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Sets, Elements, & Numbers

Slide Duration:

Section 1: Introduction
Introduction to Math Analysis

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
18:50
19:04
19:21
20:04
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
Section 2: Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
Section 3: Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
13:08
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
9:56
11:38
Derivation
11:43
Final Form
12:23
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
26:49
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
19:24
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
7:41
8:55
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
Section 4: Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
8:25
9:26
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
Section 5: Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
2:08
2:31
2:52
Half-Circle and Right Angle
4:00
6:24
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
9:43
Mnemonic: All Students Take Calculus
10:13
13:11
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Section 7: Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2

52m 52s

Intro
0:00
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
Section 8: Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
Section 9: Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
11:42
Example of Using a Point
12:41
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
Section 11: Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
Section 12: Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
Section 13: Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
Section 14: Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
11:51
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Section 15: Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
Section 16: Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
Section 17: Appendix: Graphing Calculators

10m 41s

Intro
0:00
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
9:17
9:19
9:35
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28

• ## Related Books

### Sets, Elements, & Numbers

• We learn about the idea of a set in this lesson. While many courses won't directly address these concepts, they form the foundation that a lot of math rests upon. While you can understand later concepts without these ideas, knowing them can really help, especially in the long term.
• A set is a collection of distinct (different) objects. We call each object in a set an element.
• We can denote a set with any symbol, but we commonly use capital letters like A.
• We can write out a set in various ways. We often use curly braces    { }     to denote that a set contains what is inside the braces. Here are some of the ways to write out a set.
• Directly name the elements in the set:
 A = { x,y,z}
• Clearly describe all the members of the set:
 B={the first 10 letters of the alphabet}
• Describe the quality (or qualities) each member of the set has in common:
 C = { x  |  x is an English word that rhymes with ` thing′}
• If an element is contained in a set, we show it with the symbol    ∈
For example, if x ∈ A, that means that x is an element contained in the set A.
• If an entire set is contained in another set (all the elements of one set are elements of the second set as well), we say it is a subset. If A is a subset of B, we denote this as A ⊂ B. This means for any x ∈ A, we know x ∈ B as well.
• We can consider a set that has no elements at all: the empty set (also called the null set). It is a set with nothing in it. We represent it with ∅.
Since the empty set contains nothing at all, it is a subset of all sets (after all, each of its elements trivially appears in every other set). Thus, for any set A, we have ∅ ⊂ A.
• The union of two sets is a set that contains the elements of each. We denote this with ∪.
• The intersection of two sets is a set that contains the elements (and only those) that are in both of them. We denote this with ∩.
• We can think of numbers as being elements from sets. Each set makes up a category of numbers.
• Natural Numbers: The numbers we would count objects in the real world with.
 ℕ = { 1, 2, 3, 4, 5, 6, 7, …}
• Integers: Expanding on ℕ, we also include 0 and the negatives.
 ℤ = { …, −3, −2, −1 , 0,  1,  2,  3, …}
• Rational Numbers: We take ℤ and use division to create fractions.
 ℚ = ⎧⎨⎩ m n ⎢⎢ m ∈ ℤ,  n ∈ ℕ ⎫⎬⎭
• Irrational Numbers: There are some numbers that cannot be expressed as a fraction of integers. These numbers are not rational, so we call them irrational. Some examples are π and √{47}.
• Reals: Combining the rational and irrational numbers together into a single set, we get the real numbers. We denote them with ℝ. You have been using them for years, and they are our bread and butter in math. ℝ contains any number you might normally use.
• We can express intervals of ℝ using interval notation.
• To include the end numbers, we use square brackets: [−1,  3].
• To exclude the end numbers, we use parentheses: ( −1,  3).
• If we want, we can mix these types to include one end but exclude the other end: [−1, 3).
• To talk about one end of the interval going on forever, we use −∞ or ∞ (depending on which direction). We always use parentheses with −∞ and/or ∞ because we can't actually include it in the interval: ∞ isn't actually a number, just the idea of continuing forever: [−1, ∞).

### Sets, Elements, & Numbers

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Introduction 0:05
• Sets and Elements 1:19
• Set
• Element
• Name a Set
• Order The Elements Appear In Has No Effect on the Set
• Describing/ Defining Sets 3:28
• Directly Say All the Elements
• Clearly Describing All the Members of the Set
• Describing the Quality (or Qualities) Each member Of the Set Has In Common
• Symbols: 'Element of' and 'Subset of' 6:01
• Symbol is ∈
• Subset Symbol is ⊂
• Empty Set 8:07
• Symbol is ∅
• Since It's Empty, It is a Subset of All Sets
• Union and Intersection 9:54
• Union Symbol is ∪
• Intersection Symbol is ∩
• Sets Can Be Weird Stuff 12:26
• Can Have Elements in a Set
• We Can Have Infinite Sets
• Example
• Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
• This Set Has Infinitely Many Distinct Elements
• Numbers as Sets 16:03
• Natural Numbers ℕ
• Including 0 and the Negatives ℤ
• Rational Numbers ℚ
• Can Express Rational Numbers with Decimal Expansions
• Irrational Numbers
• Real Numbers ℝ: Put the Rational and Irrational Numbers Together
• Interval Notation and the Real Numbers 26:45
• Include the End Numbers
• Exclude the End Numbers
• Example
• Interval Notation: Infinity 29:09
• Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
• Always Use Parentheses
• Examples
• Example 1 31:23
• Example 2 35:26
• Example 3 38:02
• Example 4 42:21

