concept, theory n application of sets

INSTITUT PERGURUAN PEREMPUAN MELAYU MELAKA

COURSEWORK MATHEMATICS 1 (SETS)

NAME : SYARAFINA BT MOHD SALIM

IDENTITY CARD NUMBER : 900917-06-5660

GROUP/COURSE : PPISMP MATHEMATICS JULY 2009

CODE & SUBJECT : MATHEMATICS 1

NAME : NORJAMILAH BT AB.RAHMAN




NAME : HAZIRAH BT MOHD ABDUL WAHID




Coursework mathematics 1, semester 1,2009

BORANG REKOD KOLABORASI KERJA KURSUS

NAMA PELAJAR : 1) NORJAMILAH BT AB. RAHMAN

NO MATRIK :

KUMPULAN : PPISMP AMBILAN JULAI 2009

SEMESTER : 1

MATA PELAJARAN : MATEMATIK 1

PENSYARAH : PUAN NG PEK FOONG

TARIKH PERKARA YANG DIBINCANGKAN

KOMEN TANDATANGAN PENSYARAH

21/07/2009 Tugasan diberikan beserta penerangan oleh pensyarah

23/07/2009 Pembahagian dan pembentukan kumpulan

Ahli kumpulan terdiri daripada Jamilah, Hazirah, dan Syarafina

30/07/2009 Penerangan kriteria tugasan beserta soal jawab berkaitan pelaksanaan tugasan

30/07/2009 Mengadakan perbincangan kumpulan bagi memahami dengan sepenuhnya kehendak tugasan

Tugasan di bahagikan kepada setiap ahli kumpulan

31/07/2009 Mencari bahan mengikut kehendak tugasan bersama ahli

Nota ringkas dilakukan

2


kumpulan daripada bahan rujukan di perpustakaan

1/08/2009 dan 2/08/2009

Sambung mencari bahan daripada sumber internet.

4/08/2009 Perbincangan antara ahli kumpulan tentang bahan yang dikumpulkan.

Memastikan bahan yang dikumpulkan memenuhi kehendak tugasan

6/08/2009 Penyemakan sumber oleh pensyarah

8/08/2009 Membincang dan memahami isi-isi bahan yang dikumpulkan bersama ahli kumpulan

10/08/2009 Mencari isi penting di dalam bahan dan membuat rumusan


11/08/2009 Menyambung membuat rumusan tentang tajuk yang diberikan

12/08/2009 Perbincangan antara ahli kumpulan bagi melengkapkan lagi kerja kursus

Bertukar-tukar pendapat tentang kerja kursus bagi memastikan kerja kursus siap dengan sempurna.

14/08/2009 Penyemakan terakhir tentang kerja kursus dilakukan.

17/08/2009 Tugasan dicetak dan dibukukan dengan

3


sempurna

19/08/2009 Menghantar tugasan yang telah lengkap kepada pensyarah pembimbing

Penerimaan tugasan oleh pensyarah.

4



NAMA PELAJAR : 1) SYARAFINA BT MOHD SALIM

NO MATRIK :


SEMESTER : 1











31/07/2009 Mencari bahan mengikut kehendak tugasan bersama ahli kumpulan daripada


5


bahan rujukan di perpustakaan

1/08/2009 dan 2/08/2009












17/08/2009 Tugasan dicetak dan dibukukan dengan sempurna

6




7



NAMA PELAJAR : 1) HAZIRAH BT MOHD ABDUL WAHID

NO MATRIK :


SEMESTER : 1











31/07/2009 Mencari bahan mengikut kehendak tugasan bersama ahli kumpulan daripada bahan rujukan di


8


perpustakaan

1/08/2009 dan 2/08/2009












17/08/2009 Tugasan dicetak dan dibukukan dengan sempurna

9




10


REFLECTION

By : Norjamilah Bt Ab. Rahman

On 21th July 2009, we are given a short course work about “ set ” from our

dedicated lecture, Madam Ng Pek Foong. She gave an excellent explanation to

us how to do this short course work so that this short course work will successful

finished. She divided the students into six groups which contains three member

in each group. We are given three weeks to finish and submit this short cour se

work on 19th August 2009.

According to the explanation given by our lecture, we are supposed to find

out what is the set. We have to do a research about set from many aspects. The

aspects are including history of set, set theory or set concepts and application of

set in our daily life. We try to find as many as possible about what is set through

many ways.

Then, in our group, we discuss how to do this short course work properly.

Based on a good commitment and cooperation giving by all members in our

group, we managed to plan and divided a task to each members in our group.

Through this short course work, we can refresh what we have learnt when

we was in the secondary school. Based on the experiences, we do a research

about set through many resources likes various types of books and a lots of

information from internet.

During finishing this short course work, we were confronted by a little

problem. We have to managed our time properly between study and doing this

short course work. But, thanks to God because of our commitment to our work,

we are finally can overcome this problem.

Then, while doing this short course work, we know our weakness and our

strength about set. After doing a research, we found a lot of information about set

that we do not know before this. So, by doing this short course work, we can gain 11


our knowledge. From the research, we found that set give a big implication in our

daily life.

Finally, after having a had struggle doing this short course work, we are

success to finish it completely. We hope that through this short course work, we

can gain our knowledge about set especially about its history, theory or concepts

and application to our daily life.

12


By : Hazirah Binti Mohd Abdul Wahid

On 21st July 2009, I was given a task on a short coursework about set by my

Mathematics lecturer, Ms. Ng Pek Foong. This task should be done in a group of three

members and submit it on 19th August 2009.We were asked to collect information from

books, journals, and internet about history, theory and application about sets.

