Sets

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Digitized by the Internet Archive in

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The "Reason Why" Books

SETS Irving

and Ruth Adler

A The John Day Company

New

York

The "Reason Why" Books by Irving and Ruth Adler AIR

ATOMS AND MOLECULES COAL

COMMUNICATION EVOLUTION FIBERS

HEAT HOUSES INSECTS AND PLANTS IRRIGATION: CHANGING DESERTS TO GARDENS

LEARNING ABOUT STEEL: THROUGH THE STORY OF A NAIL MACHINES

MAGNETS NUMBERS OLD AND NEW NUMERALS: NEW DRESSES FOR OLD NUMBERS OCEANS RIVERS SETS

SHADOWS STORMS TASTE, TOUCH AND SMELL THE EARTH'S CRUST THINGS THAT SPIN TREE PRODUCTS WHY? A BOOK OF REASONS WHY AND HOW? A SECOND BOOK OF REASONS YOUR EARS YOUR EYES

©

1967 by Irving and Ruth Adler may be reprinted, or reproduced in any form or by any means electronic, mechanical or other (including but not limited to photocopying, recording, or a process of information storage and retrieval), without permission in writing from the Publisher, The John Day Company, Inc., 62 West 45th Street, New York, N.Y. 10036. Published on the same day in Canada by Longmans Canada Copyright

All rights reserved.

No

part of this book

Limited.

Library of Congress Catalogue Card Number:

AC

67-10169

MANUFACTURED IN THE UNITED STATES OF AMERICA Second Impression

Contents

The

You Use

Sets

The Members Symbols

6

of a Set

Elements and Sets

for

The Empty

4

8

12

Set

Infinite Sets

12

Subsets

14

The Number

of Subsets of a Set

18

Universal Set

The Truth

16

Set of an

Open Sentence

20

Equations

22

Inequalities

24

Sets of

26

Ordered Pairs

More Equations

28

More Inequalities The Intersection of Two

30 Sets

Common Multiple Greatest Common Divisor Two Inequalities The Union of Two Sets

34

Least

The Number

of

Elements

32

35

36 38 in a Set

40

Finding n( A n B)

40

Finding n( A U B)

41

A Formula for n(AUB)

42

Note

Teachers

44

to Exercises

44

to

Answers

A

set of dishes

The

Any

Sets

You Use

collection of things

is

Sets are important because all

the time.

called a set.

you use them

When you dress, you put on a When you eat, you use a set

set of clothes.

of dishes.

At play, you

dominoes or a draw, you

may

set of checkers.

may use

When

you

a set of crayons.

You even belong to some ily is

use a set of

a set of people,

and

so

sets. is

Your fam-

your

class in

school.

^""^v^^

^ CRAYONS A

set of

checkers

A

set of crayons

A

The members

$

of things at

all.

of a set

set of clothes

may be any

The Milky

Way

is

kinds

a set of

The United Nations is a set of nations. There are sets whose members are books, and there are sets whose members stars.

are numbers.

This book will teach you some things

about

sets that will

about the

A set

sets

you

help you think clearly

use.

A

of

numbers

A

family

is a set of people

set of

books

The Members

of a Set

we must know what things are its members. We pay no attention to the way in which the members of the set may be arranged or related. For example, a collection of gears, springs and bolts may be piled up in a heap, or they may be put together to make To know

a

set,

a watch that keeps time. Either way, as a heap or as a

watch, the collection of parts has the same members. set of parts,

is

the

same

set,

because

A watch is more than a set.

with the parts arranged

we change the arrangement, we we do not destroy the set.

The same

in a special

It is

way.

it

a If

destroy the watch, but

set

There are two main ways of saying what things are

members of a set. One way is to list them, one by one. The other way is to describe them. The description must fit all the are

members of the set, and should not fit any things that not members of the set. For example, there is a set

whose members

are the letters a,

just identified the set

e,

i,

o

and

We have

u.

by listing its members. But we could

also identify this set

by describing

members

its

as the

vowels in the English alphabet.