### Transcription: Sets, Elements, & Numbers

Hi; welcome back to Educator.com.0000

Today, we are going to talk about sets, elements, and what they mean for numbers.0002

To start, we are going to talk about the idea of sets.0007

But a lot of courses don't address this stuff directly.0010

But these ideas build the foundation that the rest of math works on.0018

So, you can understand all of the concepts that will come later in this course without ever having watched this lesson.0024

But watching this first will help you see how it all fits together, which will really help your understanding, which will make it that much easier on you.0029

Also, if you want to go on and take later, more advanced math, like calculus or even much more advanced college courses0036

(like abstract mathematics), this stuff is going to be really useful to have already ingrained in your mind.0043

These ideas are great later for you; so if you are going to take advanced math, really, definitely, watch this.0049

Get an understanding of what is going on.0056

Once again, you don't really have to deeply understand any of these things; we are just going to be touching it on the surface.0057

But you want to get a glimpse of this sort of stuff, so that later on, you can really understand what is going on.0062

And also, it is just going to make things a lot smoother, especially when we are talking about functions.0068

So, you can get an abstract idea of how a function works, which will help you understand what is going on.0072

All right, let's get started!0077

A set is a collection of distinct objects; each of the objects inside of a set is called an element.0079

An example here: we have two sets: {1, 2, 3} is a set--each one of those elements is different from all the other elements.0086

1 is not 2 or 3; 2 is not 1 or 3; and 3 is not 1 or 2.0095

Similarly, for the set {cat, dog}, we have cat different than dog, and dog different than cat.0100

Also, really quickly: the way that we show that we are talking about objects inside of a set is: we have these curly braces.0107

And I am not that great at making a curly brace, but it is something like that for the left side, and something like that for the right side.0115

You can see a nice typography (typed-out font) brace in my slide here.0122

But when I am actually writing it out, I do something like these right here.0127

So, we put our elements inside of that, and we separate each of the elements with a comma.0131

That is just what is happening on a type point of view.0135

If we want to, we can also name these: we can decide, "I will name {1,2,3}; I will name that set A."0140

And if I want to, I can also name the set {cat,dog} B, because I might want to be able to talk about this;0146

and instead of having to say {1,2,3} every time I want to talk about that set, I can just say A.0152

"The set A has such-and-such property," or "the set B, when it interacts with it..."0156

That way, I don't have to say {cat,dog}, or if it was an even longer list, like 10 or 50 objects...0160

it would start to get really hard, practically impossible, to say.0166

So instead, we can just change it to using a single letter, or whatever symbol is convenient for our purposes.0168

Furthermore, the order that the elements come in has no effect on the set itself.0175

So, the order that the elements appear in doesn't matter; we don't care about the order here.0180

A = {1,2,3}, but that is the exact same thing as saying {3,2,1}, and that is the exact same thing as saying {2,1,3),0185

or any other way you have of ordering those things.0191

The important part is that it has all of those elements; the way that they come in--their places in line--that doesn't matter.0194

It is just the group that you are considering, not the specific permutation of the line.0201

All right, that is the basic idea of a set.0206

If we want to describe a set, there are a bunch of different ways to describe it.0209

Here are the three most common ways that you are going to see.0213

Directly saying all of the elements: we could go through and, like I was talking about before with the curly braces and the comma,0216

we just say each of the elements inside of the set: ice, water, steam.0222

Our set has three elements; we have just said each of the three elements; that is the most basic method.0227

We just say what is inside of the set.0232

Another way is that we can clearly describe all of the members of the set.0234

So, we also might describe it without it being inside of the curly braces; but sometimes we will actually leave it inside of the curly braces.0238

The point of it is that we are able to say, "Oh, yes, that is everything that makes it up."0243

So, we could make a set out of the first 80 elements of the periodic table, so we would know that hydrogen would be in the set;0247

helium would be in the set; lithium would be in the set; all sorts of different elements0253

are going to be inside of the set, up until the eightieth element.0258

The eightieth element would be in it; the eighty-first element would not be in it.0261

So, another way of describing it is to just say what is inside of it: here is what makes up my set, and there we go--we have a set.0265

The final way that we can do it is: we can describe the quality, or it may be qualities, that each member of the set has in common.0272

So, the way that you want to parse this--the way you want to read this--is: "x is saying this here is what our set is made up of."0279

Our set is made up of all of the x; and then, you read this vertical bar as saying "such that."0288

So, all of the x such that x is the first name of a teacher at Educator.com would be this set.0301

Another way of reading that vertical bar is the word "where"--"x where x is the first name of a teacher at Educator.com."0309

Anything will do here, so long as it is getting across the idea that this thing here, in the second part, is describing the quality0319

required of the thing in the first part; so this part, the second part, describes what happens over here in the first part.0328

So, for this set, if it is x such that x is the first name of a teacher at Educator.com, then it is going to be a bunch of first names0334

of all of the teachers who teach at Educator.com.0341

My name is Vincent; I am teaching at Educator.com (since you are watching this right now).0345

So, that means that "Vincent" is inside of this set.0349

There are going to be a bunch of other names; if you go and look at all of the teachers, you will see a whole bunch of different first names.0352

