Chapter 1

Algebra of Numbers

MATH 17. COLLEGE ALGEBRA AND TRIGONOMETRY

1. Sets and Set Operations 2. The Set of Real Numbers 3. Operations on Real Numbers and Laws of

Exponents 4. The Set of Complex Numbers 5. Operations on and Factoring of Polynomials/

Zeros of Polynomials 6. Rational Expressions and Radicals 7. Equations 8. Inequalities

Chapter Outline

Chapter 1

Section 1.1

Sets, Set Operations,

and Number Sets

Section Objectives

Describe sets using different

methods

Identify different set relations and

perform different set operations

Identify special

number sets

Set has pervaded almost all of mathematics so that it has become a fundamental concept. It becomes impossible to define set precisely in terms of more basic concepts.

Notion of a Set

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.

Notion of a Set

Example

The collection of all natural numbers between 0 and 5.

Which of the following collection of objects are sets?

SET

The collection of all buildings in UPLB.

SET

The collection of all schools near Intramuros.

NOT A SET

Example

The collection of all handsome students in this class.

Which of the following collection of objects are sets?

NOT A SET

The collection of all flying horses.

SET

The collection of all letters in the word honorificabilitudinitatibus.

SET

Definition

If an object belongs to a set, it is called an element of the set.

𝑎 ∈ 𝐴: 𝑎 is an element of the set 𝐴

Element

Otherwise, the object is not an element of the set.

𝑎 ∉ 𝐴: 𝑎 is not an element of the set 𝐴

Example

ℎ ∈ 𝐴 𝑞 ∉ 𝐴

If 𝐴 is the set of all letters in the word “Philippines”, then

3 ∈ 𝑂 22 ∉ 𝑂

If 𝑂 is the set of all odd integers, then

Roster/Enumeration method

Set is indicated by enumerating the elements of the set and enclosing them in a pair of braces.

Rule method

Set is indicated 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 belong to the set.

Describing a Set

Example

using the roster method, 𝐴 = 𝑚, 𝑎, 𝑡, ℎ, 𝑒, 𝑖, 𝑐, 𝑠

while using the rule method,

𝐴 = distinct letters of the word "mathematics"

If 𝐴 is the collection of distinct letters in the word mathematics, then

Set Builder Notation

The set-builder notation is another way of

describing sets using the rule method. This method

uses a defining property of the elements.

𝑥 𝑥 𝑖𝑠 𝑎 __________ is read as "the set of all x such

that x is a (a certain defining property of all the

elements)”.

Describing a Set

Example

𝐵 = counting number from 1 to 5 or

𝐵 = 𝑥 x is a counting number from 1 to 5 or

𝐵 = 𝑦 𝑦 is a counting number less than 6

If 𝐵 = 1,2,3,4,5 , write 𝐵 using the rule method.

Example

IMPRACTICAL and DIFFICULT

If 𝐶 = clock, blue, television , write 𝐶 using the rule method.

Example

IMPOSSIBLE

If 𝐷 = points in the line 𝑥 = 2 , write 𝐷 using the roster method.

Time to Think!

When is the use of the rule method appropriate? The roster method?

The concept and the representation of the collection is subject to the following: • the collection must be well-defined • each unique object of the set should be

uniquely represented; and • the order of representing each object of the

set is immaterial.

Remember

Definition

- denoted by ∅ or { }

- are sets having no elements

Empty Sets

Example: 𝐷 = y y is a country in Asia with no people

= ∅ 𝐴 = {positive numbers less than − 5} = { }

Empty Set is a subset of any set.

Empty Set

∅ ⊆ 𝐴, for any set 𝐴.

The empty set is a subset of itself.

∅ ⊆ ∅

Remarks:

Definition

Otherwise, the set is said to be infinite.

Finite/Infinite Sets

Loosely speaking, we say a set is finite if it is possible to write down completely in a list all the elements of the set or if its elements can be counted (and the counting process terminates)

Sets differ in sizes and kinds.

Example

The set of all rational numbers between 0 and 5.

Which of the following sets is finite? Infinite?

INFINITE

The set of all buildings in UPLB.

FINITE

The set of all flying horses. FINITE

Example

The set of all hair strands on your head FINITE

The set of points in a circle.

INFINITE

The set of counting numbers between 1 and 1,000,000,000 FINITE

The set of grains of sand in a beach FINITE

Which of the following sets is finite? Infinite?

Definition

Cardinality

The cardinality (or size) of a set is the number of elements of that set.

𝑛 𝐴 : the cardinality of set 𝐴

Example: 1. If 𝐴 is the set of vowels in the English alphabet then

𝑛 𝐴 =

2. n ∅ = 0

5.

