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8-1-1963

An introduction to the theory of sets and functionsCarlos H. VernonAtlanta University

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Recommended CitationVernon, Carlos H., "An introduction to the theory of sets and functions" (1963). ETD Collection for AUC Robert W. Woodruff Library.Paper 1362.

AH INTRODUCTION TO THE THEORY OF SITS AND FUNCTIONS

A THESIS

SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY

II PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MUSTER OF SCIENCE

BY

CARLOS H. VEHNOH

DEPARTMENT OF MILTHEMATIOS

ATLANTA, GEORGIA

AUGUST 1963

ACKNOWLEDGEMENT

I wish to express my sincere thanks to Dr. lonnie Gross

for his inspiration and critical suggestions.

ii

SYMBOLS

a6l (read "a is an element of the set A")

I ...\ (denotes a set consisting of the ele

ments ...)

\ (read "where")

a =%b (read "a implies b")

b (read "a is implied by b")

(read "a if and only if b")

&) (S) (read "the set consisting of all sub

sets of the set S")

ACB (read "A is contained in B")

ADB (read "A contains B")

A1 (read "the complement of A")

A u B (read "A union B")

A f» B (read "A intersection B")

f : a—Vb (read "under the mapping f, a_ goes into

f(a)= b (read "f at a equals b")

iii

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENT ia-

SYMBOLS Hi

Chapter

I. INTRODUCTION 1

II. PROPERTIES OP SETS 4

Sets 4Notation 5Subsets 6Equality of Sets 7On the Word "Abstract" 7

III. OPERATIONS 01 SETS 10

Complementation . . 10One-to-one Correspondences 10

Union and Intersection of Sets 11

I?. NUMBERS 14

Cardinal Number of a Set ......... 14

7. PROPERTIES OP FUNCTIONS 17

Functions 17Equality of Sanctions 19

Sequentially Applied Functions ...... 20

Inverse Functions 22Graph of a Function 22Finding the Range and Domain of a Contin

uous Function by a Graph 25

BIBLIOGRAPHY 27

iv

OH&PTER I

IUTRODUCTIOI

How old is mathematics? Mathematics is as old as man.

However, the age of man has been a subject of much specu

lation. Most scientists agree that the birthplace of man

was Africa and that centuries ago Africa was the seat of

all world progress. Contrarily, there have been scientists

who placed the origin of man in Asia. These scientists

were forced to change their minds because of discoveries

made in South Africa since 1925. Nevertheless, it was not

until 1959 that an archeological discovery of a human fossil

was made in Oldoway, Tanganyika. This fossil, called

Zinjanthropus, provided concrete evidence that the existence

of man on the face of the earth could be traced back as far

as one and three quarter million years ago (1,750,000).

Zinjanthropus is considered to be a man because he

was making tools to a regular and set pattern. Thus, it

was in Africa, because of the Zlnjanthropus, that the first

evidence of making tools to a set and regular pattern was

found. Although this evidence appears insignificant, it

marks a great step taken by man toward progress. It means

that man now was able to better defend himself and was able

to secure food from media that were not open to him before.

1

2

He now did not have to be strictly a vegetarian nor was he

confined to scavenging on parts of animals that had been

killed by truly carnivorous creatures.

Africa, the birthplace of jZinJantrojjus, also repre

sents the birthplace of mathematics, for the oldest evidence

of a number system was found there. This evidence was found

at Ishango in the Congo and has "been traced back to some

time between 9,000 B.C. and 6,500 B.C. In examination of

these findings reveals that some of the artifacts of the

Ishango man bear markings arranged in groups of notches

of three distinct columns. These notches are Indicative

of something more than decoration. In one column we find

four distinct groups of 11, 13, 17, and 19 individual

notches. In a second column we find eight distinct groups

of 3, 6, 4, 8, 10, 5, 5, and 7 notches. In the third

column there are four groups of 11, 21, 19, and 9 notches.