Another word for member of a

set

is

element

(

EL-uh-

muhnt).

Exercises 1.

The members

or elements of a certain set are de-

scribed as the whole

the elements of this 2.

The elements

numbers that are

less

than

set.

of a certain set are described as the

English colonies that

won independence and formed

United States of America. List the elements of 3.

The elements

10. List

of a certain set are the states

the

this set.

Washing-

ton, Oregon, and California. Give a good description that fits

4.

these states and no others.

The elements of a certain

dents, Lincoln, Garfield,

set are the

American

presi-

McKinley and Kennedy. Give

a good description that fits these presidents and no others.

Symbols

Your

makes a the (

for

class in school list

name

of

Elements and Sets

is

a set of pupils. Your teacher

members of the class by writing down each pupil. The name of a pupil is a symbol

of the

SIM-buhl) or sign that stands for that pupil.

To be first

able to

make

a

of the

list

attach a symbol to each

bol attached to a for that

member

members

member of the

of a set

is

like a

of

any

set,

The symname or label set.

member. To avoid confusion, we use

different

symbols for different members. For example, in the of drawings triangle,

below consisting

some

letters of the

members

we

of a circle, a square

set

and a

alphabet are used as symbols

To make a list of the members of this set we simply write down these symbols, separated by commas c, s, t. for the

of the set.

:

To write

a symbol for the whole

braces, with the

list

between the braces:

of the {

set,

members

c,s,t\.

8

we

write a pair of

of the set written

It is

as a

sometimes convenient to use a single capital

symbol for a whole

A

use the letter that the set,

two symbols

we write A —

{

t is

a

member

we may wish to c, s, t } To show

For example,

to stand for the set

A and

{

cy

{

s, t }

.

stand for the same

From this statement we can member of A, s is a member of

c,s,t}.

see at a glance that c

A,

set.

letter

is

of A,

a

and there are no other members

of A.

In some sets there are different alike.

When we

must be sure

members

that look

choose symbols for these members,

to choose different

we

symbols for them. For

example, the set shown below consists of four squares.

wouldn't do to

call

each of these squares

s,

It

because then,

we wouldn't know which square it stands for. But we may label these squares a, b, c, and d. Then, if we call the whole set B, we may say that B = {a,b,c,d}. when we

say

s,

Exercises

For each of the

sets

pictured below, a capital letter

is

given to serve as a symbol for the set and a separate

symbol

is

attached to each

symbol for the whole

set

tween braces the symbols

ment

member

of the set. Another

can be written by putting befor

its

members. Write a

that says that both symbols stand for the

6.

B:

g

10

state-

same

set.

7.

C:

8.

The

set

e, f, g,

{

h

}

and the

set

{g,f,e,h} are the same

because they have the same members. Changing the der

in

which the members

change the 9.

ways the

set

list

of

different

ways the

set

{

y, z,

x

}

members, write

in three

in exercise 5.

its

members, write

in three

C shown in exercise 7.

What capital letter,

11.

its

A shown

Using braces and a list of

10.

set

of a set are listed does not

set.

Using braces and a

different

or-

A, B, C, or

?

11

D is the name of the

The Empty Set Sometimes the members of a a

way

that there are

no things

set are described in

at all that

fit

such

the descrip-

Then the set has no members, and we call it the empty set. For example, there are no pupils in your class tion.

who

are less than

your class who are

As a symbol of braces is

0. So

:

two years

}

we may

.

So the

than two years old

less

empty

for the

{

old.

set

we

set of pupils in

the

is

write an

empty set.

empty

pair

Another symbol used for the empty

—

write

set

}.

{

Exercises 12.

Which of the sets described below is the empty set? A: The set of states in the United States whose names begin with A.

The

B:

set of states in the

names begin with C:

D

:

The

10.