Definition

Set of all elements under consideration

Universal Set

Superset of all sets under consideration

Denoted by 𝑈

Example

If 𝐴 = 𝑥 𝑥 is an even counting number 𝐵 = 𝑦 𝑦 is an odd counting number 𝐶 = 𝑧 𝑧 is a prime number

A possible universal set is

𝑈 = 𝑥 𝑥 is a counting number OR

𝑈 = 𝑦 𝑦 is an integer OR

𝑈 = 𝑧 𝑧 is a real number

Two sets 𝐴 and 𝐵 are equal if they have exactly the same elements.

𝐴 = 𝐵: Set 𝐴 is equal to set 𝐵.

Equal Sets

Otherwise, the sets are not equal.

𝐴 ≠ 𝐵:Set 𝐴 is not equal to set 𝐵.

Definition

To say 𝐴 ≠ 𝐵, we should be able to produce an element that is in 𝐴 but not in 𝐵 or an element that is in 𝐵 but not in 𝐴.

Equal Sets

Thus, to say 𝐴 = 𝐵, we should be certain that every element in 𝐴 belongs to 𝐵 and every element in 𝐵 is also in 𝐴.

Example

𝐴 = 𝐵

If 𝐴 is the set of all letters in the word “resistance”, and 𝐵 is the set of all letters in the word “ancestries”, then

𝐵 ≠ 𝐶 because 𝑖 ∈ 𝐵 but 𝑖 ∉ 𝐶.

If 𝐶 is the set of all letters in the word “ancestor”, then

Time to Think!

If 𝐴 = {2, 3, 5, 7, 11} and 𝐵 = {x|x is a prime number less than 12}

Is 𝐴 = 𝐵?

If C= {11, 2, 5, 7, 3}

Is 𝐴 = 𝐶?

Definition

We write 𝐴 ⊆ 𝐵 if and only if 𝑥 ∈ 𝐴 implies 𝑥 ∈ 𝐵.

Subsets and Supersets

𝐴 ⊆ 𝐵: Set 𝐴 is a subset of 𝐵 or Set 𝐵 is a superset of 𝐴

Set 𝐴 is said to be a subset of set 𝐵 if every element of 𝐴 is also an element of 𝐵.

Thus, if there is an element in 𝐴 which is not in 𝐵,

we say 𝐴 is not a subset of 𝐵 and we write 𝐴 ⊈ 𝐵.

Subset of a Set

If 𝐴 ⊆ 𝐵 then whenever 𝑥 is in 𝐴, 𝑥 is also in 𝐵.

and

If whenever 𝑥 is in 𝐴, 𝑥 is also in 𝐵 then 𝐴 ⊆ 𝐵

This notation is to be understood to mean two

things:

Example

If 𝐵 = 1, 2, 3, 4 then

𝐵 ⊈ 1,2, 5, 6 .

If 𝐶 = 𝑎, 𝑏, 𝑐, 𝑑 and 𝐷 = 𝑎, 𝑏, 𝑐

Yes it is.

but

𝐵 ⊆ {1, 2, 3, 4, 5}

• Is 𝐷 ⊆ C?

No it isn’t. Thus 𝐶 ⊈ 𝐷.

• Is 𝐶 ⊆ 𝐷?

Time to Think!

• If 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴, what can be said about 𝐴 and 𝐵?

• If 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐶, is 𝐴 ⊆ 𝐶?

• Is 𝐴 ⊆ 𝐴?

• Give an example for sets 𝐴 and 𝐵 such that 𝐴 ⊆ 𝐵 but 𝐵 ⊈ 𝐴

If 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐶 then 𝐴 ⊆ 𝐶.

Subset of a Set

Transitive Property of Set Inclusion

𝐴 ⊆ 𝐴, for any set 𝐴.

Reflexive Property of Set Inclusion

Definition

𝐴 ⊂ 𝐵 if and only if A is a nonempty set and 𝐴 ⊆ 𝐵 but 𝐵 ≠ 𝐴.

Proper Subsets “⊂’’

If 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴 then 𝐴 = 𝐵.

Alternative Definition of Equality of Sets

Example

If 𝐶 = 𝑎, 𝑏, 𝑐, 𝑑 and 𝐷 = 𝑎, 𝑏, 𝑐

Therefore 𝐷 ⊂ C.

𝐶 ⊈ 𝐷 since 𝑑 ∈ 𝐶 but 𝑑 ∉ 𝐷.

𝐷 ⊆ C and

Is 𝐷 ⊂ C?

Definition

Two sets 𝐴 and 𝐵 are in one-to-one correspondence if it is possible to pair each element of 𝐴 with exactly one element of 𝐵 and each element of 𝐵 with exactly one element of 𝐴.

