Introduction

BUSINESS MATHEMATICS AND STATISTICS

1
Difference between Statistics and mathematics

2
Scope and Focus:

Mathematics: Mathematics is a broad field that deals with the study

of numbers, structures, patterns, and relationships. It includes various
branches such as algebra, calculus, geometry, and number theory.
Mathematics focuses on abstract concepts, proofs, and theoretical
frameworks.

Statistics: Statistics is a branch of mathematics that deals with the

collection, analysis, interpretation, presentation, and organization of
data. It involves methods for summarizing and making inferences
from data, and it provides tools for decision-making, estimation, and
hypothesis testing.

3
Application:

Mathematics: Mathematics has applications in various fields, including physics, engineering, computer science,
economics, and more. It is used to solve problems, model real-world phenomena, develop algorithms, and make precise
calculations.

Statistics: Statistics is primarily used in the social sciences, natural sciences, business, economics, healthcare, and other
fields where data analysis and inference are crucial. It helps in understanding patterns, making predictions, drawing
conclusions, and supporting decision-making based on data.

Techniques and Methods:

Mathematics: Mathematics employs a wide range of techniques, such as mathematical equations, formulas, algorithms,
and symbolic manipulation. It involves precise calculations, logical reasoning, and deductive reasoning.

Statistics: Statistics utilizes various techniques, including data collection methods, descriptive statistics, inferential
statistics, probability theory, regression analysis, hypothesis testing, and data visualization. It involves summarizing data,
identifying patterns, estimating parameters, testing hypotheses, and making predictions.
4
What and Why ?

Set Theory
Matrix

Vector Algebra

Statistics

Probability

5
Set Theory
Topic Page No.
Introduction to Sets, Sets and their Representation 7
Tabular or Roster Method , Rule Method or Set Builder 11
Type of Sets : Empty or Void or Null Set, Finite sets and Infinite sets 13
Intervals and types of interval 26
Venn Diagrams 33
Disjoint Sets 35
Operations on Sets 35
Symmetric Difference of Sets 43
Laws of Algebra of Sets 45

6
Introduction to Sets, Sets and their Representation
In mathematics, a set is a collection of distinct elements or objects. These objects can be anything: numbers, letters,
symbols, or even other sets. Sets are fundamental in various branches of mathematics and serve as building blocks for
many mathematical concepts.

sets are represented in curly braces {}

for example:
A = {1,2,3,4,5}

Sets are usually represented with capital letter.

Thus, A is the set and 1, 2, 3, 4, 5 are the elements of the set or members of the set

The elements that are written in the set can be in any order but cannot be repeated

All the set elements are represented in small letter in case of alphabets.

Also, we can write it as 1 ∈ A, 2 ∈ A etc.

The cardinal number of the set is 5 (cardinal number refers to total number of elements in a set) 7
Some commonly used sets are as follows:

N: Set of all natural numbers

{0,1,2,3,4,5}

Z: Set of all integers

{-3,-2,-1,0,1,2,3,4,5}

Q: Set of all rational numbers

{-3,-2,-1/8,0,1,2/5,3,4,5}

R: Set of all real numbers

{-3,-2,-1/8,0,1,,3,4,5}

Z+: Set of all positive integers

{1,2,3,4,5}

8
Order of Sets
The order of a set defines the number of elements a set is having. It describes the size of a set. The order of set is also
known as the cardinality.

A ={-3,-2,-1,0,1,2,3,4,5}
order = ?

Representation of Sets
The sets are represented in curly braces, {}.
For example,
{2,3,4}
{a,b,c}
{Bat, Ball, Wickets}.

The elements in the sets are depicted in either the Statement form, Roster Form or Set Builder Form.

9
Statement Form
In statement form, the well-defined descriptions of a member of a set are written and
enclosed in the curly brackets.

For example, the set of even numbers less than 15.

In statement form, it can be written as {even numbers less than 15}.

A = {1,9,25,49,81}

Statement Form for A = ?

10
Tabular or Roster Method , Rule Method or Set Builder
There are two methods of representing a set :

Roster or tabular form: In roster form, all the elements of a set are listed, the elements are being separated by commas
and are enclosed within braces { }.