But we know for sure that Vincent is one of the names inside of the set.0356

Great! We can also symbolize things--if an element is contained in the set, and we want to talk about an element0360

being in that set, we have a convenient symbol to show it, this symbol right here: "element of," "contained in."0367

For example, if A is equal to the set {a,b,c}, then we know that a is contained in A; b is contained in A; c is contained in A,0373

because they showed up right here in our description of what the set was.0385

So, we know that a is an element in it; and we use this symbol right here to show "element of."0388

We can also talk about the idea of subsets (if a set is contained inside of another set).0395

If an entire set is contained in another set, then formally (as a formal definition) that means that every element in the first set is contained in the second set.0400

So, for every element we name in that first set, it shows up in the second set; that is how we are going to formally define it.0410

But you could just think of it as it being inside of the other set.0416

We are going to call it a subset, because it is part of the other thing; it is like a sub-part, so we call it a subset.0420

The symbol for this is this right here, "subset of."0427

So, if X is the set {3}, and Y is the set {1,3}, and Z is the set {1,2,3}, then X is a subset of Y, because 3 shows up inside of Y.0431

And then, Y is a subset of Z, because 1 and 3 both show up in Z.0444

So, we are able to see that that is a subset, because everything in here showed up in the other one.0453

Furthermore, we know that this property has to be transitive, because X is contained in Y, and Y is contained in Z;0459

then since X already lives inside of Y, it must also be inside of Z.0465

If we were to see it as sort of a picture, we would see it something like this.0469

So, X is contained in Y, is contained in Z; since Z has Y, it must also have X, so we have a transitive property--X is contained inside of Z, as well.0475

Great; we can also talk about a set that has no elements at all, the empty set.0486

And sometimes, it will also be called the null set.0492

Either way, it is a set that has nothing in it: it has no elements whatsoever.0495

We represent it with this symbol right here, "the empty set" symbol.0500

Now, this set is going to be unique, because any set that has no elements inside of it must be the empty set.0504

There is only one empty set, because there is only one way to have nothing inside of a set.0513

So, the empty set is just nothing at all; there is nothing inside of it--no elements; we have the empty set.0517

Since the empty set has nothing inside of it, it must inherently be inside of any other set.0525

All of its elements show up in every other set; each of its elements appears in every other set.0531

Now, I have the word "trivially" there, because what means is that it is trivial--it is obvious in sort of a silly way.0538

Yes, OK, sure, none of them show up...of course nothing shows up, because they don't have any there.0546

But that doesn't make it not true; it is trivially true.0552

It is kind of an obvious, silly thing, but it is still true; so that means, by our definition of subset, that the empty set is a subset to everything.0555

The set A = {walrus} must have the empty set inside of it, because that set has...in a corner...nothing; everything has a little nothing inside of it.0564

B, {17,27,47}...the exact same thing: it is also going to have the empty set inside of it.0575

A and B don't really have any connection, other than the fact that they both have empty sets inside of them,0580

because any set at all, even the empty set itself, is going to contain the empty set,0585

because containing yourself is obvious, because it means you already have yourself in there.0590

All right, union and intersection: we can create new sets through having our sets interact with each other.0595

So, if we have two or more sets, we can have an interaction between those sets and make another set that may or may not be different.0601

The union of two sets is a set that contains the elements of each.0608

We symbolize this with an open cup; that gives us our union symbol.0612

The intersection of two sets is a set that contains the elements, and only those elements, that are in both sets.0618

So, if an elements shows up in both of the sets, it is going to be symbolized with the intersection symbol, sort of like a cup pointing down.0625

A cup pointing up--we are filling it up with a bunch of things; a cup pointing down--it is cutting things off.0632

We could also see this as a Venn diagram: here we have all of the stuff in set A; here we have all of the stuff in set B.0637

What they cover together--what they both cover, here--is A intersect B; the stuff that is in A and in B is A intersect B.0649

The stuff that is in everything is going to be A union B; we can see this with the idea of a Venn diagram, as well.0660

Union adds everything from all of our sets, and makes a big set out of everything that we have.0668

And A intersect B is going to make a smaller set (generally) that is going to see where you cut into each other--0673

where you have the exact same thing--and that is all we have left.0680

Example using actual things: if A is equal to {cat,mouse}, and B is equal to {cat,dog}, then A union B is...0686

cat shows up; mouse shows up; and then, we go over to B, and cat...cat already showed up, so it is not that interesting0693

to put it in again; we can't have copies show up in our set, because everything has to be unique;0702

but dog hasn't shown up before, so we get dog in there.0706

Now, for A intersect B, we ask, "Well, what is the thing that shows up in both of them?"0710

Cat, cat...yes, cat showed up in both of them, so it gets to go here.0714

But mouse doesn't show up over there; dog doesn't show up in A; so it doesn't show up either.0720

You have to be in both of the two sets; intersection is if you were in both of them--you get to go on to the intersection.0728

If you are only in one of them, that is not good enough.0735

But union is where you only have to be in one of them, and you automatically make it in.0738

You can be in both of them, and that is great; you still get in that way, as well.0741

Sets can be weird stuff: we have talked about fairly simple stuff so far, that has been finite--just a couple of elements at a time.0746

And there have been some numbers; there have been some words; but we haven't encountered anything that crazy.0754

Now, the sets you are going to see for math, at least for the next couple of years, are going to generally just be sets of numbers.0759