It is evident, by the arrangements of the notches in

groups, that the Ishango man had a well developed know

ledge of numbers and relations between numbers. Examining

column one, we see that it is composed of the numbers

11, 13, 17, and 19. These numbers are all prime and,

more precisely, the only primes between 10 and 21. The

second column shows a knowledge of multiplication by two.

Finally, examination of the third column reveals a definite

knowledge of a number system to the base 10, for 11 = 10 -+- 1;

21 = 10 +10-H 1; 19 =10 + 10 — 1. After the ishango

3

man,, the next mathematicians to appear were the Egyptians.

Prom the time of the Egyptians up to the present day the

history of mathematics is well known.

CHAPTEE II

PROPEHTIES OF SETS

Sets.--Although the word "set" is relatively new, the

idea of sets is very ancient. This idea can be traced back

to the beginning of the existence of man. The principle

of sets begins with the ability to group a collection of

items together and is a quality not peculiar to man. for

example, birds collect twigs to build a nest; ants collect

food for the winter, beavers collect twigs to make a dam.

Although in many cases, the collection of objects is a

result of instinctive ability, it is the basis for forming

sets. A set may be defined as follows:

Definition: A set is a collection of well distinguishable

objects.

A set is denoted by capital letter, i.e., A,B,C,....

The objects which constitute a set are called elements

or members of the set. The elements or members of a set

are denoted by small letters, i.e., a, b, c,....

There are two methods used to describe a set. The

first consists in naming each element of the entire set.

This method is called the roster method.

Example: Let the letters a, b, and c, be the elements

of a set, say A.

5

The second method used in describing a set is to des

ignate the members of a set by a general rule.

Example i Let one be interested in grouping the males

in the United States. This would be a very tedious Job

if an attempt were made to group all males in the United

States. If, instead, one were to let a set, say A, be the

set of all males in the United States, this designation

would be as accurate as listing all male individuals.

Moreover, on certain occasions, it is more accurate to

designate a set by a rule. Let us take the above example

of males. By the time one would have listed all males

in the United States, some that have already been listed

would have died and hence, give rise to an inaccurate

listing of the set. However, by the rule, when a person

dies, he automatically drops from the set.

Another example of the superior accuracy of listing

sets by a rule is that of being able to list every element

of the set. If one were to try to list a set containing

an infinite amount of elements, he could never do it.

Example: The set consisting of all real numbers

between 1 and 2.

dotation..—How that the idea of set is understood, an

attempt can be made to express a set in terms of symbols.

Before focusing our attention on the notation of sets, it

Is necessary to show that the definition of a set is not

unique,- i.e., a set can be defined in several ways. Con

sider a set whose elements are all four-footed animals.

6

The set would be the same if the members were considered to

be quadrupeds.

Another example would be to consider a set whose elements

are x2. The elements would be the same if they were equal to

Let A be a set, and a an element of the set A. Then

'is an element of

means that a is a member of V the set A.

kbelongs to

The set !••■•! denotes a set consisting of the elements....

The set consisting of no elements is called the null or empty

set and is denoted by 0.

Example: A=^a,b,c}. This is read the set A consisting

of the elements a,b, and £.

A set may also be denoted B =■ ^x\x e J+j. This set is

read the set B whose elements are all x where x belongs to

J+ (positive integers). Whatever follows the vertical bar

(I) inside the braces is called the definition of the set.

Subsets.—-From any given set it is possible to form

another set. If a set, say A, contained the elements a, b,

and £, by talcing a and b, or £ and a we are able to form sets

from the given set A. The set formed from a given set is

known as a subset and is defined as follows:

Definition? A set S is called a subset of a set A, if and

only if, each element £eS, is also an element of A.

In symbolic form:

sci

7

We denote this by ScA (read"S, is contained in A") or A^S

(read"A contains S").

If a set contains a finite number of elements, then the

process of forming distinct subsets of the original set Is

limited,i.e., only a finite number of subsets can be formed

from any given finite set. The set of all subsets of a given

set is denoted by£9 (S) and is read the set consisting of all

subsets of the set S3.