After the task was briefing by Ms. Ng Pek Foong, I started to find my group

members. Finally, Syarafina and Jamilah and I decided to become a group and we

agree to take Jamilah as our group leader who will responsible in assigning the tasks

and held group discussion.

Then, we held a group discussion and began to divide the works fairly to each of

the group’s member, I was asked to gather information on application of sets and make

reviews on that topic. Actually, this is my first experience on making a coursework. So,

I’m quite nervous and blur on how to make an excellent and impressive coursework. My

group and I decided to ask everything about this task that we don’t know to Ms. Ng Pek

Foong and she gave a good explanation about the task which helps us a lot.

After that, we started to collect information as many as we can about sets. My

group’s member and I went to the library and it’s a relief that we had found a lot of

books and journals about sets at the library. We collected all of them, photostat, and

made a review. Even, I had to use a lot of money in order to copy the information but it

is worth for me because I want to give the best commitment as I can to this task.

However, we know that there must be a great challenge in making a good thing. In

my case, my group’s member and I had a difficulty in finding information on the internet

because there is no wireless connection in the hostel except at the library.

We had to find it on the computer at the library and save it on the pen drive. This

way is quite hard as the wireless connection is really slow and the computer that is

provided is in a small amount and many students want to use too. Finally we found the 13


solution to this problem and agree to borrow the broadband from the senior to find the

information.

For this problem, I want to suggest the institute to provided the wireless connection

at the hostels as we know that internet is really important for the students to find

information about a lot of thins in order to make it easier for us to do our assignment.

Besides, my group and I also had a problem in managing our time well. There are

a lot of jobs to do and we had to steal sleeping time to finish this coursework. However,

we got a great learning from this experience that is we have to make a timetable from

the beginning to make sure our jobs are done smoothly.

Although, I had to sacrifice my time for rest and sleep, my money and my energy to

complete this task it is worth when eventually I managed to accomplish my goal of

finishing this works. One of the reasons is because I had works with such an amazing

persons, my group members, Syarafina and Jamilah who are ready to give the best

cooperation and go through all of the problems in making this task together with me.

Moreover, I had learnt a lot from my experience in making this task such as we

have to be patient and try to find the solutions when the problems come rather than

make it as an excuse for us. We must also give our best and our commitment when do

a thing so that we will satisfied with our jobs at the end.

14


By : Syarafina Binti Mohd Salim

Thanks to God, after 1month finally, my group and I had finish completely our

mathematic 1 coursework. Since this is my first coursework, I found many problem,

our weakness and also strength during doing it.

First and foremost my group and I faced a little problem when searching and

collect data about the title, sets especially the history of sets. Actually, it is easy to find

the information about sets. Maybe because this is famous and popular topic . the

problem happen when we didn’t focus to the subtopic. However, our senior had help

us a lot. They had show us the actual way in finding data.

After we had collected data, we cant make a review properly because our data is

included somethings which is not important. Fortunately ,our lecturer, madam Ng Pek

Foong had teach us a lot.

Besides that, I found that my group didn’t organized and planned when and how

to do the work properly. We do it suddenly and without plan first. Sometimes it is quit

messy. We also didn’t divide work among the group members. This make some of us

work harder and some also do it but only a little. Finally, we had done the coursework

but only at the last minute.

From the problem I had identified above, I learned that if we get another

coursework in the future, we must plan it first. We also must choose one leader in our

group to make sure the work is always done and to divide work fairly. Even this is a

group work, we should divide work to ensure all members contribute their ideas, times

and energy to get a quality coursework.

Absolutely there is not only the problem and weaknesses I found when spending

my time to do this coursework. I noticed that is a lot of benefits and strength in it. One

of them is I had improved my knowledge about sets. My friends and I had do research

about sets. It is quit exciting because there is a lot of things we don’t know before. For

15


example, we had knew about the history of Geog Cantor, a famous mathematician

which we didn’t ever heard his name before.

The most sweet things is my relationship between me and my group had

become closer because we had spent many time together to discuss about our

coursework. We also had ask a lot question to our lecturer, madam Ng Pek Foong.

This is due to our misunderstanding about the real way to do this coursework. I know

that this is very important so I don’t want we do it completely but wrongly.

Furthermore, because of this coursework I learned that how cooperation is very

important in a group. I also learned how to communicate with teamwork and be

rational. We must discuss between our members to achieve agreement when decide

something.

In conclusion, I found that this coursework is good and I am pleasure to do

another coursework in the future. Thank you.

16


HALAMAN PENGAKUAN“ Saya akui karya ini adalah hasil kerja saya sendiri kecuali nukilan dan ringkasan yang

tiap-tiap satunya telah saya jelaskan sumbernya”

Tandatangan :

Nama penulis: NORJAMILAH BT AB RAHMAN

Tarikh : 18/08/2009

“ Saya akui karya ini adalah hasil kerja saya sendiri kecuali nukilan dan ringkasan yang tiap-tiap satunya telah saya jelaskan sumbernya”

Tandatangan :

Nama penulis: SYARAFINA BT MOHD SALIM

Tarikh : 18/08/2009

“ Saya akui karya ini adalah hasil kerja saya sendiri kecuali nukilan dan ringkasan yang tiap-tiap satunya telah saya jelaskan sumbernya”

Tandatangan :

Nama penulis: HAZIRAH BT ABDUL WAHID

Tarikh : 18/08/2009

17


CONTENTS

CONTENTS PAGE

INTRODUCTION

ACKNOWLEDGEMENT

REVIEW

TOPICS

HISTORY OF SETS

THEORY OF SETS :

-CONCEPT OF SETS

-SUBSETS, UNIVERSAL SETS, AND

COMPLEMENT OF SETS

-OPERATIONS OF SETS

APPLICATION OF SETS

CONCLUSION

BIBILOGRAPHY

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INTRODUCTION

The human mind likes to create collections. Instead of seeing a group of five

stars as five separate items, people tend to see them as one group of stars. The mind

tries to find order and patterns. In mathematics this tendency to create collections is

represented with the idea of a set. A set is a collection of objects. The objects

belonging to the set are called the elements or members of the set.