The

set of

than

E

:

set of

than

The

1

B.

even whole numbers that are

odd whole numbers

and

set of

United States whose

less

than

less

that are greater

3.

whole numbers

different

from

1

and

13 that are divisors of 13.

Infinite Sets

You cannot write down ing numbers.

No

matter

a complete

how many

list

of

all

the count-

counting numbers

12

:

you may write

in a

there

list,

is

a greatest one in the

Then the next higher counting number

after that

So your

numbers

not in the

list.

list

of counting

list.

one

is

not

is

complete.

A

set like the set of

counting numbers that has more

members than can ever be (IN-fuh-nit)

set.

called a finite

To show an

(

A

set that

FY-nite )

three dots.

The

that there are

than are

listed.

called an infinite

can be completely

we

listed

is

put inside a pair of braces

members

They show

is

set.

infinite set,

the symbols for a few

listed

of the set, followed

by

and so

on.

three dots are like the words

many more members

Here are some examples

showing an

infinite set:

the set of

all

counting numbers

the set of

all

even counting numbers

the set of

all

=

in the set

of this

{ 1, 2, 3, 4,

.

.

.

way

of

};

= 2, 4, 6, 8, odd counting numbers = {1,3,5,7, {

.

.

.

.

.

.

}; }.

Exercises State whether each of these sets finite, list all it is

of

infinite, list

is

finite or infinite. If

it is

members between a pair of braces. some of them followed by three dots.

its

If

The set of all counting numbers less than 5. The set of all multiples of 3. (A multiple of 3 is a number you get by multiplying 3 by a counting number.) 15. The set of all multiples of 6. 16. The set of all multiples of 9. 17. The set of all divisors of 12. 13.

14.

13

.

Subsets

We

show below

are a circle

c,

a picture of a set A,

a square

5,

and a

triangle

whose elements

£

o A c

s

We

can form a

either

all,

new

set

by choosing

way

formed

Let us fonn

all

the subsets of A.

we we

in this

we get A

itself.

we may

So there are three different

two elements is

for

elements

its

So

A is

new

set all of the

a subset of A.

new

set only

They

sets

each of which has

O o A A s

C

C

t

t

s

B-JM|

C=|c,t|

D=

t

.

just

and D.

are pictured below.

14

two

leave out either c or s or

of A. Let us call these sets B, C,

a subset of A.

and no

called a subset of A.

choose for the elements of the

of the elements of A,

Each

is

choose for the elements of the

elements of A, If

f

or some, or none of the elements of A, set

If

jc s,t

t

Any

others.

A=

|c,s(

we

If

choose for the elements of the

we may

of the elements of A,

new

set

choose either c or

only one s or

So

t.

there are three different sets each of which has just one

element of A. Let us subset of A.

They

call

these sets E,

•

E-jet

F-]s|

we is

The

of

new

set

is

new

the

set

none

empty

of

set 0.

subsets of any set can be obtained in the all,

some or none

You can recognize

two

G=|t|

a subset of A.

way, by taking set.

a

is

A'

choose for the elements of the

the elements of A, then the

So

G. Each

are pictured below.

o If

F and

same

of the elements of the

a subset of any given set

by means

rules:

1.

The empty

2.

If

a set

is

set

is

a subset of any given

not empty, and

ments of a given

set,

then

it is

all its

set.

elements are also

a subset of the given

ele-

set.

Exercises 18.

A set

{

w, b

black checker b subsets of this

}

is

consisting of a white checker

pictured below.

set.

w 15

Form

all

w

and

a

the possible

: ,

.

the set

19.

Is

20.