For Example:
Z=the set of all integers={…,−3,−2,−1,0,1,2,3,…}

Set-builder form: In the set builder form, all the elements of the set, must possess a single property to become the
member of that set.
The general form is, A = { x : property }

For Example:
Z={x:x is an integer}

You can read Z={x:x is an integer}

as "The set Z equals all the values of x such that x is an integer."
11
A={2, 4, 6, 8}

A=?

B = {1, 3, 5, 7, 9}

B=?

C = {1, 3, 5, 9}

C=?

A = {x: x=2n, n ∈ N and 1 ≤ n ≤ 4}

B = {x : x = 2n - 1, n ∈ N, 1 ≤ n ≤ 5}
C = {x : x = 2n - 1, n ∈ N, 1 ≤ n ≤ 5, n ≠ 4}

12
Types of Set
Types of Sets
We have several types of sets in Math's. They are empty set, finite and infinite sets, proper set, equal sets, etc.

we will go through each type of set in detail.

13
Empty Set
A set which does not contain any element is called an empty set or void set or null set. It is denoted by { } or Ø.

A set of apples in the basket of grapes is an example of an empty set because in a grapes basket there are no apples
present.

14
Singleton Set
A set which contains a single element is called a singleton set.

Example: There is only one red apple in a basket.

15
Finite set
A set which consists of a definite number of elements is called a finite set.

Example: A set of natural numbers up to 10.

A = {1,2,3,4,5,6,7,8,9,10}

16
Infinite set
A set which is not finite is called an infinite set.

Example: A set of all natural numbers.

A = {1,2,3,4,5,6,7,8,9……}

17
Equal sets
The two sets A and B are said to be equal if they have exactly the same elements, the order of elements do not matter.

Example: A = {1,2,3,4} and B = {4,3,2,1}

A=B

18
Equivalent set
If the number of elements is the same for two different sets, then they are called equivalent sets. The order of sets does
not matter here. It is represented as:

n(A) = n(B)

where A and B are two different sets with the same number of elements.

Example: If A = {1,2,3,4} and B = {Red, Blue, Green, Black}

In set A, there are four elements and in set B also there are four elements.
Therefore, set A and set B are equivalent.

19
Universal Set
A set which contains all the sets relevant to a certain condition is called the universal set. It is the set of all possible
values.

Example: If A = {1,2,3} and B {2,3,4,5}, then universal set here will be:

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

20
Subsets
A set ‘A’ is said to be a subset of B if every element of A is also an element of B, denoted as A ⊆ B. Even the null set is
considered to be the subset of another set. In general, a subset is a part of another set.

Example: A = {1,2,4}

Then {1,2} ⊆ A.

Similarly, other subsets of set A are: {1},{2},{4},{1,2},{2,4},{1,4},{1,2,4},{}.

A is said to be a subset of B

21
Proper Subset
If A ⊆ B and A ≠ B, then A is called the proper subset of B and it can be written as A ⊂B.

Example: If A = {2,5,7} is a subset of B = {2,5,7} then it is not a proper subset of B = {2,5,7}

But, A = {2,5} is a subset of B = {2,5,7} and is a proper subset also.

22
Superset
Set B is said to be the superset of A if all the elements of set A are the elements of set B. It is represented as B ⊃ A.

For example, if set B = {1, 2, 3, 4} and set A = {1, 3, 4}, then set B is the superset of A.

B is the superset of A

23
Proper Superset
A proper superset of a set A is a superset of A that is not equal to A . In other words, if B is a proper superset of A , then
all elements of A are in B but B contains at least one element that is not in A.

For example, if A={1,3,5} then B={1,3,4,5} is a proper superset of A.

The set C={1,3,5} is a superset of A, but it is not a proper superset of A since C=A .

The set D={1,3,7} is not even a superset of A, since D does not contain the element 5.

24
Power set
The power set is a set which includes all the subsets including the empty set and the original set itself. It is usually denoted
by P. Power set is a type of sets, whose cardinality depends on the number of subsets formed for a given set.

If set A = {x, y, z} is a set,

then all its subsets {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z} and {} are the elements of power set,

such as:
Power set of A, P(A) = { {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z}, {} }

Where P(A) denotes the power set.

25
Intervals
An interval is a set that consists of all real numbers between a given pair of numbers. It can also be thought of as a
segment of the real number line. An endpoint of an interval is either of the two points that mark the end of the line
segment. An interval can include either endpoint, both endpoints or neither endpoint. To distinguish between these
different intervals, we use interval notation.

