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CHAPTER 1

ALGEBRA AS THE

STUDY OF

STRUCTURES

MATH 17 College Algebra and

Trigonometry

Chapter Outline

1. Sets, Set Operations and Number Sets

2. The Real Number System

3. The Complex Number System

4. The Ring of Polynomials

5. The Field of Algebraic Expressions

6. Equations

7. Inequalities

Chapter 1.1

Sets, Set Operations,

and Number Sets

Objectives

At the end of the section, we should be able to:

1. Identify special number sets

2. Perform set operations on number sets

3. Draw Venn diagrams

4. Identify finite and infinite sets of

numbers and how to represent them

Set and Set Notations Set and Set Notations

A set is a well-defined collection of

objects.

It should be possible to determine (in

some manner) whether an object

belongs to the given collection or not.

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Example 1.1.1

Which of the following collection of objects

are sets?

The collection of all:

1. colleges in UPLB.

SET

2. counting numbers from 1 to 100

SET

3. provinces near Laguna.

NOT A SET

4. planets in the solar system.

SET

5. handsome instructors in UPLB.

NOT A SET

6. letters in the word “algebra.”

SET

7. points in a line.

SET

8. MATH 17-Z students who can fly.

SET

Element

If an object belongs to the set, it is called

an element of the set.

Otherwise, the object is not an element of

the set.

: is an element of set .a A a A

: is not an element of set .a A a A

Example 1.1.2

If is the set of letters in the word "mathematics"A

t A

z A

If is the set of even numbers then

1

10

E

E

E

Equal Sets

Two sets and are if they have

exactly the same ele

eq

ments.

ualA B

Symbolically, we write .A B

Otherwise, we write .A B

Example 1.1.3

If is the set of letters in the word

"mathematics"

is the set of letters in the word

"mathetics"

A

B

A B

If is the set of letters in the word

"math"

C

since but A C s A s C

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Example 1.1.4

If the elements of are 1,2,3,4, and 5

and the elements of are 1,1,2,2,2,3,4, and 5

Is E? Y S

A

B

A B

If the elements of are 5,4,3,2, and 1

Is Y? ES

C

A C

Finite/Infinite Sets

A set is if it is possible to write down

completely in a list all the elements of the

finite

set.

Otherwise, the set is said to be infinite.

Example 1.1.5

Determine if the following sets are finite or

infinite.

1. Set of counting numbers from 1 to 5

FINITE

2. Set of all professors in UPLB.

FINITE

3. Set of points in a circle.

INFINITE

4. Set of counting numbers between 1 and

1,000,000,000

FINITE

5. Set of grains of sand in a beach

FINITE

6. Set of counting numbers greater than 1

INFINITE

REVIEW:

SET: A well-defined collection of objects

ELEMENT: An object that belongs to a set.

EQUAL SETS: Sets with same elements.

FINITE SET: Elements can be completely listed

down.

INFINITE SET: Not finite.

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Describing Sets

indicate a set by enumerating the

elements of the set and enclosing them

in a pai

Rost

r of

er Meth

bra

od

ces.

Describing Sets

indicate a set by enclosing in a pair of

braces a phrase describing the elements of

the set with the condition that those objects,

and only those, which have the described

property be

Rule Method

long to the set

Example 1.1.6

If F is the set of distinct letters of the

word "FILIPINO," write F using

a. roster method

, , , , ,M F I L P N O

distinct letters o

b. the rule method

f the word FilipinoF

Example 1.1.7

If 5,4,3,2,1 , write using

the rule method.

C C

such that is a counting number from 1 to 5C x x

is a counting number less than 6z z

is a counting number from 1 to 5x x

Example 1.1.8

If bread,butter,coffee,rice , write using

the rule method.

D D

DIFFICULT/IMPRACTICAL

Example 1.1.9

If is a point in a plane ,D x x

It is IMPOSSIBLE to use roster method.

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One-to-one

Correspondence

Two sets and are in

if it is possible to pair

each element of with exactly one

element of and each element of

one-to-one

correspo

with

exactly one element of .

ndence

A B

A

B B

A

Example 1.1.10

Is there a one-to-one correspondence

between the set of days in a week and

the set of counting numbers from 2 to 8?

M T W Th F Sa Su

2 3 4 5 6 7 8

YES

Example 1.1.11

Is there a one-to-one correspondence

between

the set of days in a week and

the set of months in a year.

NO

Example 1.1.12

Let A = { 1, 2, 3, 4 }

B = { 3, 6, 9, 12 }

C = { -4, -3, -2, -1, 1, 2, 3, 4 }

Is there a one-to-one correspondence

between set A and set B?

Is there a one-to-one correspondence

between set A and set C?

YES

NO

Example 1.1.13

even

Is t

cou

here a o

nting nu

ne-to-one correspondence between

the set of anmbers

odd count

d the set

ing numb

of

ers.

2 1

4 3

6 5

E O

M

1,000,000 999,999

YES

Equivalent Sets

Two sets are or of the same size

if they are in one-to-one corres

equ

pon

ivalent

dence.

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Example 1.1.14

True or False

1. Equal sets are equivalent.

2. Equivalent sets are equal.

3. If set A is equivalent to set B and set

B is equivalent to set C, then A is

equivalent to C.

TRUE

FALSE

TRUE

Subsets

Set is said to be a of set if every

element of is also an eleme

s

n

ubse

t o .

t

f

A B

A B

: is a of .

is a

subset

supe ofrs .t e

A B A B

B A

if and only if implies .A B x A x B

Subsets

If there is an element in which is

not in , we say is not a subset of and

we write .

A

B A B

A BÖ

Example 1.1.15

If , , , and , , , , ,

a. Is YES ?