An example of a set is the set is the set of the days of the week, whose elements

are Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday.

Capital letters are generally used to name sets. Lets use W to represent the set

of the days of the week.

Three methods are commonly used to designated a set. One method is a word

description. We can describe set W as set of the days of the week. A second method is

the roster method. This involves listing the elements of a set inside a pair of braces, { }.

The braces at the beginning and end indicate that we are representing a set. The roster

form uses commas to separate the elements of the set. Thus, we can designated the

set W by listing its elements :

W= { Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

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ACKNOWLEDGEMENT

In the name of ALLAH Most Gracious Most Merciful. Firstly, thank to God

because we are managed to finish this short course work. We would like to say thank

you to our lecturer, Madam Ng Pek Foong for her guidance and advises to complete

this short course work. It help us a lot to complete this work as she wish. Besides, we

also would like to say thank you to our adorable parents for their encouragement,

supportive and understanding with us. Their moral support gives us determination to do

it best of the best. Last but not least, to our fellow friends, thanks a lot for your

commitment and cooperation. We learnt many things through this short course work.

Moreover it gain united among the members in-group because we were work for this

short course work together and finally completed this assignment. We hope our lecturer

will be satisfied with our effort after there are some little problem during finishing our

short course work . Thank you….

Yours faithfully,

Syarafina bt Mohd Salim

Hazirah bt Mohd Abd Wahid

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Norjamilah bt Ab Rahman

21


HISTORY OF SETS THEORY

The history of set theory is rather different from the history of most other areas of

mathematics. For most areas a long process can usually be traced in which ideas

evolve until an ultimate flash of inspiration, often by a number of mathematicians almost

simultaneously, produces a discovery of major importance. Set theory however is rather

different. It is the creation of one person, Georg Cantor(1845-1918) in about 1875.

Some of the things he proved flew in the face of accepted mathematical beliefs of the

times. Cantor created a new field of theory and at the same time continued the long

debate over infinity that began in ancient times. He developed counting by one-to-one

correspondence to determine how many objects are contained in a set. Infinite sets

differ from finite sets by not obeying the familiar law that the whole is greater than any of

its parts.

The set of all the natural numbers could be conceived to be a potentially infinite set,

formed by extending the finite set {1,2,3,…,n}.

But, consider a phrase such as, “the set off all rational numbers”, essential to the

Dedekind definition of real number. We might suppose this, too, is a potentially infinite

set, but how can an existing set be extended so it would include all the rational

numbers? For example, if the set is {1,1/2,1/3,…,1/n}, then any extension of the set

would omit all non unit fractions. Furthermore, while it is obvious whether or not a set

contains all natural numbers, it is not as clear whether a given set will contain all rational

number. It seems that the only way to create the infinite set of all rational numbers is to

begin with the infinite set of all rational numbers. It was questions such as these that

Cantor addressed between 1874, and doing so, Cantor founded modern set theory.

Bolzano was a philosopher and mathematician of great depth of thought. In 1847

he considered sets with the following definition

22

http://www.gap-system.org/~history/Mathematicians/Bolzano.html

http://www.gap-system.org/~history/Mathematicians/Cantor.html


an embodiment of the idea or concept which we conceive when we regard the

arrangement of its parts as a matter of indifference.

Bolzano defended the concept of an infinite set. Bolzano gave examples to show

that, unlike for finite sets, the elements of an infinite set could be put in 1-1

correspondence with elements of one of its proper subsets. This idea eventually came

to be used in the definition of a finite set.

It was with Cantor's work however that set theory came to be put on a proper

mathematical basis. Cantor's early work was in number theory and he published a

number of articles on this topic between 1867 and 1871. These, although of high

quality, give no indication that they were written by a man about to change the whole

course of mathematics.

Cantor moved from number theory to papers on trigonometric series. These papers

contain Cantor's first ideas on set theory and also important results on irrational

numbers. Dedekind was working independently on irrational numbers and Dedekind

published Continuity and irrational numbers.

In 1874 Cantor published an article in Crelle's Journal which marks the birth of set

theory. A follow-up paper was submitted by Cantor to Crelle's Journal in 1878 but

already set theory was becoming the centre of controversy. Kronecker, who was on the

editorial staff of Crelle's Journal, was unhappy about the revolutionary new ideas

contained in Cantor's paper. Cantor was tempted to withdraw the paper but Dedekind

persuaded Cantor not to withdraw it and Weierstrass supported publication. The paper

was published but Cantor never submitted any further work to Crelle's Journal.

In his paper, the one that Cantor had problems publishing in Crelle's Journal, Cantor

introduces the idea of equivalence of sets and says two sets are equivalent or have the

same power if they can be put in 1-1 correspondence. The word 'power' Cantor took

from Steiner. He proves that the rational numbers have the smallest infinite power and

also shows that Rn has the same power as R. He shows further that countably many

23

http://www.gap-system.org/~history/Mathematicians/Steiner.html



http://www.gap-system.org/~history/Mathematicians/Crelle.html




http://www.gap-system.org/~history/Mathematicians/Weierstrass.html


http://www.gap-system.org/~history/Mathematicians/Dedekind.html




http://www.gap-system.org/~history/Mathematicians/Kronecker.html














copies of R still has the same power as R. At this stage Cantor does not use the word

countable, but he was to introduce the word in a paper of 1883.