Example; Let S = (1,2,3} • Then

ft (S)=^J,(i},$2],$ .(2,3}. fc,i}, ^1.3}. (1.2.3)}.Mote J_: The order in which the elements are situated in a

given set is immaterial. Thus, in dealing with a

set, the alteration of positions does not change

the set in any way.

Equality of Sets.,—

Definition; Two sets A and B are equal, if and only if,

they contain exactly the same elements. In symbolic form:

JL= B4=4 x * A ■=). x i B and £ t B =4. y_ e A.

From this definition we have that two sets A and B are equal,

if and only if, A is a subset of B and B is a subset of A.

In symbols:

i = B^=>AcB and BCA.

Example: Let A=(m,i,s,s,i,s,s,i,p,p,i} and B=(m,i,s,p}.

. . since A_cB and Be A, A = B.

On the Word "Abstract".—The word "abstract" has been

the subject of much misinterpretation and- as a result has

caused a great deal of unjust fear of mathematics. It is

8

true that a great deal of mathematics is based on abstractions.

The difficulty lies, however, in the fact that the word "ab

stract" has been given too many meanings.

Some people, when they hear the word "abstract," immedi

ately think about "something spookish," others about "some

thing that cannot be seen." These people, unfortunately,

form the vast majority of those who come in contact with

mathematics. In reality, the interpretations of these people

are far from the real meaning. This misinterpretation is

a problem that has long plagued mathematics. In order to

eliminate the plague, we shall try to approach the word from

its origin.

The word "abstract" is an adaptation of the Latin word

abstraotus to the English language. Abstractus is a compound

word made up of abs_ meaning "off," "away;" and tractus which

is the past participle of trahere meaning "to draw."

"Abstract" in English, like abstractus in Latin, means

"drawn from." In mathematics it is used as a substantive

denoting the process of. drawing from a group that which is

common to the entire group. This is the only meaning that

will be assumed by the word "abstract" in this text.

Let us consider an example. Suppose we had a piece of

coal, a piece of tar, some crude oil, a panther, and some

smut. Certainly this collection is very odd, but carefully

observing it, we would find that there is one characteristic

which they all share with each other and many, many other

9

objects. This characteristic is that of "blackness."

Since "blackness" is the common characteristic drawn

from the entire group,1 "blackness1' is an abstraction. The

object from which an abstraction is made is called the con

crete representation of an abstraction. With this in mind

we can say that tar is a concrete representation of "black

ness. " .

CHAPTER III

OPERATIONS ON SETS

Oomplementation.—Before speaking of complementation

of sets it is necessary to define a universal set.

Definition: A universal set, denoted by E or by U, is the

set consisting of all the elements which are to be consi

dered in a particular discussion.

Example; If in our discussion we were considering only

the chairs of a particular school, then the set of all chairs

that are in the school is the universal set.

Now that the idea of a universal set is clear, an attempti

to define the complement of a set may be made.

Definition: The complement of A,, written <jA or A1, with

respect to the universal set, Is. the set of all elements

which do not belong to A, but belong to the universal set.

In symbols:

This could also be expressed as follows:

A== A* = S—A=XxlxeS, xi k\ .

One-to-one Correspondences.—Although the term "one-to-

one correspondence" is a relatively new name for an old

association, this association is as old as the beginning of

10

11

time. In the very early days , if a man had more than one

wife and wanted to check each night to see if all were

accounted for, he would make marks on the side of his cave

and associate a given wife with each mark. If, when making

his nightly check, he would find that there was a wife corre

sponding to each mark, he knew that they were all present.

Moreover, if there were more or less wives than the amount

of marks this would tell him that there was an additional

woman or a missing woman respectively (fig. 1).

a wife missingAll wives present

one wife too many.

Figure 1

The process of pairing one object with another one

became known as a one-to-one correspondence and may be

defined as follows:

Definition: Let A and B be two sets. The set A and B are

in one-to-one correspondence, if and only if, to each element

a_€A there exists one and only one element beB which corre

sponds to a and vice versa.