Types of Interval :
Open Interval
Close Interval
Semi Open or Semi Closed Interval
Infinite Interval

26
Open Intervals
The set of real numbers {x : a < x < b} is called an open interval and is denoted by (a, b). Open intervals contain all the
points between a and b belonging to (a, b), but a, b themselves do not belong to this interval.

This can be represented on the real number line as:

{x : a < x < b}

The hollow circles denote that the points at these circles are not included in the set of numbers of that interval.

A = {x : 1 < x < 5}

A = {2,3,4}
27
Closed Intervals
The interval containing the endpoints is also called the closed interval and is denoted by [a, b], and it is written as [a, b] =
{x : a ≤ x ≤ b}.

Closed interval [a, b] can be described on a real number line as:

{x : a ≤ x ≤ b}.

The solid circles denote that the points at these circles are included in the set of numbers of that interval.

A = {x : 1 ≤ x ≤ 5}

A = {1,2,3,4,5}
28
Degenerate Interval
A set consisting of a single real number or an interval of the form a to a, i.e. [a, a] is called a degenerate interval.

Bounded and Unbounded Intervals

An interval is said to be left-bounded if there is some real number that is smaller than all its elements and is called a
right-bounded if there is some real number that is larger than all its elements. So, an interval is said to be bounded if it
is both left- and right-bounded; otherwise, it is called an unbounded interval.

Intervals that are bounded at only one end are called half-bounded. However, bounded intervals are also known as
finite intervals.

29
Semi-Open or Semi-Closed interval
Also called as Half-open intervals mean the intervals that are closed at one end and open at the other. These can be
represented as:

[a, b) = {x : a ≤ x < b} is an open interval from a to b, including a but excluding b.

(a, b] = {x : a < x ≤ b} is an open interval from a to b including b but excluding a.

These intervals can be represented on the real number line as shown in the below figure:

{x : a ≤ x < b} {x : a < x ≤ b}

30
Infinite intervals
Infinite intervals are those that do not have an endpoint in either the positive or negative direction, or both. The interval
extends forever in that direction. Infinite intervals are summarized in the table below.

31
32
Venn Diagrams
Venn diagrams are the diagrams that are used to represent the sets, relation between the sets and operation performed
on them, in a pictorial way.

Venn diagram, introduced by John Venn (1834-1883), uses circles (overlapping, intersecting and non-intersecting), to
denote the relationship between sets.

A Venn diagram is also called a set diagram or a logic diagram showing different set operations such as the intersection of
sets, union of sets and difference of sets.

It is also used to depict subsets of a set.

The universal set (U) is usually represented by a closed

rectangle, consisting of all the sets.

The sets and subsets are shown by using circles or oval shapes.

33
In the above figure, we can see a Venn diagram, represented by a rectangular shape about the universal set, which has
two independent sets, X and Y. Therefore, X and Y are disjoint sets. The two sets, X and Y, are represented in a circular
shape. This diagram shows that set X and set Y have no relation between each other, but they are a part of a universal
set.

For example: set X = {Set of even numbers}

set Y = {Set of odd numbers}
Universal set, U = {set of natural numbers}
34
Disjoint Sets
A pair of sets which does not have any common element are called disjoint sets.

For example, X = {2,3}

Y = {4,5}

Then set X and Y are disjoint sets.

C = {3,4,5}
D = {3,6,7}

Then Set C and D are not disjoint as both the sets C and D are having 3 as a common element.

Another definition: When the intersection of two sets is a null or empty set, then they are called disjoint sets. Hence, if A
and B are two disjoint sets, then;

A∩B=ϕ

35
Operations on Sets
The set operations are performed on two or more sets to obtain a combination of elements as per the operation
performed on them. In a set theory, there are three major types of operations performed on sets, such as:

Union of sets (∪)

Intersection of sets (∩)
Difference of sets ( – )

We also have one operation which we can perform on a single set i.e.
compliment of a set ( ‘ )

We will see each operation in proper detail.

36
Union of Sets
If two sets A and B are given, then the union of A and B is equal to the set that contains all the elements present in set A
and set B. This operation can be represented as;

A ∪ B = {x: x ∈ A or x ∈ B}

Where x is the elements present in both sets A and B.

