2. Introducing…
Group members
Serial Name ID
1 Md.Saffat-E-Nayeem (Group
Leader)
EV 1406009
2 Md. Shamim Ahmed EV 1406013
3 Fahmida Zaman EV 1406045
4 A M Nazmul Huda EV 1406053
5 Md Rakib Hasan EV 1406081
4. Introduction
Sets are used to define the
concepts of relations and functions.
The study of geometry, sequences,
probability, etc. requires the
knowledge of sets.
Studying sets helps us categorize
information.
It allows us to make sense of a
large amount of information by
breaking it down into smaller groups.
5. Set
Definition: A set is any collection of objects specified in
such a way that we can determine whether a given
object is or is not in the collection.
In other words A set is a collection of objects.
These objects are called elements or members of the
set.
The following points are noted while writing a set.
Sets are usually denoted by capital letters A, B, S, etc.
The elements of a set are usually denoted by small
letters a, b, t, u, etc .
Examples: A = {a, b, d, 2, 4}
B = {math, religion, literature, computer
science }
6. History of Set
The theory of sets was developed by German
mathematician Georg Cantor (1845-1918). A single
paper, however, founded set theory, in 1874 by Georg
Cantor: "On a Characteristic Property of All Real
Algebraic Numbers".
He first encountered sets while working on “problems
on trigonometric series”.
Cantor published a six-part treatise on set theory from
the years 1879 to 1884. This work appears
in Mathematische Annalen and it was a brave move by
the editor to publish the work despite a growing
opposition to Cantor's ideas.
The next wave of excitement in set theory came around
1900, when it was discovered that Cantorian set theory
gave rise to several contradictions,
7. The History of Set (continued)
Bertrand Russell and Ernst
Zermelo independently found the simplest and
best known paradox, now called Russell's
paradox: consider "the set of all sets that are not
members of themselves“.
The 'ultimate' paradox was found by Russell in
1902 (and found independently by Zermelo). It
simplify defined a set A = { X | X is not a member
of X }.
Russell used his paradox as a theme in his
1903 review of continental mathematics in
his The Principles of Mathematics.
Zermelo in 1908 was the first to attempt an
axiomatisation of set theory.
Gödel showed, in 1940, that the Axiom of
Choice cannot be disproved using the other
8. Importance in Business Organization
Set Theory does have its place in a Business
Organization. Any organization essentially
comprises of various types of resources such
as men, machines, money, materials, etc.
To equate assets of one kind with assets of
another kind.
An impact on the overall productivity of the
organization through skilled workers within the
set of all workers.
Subsets of products, materials, etc. which
when inquired into analytically could pave the
way for:
A. effective decision making
B. sound organizational plans, policies and
procedures.
9. Denoting a set as an object
Where it is desirable to refer to a set as
an indivisible entity, one typically denotes
it by a single capital letter. By convention,
particular symbols are reserved for the
most important sets of numbers:
∅ – empty set .
C – complex numbers .
N – natural numbers .
Q – rational numbers (from quotient)
R – real numbers
Z – integers(from Zahl, German for
integer).
11. Empty Sets
A set that contains no members is called the empty set or null set .
The empty set is written as { } or ∅.
Finite Sets
A set is finite if it consists of a definite number of different elements.
If W be the set of people living in a town, then W is finite.
Infinite Sets
An infinite set is a set that is not a finite set. Infinite sets may be countable
or uncountable.
The set of all integers, {..., -1, 0, 1, 2, ...}, is a count ably infinite set;
Equal Sets
Equal sets are sets which have the same members.
If P ={1,2,3}, Q={2,1,3}, R={3,2,1} then P=Q=R.
12. Subsets
Sets which are the part of another set are called
subsets of the original set.
For example, if A={1,2,3,4} and B ={1,2} then
B is a subset of A it is represented by .
Power Sets
If ‘A’ is any set then one set of all are subset of
set ‘A’ that it is called a power set.
Example- If S is the set {x, y, z}, the power
set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y,
z}, {x, y, z}}.
Universal Sets
A universal set is a set which contains all
objects, including itself.
Example- A={12345678} B={1357} C={2468}
D={2367} Here A is universal set and is denoted
by U.
13. Operation Of Sets
Union of sets
Intersection of sets
Compliments of sets
14. Union
The union of two sets would be wrote as A
U B, which is the set of elements that are
members of A or B, or both too.
Example:
16. Complements
If A is any set which is the subset of a given
universal set then its complement is the set
which contains all the elements that are in
but not in A.
Example:
17. Venn Diagrams
• Venn diagrams are
named after a
English logician,
John Venn.
• It is a method of
visualizing sets
using various
shapes.
• These diagrams
consist of
rectangles and
circles.
18. Application
Problem:
Out of forty students, 14 are taking English Composition
and 29 are taking Chemistry.
a) If five students are in both classes, how many students
are in neither class?
b) How many are in either class?
c) What is the probability that a randomly chosen student
from this group is taking only the Chemistry class?
19. here are two
classifications in this
universe: English
students and Chemistry
students.
First I'll draw my universe
for the forty students, with
two overlapping circles
labeled with the total in
each:
Since five students are
taking both classes, I'll
put "5" in the overlap:
20. I've now accounted for five
of the 14 English students,
leaving nine students taking
English but not Chemistry,
so I'll put "9" in the "English
only" part of the "English"
circle:
I've also accounted for five
of the 29Chemistry
students,
leaving 24 students taking
Chemistry but not English,
so I'll put "24" in the
"Chemistry only" part of the
21. This tells me that a total
of 9 + 5 +
24=38 students are in
either English or
Chemistry (or
both). This leaves two
students unaccounted
for, so they must be the
ones taking neither
class.
22. Findings from Set Theory
Set theory is used in almost every discipline
including engineering, business, medical and related
health sciences, along with the natural sciences.
In business operations, it can be applied at every
level where intersecting and non-intersecting sets are
identified.
For example, the sets for warehouse operations
and sales operations are both intersected by the
inventory set. To improve the cost of goods sold,
the solution might be found by examining where
inventory intersects both sales and warehouse
operations.
For Security purpose, risk management and system
analysis.
23. Recommendation
Scientific analysis of decision problems aims at giving the decision
maker (DM) a recommendation concerning a set of objects (called also
alternatives, solutions, acts, actions, cases, candidates). For example,
a decision can regard:
Diagnosis of pathologies for a set of patients, being the objects of the
decision, and the attributes are symptoms and results of medical
examinations.
Assignment to classes of risk for a set of firms, being the objects of the
decision.
The attributes are ratio indexes and other economic indicators such as
the market structure, the technology used by the enterprises, the
quality of the management and so on.
Selection of a car to be bought from a given set of cars, being the
objects of the decision, and the attributes are maximum speed,
acceleration, price, fuel consumption and so on,
Ordering of students applying for a scholarship, being the objects of
the decision, and the attributes are scores in different disciplines.