Union and Intersection of Sets.—Consider a town in which

no two distinct persons have the same name. If in that town

there were a baseball team consisting of the players Prank,

12

Joseph, Harry, James, Leroy, Egbert, Jack, Charles, and

Anthony; and a basketball team made up by Ronald, Vincent,

Harry, Leroy, and Teddy. Now, if one were to combine the

players of both teams to form a socoer team, then the soccer

team would contain the players Frank, Harry, James, Leroy,

Egbert, Jack, Joseph, Charles, Anthony, Ronald, Yincent, and

Teddy. The combining of the basketball and baseball teams to

form the soccer team is known as the union of the baseball

and basketball teams. In general, the union of sets is de

fined as follows:

Definition; Let A and B be two sets. Aw B (read "A union B")

is the set consisting of all the elements which are either

in set A or set B or in both.

In symbols:

Let A=* { a,b,c,d \ and B= \a,b,c] .

Then AuE =[a,b,c,d| .

A more general notation is

Avj B = ^x\xg A or x e b] .

fote ±: It should be observed that when the combining of the

baseball and basketball teams was made, the players

Harry and Leroy appeared only once. Hence, if an ele

ment occurs more than once in a set, it is written

only once.

How consider the same baseball and basketball teams

above. If one were interested in forming a ping-pong team

made up by the players that are common to both teams, the

team would contain only Harry and Leroy. This ping-pong

team formed by the players that are common to both teams is

called the intersection of the baseball and basketball teams.

In general, the intersection of sets is defined as follows:

Definition: Let A and B be two sets. An B (read "A inter

section B") is the set consisting of all the elements which

are in both set A and set B.

In symbols s

Let A= X a,b,c,dj and B =ia,b,c\ .

Then An B== {a,b,c} .

A more general notation is

A o B =( x |x e A and x e

CHAPTER IV

NUMBERS

Cardinal limber of a Set.—During the early years of

life, one begins to become familiar with the process of

counting. In this process one is concerned chiefly with enu

merating objects. When a set has a single element, the num

ber of elements of the set is said to be one; if the set has

one and only one more element than one element, the number

of the elements is called two; if the set has one and only

one more element than two elements, the number of the ele

ments Is called three. This process can be continued until

every finite set is numbered. Thus, it is possible to define

the cardinal number of a set as follows:

Definition _Is One (1) is the cardinal number of a set which

contains one and only one element.

Two (2) is the cardinal number of a set which

contains two and only two elements.

Three (3) is the cardinal number of a set which

contains three and only three elements.

In general, a number n is the cardinal number

of a set which contains n and only n elements.

However, 1, 2, 3, ... » n, ... are abstractions made

from sets having the same number of elements. Hence, a

14

15

cardinal number can also "be defined as follows:

Definition II: The cardinal number of a set or number of a

set is an abstract property which this set shares with all

sets which can be put into one-to-one correspondence with it.

The cardinal number©?, 2, 3, ..., n, ... which are ob

tained by continuing the process in definition I are called

natural numbers.

How it can be observed that every element, except the

first has a successor, i.e., given the first natural number

1, it is possible to obtain all other natural numbers by re

peatedly adding one to each succeeding cardinal number. This

process leads us to a very interesting and important princi

ple, the Principle of Induction. By way of this principle,

if a statement is true for every natural number n, then it

must be true for n -+- 1; and if it is true for an arbitrary

natural number k, it must also be true for the succeeding

natural number k -f- 1. Formally the principle of induction

states:

If Sn is a statement associated with each natural num

ber n, then Sn is true for each natural number, if the fol

lowing holds:

i) S1 is true.

11) If k is an arbitrary natural number such that Sk is

true, then S^^ j is also true.

The set of natural numbers Is infinite. This means that

the set contains no definite amount of elements. Generally

sets can be divided into two classes: finite and infinite.