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.Even Kronecker's critic

his work, Cantor however continued with his work. His fifth work in the six part treatise

was published in 1883 and discusses well-ordered sets. Ordinal numbers are

introduced as the order types of well-ordered sets. Multiplication and addition of

transfinite numbers are also defined in this work although Cantor was to give a fuller

account of transfinite arithmetic in later work. Cantor takes quite a portion of this article

justifying his work. Cantor claimed that mathematics is quite free and any concepts may

be introduced subject only to the condition that they are free of contradiction and

defined in terms of previously accepted concepts. He also cites many previous authors

who had given opinions on the concept of infinity including Aristotle, Descartes,

Berkeley, Leibniz and Bolzano.

Although not of major importance in the development of set theory it is worth noting

that Peano introduced the symbol for 'is an element of' in 1889. It comes from the first

letter if the Greek word meaning 'is'.

In 1885 Cantor continued to extend his theory of cardinal numbers and of order

types. He extended his theory of order types so that now his previously defined ordinal

numbers became a special case. In 1895 and 1897 Cantor published his final double

treatise on sets theory. It contains an introduction that looks like a modern book on set

theory, defining set, subset, etc. Cantor proves that if A and B are sets with A equivalent

to a subset of B and B equivalent to a subset of A then A and B are equivalent. This

theorem was also proved by Felix Bernstein and independently by E Schröder.

The dates 1895 and 1897 are important for set theory in another way. In 1897 the

first published paradox appeared, published by Cesare Burali-Forti. Some of the impact

of this paradox was lost since Burali-Forti got the definition of a well-ordered set wrong!

24

http://www.gap-system.org/~history/Mathematicians/Burali-Forti.html

http://www.gap-system.org/~history/Mathematicians/Burali-Forti.html

http://www.gap-system.org/~history/Mathematicians/Schroder.html

http://www.gap-system.org/~history/Mathematicians/Bernstein_Felix.html




http://www.gap-system.org/~history/Mathematicians/Peano.html


http://www.gap-system.org/~history/Mathematicians/Leibniz.html

http://www.gap-system.org/~history/Mathematicians/Berkeley.html

http://www.gap-system.org/~history/Mathematicians/Descartes.html

http://www.gap-system.org/~history/Mathematicians/Aristotle.html










However, even if the definition was corrected, the paradox remained.In 1899 Cantor

discovered another paradox which arises from the set of all sets. It began to look as if

the criticism of Kronecker might be at least partially right since extension of the set

concept too far seemed to be producing the paradoxes. The 'ultimate' paradox was

found by Russell in 1902 (and found independently by Zermelo).

By this stage, however, set theory was beginning to have a major impact on other

areas of mathematics. Lebesgue defined 'measure' in 1901 and in 1902 defined the

Lebesgue integral using set theoretic concepts. Analysis needed the set theory of

Cantor, it could not afford to limit itself to intuitionist style mathematics in the spirit of

Kronecker. Rather than dismiss set theory because of the paradoxes, ways were sought

to keep the main features of set theory yet eliminate the paradoxes.

Again in 1902 it was mentioned by Beppo Levi but the first to formally introduce the

axiom was Zermelo when he proved, in 1904, that every set can be well-ordered. This

theorem had been conjectured by Cantor. Émile Borel pointed out that the Axiom of

Choice is in fact equivalent to Zermelo's Theorem.

Zermelo in 1908 was the first to attempt an axiomatisation of set theory. Many

other mathematicians attempted to axiomatise set theory. Fraenkel, von Neumann,

Bernays and Gödel are all important figures in this development. Gödel showed the

limitations of any axiomatic theory and the aims of many mathematicians such as Frege

and Hilbert could never be achieved.

From other resources,we find out that at the beginning of his Beiträge zur Begründung

der transfiniten Mengenlehre, Georg Cantor, the principal creator of set theory, gave the

following definition of a set]

By a "set" we mean any collection M into a whole of definite, distinct objects m

(which are called the "elements" of M) of our perception [Anschauung] or of our thought.

In formal mathematics the definition given above turned out to be inadequate;

instead, the notion of a "set" is taken as an undefined primitive in axiomatic set theory,

25

http://www.gap-system.org/~history/Mathematicians/Hilbert.html

http://www.gap-system.org/~history/Mathematicians/Frege.html

http://www.gap-system.org/~history/Mathematicians/Godel.html

http://www.gap-system.org/~history/Mathematicians/Godel.html

http://www.gap-system.org/~history/Mathematicians/Bernays.html

http://www.gap-system.org/~history/Mathematicians/Von_Neumann.html

http://www.gap-system.org/~history/Mathematicians/Fraenkel.html

http://www.gap-system.org/~history/Mathematicians/Zermelo.html


http://www.gap-system.org/~history/Mathematicians/Borel.html





http://www.gap-system.org/~history/Mathematicians/Lebesgue.html


http://www.gap-system.org/~history/Mathematicians/Russell.html




and its properties are defined by the Zermelo–Fraenkel axioms. The most basic

properties are that a set "has" elements, and that two sets are equal (one and the

same) if they have the same elements.

Mathematical topics typically emerge and evolve through interactions among

many researchers. Set theory, however, was founded by a single paper in 1874 by

Georg Cantor: "On a Characteristic Property of All Real Algebraic Numbers".

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. Russell and Zermelo independently found the simplest and

best known paradox, now called Russell's paradox and involving "the set of all sets that

are not members of themselves." This leads to a contradiction, since it must be a

member of itself and not a member of itself. In 1899 Cantor had himself posed the

question: "what is the cardinal number of the set of all sets?" and obtained a related

paradox.The momentum of set theory was such that debate on the paradoxes did not

lead to its abandonment. The work of Zermelo in 1908 and Fraenkel in 1922 resulted in

the canonical axiomatic set theory ZFC, which is thought to be free of paradoxes. The

work of analysts such as Lebesgue demonstrated the great mathematical utility of set

theory. Axiomatic set theory has become woven into the very fabric of mathematics as

we know it today.