But, as we have seen, we can also contain a lot of different ideas.0766

We don't just have to be stuck with numbers; we can also have elements other than numbers,0769

like words, or maybe even symbols or faces; we could have a bunch of different things inside of our set.0773

The important thing is that they are distinct objects.0778

We have also only talked about the idea of finite sets; a finite set means that it has a limited number of elements--it doesn't just keep going forever.0781

But we can also have an infinite set; that is going to be a set where the elements just keep going forever.0789

So, an infinite set means the elements keep going forever--they never stop; there is an unlimited number of elements.0794

So, how can we see an infinite set?0801

Well, let's just start counting and never stop: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17...0803

There is no reason I have to stop; I am going to stop, because I am mortal and I am not going to be able to count forever.0815

But we get the idea that, even though I can't count forever,0822

even though there is no way to literally write an infinite number of things, it still exists as an idea.0825

And so, as an idea, it is a perfectly fine set.0832

All of the numbers--all of the counting numbers, just listed out forever and ever and ever and ever--that gives us a set.0836

It is an infinite set, because it has an unlimited number of elements; but it is a perfectly reasonable set.0842

We can make even weirder, more interesting, stranger infinite sets if we want.0848

Consider the set where we take a word, and then we repeat it an ever-increasing number of times:0854

word, and then wordword, and then wordwordword, and then...etc., etc., etc.0860

So, each time we do this, we will make a new "word"; it is not really a word in English, but it is a word in our sense of making up a new thing.0866

And if we keep doing this forever, we are going to have an infinite number of words.0874

For example, what if we took the word cat?0877

Then we would have cat, catcat, catcatcat, catcatcatcat, catcatcatcatcat (I think I said cat 5 times)...0880

and we could keep going and going and going and going; this set has infinitely many distinct elements.0889

No matter what number you say, there is an element in the set that is going to have that word, "cat," repeated that many times.0895

If you say 72, inside of this set right here, there is somewhere (not on this slide, but somewhere)--0901

if you just keep going, you are going to be able to imagine the idea of "cat" being repeated 72 times in a row.0909

So, we are creating new elements out of doing this; we build a set out of this idea.0917

And each one of these is distinct from the others: "cat" is not the same as "catcat," which is not the same as "catcatcat."0923

So, each one of these elements is distinct from the others, and there is an unlimited number of them; we have an infinite set.0929

We can make really interesting, weird things in set theory--it is really, really cool stuff: we have just scratched the surface of how cool this stuff can get.0935

I love set theory personally; but it is something you will have to study in college if you are really, really interested in it.0943

So, I just want to finish by saying that sets can be strange and beautiful things, and that there is a whole bunch of stuff out there.0949

Now, let's start talking about how all of this set theory stuff applies to what we are going to be seeing,0955

in the near future in Precalculus, and then hopefully one day in calculus.0959

We can talk about numbers as sets: we understand the notion of "set" now, and that is great;0963

so we can now look at sets that make up numbers that we are going to use in math.0968

We have already seen one of the most essential sets: it was our first example of an infinite set, the natural numbers:0972

N = {1,2,3,4,5,6,7...}; this is just starting at 1 and counting on forever.0978

This is our first, most basic infinite set, in many ways.0985

We get this idea of counting and never stopping from the age of 3 on, if not even earlier.0990

We are getting this idea where you start counting, and you just never stop.0996

And of course, as a child, you realize, "Eventually I have to stop--I will say I will count to 100, and then I will not count any further."0999

But you could just keep going; and that is the idea of the natural numbers--1006

you just keep going forever and ever and ever, and you have an infinite number of elements.1009

One thing to note is that some teachers will define the natural numbers as starting with the 0.1015

So, you might, instead, have N be {0,1,2,3,4...}; so it is the exact same thing on the latter part of it.1020

I prefer the version without 0, that starts with 1; but some teachers make a distinction,1030

and will call the one starting with 1 the counting numbers, and the one starting with 0 the natural numbers.1036

The point is to just pay attention to what your teacher is teaching you, if you are taking an outside class.1040

And make sure you are using their definition, so that you get everything right on the homework,1044

and that you understand what they are trying to teach you.1047

There is nothing better or less good about one or the other; it is just sort of a taste thing.1050

And I happen to prefer the version without 0.1055

Also, I just really quickly want to talk about this symbol that we use: that is N in blackboard bold,1058

which is to say what we would write out if we were writing the symbol by hand.1064

However, it is kind of hard to make that symbol by hand, since it is such a fancy typography symbol.1068

Instead, if you were writing this out by hand, the symbol that you write is like this.1073

You start out with an N, and then you just drop another line down here; and that is seen as ℕ if you want to write it by hand.1078

Probably, you are not going to have to write this stuff by hand for at least a while, and maybe not ever.1086

But I want you to know, in case you were interested in doing that.1090

So, let's keep expanding on these ideas.1094

We can take the natural numbers and say, "Well, we have positive numbers; but we also have negative numbers."1096

So, let's count, not just forward, but let's count backward, as well.1101

We will hit 0, and then we will just drive on into the negatives.1104

This gives us the integers: we start at 0, and then we go forward for positive, but we also go backward for negative.1107

0 forward is 1, 2, 3, 4, 5, 6... 0 backward is -1, -2, -3, -4, -5, -6, -7, -8.1114