One-to-one Correspondence

Example

Is there a one-to-one correspondence between the set of days in a week and the set of colors in the rainbow?

YES

M T W Th F Sa S

red orange yellow green blue indigo violet

Example

Is there a one-to-one correspondence between the set of days in a week and the set of months in a year? NO

Jan

Feb

Mar

Ap

r

May

Jun

Jul

Au

g

Sep

Oct

No

v

Dec

S M T W Th F Sa

Example

Let 𝐴 = 1,2,3,4 𝐵 = 3,6,9,12 𝐶 = −4, −3, −2, −1,1,2,3,4

NO

Is there a one-to-one correspondence between set 𝐴 and set 𝐶?

YES

Is there a one-to-one correspondence between set 𝐴 and set 𝐵?

Example

Is there a one-to-one correspondence between the set of even counting numbers and the set of odd counting numbers?

YES

E 2 4 6 8 34290

…

O 1 3 5 7 34289

Time to Think!

Is there a one-to-one correspondence between the set of points in a line and the

set of all counting numbers?

Definition

Two sets are equivalent (or of the same size) if they are in one-to-one correspondence.

Equivalent Sets

Time to Think!

• When are two sets “not equivalent”?

• If 𝐴 and 𝐵 are equivalent and 𝐵 and 𝐶 are equivalent, are 𝐴 and 𝐶 equivalent?

• Are equivalent sets equal?

• Are equal sets equivalent?

Number Sets

ℕ = set of natural (counting) numbers ={1,2,3, … }

𝕎 = set of whole numbers = {0,1,2,3, … }

ℤ = set of integers = {… , −3, −2, −1,0,1,2,3, … }

Number Sets

N- = set of negative counting numbers E = set of even integers

O = set of odd integers

E+ = set of positive even integers

E- = set of negative even integers

P = set of prime numbers C = set of composite numbers

𝑘ℤ= set of multiples of k, k is positive

Example

2ℤ = … , −6, −4, −2,0,2,4,6, …

3ℤ = … , −9, −6, −3,0,3,6,9, …

4ℤ = … , −12, −8, −4,0,4,8,12, …

Venn Diagram

𝐴 ⊆ 𝐵 𝐵 ⊆ 𝐴

𝐴

𝐵 𝐴

𝐵

Example

Draw a Venn Diagram such that A is not a subset of B.

𝐴 𝐵

𝐴

𝐵 or

Example

Draw a Venn Diagram satisfying 𝐴 ⊆ 𝐵, 𝐴 ⊆ 𝐶 and 𝐵 ⊆ 𝐶.

𝐴

𝐵

𝐶

Definition

Two sets are disjoint if they have no element in common.

Disjoint Sets

𝐴 and 𝐵 are disjoint:

If 𝑥 ∈ 𝐴 then 𝑥 ∉ 𝐵.

Disjoint Sets

𝐴

𝐵

𝐴

𝐵

𝐴 and 𝐵 are disjoint 𝐴 and 𝐵 are not disjoint

Definition

Union The union of two sets 𝐴 and 𝐵 is the set of elements that belong to 𝐴 or to 𝐵.

𝐴 ∪ 𝐵: 𝐴 union B

UA B

𝐴 ∪ 𝐵 = 𝑥 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵

Example

If 𝐴 = 1,3,5 and 𝐵 = 𝑎, 𝑏, 𝑐 then 𝐴 ∪ 𝐵 = 1, 3, 5, 𝑎, 𝑏, 𝑐

Moreover, 𝑛 𝐴 = 3, 𝑛 𝐵 = 3 and

𝑛 𝐴 ∪ 𝐵 = 6

Time to Think!

Is it always true that 𝑛 𝐴 ∪ 𝐵 = 𝑛 𝐴 + 𝑛 𝐵 ?

Definition

Intersection

The intersection of two sets 𝐴 and 𝐵 is the set of elements that belong to 𝐴 and to 𝐵.

𝐴 ∩ 𝐵: 𝐴 intersection B

𝐴 ∩ 𝐵 = 𝑥 𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵

UA B

Example

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

If 𝑃 is the set of all prime numbers and 𝐶 is the set of all composite numbers, then

𝑃 ∩ 𝐶 = ∅

Definition

Alternative Definition

Two sets 𝐴 and 𝐵 are disjoint if and only if 𝐴 ∩ 𝐵 = ∅.