26

http://en.wikipedia.org/wiki/Henri_Lebesgue

http://en.wikipedia.org/wiki/Real_analysis

http://en.wikipedia.org/wiki/ZFC

http://en.wikipedia.org/wiki/Abraham_Fraenkel

http://en.wikipedia.org/wiki/Zermelo

http://en.wikipedia.org/wiki/Cardinal_number

http://en.wikipedia.org/wiki/Russell's_paradox

http://en.wikipedia.org/wiki/Zermelo

http://en.wikipedia.org/wiki/Bertrand_Russell

http://en.wikipedia.org/wiki/Paradox

http://en.wikipedia.org/wiki/Georg_Cantor

http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_axioms


Geog cantor bibliography :

Georg Ferdinand Ludwig Philipp Cantor

Actually, Georg inherited considerable musical and artistic talents from his

parents being an outstanding violinist. He has no mathematical blood. In 1862 Cantor

had sought his father's permission to study mathematics at university. His studies at

Zurich, however, were cut short by the death of his father in June 1863. Cantor moved

to the University of Berlin. He spent the summer term of 1866 at the University of

Göttingen, returning to Berlin to complete his dissertation on number theory De

aequationibus secondi gradus indeterminatis in 1867.

In 1873 Cantor proved the rational numbers countable, i.e. they may be placed in

one-one correspondence with the natural numbers. He also showed that the algebraic

numbers, i.e. the numbers which are roots of polynomial equations with integer

coefficients, were countable. However his attempts to decide whether the real numbers

were countable proved harder. He had proved that the real numbers were not countable

by December 1873 and published this in a paper in 1874. It is in this paper that the idea

of a one-one correspondence appears for the first time, but it is only implicit in this work.

At the end of May 1884 Cantor had the first recorded attack of depression. He

recovered after a few weeks but now seemed less confident. Mathematical worries

began to trouble Cantor at this time, in particular he began to worry that he could not

prove the continuum hypothesis, namely that the order of infinity of the real numbers

27

http://info.math.nankai.edu.cn/navigate/math/history/PictDisplay/Cantor.html


was the next after that of the natural numbers. In fact he thought he had proved it false,

then the next day found his mistake. Again he thought he had proved it true only again

to quickly find his error.

In 1886 Cantor turned from the mathematical development of set theory towards

two new directions, firstly discussing the philosophical aspects of his theory with many

philosophers (he published these letters in 1888) and secondly taking over after

Clebsch's death his idea of founding the Deutsche Mathematiker-Vereinigung which he

achieved in 1890. His last major papers on set theory appeared in 1895 and 1897,

again in Mathematische Annalen under Klein's editorship, and are fine surveys of

transfinite arithmetic. He hoped to include a proof of the continuum hypothesis in the

second part. However, it was not to be, but the second paper describes his theory of

well-ordered sets and ordinal numbers.

In 1897 Cantor attended the first International Congress of Mathematicians in

Zurich. By the time of the Congress, however, Cantor had discovered the first of the

paradoxes in the theory of sets. He discovered the paradoxes while working on his

survey papers of 1895 and 1897 and he wrote to Hilbert in 1896 explaining the paradox

to him. Burali-Forti discovered the paradox independently and published it in 1897.

Cantor began a correspondence with Dedekind to try to understand how to solve the

problems but recurring bouts of his mental illness forced him to stop writing to Dedekind

in 1899.

Whenever Cantor suffered from periods of depression he tended to turn away

from mathematics and turn towards philosophy and his big literary interest which was a

belief that Francis Bacon wrote Shakespeare's plays. He did continue to work and

publish on his Bacon-Shakespeare theory and certainly did not give up mathematics

completely. He lectured on the paradoxes of set theory to a meeting of the Deutsche

Mathematiker-Vereinigung in September 1903 and he attended the International

Congress of Mathematicians at Heidelberg in August 1904.

28

http://info.math.nankai.edu.cn/navigate/math/history/Mathematicians/Dedekind.html

http://info.math.nankai.edu.cn/navigate/math/history/Mathematicians/Dedekind.html

http://info.math.nankai.edu.cn/navigate/math/history/Mathematicians/Burali-Forti.html

http://info.math.nankai.edu.cn/navigate/math/history/Mathematicians/Hilbert.html

http://info.math.nankai.edu.cn/navigate/math/history/Mathematicians/Klein.html

http://info.math.nankai.edu.cn/navigate/math/history/Mathematicians/Clebsch.html


Cantor retired in 1913 and spent his final years ill with little food because of the

war conditions in Germany. A major event planned in to mark Cantor's 70 th birthday in

1915 had to be cancelled because of the war, but a smaller event was held in his home.

In June 1917 he entered a sanatorium for the last time. He died of a heart attack.

29


SET CONCEPTS

THEORY ABOUT SET

-Set theory is the branch of mathematics that studies sets. Although any type of object

can be collected into a set, set theory is applied most often to objects that are relevant

to mathematics.

Definitions :

1) George Cantor

By a “set”, we mean any collection M into a whole of definite, distinct object M

(which are called the “elements” of M) of our perception (Anschauung) or of our

thought.

2) A set is a collection or group of objects whose contents can be clearly

determined. The objects in a set are called the elements or members of the set.

Set of numbers :

Natural or counting number {1, 2, 3, 4, …..}

Whole numbers {0, 1, 2, 3, 4, …..}

Rational numbers { p / q \ p and q are integers, and q ≠ 0}. Some examples of

rational numbers are 3/5, -7/9, 5, and 0.