16

In attempting to define finite, and Infinite sets, we

shall define one, and all sets that do not satisfy the given

definition shall "be placed in the other set. Hence, we may-

define an infinite set and define a finite set to be a set

that is not infinite, or we may define a finite set and de

fine an infinite set to "be a set that is not finite.

We shall choose to define a finite set, for it Is the

type with which we are more familiar in everyday life. Roughly

speaking, a set is called finite if the counting of its ele

ments comes to an end. We say "roughly" speaking, because

the set of air molecules does not permit itself to be counted.

In order to avoid misunderstanding, we shall try to give a

more precise definition of a finite set.

Definition: Let K be the set of natural numbers. Let

I - f n|k,n*I and U n*k}. Then a set S, is said to be finite

if it is empty, or if there exists a one-to-one correspondence

between the elements of the set 53 and the set Ifc, where k

is a definite natural number.

Having defined a finite set, we now proceed to define

an infinite set as a set that is not finite.

Let N denote the natural numbers, i.e., N = 1, 2, 3» ....

The set 1?Q = ^0,1,2,3,...^ Is called a set of finite cardinal

numbers. ^0 (called aleph null) is the cardinal number of

the infinite set I and of any set which can be put into one-

to-one correspondence with 11.

CHAPTER V

PROPERTIES OP PUICTIOIS

Functions,—Consider the correspondences;

A one-to-one

correspondence

John

James

Ben

Harry

Howard

Glarenc

Rober

A one-many

correspondence

1 many-one

correspondence

A many-many

correspondence

17

18

The reader is already familiar with the first corre

spondence above, but that does not mean that he is not in

some manner familiar with the other correspondences. The

second type of correspondence is common to a polygamous

society; the third is common to a polyandrous society; the

fourth is common to a society that is both polygamous and

polyandrous. These different correspondences are functions

and may be defined as follows:

Definitions A function is defined when:

i) A set D is given,

ii) To each element in E there is assigned some other

object.

The set D is called the domain of the function. Con

sider f : a—>b (read "under the mapping f, a goes into b").

This same idea can also be expressed: f(a) = b (read "f at a

equals h).

The object j> which is assigned or associated with a by the

given rule is called the image of a and may be denoted by

f(a). Sometimes we say f(a) is the image of a under the

mapping f.

Let R be the set of all images of x feD, that is,

S=^y\y= f(a), a fell). Then we call the set R the range

(of values) of the function P.

Example: Consider

f (x) ss 2x, x<i.I (natural numbers). Here a function f

is defined, let N be the set of all natural numbers and

let f be a rule which assigns to each x* IT the value 2x.

In symbols:

f i x—» 2x, xe-1 or f (x) = 2x.

The range will be the set of even natural numbers.

If f(a) = b, a fe D, there is assigned one and only one

object by f, we say that f is single-valued; otherwise f is

said to be multiple-valued. If f is single-valued and each

element of b€H has one and only one pre-image in D, we say

that f is one-to-one. If f(x) consists of exactly one element,

f is said to be a constant function.

Equality of Functions.—

Definition: Let f1 and fg be two functions with domain D.

Then fj= fg if and only if each element of the domain has

the same image under f1 and f .

In symbols:

f -f <L V.-P ft\ f (v\ veT|1 — oi /■L1V'2W — J-o*1*1/* •*• * si'

Example: Let f1 and fg have domain D = (l,2].

Let f^ and f be defined on D as follows:

f t(x) = x2 -4- 1 and

f2(x) = 3x - 1.

Observe:

f.(O = 2(D = fo0>

2m - o .

» ()f (1) = 2 ' 1 2

> =^f (2)=,= 5 j 1

f2(2)fg(2) =5 J ' d

Hence

= f2(x), x €D

1-^2 on D.

