: A = {2k | k is a positive integer } and B is setcontaining the pair wise sum of all prime numbers (________ conjecture !!!)

Set Notations and Operations

I A is a subset of B (written A⊆B) if every element of A is anelement of B

I Example: A = {Sachin, Dravid, Ganguly} and B is a collectionof all cricketers who scored 10,000 or more runs in ODIs

I Sets A and B are disjoint if no element of A is an element of Band vice-versa

I Sets A and B are equal if every element of A is an element ofB and vice-versa

I Exercise: A = {2k | k is a positive integer } and B is setcontaining the pair wise sum of all prime numbers (________ conjecture !!!)

Set Notations and Operations

I A is a subset of B (written A⊆B) if every element of A is anelement of B

I Example: A = {Sachin, Dravid, Ganguly} and B is a collectionof all cricketers who scored 10,000 or more runs in ODIs

I Sets A and B are disjoint if no element of A is an element of Band vice-versa

I Sets A and B are equal if every element of A is an element ofB and vice-versa

I Exercise: A = {2k | k is a positive integer } and B is setcontaining the pair wise sum of all prime numbers (________ conjecture !!!)

Set Notations and Operations

I The complement of a set A (denoted as Ac) is the collectionof all elements not in A

I Exercise: What is the complement of the set A ={2, 3, 5, 7, 11, 13, ...}

I The union of sets A and B (denoted by A∪B) is the set ofelements of A and B

I The intersection of A and B (denoted A∩B) is the set ofelements common to A and B

I The di�erence of A and B (denoted by A\B or A-B) is the setof elements of A but not in B

I The Cartesian product of A and B is the exhaustive set ofordered pairs shown below

I A x B = {(a, b) | aε A, bε B}

Set Notations and Operations

I The complement of a set A (denoted as Ac) is the collectionof all elements not in A

I Exercise: What is the complement of the set A ={2, 3, 5, 7, 11, 13, ...}

I The union of sets A and B (denoted by A∪B) is the set ofelements of A and B

I The intersection of A and B (denoted A∩B) is the set ofelements common to A and B

I The di�erence of A and B (denoted by A\B or A-B) is the setof elements of A but not in B

I The Cartesian product of A and B is the exhaustive set ofordered pairs shown below

I A x B = {(a, b) | aε A, bε B}

Set Notations and Operations

I The complement of a set A (denoted as Ac) is the collectionof all elements not in A

I Exercise: What is the complement of the set A ={2, 3, 5, 7, 11, 13, ...}

I The union of sets A and B (denoted by A∪B) is the set ofelements of A and B

I The intersection of A and B (denoted A∩B) is the set ofelements common to A and B

I The di�erence of A and B (denoted by A\B or A-B) is the setof elements of A but not in B

I The Cartesian product of A and B is the exhaustive set ofordered pairs shown below

I A x B = {(a, b) | aε A, bε B}

Set Notations and Operations

I The complement of a set A (denoted as Ac) is the collectionof all elements not in A

I Exercise: What is the complement of the set A ={2, 3, 5, 7, 11, 13, ...}

I The union of sets A and B (denoted by A∪B) is the set ofelements of A and B

I The intersection of A and B (denoted A∩B) is the set ofelements common to A and B

I The di�erence of A and B (denoted by A\B or A-B) is the setof elements of A but not in B

I The Cartesian product of A and B is the exhaustive set ofordered pairs shown below

I A x B = {(a, b) | aε A, bε B}

Set Notations and Operations

I The complement of a set A (denoted as Ac) is the collectionof all elements not in A

I Exercise: What is the complement of the set A ={2, 3, 5, 7, 11, 13, ...}

I The union of sets A and B (denoted by A∪B) is the set ofelements of A and B

I The intersection of A and B (denoted A∩B) is the set ofelements common to A and B

I The di�erence of A and B (denoted by A\B or A-B) is the setof elements of A but not in B

I The Cartesian product of A and B is the exhaustive set ofordered pairs shown below

I A x B = {(a, b) | aε A, bε B}

Venn Diagrams

I Geometric interpretation tool

I Represent the universal set as a rectangle

I Represent each set as a circle

I Combine the circles to depict the result of unary and binaryoperations

Venn Diagrams - Examples

Cardinality

I A set's cardinality is a measure of its size. If A is the set used,it's cardinality is represented by n(A)

I Intuitively, if we can exhaustively count the elements of a set,it is �nite

I If a set is in�nite we use ellipsis (. . . )

I Trivia: Did you know that the set of natural numbers is

countably in�nite ?

I Exercise: Try to prove using Venn diagrams

I n(A∪B) = n(A) + n(B) - n(A∩B)I If A ⊆ B, n(B) = n(A) + n(B\A)

Thank You