Real numbers { x | x is a number that can be written as a decimal}

Irrational numbers { x | x is a real number and x cannot be written as a quotient

of integers}.

30


(A)SET USING THREE DESIGNATIONS

The are three methods which commonly used to designate a set.

Sets can be designed by word descriptions, the roster method ( a listing with

braces, separating elements with commas), or set-builder notation.

WORK DESCRIPTION ROSTER METHOD SET-BUILDER NOTATION

MEAN: describe the set MEAN: listing the elements

inside a pair of braces

MEAN: set is expressed as

W= { x | condition (s) }

Exp 1: B is the set of

members of the Beatles

in 1963

Exp1:B={George Harrison,

John Lennon, Paul

McCartney, Ringo Starr}

Exp 1:B={ x| x is a member of

the Beatles in 1963

Exp 2: S the set of states

whose names begin with

the letter A

Exp 2: S={ Alabama,

Alaska, Arizona, Arkansas}

Exp 2:S={ x| x is a state whose

name begins with the letter A}

Exp 3: P is the set of the

first five presidents of the

United States.

Exp 3:P={ Washington,

Adams, Jefferson, Madison,

Monroe }

Exp 3:P={ x | x is a state

whose was the first five

president of the USA}

Table 1 : Methods used to designate set

31


(B) ELEMENTS OF SET

The symbol ∈ means that an object is an element of a set. The symbol ∉ means

that an object is not an element of a set.

Adapted from: thinking mathematically

A set must be well defined and each element in the set must satisfy the

conditions given in the definition of the set.

If set A is defined as the set of insects, then an ant is an elements of set A

because an insect is defined as a small creature with six legs and a body

divided into three parts. We abbreviate the phrase “is an elements of” or “is a

member of” by using the Greek epsilon, ∈. Hence, “ant ∈ A”.

On the other hand, the symbol is used to denoted the phrase “ is not an ∉

elements of”. Hence, ‘spider A’.∉

Example 1: If A={factors of 30} and B={prime numbers less than 18}, then

(1) 6 ∈ A

(2) 15 ∉ B

Example 2: Complete each of the following statements using the symbol ∈ or ∉

(1) U (∈) {vowels in the English alphabet}

(2) Putrajaya ( ) {states in Malaysia}∉

(3) Rhombus (∈) { regular polygon}

(4) Mercury (∈) { metals that are in liquid state at room temperature}

(5) Black ( ) {one of the major colours in a rainbow}∉

(C) VENN DIAGRAM

32


-Firstly, Venn Diagrams are developed by an English logician, John Venn (1834-

1923). In these diagrams, sets are represented by ovals, circles and other

geometrical shapes.

Besides the methods of description and set notation, we can use geometrical

diagrams to represents sets. The closed geometrical shapes such as ovals and

circles that are used to represent sets are called Venn Diagrams.

There are some cases of Venn Diagram such as disjoint set, subsets, equal sets

and overlapping set.

Ɛ

Case 1 : Disjoint set

-No A are B

-A and B are disjoint

-When set A and B are disjoint, they have no elements in common.

Ɛ

33


Case 2 : Subsets

-All A are B

-A ⊂ B

-When A ⊂ B ,every elements of set A is also an element of set B

Ɛ

Case 3 : Overlapping set

-Some ( at least one ) A are B

Ɛ

34


Case 4 : Equal set

-A and B are equal set.

-When set A = set B, all the elements of set A are elements of set B

and vice versa.

35


(D) NUMBER OF ELEMENTS

The notation n(P) denotes “ the number of elements” in set P

P

n(P)=5

In a Venn Diagram, a number without dot beside it denotes the number of

elements in the set A. Hence, we say that n(A) = 6.

A

Example 1: For each the following sets, find the number of elements

(a) E = { states in Malaysia whose names begin with the letter P }

So, n(E) = 4

(b) F = { factors of 100 }

So, n(F) = 9

(c) G = { x : x is a prime number between 10 and 40 }

So, n(G) = 8

36


(E) NULL OR EMPTY SET

Some sets do not contains no elements is called the empty set or null set and is

symbolized by { } or ø

Note that { ø } is not the empty set. This set contains the elements ø and has a

cardinality of 1. The set { ø } is also not the empty set because it contains the

elements 0. It also has a cardinality of 1.

Thus, n(ø) = n({ }) = 0

Example 1: Determine whether each of the following sets is an empty set

(a) P = { cuboid with four surfaces }

So, P is an empty set / P= ø

(b) Q = { x : x is a root of x2 – 3x – 4 = 0 and x ˃ 0 }

So, Q is not an empty set / Q ≠ ø

(c) R = { integer between 9 and 10 }

So, R is an empty set / R= ø

(d) S = { x : x is a prime number and 83 ˂ x ˂ 89

So, S is an empty set / S= ø

(e) T = { x : x is a prime number and also an even number }

So, T is not an empty set / T≠ ø

(f) U = { quadrilaterals having five vertices }

So, T is an empty set / U= ø

37


(F) EQUAL SETS

Set A is equal to set ,symbolized by A=B, if and only if set A and set B contain

exactly the same elements.

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

Then, A=B because they contain exactly the same elements

Note :The order of elements in sets is not important. If two sets are equal, both

must contain the same number of elements.