20

Sequentially Applied Functions.—Let us consider two

functions: f1 with domain D1 and range R^; and f2 with domain

D2 and range Eg. If the domain of f2 is a subset of the range

of fj, then f1 and f2 can be applied sequentially. Consider

the graph

^ R2 •1 >. 2

f1 f2

Let x = fj(x). Let x e D1 and under the mapping f-, x goes

into H1# If x<=D , then under the mapping f2, f 1 (x) goes

into

Consider the following graph.

let x^f^x). LetxeD,. If f,(x)tD2> then x under the

mapping fgf1 goes into Eg. Hence we san Say that two func

tions are sequentially applied if

Example J_: Let D1 and D2 be the set of natural numbers

and let

x2,

f2(x)=

Then f^(x) == f2[^1 (x)J , so that

f2fi(x)sf2[fiUT]«f2(i) « 2;

f1f2(x)sf1[f2(x)] , so that

f1f2(1)=Bf1f2(x)]-f1(2)— 4.

Example 2s f 1 (x) ==r 2x2 - 1,

f2(x) —

21

f^jO) 5 f2[fi(i)]-f2(1) * 2

(b) f,f2(2)a f1[f2(2)]=f1(5) - ?

Thus, f^S) and f2(7) are undefined since 5»74fv»21 the

domain of definition of f1 and fg.

It is very important in the study of sequentially applied

functions to remember that if a function f has domain D£ and

range Rf, then f(Df) equals Rj, where Rf =-f y\y fc f (x), xfeDf J- .

Hence,

gf(x) = g[f(xl]

is defined only if f(x) is an element of D , i.e., only ifo

the domain of D contains the range of R-. In symbols:

If f is a function with domain D^ and range H^, then

Rf ~f(Df)=fy|yef(x), x 6 Df)

is defined only if f(x)€L , i.e., only if D dr . On the

other hand:

fg(x) = f [gCxJ] is defined for each x eDg only if g

Pf, i.e,, only of DfO Re.

A graph might enable us to get a better understanding of

sequentially applied functions. Let the circle D be the do

main of the function f; let the circle .R be the range of f;

let the circle R1 be the domain of g; and let the circle S

be the range of g:

22

If f(x) were to have a value that would fall in R and

also in Rj, say y, then g(y) is defined. But if f(x) were to

have a value, say w, in R but not in R1, then g(w) is undefined,

It is undefined because w, although in the range of f, it is

not in the domain of g.

Inverse Functions.—-Consider a function f, and under f

--^ikbarV ■ -*■ f wise

Abdul \ ^- ysmart

Ahmed / • • s>- \4ntelligent

Under the inverse of f,

Akbar )< ■ • ■ --• \ wise

Abdul \-4 / smart

Ahmed U •—■ vintelligent

In order to have an inverse function we must have a one-

one function with domain D and range R.

Definition: Let f(a) =• b be a function. The inverse function

associates with each image a unique pre-image through the

mapping f"1 (read "f inverse").

In symbols f~1 is defined by

f"1(b)« a,

where a is the unique pre-image of b_ under the mapping.

Graph of a Function.—Let us consider a real-real function

f with domain V and range R. Then such a function is graphable

and hence we can define a graphable function as follows:

Definition: A function is graphable if

»y) xeD, y e £(x)l c E x E,

23

where E represents the set of real numbers.

Example; let f be defined on the set {1,2,3} by f(x)=x2-1

Find the graph of f.

By definition

Gf= £(x,y) |x* {1,2,3} , 7- x2 - 1 J. Therefore,

Gf = {(1,0),(2,3),(3,8)} .

The graph of f is

7

C

5

3

• 2 » A S (.

Is can be seen, the graph of the example, consisted of

only a finite number of points. However, this is not always

the case, for the domain of definition could be an infinite

number of points. These points would make up an Infinite set

ans as a set they can be defined and denoted in several equiva

lent ways.

Example; Let S be a set of real numbers x, where 0 is

less than or equal to x and x is less than or

equal to 1. This very same idea can be expressed

as follows:

Iotation J_: S = {x\o £x i1 } (read "the set S = x where 0

is less than or equal to x and x is less than

24

or equal to 1").