Each

{

a,

set in

x

a subset of the set

}

column

the sets in column

II.

a subset of one or

I is

Which ones

=

the set of

mont E

=

U

the set of

T

X

= = = =

R

the set of

all

divisors of 6

the set of

all

divisors of 15

The b

Q

all

B

}

chart below shows

and

{ a,

The subsets

b, c

all

the set of

all

the set of

whole numbers

all

citizens of the

= = = =

the set of

all

rectangles

the set of

all

divisors of 60

the set of

all

quadrilaterals

the set of

all

divisors of 12

the subsets of the sets

{

a

}

}

of ja(:

The subsets 1.

of ja, b, c

ja,b,c(

2.|b,c|

1

3.ja,cj

The subsets

of

of Subsets of a Set

i.ja{ 2.J

more

squares

The Number { a,

A

the empty set the set of

Why?

U.S.A.

bers

S

=

even whole num-

all

?

II

W =

citizens of Ver-

all

}

are they?

I

V

c

a, b,

{

4.ja,b|

of ja, b\:

l.|a,b|

5.jaj

2.{a|

6.|b|

3.jb|

7.|cj

4-1

8-1

I

16

1

•

Exercises 21.

Prepare a chart showing

all

the subsets of the sets

{*}, {x,t/}and{x,t/,z}. 22.

How many

subsets can be formed from a set with

member? b) two members? c) three members? 23. Take a guess: How many subsets can be formed from a set with four members? See if your guess is a) one

right

by forming and counting

all

the subsets of the set

{a,b,c,d}. 24.

If

you increase by one the number

finite set,

what happens

to the

number

of elements in a

of subsets

it

has?

The table below shows how the number of subsets of a finite set is related to the number of elements in the set: Finite Sets

Number

of

Number

Elements 2

1

4

2

of Subsets

=2 = 2x2

8=2x2x2 16 = 2x2x2x2 32 = 2x2x2x2x2

3

4 5

The numbers in this table follow a simple pattern. To find the number of subsets in a finite set, you multiply some twos. You use as many twos as there are elements in the set.

25. a)

number

How many

twos would you multiply to find the

of subsets in a six-member set? b)

subsets does a six-member set have? 17

How many

Universal Set

In a discussion

about

people or things.

sets of

they are

we often talk about people or things,

members

all

set for the discussion.

If

one

is

we say it is

or subsets,

We use

there

17 as

set of

or

which

the universal

a symbol for the uni-

versal set.

For example,

in a certain discussion these statements

were made: "In 1963, the Mets won 51 games and 111 games. The Dodgers

won 99 games and

Mets and the Dodgers mentioned

lost 63."

lost

The

in these statements are

baseball teams, and the universal set of which they are

members

the National League.

is

Sometimes, although the universal is

not mentioned,

we can tell what

of the statements that are says, "Mr.

gave

me is

from the meaning

it is

made. For example,

if

a boy

Frank gave me an A in French, and Miss Glenn

C

a

versal set of

bers

set for a discussion

in mathematics,"

we

can

tell

that the uni-

which Mr. Frank and Miss Glenn are mem-

the set of teachers in his school.

Sometimes discussion

we

cannot

tell

what the universal

know what the

statements

the universal set

is.

from the statements set

mean

is.

We may

unless

we are

in a

not even told

what

For example, suppose you hear some

boys but cannot see them, and you hear one of them say, "Let's send

it

up now." You cannot

18

tell

from

this state-

ment what kind

of thing

is

statement has one meaning it

belongs

meaning

is

if

meant by the word

The

the universal set to which

if

The statement has another set is a set of model planes.

a set of kites.

the universal

There are many other meanings that have, too. In cases like

this

give a clear

this, to

statements

we make, we must

versal set

that contains the

is

it.

statement

meaning

may

to the

always say what the uni-

members

or subsets that

we

are talking about.

Exercises In each statement below, the words in bold face

26.

Choose from the

refer to a person, a thing, or a set. sets the universal set in

which

it is

a

member

Statements a)

A

he crossed the street by himself.

He

c)

It

hit

49 home runs

was one

in

B

1962.

C

of the original thirteen

He swallowed two

makes

the earth

=

the set of states

in

the United

= a set of pills = the set of

children

in

Smith's family of

them with

D

some water. e) It

or a subset.