Example: If set A = {1,2,3,4,5} and B {5,6,7}

Then, Union of sets, A ∪ B = {1,2,3,4,5,6,7}

37
Intersection of Sets
If two sets A and B are given, then the intersection of A and B is the subset of universal set U, which consist of elements
common to both A and B. It is denoted by the symbol ‘∩’. This operation is represented by:

A∩B = {x : x ∈ A and x ∈ B}
Where x is the common element of both sets A and B.

The intersection of sets A and B, can also be interpreted as:

A∩B = n(A) + n(B) – n(A∪B)

Where,
n(A) = cardinal number of set A,
n(B) = cardinal number of set B,
n(A∪B) = cardinal number of union of set A and B.

Example: Let A = {1,2,3} and B = {3,4,5}

Then, A∩B = {3};
Because 3 is common to both the sets
38
Difference of Sets
If there are two sets A and B, then the difference of two sets A and B is equal to the set which consists of elements present
in A but not in B. It is represented by A-B.

Example: A = {1,2,3,4,5,6,7}
B = {6,7} are two sets.

Then, the difference of set A and set B is given by;

A – B = {1,2,3,4,5}

We can also say, that the difference of set A and set B is equal
to the intersection of set A with the complement of set B.
Hence,

A−B = A ∩ B’

39
Complement of Set
If U is a universal set and A is any subset of U then the complement of A is the set of all elements of the set U apart from
the elements of A.

A′ = {x : x ∈ U and x ∉ A}

Example: U = {1,2,3,4,5,6,7,8}
A = {1,2,5,6}

Then, complement of A will be;

A’ = {3,4,7,8}

40
Cartesian Product of sets
If set A and set B are two sets then the cartesian product of set A and set B is a set of all ordered pairs (a,b), such that a
is an element of A and b is an element of B. It is denoted by A × B.

We can represent it in set-builder form, such as:

A × B = {(a, b) : a ∈ A and b ∈ B}

Example: set A = {1,2,3} and set B = {Bat, Ball}, then;

A × B = {(1,Bat),(1,Ball),(2,Bat),(2,Ball),(3,Bat),(3,Ball)}

41
42
Symmetric Difference of Sets
The set which contains the elements which are either in set A or in set B but not in both is called the symmetric difference
between two given sets. It is represented by A ⊝ B and is read as a symmetric difference of set A and B.

43
Some of the properties related to difference of sets are listed below:
Suppose two sets A and B are equal then,
A – B = A – A = ∅ and B – A = B – B = ∅. (empty set)

The difference between a set and an empty set is the set itself, i.e,
A – ∅ = A.

The difference of a set from an empty set is an empty set, i.e,

∅–A=∅

The difference of a set, say A from universal set U is equal to empty set, i.e.
A–U=∅

When a superset is subtracted from a subset, then result is an empty set, i.e,
A – B = ∅ if A ⊂ B

If A and B are disjoint sets (no common elements for A and B),
then A – B = A and B – A = B.
44
Example 1:

If A = {1, 2, 3, 4, 5, 6} and B = {3, 4, 5, 6, 7, 8},

then find A – B and B – A.

Example 2:

If X = {11, 12, 13, 14, 15}, Y = {10, 12, 14, 16, 18} and Z = {7, 9, 11, 14, 18, 20}, then find the following:
(i) X – Y – Z
(ii) Y – X – Z
(iii) Z – X – Y

45
Laws of Algebra of Sets.
Sets under the operations of union, intersection, and complement satisfy various laws (identities) which are listed
below :

Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
De Morgan's Laws
Identity Laws
Complement Laws
Involution Law

We will see each law in detail

46
Idempotent Laws

A∪A=A
A∩A=A
U

47
Associative Laws
U
(A ∪ B) ∪ C = A ∪ (B ∪ C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)
A B

48
Commutative Laws

A∪B=B∪A
A∩B=B∩A

49
Distributive Laws

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) =(A ∩ B) ∪ (A ∩ C)

50
De Morgan's Laws

(A ∪B)’= A’∩ B’
(b) (A ∩B)’= A’ ∪ B’

51
Identity Laws

A∪∅=A
A∪U=U
A ∩ U =A
A∩∅=∅

52
Complement Laws

A ∪ A’= U
A ∩ A’= ∅
U’= ∅
∅’ = U

53
Involution Law

(A’)’ = A

54
55
Email Id: imran.wadkar@isdcglobal.org.uk
Contact Number: +919834241631
56