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Our Presentation

Introducing…

Group membersSerial

Name ID

1 Md.Saffat-E-Nayeem (Group Leader)

EV 1406009

2 Md. Shamim Ahmed EV 14060133 Fahmida Zaman EV 14060454 A M Nazmul Huda EV 14060535 Md Rakib Hasan EV 1406081

Set TheoryIts importance and Application

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.

SetDefinition: 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 }

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, called antinomies or paradoxes. 

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 axioms of set theory. It was not until 1963

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 makingB. sound organizational plans, policies and

procedures.

Denoting a set as an objectWhere 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).

Types Of Sets

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.

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.

Operation Of Sets Union of sets Intersection of sets Compliments of sets

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:

Intersection Intersection are written as A ∩ B, is

the set of elements that are in A and B. For example:

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:

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.

ApplicationProblem: 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?

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:

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 "Chemistry" circle:

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.

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. For experimental result between two or several products.

RecommendationScientific 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.