We are going off in both directions, and that gives us the integers.1122

We use this symbol here with the special Z...if you want to make this ℤ, you start off with just a fairly normal Z;1125

and then you drop down this extra diagonal and join it with your Z, and you have the symbol for the integers in handwritten form.1131

Now, I think you are wondering, "Wait, that made sense with the natural numbers; that was N; but why Z?"1139

Z comes from German; I believe zollen is the word for integer...no, zollen is just numbers;1145

and so, zollen is numbers, and because German mathematicians were doing this work1151

around the same time that English-speaking mathematicians were, and it was all being codified into symbols,1156

we ended up using Z for the German version of numbers, because those mathematicians did a lot of great work when setting up set theory.1160

All right, the next idea: we can add yet another layer of depth by including the idea of division.1167

So, we have 1, 2, 3...-1, -2, -3...these are great, and they get us a good idea of what is in the real world.1172

But what if I want to talk about wanting half of a pie, or if I want to talk about..."He got one and a half dollars,"1178

or something where I want to break a number into pieces?1186

Now, we have to be able to talk about fractions.1189

To do that, we use the rational numbers.1192

So, here we have that interesting format where we have the middle bar meaning "where," "such that," something like that.1194

So, what this means is that we have m/n, where m comes from the integers (m is one of these integers);1201

it can be a negative number; it can be a positive number; but it is going to be a whole number.1209

And n has to be contained in the natural numbers, which is good, because we certainly don't want to be able to divide by 0,1214

and because of my definition of the natural numbers, we are not allowed to have 0 in the naturals.1220

That means we can't divide by 0, so we are safe there.1225

This gives us the ability to have any number up top, divided by any whole number that is positive on the bottom,1228

which lets us make any fraction that we want to.1234

You give me any fraction (like, say, 47/9), and look: we have 47 (which belongs to the integers); 9 belongs to the natural numbers.1237

If we want to talk about the fraction -52/101, well, we can turn that into being equivalent to -52/101;1247

and so, we have -52...well, that is an integer; and 101 is a natural number; so right there, we have the natural numbers.1260

We are able to build any fraction that we are used to seeing in normal circumstances.1270

Any sort of normal fraction that we would talk about, we can make now with the rational numbers.1274

This gives us a lot of ability to make numbers.1280

We can get pretty much anywhere we want to be by using the rational numbers.1282

Also, you might wonder why it is Q; really quickly, if we wanted to write this by hand, you make a Q first;1288

and then, you drop a vertical line like that; so you have ℚ; that gives us our blackboard bold once again,1294

which is just to say something we can write by hand that makes it other than just writing the letter Q.1301

So, that lets us talk about that set of all the rational numbers.1306

And why do we use the letter Q? Because a fraction is connected with the idea of quotients.1309

So, as opposed to using F (which we kind of use for functions a lot, as we will talk about later),1314

we use Q to talk about quotients; so that is where we get the letter Q from.1319

All right, onward: we can also talk about rational numbers as a decimal expansion.1324

We have this idea of expanding a rational number into a decimal version; there is nothing wrong with decimal versions.1330

And we can have pretty much any number turn into a decimal version of itself.1336

So, the decimal expansion of every rational number (you probably learned this in grade school)...1340

every rational is either going to terminate (which means it ends), or it continues with repeating digits.1344

For our first example, something terminating, we have 0.09375; that is what we get from 3/32.1351

And see how it just ends right here: if we were to keep going, it would be 00000...we would just have 0's forever.1357

So, we just cut it off, and it terminates--it stops at a certain point.1364

If, on the other hand, it continues with repeating digits, then that means there is some block of digits that will keep repeating forever.1368

So, with 77/270, we get .2851851851...we realize that 851851851...point 2 happens first, and then our repeating block shows up: 851851851.1376

And it is just going to march out forever and ever and ever.1391

So, if we have a rational number, it is going to do one of these two things.1395

It either terminates (it ends), or it repeats.1398

Every rational number, anything that can be expressed as an integer divided by an integer, by whole numbers over whole numbers,1401

with maybe a positive or a negative sign--that is going to have either the decimal ending or the decimal going forever, but repeating.1408

Why is this important? This idea of the rational numbers is really great, but there are still some numbers we can't express.1417

So, you might remember that decimal expansions of all the rationals either terminate, or they go into repetition.1423

There is at least one number you have heard of by now that keeps changing: pi.1431

You have learned about the number π for probably quite a few years now.1437

And you know that it just keeps shifting around: 3.1415...and you can memorize a bunch of digits, if you want.1440

But it is never going to just lock down and turn into something where you are done memorizing it.1447

There are always going to be infinitely many more digits to remember.1452

So, π never stops--it never repeats; it is not a rational number.1455

You have probably also heard that √2 is also not a rational number.1462

These turn out to be true; we can't express them as rational numbers--we can't express them as a fraction of integers.1467

The decimal expansion of an irrational number, unlike a rational, never stops, and it always keeps changing.1474

They are these sort of shifting, mixed-up numbers that just always keep doing interesting things.1480

They keep us working hard, unlike the rational numbers.1486

So, if we want to really be able to describe everything that is out there--all of the numbers we might encounter--1489

we need to be able to talk about the irrationals, in addition to the rationals.1493

Also, why do we call them irrationals?1497

It is nothing because they are crazy and they are something weird; it is because they are just not rational--they are irrational.1499