Notation 2; S= /0. . i) (read "the set consisting of

all reals from zero to one inclusive").

If m and n are two distinct real numbers and m is less

than n, then the first and second notations can be respectively

generalized as follows?

S= f x\m s

S= {m.

An example of a function whose domain (of definition)

is an infinite set is as follows:

Let f be defined on (have domain) ^0, 3} .

Between the reals 0 and 3 there is an infinite number of reals.

Hence, it would be impossible to plot all the points of the

graph of such a function on a real plane. In such a case we

choose: 1°) to plot a finite number of points, 2°) to observe

the general pattern of the graph of a finite number of points,

and 3°) join the finite number of points by a smooth line.

For the domain defined above, let f(x)- x2 - 1. Find

the graph of f.

Prom the general definition of the graph of a function,

Gf = {(x,y)|xeD, y= f(x)} , we obtain

Gf= |(x,y)\xe(o. .3} , y - x2 - 1 ] .

Malting a table of values for a finite number of points, we

obtain

X

0

1

2

3

f(x)-1

0

38

25

Consequently {(0,-1), (1,0), (2,3) i (3»8))-c &f. Hence the graph

is

f ■7 ■

II

3

-i

-a

I a, 4 5

At this point, it is necessary to say that the number of

points that we choose to obtain the general behavior of a func

tion whose domain is infinite is very important and may be

misleading.

Finding the Bange and Domain of a Continuous Function by_

a Graph.—The graphs of functions considered so far. give us.

an intuitive idea of a continuous function. Houghly speaking

we say that a function is continuous if its graph is a single

curve. To be precise, we define a continuous function as

followsi

Definition; A function f is continuous at x0 if for every

open set 33 containing f(x0), there exists an open set A con

taining xQ, such that the image of every point in A lies in B.

If f is continuous at every xeA, i.e., if f is continuous

at every x which is a member of its domain, then f is said

to be a continuous function.

26

Given the graph of a continuous function, it is possible

to find the range and domain of the given function. The range

can be found by 1°) extending a line from the highest point

of the graph perpendicular to the y axis. Call the point of

intersection of the line with the y axis r2. 2°) extending a

line from the lowest point of the graph perpendicular to the

y axis. Call the point of intersection of the line with the

y axis r1. Then the range of the continuous function would

consist of all reals from r1 to r2.

low to find the domain of a given graph, we use a similar

method to that of finding the range. 1°) Extend a line from

the point of the graph nearest to the y axis perpendicular

to the x axis. Call the point of intersection of the line

with the x axis dj. 2°) Extend a line from the point of the

graph farthest away from the y axis perpendicular to the x

axis. Call the point of intersection of the line with the

x axis d2. Then the domain of the continuous function con

sists of all reals from d1 to d2 (see figure below).

4.

i, h

BIBLIOGRAPHY

Books

Courant, Richard and Robbins, Herbert. An Elementary Approach

"to Ideas and Methods. London* Oxford University Press.1960" ~" "

. What Is Mathematics? London: Oxford Univer

sity Press, 1960.

Fuji, John N. An Introduction ibo the Elements of_ Mathematics.New York: John Wiley & Sons, Inc., 1961.

Insights into Modern Mathematics. The National Council of

TeacHers of Mathematics, District of Columbia, 1960.

Leakey, L.S.B. The Progress and Evolution of fen in Africa.London: Oxford University Press, 1961.

May, Kenneth 0. Elements of Modern Mathematics. Massachusetts:Addison-Wesley Publishing Company, Inc., 1959.

Rose, Israel H. in Introduction t_o College Mathematics. lew

York: John Wiley & Sons, Inc., 1960.

Wilder, Raymond L. in Introduction t£ the Foundations of Mathematics, lew York: John Wiley & Sons, Inc., 1952.

Periodical

Heinzelin, Jean de. "Ishango," Scientific American, CC1/T (June,1962), 105-119.

27