Ɛ

Equal set where set A = set B

Exp 1 : Determine whether the sets in each of the following pairs are equal

(a) P = { 1, 2, 3 } and Q = { 2, 3, 4 }

Answer : P ≠ Q

(b) A = { 1, 5, 7 } and B = { digits in the number 755117 }

Answer : A = B

(c) K = { 0, 1 } and L = { x : x is an integer that satisfies the equation x2 –x=0}

Answer : K = L

38


SUBSET, UNIVERSAL SETS AND COMPLEMENT OF

SET

(A)SUBSETS

Sets A is a subset if every elements of set A is also an elements of set B

The symbol for “ is a subset of” is ⊂. Hence, A⊂ B.

The notation A ȼ B means that set A is not a subset of set B, so there is at

least one elements of set A that is not an elements of set B.

The empty set is a subset of every set.

Ɛ

Subset where set A ⊂ B Example : State whether each of the following statements is true or false

(a) { blue } ⊂ { major colour of rainbow }

-True

(b) { multiples of 3 } ȼ { multiples of 6 }

-True

(c) { 2, 3 } ȼ { prime number }

- False

39


(B)UNIVERSAL SET

A set which contains all elements under discussion is called the universal

set.

The symbol Ɛ denotes a universal set.

Ɛ

Universal set where A U B U C

Exp : Determine whether each of the following sets can be a universal set of

{ 1, 2, 4 }

(a) { 0, 1, 2, 3 } Cannot

(b) { 1, 2, 3, 4, 5 } Can

(c) { factor of 6 } Can

(d) { multiples of 2 } Cannot

A

CB

40


(C)COMPLEMENT OF A SET

The complement set A, symbolized by A’, is the set of a all the elements in

the universal set that are not in set A.

Complement: things that are not in A

Note: The shaded region outside of set A within the universal set represents the

complement of set A or A’.

Example : Given that the universal set Ɛ = { 1, 2, 3, 4, 5, 6, 7, 8 }. Find the complement

each of the following sets.

(a) R = { prime number }

So, R = { 2, 3, 5, 7 }, therefore R’ = { 1, 4, 6, 8 }

(b) S = { positive number less than 6 }

So, S = { 1, 2, 3, 4, 5 }, therefore S’ = { 6, 7, 8 }

41

http://en.wikibooks.org/wiki/File:VennComplement.jpg

http://en.wikibooks.org/wiki/File:VennComplement.jpg


OPERATION OF SETS

(A) INTERSECTION

The intersection of sets A and B, written as set A n B, is the set of

elements which are common to both sets A and B

Intersection of A and B

Note : The overlapping regions in above figure represents the intersection of

sets A and B

BA

42


Example : Find the intersection of the following pairs of sets.

(a) A = { d, e, f, g, h } and B = { e, g, k, m, n } So, A n B = { e, g }

(b) P = { 3, 4, 5 } and Q = { 1, 2, 3, 4, 6, 12 } So, P n Q = { 3, 4 }

(c) E = { p, L, R, H } and F = { R, m, O, S } So, E n F = { R}

(B)RELATIONSHIP BETWEEN A n B, A AND B

We have learnt that empty set ø is a subset of any set. Therefore, for any

two disjoint sets A and B where ( A n B ) = ø , ( A n B ) A and ( A n B ) ⊂ ⊂

B.

For any two sets A and B, ( A n B) ⊂ A and (A n B) ⊂ B

The set which does not contain the contain the elements of A n B is called

the complements of the intersection of set A and B is denoted as (A n B)’.

43


(C)UNION OF SETS

The word union means to unite or join together as in marriage, and that is

done when we perform the operation of union.

Each member of the combined set is a member of set A or set B or both.

We call this set the union of sets A and B.

The union of set A and B, written as set A U B, is the set all elements belonging

to either of the sets.

A U B

Note: The three shaded region together represent the union sets A and B

Example : Given Ɛ = { 1, 2, 3, 4, 5, 6 ,7 ,8 ,9 }, P = { 2, 5, 8, 9 }, Q = { 1, 5, 6 , 7,

8 } and R = { 2, 4, 6 , 8}, find the following sets.

(a) P U Q

= { 1, 2, 5, 6, 7, 8, 9 }

(b) P U Q’44

http://en.wikipedia.org/wiki/File:Venn0111.svg


= { 2, 3, 4, 5, 8, 9 }

(c) P U Q U R

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

(D)RELATIONSHIP BETWEEN A U B, A AND B

We have learnt that union of two sets is the set of all elements belonging to

both or either of the sets. Now, we shall determine the relationship between

sets A U B, A and B.

For any two sets A and B , A ⊂ (A UB) and B ⊂ (A U B)

The set which does not contain the elements of A U B is called the

complements of the unions of set A and B and is denoted as (A U B)’

45


APPLICATION OF SETS

Scientist use sets to classify and categorize knowledge. In biology, the science of

classifying all living things is called taxonomy. Over 2000 years ago, Aristotle formalized

animal classification with his ‘’ladder of life’’: higher animals, lower animal, higher plants

and lower plants.

Contemporary biologists use a system of classification called the Linnaen

system, named after Swedish biologist Carolus Linnaeas (1707-1778). The Linnaean

system starts with the smallest unit (member) and assigns it to a specific genus (set)

and species (subset).

In addition, a financial planning company uses the Venn diagram to categorize

the financial planning services the company offers such as investment, retirement and

college planning. From the diagram, we can see that the company offers financial

planning in an ‘’intersection’’ of the areas investment, retirement and college planning.

We categorize items on a daily basis, from filing items to planning meals to

planning social activities. Children are taught how to categorize items at an early age

when they learn how classify items according to color, shape, and size. Biologists

categorize items when they classify organisms according to shared characteristics. A

Venn diagram is a very useful tool to help order and arrange items and to picture the

relationship between sets.

Moreover, sets can be used by the students, worker and any centre of education

in order to summarize data or a pictorial presentation to make it easier to understand.