Irrational numbers...it is just because they are not rational, not because there is anything wrong with them,1506

but just because they are not that set that we call the rationals; that is it.1510

So, if we want to put the rational and irrational numbers together to get something1516

where we can really have all the numbers we work with, we have a great set.1519

That will give us the real numbers: we put them together, and we get the real numbers.1524

These are our bread and butter in mathematics.1529

You are going to be using them for years; you have been using them for pretty much everything you have ever done,1532

unless you have worked on the complex numbers for a little while.1536

And even if you did work on the complex numbers before, it was still using real numbers as part of those complex numbers.1539

The only thing was that i, and it still had a real number right next to it.1545

So, real numbers make up a huge portion of mathematics.1549

And unless you go for a whole bunch more math in college (which I would recommend--I really like math),1553

you are not going to end up seeing, probably, anything other than the real numbers,1559

until you get to some really abstract, interesting math.1563

But it is going to take a while before you see anything other than the reals.1565

They are great things to get at home with, and settle down with, and get a good understanding of.1569

And the purpose of all these set concepts, beforehand, is to be able to get a sense of how this work--1573

"Where do the reals live when we are not moving them around and working with them and doing things with them?"1578

We express them...if we want to be able to talk about them with this nice, simple symbol, we use ℝ, in this blackboard bold font.1584

If we want to be able to write this by hand, we make a normal R, and then we throw down this extra vertical line right here.1592

And that is the symbol for the real numbers (and R stands for real numbers; it makes a lot of sense, unlike some of the other ones).1598

If we want to talk about an interval of the real numbers, if we want to go into that home of real numbers and say,1606

For example, we might want to talk about everything from -1 to 3.1615

We don't want to talk about 100; we don't want to talk about negative one billion; we just want to talk about everything from -1 to 3.1618

So, we use interval notation; if we want to include the end numbers (-1 and 3), we use square brackets.1625

So, square brackets here give us inclusion; they keep those endpoints in it.1633

We go from -1 up until 3, and those points will be there; they are actually going to be part of our interval: -1 and 3 show up.1640

If we want to exclude them (we want everything in between them, but we don't want the end things),1653

then we exclude them by using parentheses; parentheses give us exclusion.1659

That gets us -1 to 3, but without actually having -1 and 3.1665

So, -1 does not show up; 3 does not show up.1671

We use, if we want to symbolize it in a graphical manner (as a picture), open circles like this right here to show exclusion.1677

We use filled-in dots to show inclusion.1685

Exclusion is with parentheses, a curve, empty circle; and inclusion is with a filled-in dot or a nice square, solid bracket.1688

But in either case (-1 to 3 with square brackets or -1 to 3 with parentheses), we are going to always include everything between those.1697

It is just a question of whether or not we are going to include the ends of the interval.1704

If we want to talk about 4 to 7, but we want to not include 4, and we want to include 7, we have (4,7].1710

So, that is going to be all of the real numbers between 4 and 7, of course;1722

but it will keep the number 7 (because we have the square bracket);1726

but it is going to not include 4 (because we have the parenthesis).1729

So, the parenthesis next to the 4 will exclude it--will keep it out; but the square bracket next to the 7 will keep it in.1734

So, we can talk about intervals where one end gets left out, and one end gets kept in, by mixing up how we use this interval notation.1741

If we want to talk about the idea of infinity, then we can talk about going on forever.1750

So, the symbol for infinity--that nice infinity sign--gives us a nice, convenient way to talk about going on forever.1754

So, if we want to talk about the interval going forever in one direction or the other, we will use -∞ or positive ∞.1763

And keep in mind: when there is no symbol in front of it, we just assume that it is positive.1769

So, negative infinity has the negative sign; positive infinity doesn't have anything.1773

If you absolutely had to symbolize that it was the positive version, you could put a little plus sign in front of it.1777

So, that will show us which direction we are going to go forever.1782

Depending on the direction that we want to talk about going forever, we will choose the appropriate infinity, negative or positive.1785

Now, keep in mind: you are always going to use parentheses with negative infinity or infinity.1791

Why is it that we always use parentheses when we are talking about them in interval?1795

It is because we can't actually include infinity: infinity isn't a number.1798

Infinity is just the idea of continuing forever.1803

So, since infinity is an idea of just keeping going, it is not an actual place; so we can't end on it.1806

To have a square bracket implies that we end on it, and it is there.1812

The parenthesis, on the other hand, will just show the idea of keeping going, keeping reaching towards it.1815

You will never actually reach it, but the interval will just keep going towards that notion of infinity.1821

So, for example, we could have -∞ to 2, with a square bracket on the 2.1827

That is going to be all numbers less than or equal to--everything starting at negative infinity, and working all the way up until 2.1831

And we will actually get to 2, and we will achieve 2.1838

(3,∞) is going to be all of the numbers greater than 3, but we won't include 3,1841

because we don't have a bracket on it; we have a parenthesis on the 3.1847

So, it is going to be everything from 3, but not actually including 3.1850

So, we will get really, really, really close to 3, but we will never actually touch it; we will never actually achieve 3.1854

And finally, if we want to just talk about the entire real line, that is the same thing as saying -∞ to positive ∞,1860

because that is everything that the real numbers have.1866

Start all the way from the very beginning; reach all the way to the beginning, and reach all the way to the end.1868

Just keep reaching forever and ever; go all the way to negative infinity; go all the way to positive infinity.1874

