CHAPTER 1ALGEBRA AS THE STUDY OF STRUCTURESMATH 17College Algebra and Trigonometry

Chapter Outline1.Sets, Set Operations and Number Sets2.The Real Number System3.The Complex Number System4.The Ring of Polynomials5.The Field of Algebraic Expressions6.Equations7.Inequalities

Chapter 1.1Sets, Set Operations, and Number Sets

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

1.Identify special number sets2.Perform set operations on number sets3.Draw Venn diagrams4.Identify finite and infinite sets of numbers and how to represent them

Set and Set NotationsA 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 1.1.1Which of the following collection of objects are sets?

The collection of all:1.colleges in UPLB.SET2.counting numbers from 1 to 100SET3.provinces near Laguna.NOT A SET

4.planets in the solar system.SET5.pretty instructors in UPLB.NOT A SET6.letters in the word algebra.SET7.points in a line.SET8.MATH 17-A students who can fly.SET

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

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

Example 1.1.2

Equal Sets

Example 1.1.3

Example 1.1.4

Finite/Infinite Sets

Example 1.1.5Determine if the following sets are finite or infinite.

1.Set of counting numbers from 1 to 5FINITE

2.Set of all professors in UPLB.FINITE

3.Set of points in a circle.INFINITE

4.Set of counting numbers between 1 and1,000,000,000FINITE

5.Set of grains of sand in a beachFINITE

6.Set of counting numbers greater than 1INFINITE

Describing Sets

Describing Sets

Example 1.1.6

Example 1.1.7

Example 1.1.8

Example 1.1.9

One-to-one Correspondence

Example 1.1.10

Example 1.1.11Is there a one-to-one correspondence between

the set of days in a week and

the set of months in a year.

NO

Example 1.1.12Let 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

Equivalent Sets

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

Subsets

Example 1.1.15

Subsets

Subsets

Equal Sets (Alternative Definition)

Proper Subsets

Example 1.1.16

Empty Sets

Example 1.1.17

Empty Sets

Venn Diagram

Example 1.1.18

Example 1.1.18

Disjoint Sets

Disjoint Sets

Universal Set

Example 1.1.19

Complement

ComplementExample 1.1.20

Complement

Complement

Cardinality

CardinalityExample 1.1.21

Power SetExample 1.1.22

Example 1.1.22

Union

Union

Example 1.1.23

Intersection

Intersection

Example 1.1.24

Example 1.1.24

Alternative Definition

n(A U B)

Example 1.1.25

Example 1.1.26

Example 1.1.26

Example 1.1.26

Example 1.1.26

Example 1.1.26

Example 1.1.26

Example 1.1.27

Cross Product

Example 1.1.28

Number Sets

Number Sets

Number Sets

Number Sets

Example 1.1.29

End of Chapter 1.1

How to say that sets are not equal.******Larger sets - Subsets*******************************