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Chapter 1Chapter 1Algebra as the Study ofAlgebra as the Study of
StructuresStructures
MATH 17MATH 17CollegeCollegeAlgebra andAlgebra and
TrigonometryTrigonometry
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
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.
Example 1.1.1
Which of the following collection of objects are
sets?
The collection of all:
1. colleges in UPLB.
SET2. counting numbers from 1 to 100
SET
3. provinces near Laguna.
NOT A SET
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4. planets in the solar system.
SET
5. handsome instructors in UPLB.
NOT A SET
6. letters in the word “algebra.”
SET
7. of points in a line.
SET
8. of MATH 17-S students who can fly.
SET
Element
If an object belongs to the set, it is called anelement 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"
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.
ualB
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
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
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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
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
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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 6 z z
is a counting number from 1 to 5 x 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.
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
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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? YES
Is there a one-to-one correspondence between
set A and set C? 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
1,000,000 999,999
Equivalent Sets
Two sets are or of the same size
if they are in one-to-one corres
equ
pon
ivalent
dence.
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 .
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 of rs .t e
A B A BB 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 .
B A B
A B
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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 ?B B C A C
3. If and , what can be said
about and ?
B B A
A B
4. Give examples of sets and such that
but .
A B
A B B A
Subsets
Reflexive Property:
A A
Transitive Property:
If and , then . A B B C A C
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 A
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
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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 S
Hence, .S
Empty Sets
1. A
2.
Venn Diagram
A B
A
B A
B
B A
Example 1.1.18
Draw a Venn diagram satisfying
and A B B A
B
or
B
Example 1.1.18
Draw a Venn diagram satisfying
, , andB A C B C
B
C
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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 disjoint A B
A
B
and are not disjoint B
B
Universal Set
- set of all elements under consideration.
- superset of all sets under consideration.
- denoted by U
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
U A
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
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Complement
'U
' U
Complement
' ' A
U
A
'
U
A
' ' 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.
Cardinality
: number of elements of setn A A
Example 1.1.21
If is the set of all vowels in the alphabet,
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
: In general, the cardinality of the
power set of any set ,
Remark
.2n A
n A A
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Union
The of two sets and is the set of
elements that belong to
unio
n
.ro
B
A B
: union A B B
Union
orB x x A x B
U A B
Example 1.1.23
If 1,3,5 and 2,4,6 A 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 .dan
B
A B
: intersection A B A B
Intersection
andB x x A x B
U A B
Example 1.1.24
If , , , , and , , , , A a e i o u B a b c d e
,B a e
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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
B
B
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 3,6,9,12 A B
then 6,12 . A B
n A n B n A B 6 4 2
n A B 6 4 2 8
2,3,4,6,8,9,10,12 A B
Example 1.1.26
Illustrate the following sets using Venn
diagrams.
1. ' A B
U A B
B 'B
Example 1.1.26
2. ' ' A B
U A B
' A
U A B
'B
' 'B
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Example 1.1.26
'B ' 'B
' ' 'B A B
Example 1.1.26
3. A B C
U A B
C
B C
A B C
Example 1.1.26
4. A B A C
U A B
C
A B
U A B
C
A C
Example 1.1.26
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 x y B x
, and x x A A y y B B
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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
Number Sets
set of natural (counting) numbersN = 1,2,3,...
set of whole numbersW
= 0,1,2,3,...
set of integers Z
= ..., 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
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End of Chapter 1.1