46


For example, when a survey is carry out to learn whether a pregnant mother’s status as

a smoker or non smoker affects whether she delivers a low or normal birth weight baby.

We can see that there are four combinations of birth weight and smoking status,

so we can use a Venn diagram with two overlapping circles (creating four regions on the

diagram).The Venn diagram makes it easy to see how smoking affected babies in

study. Notice that normal birth weight babies were much more common than low birth

weight babies among both smokers and nonsmokers.

However, the smoking mothers had a low proportion of normal birth weight

babies and a higher proportion of low birth weight babies. This suggest that smoking

increases the risk of having a low birth weight baby, a fact that has been borne out by

careful statistical analysis of this and other studies.

Furthermore, we tend to place things in categories, allowing us to order and

structure the world. For example, we can categorize ourselves by our age, ethnicity,

academic in major, hobby, height, weight, country, and our gender.

One of the main applications of set theory is constructing relations. A relation

from a domain A to a codomain B is a subset of the Cartesian product A × B. Given this

Set theory is seen as the foundation from which virtually all of mathematics can be

derived. For example, structures in abstract algebra, such as groups, fields and rings,

are sets closed under one or more operations.

Concept, we are quick to see that the set F of all ordered pairs (x, x2), where x is

real, is quite familiar. It has a domain set R and a codomain set that is also R, because

the set of all squares is subset of the set of all reals. If placed in functional notation, this

relation becomes f(x) = x2. The reason these two are equivalent is for any given value, y

that the function is defined for, its corresponding ordered pair, (y, y2) is a member of the

set F.

Nearly all mathematical concepts are now defined formally in terms of sets and

set theoretic concepts. For example, mathematical structures as diverse as graphs,

manifolds, rings, and vector spaces are all defined as sets having various (axiomatic)

47


properties. Equivalence and order relations are ubiquitous in mathematics, and the

theory of relations is entirely grounded in set theory.

Set theory is also a promising foundational system for much of mathematics.

Since the publication of the first volume of Principia Mathematica, it has been claimed

that most or even all mathematical theorems can be derived using an aptly designed set

of axioms for set theory, augmented with many definitions, using first or second order

logic (see Metamath). For example, properties of the natural and real numbers can be

derived within set theory, as each number system can be identified with a set of

equivalence classes under a suitable equivalence relation whose field is some infinite

set.

Set theory as a foundation for mathematical analysis, topology, abstract algebra,

and discrete mathematics is likewise uncontroversial; mathematicians accept that (in

principle) theorems in these areas can be derived from the relevant definitions and the

axioms of set theory. Few full derivations of complex mathematical theorems from set

theory have been formally verified, however, because such formal derivations are often

much longer than the natural language proofs mathematicians commonly present. One

verification project, Metamath, includes derivations of more than 10,000 theorems

starting from the ZFC axioms and using first order logic.

48


Another application of set theory in our daily life is shown in the example below :

Blood types

Human blood is often classified according to whether three antigens A,B, and Rh

are present or present. Blood type is stated first in terms of the antigens A and B : Blood

containing only A is called type A, blood containing only B is called type B, blood

containing both A and B is called type AB, and blood containing neither A nor B is called

type O. The present or absence of Rh is indicated by adding the positive (present) or

negative(absent) or its symbol. Table 1 shows the eight blood types that result and the

percentage of people with each type in the U.S. population.

Blood Type Percentage of Population

A positive 34%

B positive 8%

AB positive 3%

O negative 35%

A negative 8%

B negative 2%

AB negative 1%

O negative 9%

49


Table 1

Solution

We can think of the three antigens as three sets A, B and Rh (positive) and draw

a Venn diagram with three overlapping circles. Figure below shows eight regions each

labeled with its type and percentage of the population. For example, the central region

corresponds to the presence of all three antigens (AB positive), so its labeled 3%. You

should check that all eight regions are labeled according to data from Table 1.

B

O A

AB%

3%

B+ 8%

B-

2%

AB-

1%

34%

50


CONCLUSION

Based on the research of set, we found that set is the branch of mathematics

that studies sets, which are collections of objects. Although any type of object can be

collected into a set, set is applied most often to objects that are relevant to

mathematics.

Set theory, however was founded by a single paper in 1874 by Georg Cantor :

“ On a Characteristic Property of All Real Algebraic Numbers”

There are a few section in set like elements of set, null or empty set, venn

diagram, number of elements and equal set. There are also has subsets, universal set,

complement of set, intersection and union of sets.

Finally, in this research, it is found that scientists apply this study of set to

classify and categorize knowledge. In biology, the science of classifying all living thing is

called taxonomy. A financial planning company uses the venn diagram to categorize the

financial planning services the company offers

51


BIBLIOGRAPHY

Robert Blitzer (2003,2000) : Thinking Mathematically , Upper Saddle River

New Jersey : Prentice Hall.

Jeffrey Bennett and William Briggs (2008) : Using and Understanding

Mathematics A Quantitative Reasoning Approach , New York : Pearson

Addison Wesley.

Charles D. Miller and Vern E. Heeren (2004) : Mathematical Ideas, United

States of America : Pearson Addison Wesley.

Jeff Suzuki (2002) : A History of Mathematics Upper Saddle River New

Jersey : Prentice Hall

Angel, Abbot, Runde (2005) A Survey Of Mathematics With Applications –

7th edition. United States America : Greg Tobin

http://en.wikipedia.org/wiki/Set_(mathematics)

www.gap-system.org/~history/.../Beginnings_of_set_theory.html

en.wikibooks.org/wiki/Discrete_mathematics/Set_theory

52

http://www.gap-system.org/~history/.../Beginnings_of_set_theory.html

http://en.wikipedia.org/wiki/Set_(mathematics)

concept, theory n application of sets

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