That is going to be the same thing as just saying "all the real numbers at once."1879

All right, let's do some examples.1883

We have the set X = {a,b,c}, the set Y = {b,c,d}, and the set Z = {c,d,e}.1885

Let's figure out a couple of different ways to talk about unions and intersections.1892

First, X ∪ Y ∪Z: that is going to be equal to...X ∪ Y is going to be all of the elements included in X and Y.1896

And then, we add "union Z" on that; it is going to be in addition to all of the units with Z.1906

So, it is going to be all of the elements that show up in all of them: a shows up; b shows up;1910

c shows up; well, b already showed up; c already showed up; but d is new.1916

c already showed up; d already showed up; but e is new.1921

So, it is going to be {a,b,c,d,e}: there we go.1924

If we want to talk about X ∩ Y ∩ Z, then that is going to be...what is the only place that they all have in common?1932

What are the elements that are in each and every one of them?1940

Well, a does not show up in Z, nor does it show up in Y.1943

b does not show up in Z; it does show up in Y, but it has to show up in all three of them.1949

c does show up in Y and does show up in Z, so c is in.1954

And since everything else must not show up in X, it must be that the only thing inside of it is c.1957

We can also break this down into two pieces: we can say, "Well, what is X ∩ Y, first?"1963

X ∩ Y would be b and c, because those are the elements X and Y share in common.1967

And then, we intersect that with Z, as well; the only thing that {b,c} shares with Z is the c right here, so we get {c} as our answer to all of them intersecting.1974

If they are all unions, and they are all intersections, it doesn't really matter the order that we choose--1985

which ones to intersect, which ones to "union" first...it is going to be a question of how they all interact.1990

What if we put all the elements in all of them together, or what element is inside of every single one of these sets?1996

So, it doesn't matter about the order; it doesn't matter about how we approach doing it.2003

But it does sometimes matter, if we talk about intersection and union working together.2006

So, for example, if we had (X ∩ Y), and then union Z, well, we have parentheses around it.2012

While we haven't explicitly reminded you of the order of operations, I am sure you remember to do things inside of parentheses first.2019

So, if X ∩ Y is inside of parentheses, then we have to do it first.2025

So, X ∩ Y gives {b,c}; and now we are going to do union Z.2028

Z is going to be c, d, and e; so that gives us a total of {b,c,d,e} in our set.2035

So, {b,c,d,e}: but compare--what if we did it a different way--if we had X being "unioned" with the intersection of Y and Z?2044

Now, we need to start by asking, "Well, what is the intersection of Y and Z?"2054

Well, c and d show up in both of them; e does not show up; b does not show up in both of them.2059

So, c and d make up the intersection of Y and Z.2066

So, X ∪ {c,d} is going to be a and b (because they are new), and c and d (were already there).2070

So, {a,b,c,d) is (X ∪ Y) ∩ Z; but we get a different one if we do (X ∩ Y) ∪ Z: we get {b,c,d,e}.2079

Notice: these two things are not the same--there is not an equivalence between those two sets; they are not equal sets.2088

They aren't the same set, because how we approach putting these things together matters.2096

It is not like 3 times 4 times 5, which is the exact same thing as 4 times 3 times 5, which is the exact same thing as 5 times 4 times 3.2101

It matters how we put these together, because we have different things going on.2109

It is not just multiplication; in a way, it is multiplication and addition--it matters the order that we do it in.2113

So, intersection and union--we can't just do it in any order; we have to pay attention to the order that it has been put together in.2118

The next example: we have ℕ, ℤ, ℚ, and ℝ; we have all of those big number sets that we talked about before.2126

Which one of them will be subsets to the others? How will the subsets work?2133

ℕ is everything from 0...oops, not from 0--I don't believe in that one!...I said that one wrong: 1, 2, 3, 4...just keep going forever.2138

The integers are going to be going off in the negative direction and the positive direction.2151

We have ... up until...and then we meet up...and then we just keep going that way.2156

And if we talk about the rationals, that is the way of saying all integer fractions--fractions made up with integers on the top and bottom.2164

So, that is going to give us the rationals.2173

And the reals are just all numbers--what we are used to as thinking of all the possible numbers--all numbers are the reals.2176

Well, with that in mind, it is pretty easy to see that the natural numbers...2184

Well, since the integers...not equal...subset is what I meant to write...2188

Since the natural numbers are {1,2,3,4...}--they are all the positive integers--they must show up in the integers,2194

because the integers are the positive integers, and the negative integers, and 0.2201

So, ℕ is a subset of ℤ.2205

Now, ℤ shows up in the rationals; how is that possible?2209

Well, if you give me any integer number, I can very easily make a rational number out of it.2212

If you give me -5, well, -5/1 is the same thing as -5; and -5/1 is very clearly contained inside of the rationals: -5/1 is very clearly an element of the rationals.2218

You give me any integers (like -572), and I just put it over 1, and once again, we are back inside of the rationals.2232

So, whatever integer you give me, pretty clearly, has a rational version, as well.2239

We can keep going and now include the reals; we can talk about the reals.2244

And the reals are going to have everything, because we define the reals as having all of the rationals and all of the irrationals.2247

So, the rationals fit inside of the reals, as well; so we have subsets going up:2253

ℕ is a subset of ℤ, is a subset of ℚ, is a subset of ℝ.2257

That also means that, because this is transitive, ℕ is also a subset of ℚ, and ℕ is also a subset of ℝ.2261