Cardinality of 𝐴 ∪ 𝐵

If 𝐴 and 𝐵 are disjoint 𝑛 𝐴 ∪ 𝐵 = 𝑛 𝐴 + 𝑛 𝐵

In general, 𝑛 𝐴 ∪ 𝐵 = 𝑛 𝐴 + 𝑛 𝐵 − 𝑛 𝐴 ∩ 𝐵

Example

If 𝐺 = 1,3,5,7,9,10 and 𝐻 = 3,6,9,12 , then

𝑛 𝐺 = 6 𝑛 𝐻 = 4

𝑛 𝐺 ∩ 𝐻 = 2

𝑛 𝐺 ∪ 𝐻 = 8

Then

Definition

Set Difference

𝐴 𝐵

A-B= 𝑥 𝑥 ∈ 𝐴, 𝑥 ∉ 𝐵

Definition

Set Difference A-B= 𝑥 𝑥 ∈ 𝐴, 𝑥 ∉ 𝐵

A= {1,2,3,4,5,6} B= {2,5,7,9,10}

{1, 2, 3, 4, 5, 6}

{2, 5, 7, 9, 10}

A−B =

B−A =

Definition

The complement of 𝐴 denoted by 𝐴′, is the set of all elements of 𝑈 that are not in 𝐴.

Complement

𝑈 𝐴

𝐴′ = 𝑥 𝑥 ∈ 𝑈, 𝑥 ∉ 𝐴

Example

If 𝑈 = 1,3,5,7,9 and A = 5, 9 then

𝑈 𝐴

5 9 1

7

3

𝐴′ = 1,3,7

Complement of a Set

𝑈′ =

∅′ =

Remark:

𝐴′ = 𝑈 − 𝐴

∅

𝑈

Complement of a Set

U U

𝐴′ ′ = 𝐴

𝐴′ 𝐴′ ′

𝐴 𝐴

Example

UA B

A B 'A B

Illustrate using the Venn diagrams

𝐴 ∪ 𝐵 ′

Example

Illustrate using the Venn diagrams

𝐴′ ∩ 𝐵′

UA B

'A

UA B

'B

' 'A B

Example

'A B ' 'A B

' ' 'A B A B

Definition

Cross Product

The cross product (or Cartesian product) of two sets 𝐴 and 𝐵 is the set of all possible ordered pairs 𝑥, 𝑦 where 𝑥 ∈ 𝐴 and 𝑦 ∈ 𝐵.

𝐴 × 𝐵: 𝑥, 𝑦 𝑥 ∈ 𝐴 and 𝑦 ∈ 𝐵

Example

If 𝐴 = 1,2 and 𝐵 = 𝑎, 𝑏

On the other hand 𝐵 × 𝐴 = 𝑎, 1 , 𝑎, 2 , 𝑏, 1 , 𝑏, 2

then 𝐴 × 𝐵 = 1, 𝑎 , 1, 𝑏 , 2, 𝑎 , 2, 𝑏

Time to Think!

What is the cardinality of 𝐴 × 𝐵?

Definition

Power Set

The power set of any set, denoted by ℘ 𝐴 , is the set of all subsets of set 𝐴.

Let 𝐴 = 𝑎, 3, # , then ℘ 𝐴 =

𝑎 , 3 , # , 𝑎, 3 , 𝑎, # , 3, # , 𝐴, ∅

What is 𝑛 ℘ 𝐴 ? 8

Time to Think!

What is the cardinality of the power set of any set 𝐴?

Time to Think!

Let 𝑈 be a universal set and A ⊆ 𝑈

𝐴 ∪ 𝑈

𝐴 ∪ 𝐴′

𝐴 ∪ ∅

𝐴 ∩ 𝑈

𝐴 ∩ 𝐴′

𝐴 ∩ ∅

Time to Think!

Let U = ℤ. Find the following:

1. ℕ ∪ 𝕎 = 𝕎 2. ℕ ∩ 𝕎= ℕ 3. 𝐸 ∪ 𝑂 = ℤ 4. 𝐸+ ′ = {0}

∪O ∪ 𝐸− 5. 3ℤ ∪ 2ℤ

6. 3ℤ ∩ 2ℤ= 6ℤ 7. ℕ′ ∩ 𝕎 = {0} 8. ℕ ∪ 𝑃′ = ℤ 9. 𝐶 ∩ 𝑃 × 𝐸− ={} 10. ℤ′ = {}

Example

If 𝑛 𝑈 = 𝑟 and 𝑛 𝐵 = 𝑗, then 𝑛 𝐵′ =

𝑟 − 𝑗

Example

Illustrate using the Venn diagrams

𝐴 ∩ 𝐵 ∪ 𝐶

B C A A B C

Example

Illustrate using the Venn diagrams 𝐴 ∩ 𝐵 ∪ 𝐴 ∩ 𝐶

UA B

C

A B

UA B

C

A C

Example

A B A C A B C

A B C A B A C