States

states. d)

in

E a

complete

of

Sets

Mrs. Smith spanked him because

b)

list

trip

around

90 minutes.

19

= =

a set of earth satellites

a set of baseball players

Mrs.

.

.

The Truth Set

The and

sentence, "The letter a

it is

The

is

a vowel,"

is

a statement,

is

a vowel,"

is

a statement,

true.

sentence, "The letter b

and it is

The

an Open Sentence

of

false.

sentence, "The letter

statement, because there

should be.

is

is

a blank space

where a

letter

if

we

it.

may be used

A

an open

Any one to

fill

sen-

we have

sentence,

universal set connected with

the

sentence

filled is called

Whenever we use an open

elements of the universal set

fill

either true or false.

is

tence that has a blank space to be

mind a

not a

letter of the alphabet, the

becomes a statement that

in

is

neither true nor false. But

It is

blank space with a

sentence.

a vowel,"

of the

the blank

space and turn the open sentence into a statement that either true or false. In the

versal set

is

the set of

is

example given above the uni-

all letters

of the alphabet.

In the universal set connected with an open sentence there

is

a subset

made up

of

all

members

the

that

the open sentence a true statement. This subset

is

make called

the truth set of the open sentence. In the example given

above, the truth set

is

the set

{

a, e,

i,

o,

u

}

Example Find the truth set of the open sentence, "The word contains the letter p," if the universal set is U = { skip, walk, hop } :

By

filling

members

of

the blank space in turn with each of the 17,

we

get these statements:

contains the letter p.

The word hop

The word walk

contains the letter p.

20

The word

skip

contains the letter p.

Two

of these state-

.

ments are {

skip,

hop

true.

The

open sentence

truth set of the

is

}

Exercises

In the exercises below, the letter

open sentence that

versal set of the

find the truth set of the

=

27.

U

28.

U = the set of

the set of

U is

stands for the unigiven. In each case

open sentence.

all states in

the U.S.A.

has no other state on all letters

its

boundary.

of the alphabet.

sometimes stands for a hissing sound made without using your voice. 29.

U

= the set of cards in a bridge deck. is

30.

a card with a picture on

it.

U = the set of New England states. The Connecticut River

passes through or touches

The New England states

CONNECTICUT RIVER

31.

U

=

{

Frank, John, George, Harold}.

The name

begins with T. 21

.

Equations

+ 5 = 8.

Consider the open sentence: Let the universal

be the is

set of all

connected with

set

whole numbers.

put into the blank space,

this

this

When

open sentence

number

a whole

open sentence becomes a

statement that says that two numbers are equal. For this reason the open sentence

is

called an equation

KWAY-zhun). The statements formed from

this

(ee-

open

+ 5 — 8; 1 + 5 = 8; 2 + 5 = 8; = = 3-\-5 8;4-\-5 8; and so on. Of all these statements, the only one that true 3 5 = 8. So the truth set for the open sentence, + 5 = 8 the set 3 which sentence are these:

is

is

-f-

is

has only one element in

When

},

{

it.

an open sentence

is

about numbers,

we

usually

write a letter of the alphabet instead of a blank space.

For example, instead of x

-f-

5

—

:l:\::7...^ 5$ 7 8-9 Jfi

1

12

3

4

Ul$ tj

14

W

f*

£

B The

set

A

n B

in that part of the

is

number

line that

is

both red and gray: 1

I

1

12

1

3

l:^kl-

4

'

l^^ 6

5

8

7

9

10 It 12

1.3

14

J

5 16

>

AOB

Exercises

Draw

a picture on the

number

line to help

you solve

each of these problems 53.

two 54.

two 55.

two 56.

two

Find the

set of all

inequalities x

Find the

< >

set of all

inequalities x




satisfy the

7.

whole numbers that

5 and x

satisfy the

10.

whole numbers that

5 and x

satisfy the

> 5.

whole numbers that

8 and x

set of all

inequalities x

Find the

10 and x

set of all

inequalities x

Find the