L a b c d M a b c d e

L M

NOb. Is ?M L M L

Subsets 1. Is ?A A

2. If and , is ?A B B C A C

3. If and , what can be said

about and ?

A B B A

A B

4. Give examples of sets and such that

but .

A B

A B B AÖ

Subsets

Reflexive Property:

A A

Transitive Property:

If and , then .A B B C A C

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Equal Sets

(Alternative Definition)

if and only if and .A B A B B A

Proper Subsets

if and only if but .A B A B B AÖ

Example 1.1.16

If , , , and , , , , ,

is ?

since but .

Therefore, .

L

L a b c d M a b c d

M

M

e

L M

e M e L

L

L

M

Ö

Empty Sets

- sets having no elements

- denoted by

Example 1.1.17

Let is a town in the Laguna and

is a town in the Laguna with only 4 voters .

T x x

F x x

is an empty set so .F F

Also, .F ST.

Hence, .ST.

Empty Sets

1. A

2.

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Venn Diagram

A B

A

B A

B

B A

Example 1.1.18

Draw a Venn diagram satisfying

and A B B A

A B

or

AB

Example 1.1.18

Draw a Venn diagram satisfying

, , and A B A C B C

A

B

C

Disjoint Sets

Two sets are if they have no

element in

disjoin

com

t

mon.

and are disjoint: If , then

If , then

A B x A x B

x B x A

Disjoint Sets

and are disjointA B

A

B

and are not disjointA B

A

B

Universal Set

- set of all elements under consideration.

- superset of all sets under consideration.

- denoted by U

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Example 1.1.19

If is an even counting number

is an odd counting number

is a prime number

is a composite number

A x x

B y y

C z z

D w w

A possible universal set is

is a counting numberU x x

Complement The complement of , denoted by ',

is the set of all elements of

that are not in .

A A

U

A

UA

Complement

' ,A x x U x A

Example 1.1.20

If 2,4,6,8,10 and 2,6 ,U A

then ' 4,8,10 .A

Complement

'U

' U

Complement

' 'A

UA

'A

UA

' 'A

A

Cardinality

The cardinality (or size) of a finite set

is the unique counting number such

that the elements of are in one-to-one

correspondence with the set 1,2,..., .

A

A

n

n

The cardinality of the empty set is 0.

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Cardinality

: number of elements of set n A A

Example 1.1.21

If is the set of all vowels in the alphabet,A

then 5.n A

If and = what is ' ?n U k n A m n A

' .n A k m

Power Set

The power set of any set , , is the set

of all subsets of set .

A A

A

Let = , , .A a b c

Example 1.1.22

A

, , , , , , , , , , , ,a b c a b a c b c a b c

Example 1.1.22

What is ?n A 8

Remark: In general, the cardinality of the

power set of any set A, n( (A)) = 2n(A)

Union

The of two sets and is the set of

elements that belong to

unio

n

.r o

A B

A B

: union A B A B

Union

or A B x x A x B

UA B

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Example 1.1.23

If 1,3,5 and 2,4,6A B

then 1,2,3,4,5,6 .A B

Intersection

The of two sets and is the

set of elements that belon

intersection

g to .d an

A B

A B

: intersection A B A B

Intersection

and A B x x A x B

UA B

Example 1.1.24

If , , , , and , , , ,A a e i o u B a b c d e

,A B a e

Example 1.1.24

If is the set of all prime numbers and

is the set of all composite numbers,

What is ?

P

C

P C P C

Alternative Definition

disjoTwo sets and are if and onl

.

t y

if

in

A

A B

B

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n(A U B)

If and are disjoint, A B n A B n A n B

In general, n A B n A n B n A B

Example 1.1.25

If 2,4,6,8,10,12 and 3,6,9,12A B

then 6,12 .A B

n A n B n A B 6 4 2

n A B 6 4 2 8

2,3,4,6,8,9,10,12A B

Example 1.1.26

Illustrate the following sets using Venn

diagrams.

1. 'A B

UA B

A B 'A B

Example 1.1.26

2. ' 'A B

UA B

'A

UA B

'B

' 'A B

Example 1.1.26

'A B ' 'A B

' ' 'A B A B

Example 1.1.26

3. A B C

UA B

C

B C

A

A B C

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Example 1.1.26

4. A B A C

UA B

C

A B

UA B

C

A C

Example 1.1.26

A B A C A B C

A B C A B A C

Example 1.1.27

If is the universal set and ,

find the following by visualizing the

Venn diagrams.

a. d. '

b. e.

c. ' f.

U A B

A B B A A

A B A A A

A A U A

Cross Product

The (or Cartesian product)

of two sets and is the set of all possible

ordered pairs whe

cross produ

re and .

ct

,

A B

x A xy Bx

, and x x AA y yB B

Example 1.1.28

Let 1,2 and , .

What is ?

1, , 1, , 2, , 2,

,1 , ,2 , ,1 , ,2

A B p q

A B

A B p q p q

B A p p q q

A B B A

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Number Sets

set of natural (counting) numbersN

= 1,2,3,...

set of whole numbersW

= 0,1,2,3,...

set of integersZ

= ..., 2, 1,0,1,2,...

Number Sets

set of negative counting numbersN

set of even integersE

set of odd integersO

set of positive even integersE

set of negative even integersE

Number Sets

set of prime numbersP

set of composite numbersC

set of multiples of ,

is positive

kZ k

k

Number Sets

2 ..., 6, 4, 2,0,2,4,6,...Z

3 ..., 9, 6, 3,0,3,6,9,...Z

4 ..., 12, 8, 4,0,4,8,12,...Z

Example 1.1.29

If , find the following

1. 6. 5 4

2. 7. '

3. 8. '

4. ' 9.

5. 3 2 10. '

U Z

N W Z Z

N W W N

E O N P

E C P E

Z Z Z

End of Chapter 1.1