ℤ is also a subset of ℝ, as well; and those are all of the relations that we can get out of this.2270

ℕ is a subset, and ℤ is a subset, and ℚ is a subset, inside of ℝ.2277

The third example: if we let A be the set of all titles of all published written works;2282

and B is all of the phrases that are precisely three words long; let's talk about what would be some elements inside of A ∩ B.2288

There are many more answers than I am aware of.2301

Let's also just rephrase this, so we have another way of thinking about it.2308

A is the same thing as talking about...A is every title of books and magazines and poems...2311

it is everything that is a written piece of work that has been published, that we could have actually2328

gone to a store and bought, or found in a published book; A is every title of books, etc., etc., etc.--2332

everything written, that is published--that is what A makes up.2338

Now, B is everything (from the way we are writing this) that is three words long.2342

So, what we are looking for: if we want to find the intersection of A and B, then A ∩ B is going to be things that are in both.2361

So, if you are in both, then to be inside of A ∩ B...that is the same thing as saying "titles that are three words long."2373

So, A ∩ B is just titles that are three words long.2386

To be able to answer this question, we just need to figure out what are some titles that are three words.2397

So, we start thinking, and here are some of the ones that I thought of.2404

We could say Romeo and Juliet, right? Almost everyone is going to know Romeo and Juliet, so that is a good one to start with.2408

Romeo and Juliet: there is a title that is three words long, written by Shakespeare, and it is a published piece of work.2414

We have all been able to find a copy of Romeo and Juliet if we have been looking for it.2424

So, Romeo and Juliet is one.2427

Or we could also talk about something by Kurt Vonnegut: Kurt Vonnegut wrote Breakfast of Champions.2438

So, Breakfast of Champions is another example of something where we have a phrase that is 3 words long,2446

and is the title of something that is a written work.2462

We could also talk about To the Lighthouse by Virginia Woolfe; To the Lighthouse is another example.2466

There are a whole bunch of examples out there; I can't list all of these, because we would be here for days and days and days and days.2475

And I don't know them; but it is going to be anything that is written and has three words in it...2482

3 words...not just in it, but 3 words for the title--precisely 3 words.2488

As much as I would like to be able to say Cannery Row, or Of Mice and Men, or 1984,2493

I can't talk about those, because they are not precisely 3 words long.2502

There are a lot of books out there that aren't 3 words long in the title.2507

And there are lots of phrases that are three words long, like "hot in here" (sorry, I didn't come up with any brilliant phrases in that period of time).2512

But any phrase that is three words long would be in B, and any title would be in A.2523

But what we are looking for is the intersection of A and B--titles that are three words.2528

Romeo and Juliet, Things Fall Apart, Breakfast of Champions,2533

To the Lighthouse: these are all some examples from various different authors.2536

The final example, Example 4: List all of the subsets of {x,y,z}.2542

The very first subset that we have to remember is the empty set: the empty set shows up as a subset for everything.2546

The empty set is our very first subset.2553

The next one--well, let's look at all of the subsets that have one element inside of them.2556

{x} (oops, I made a really bad bracket there) is going to be a set, just on its own; and that is a subset.2561

Another one would be {y}; that is another subset.2570

Another one would be {z}; those are all of the sets that are one element long, and are subsets of {x,y,z}.2574

Now, we can go with the two-element ones, and we can say, "All right, well, {x,y}--that is going to be a subset."2582

What about {x,z}? And then, finally, there is {y,z}.2593

And we think about that for a little while, and we realize that those are all the sets I can possibly make out of {x,y,z}2600

that have 2 elements precisely in them: x and y, x and z, y and z.2606

You could rearrange them in different orders, but remember, since it is a set we are talking about, order is not important.2612

It doesn't matter the order that it shows up in--just that it did show up at all.2617

Those are all of the sets that are going to be two elements long, and are subsets of {x,y,z}.2621

And then, finally, we have {x,y,z} itself; it is a subset of itself, because remember, by the formal definition2625

of being a subset, it just means that all of the elements inside of your set show up in the other set.2632

And every element {x,y,z} shows up inside of {x,y,z}; it makes sense; so every set is a subset of itself.2638

It is kind of obvious, and not that really interesting; but it is another trivial assertion.2646

It is interesting to think about, but not something that really gains us a lot of knowledge of any specific thing.2651

But it is still an interesting idea, and might have other connections later on, if we think about it a lot.2656

All right, so that gives us a total of 8 subsets; and those are all of them.2661

All right, I hope you enjoyed this; I hope you learned something about sets.2667

Like I said before, we are not going to really focus on the ideas that we had here.2670

But what we just did was built the foundation of pretty much everything else that you are going to end up ever seeing in math.2674

Virtually all of modern mathematics is built upon the idea of set theory.2679

It can be explained through the idea of set theory.2683

So, I just wanted you to get some exposure to this foundation, so that later things we talk about,2685

like when we talk about functions and a whole bunch of things, in fact, we have some idea of being able2689

to refer back to these sets, pulling things out from sets, going to other sets.2693

There is really cool stuff here; set theory is really fascinating; I totally recommend studying it sometime, if you get the chance.2697

I am glad that you managed to get here, and that you have some idea of how sets work.2704

And we will see you in the next lesson--goodbye!2707

Talk to you later at Educator.com!2709