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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*******************************