SlideShare a Scribd company logo

Sets Notes

L
Lori Rapp

The document describes sets and Venn diagrams using data about members of math, science, and chess clubs. It provides examples of representing sets using brackets and defining the intersection, union, and relationship between sets visually in a Venn diagram. Key points covered include using set notation to represent membership of each club, observing relationships like some students belonging to multiple clubs, and how intersection, union, and Venn diagrams can model relationships between sets.

EducationEntertainment & Humor
1 of 127
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
Venn Diagrams & Sets
• Adam, Barb, Ciera, Dante, and Ed are members of the Math Club.
• Adam, Barb, Ciera, Dante, and Ed are members of the Math Club. • Dante, Ed, Farrah, Gabby, Hali, and Ian are members of the Science Club.
• Adam, Barb, Ciera, Dante, and Ed are members of the Math Club. • Dante, Ed, Farrah, Gabby, Hali, and Ian are members of the Science Club. • Adam, Ed, Farrah, and Gabby are members of the Chess Club.
• Adam, Barb, Ciera, Dante, and Ed are members of the Math Club. • Dante, Ed, Farrah, Gabby, Hali, and Ian are members of the Science Club. • Adam, Ed, Farrah, and Gabby are members of the Chess Club. • Your task is to organize this data on paper in any manner you choose (such as a diagram, table, etc.). Then you will answer questions about the data.
Possible Table Organization Adam Barb Ciera Dante Ed Farrah Gabby Hali Ian Math X X X X X Science X X X X X X Chess X X X X
Possible Diagram Organization Sc l ub ien h C ce Mat Cl Barb ub Dante Hali Ciera Ian Ed Adam Farrah Gabby Chess Club
Venn Diagram
Venn Diagram • The last data organization is called a Venn Diagram.
Venn Diagram • The last data organization is called a Venn Diagram. • Useful for organizing 2 or more sets.
Venn Diagram • The last data organization is called a Venn Diagram. • Useful for organizing 2 or more sets. • Can you determine the sets we used based on the Venn Diagram?
Venn Diagram • The last data organization is called a Venn Diagram. • Useful for organizing 2 or more sets. • Can you determine the sets we used based on the Venn Diagram? • The sets we used are Math Club, Science Club, and Chess Club.
Use the Venn Diagram to answer...
Use the Venn Diagram to answer... • What observation can you make about each student?
Use the Venn Diagram to answer... • What observation can you make about each student? • What observation can you make about each club?
Use the Venn Diagram to answer... • What observation can you make about each student? • What observation can you make about each club? • What are the benefits to visualizing the relationship between the clubs and students?
• Organizing the information helps us see the relationships between the students and clubs.
• Organizing the information helps us see the relationships between the students and clubs. • All members of the Chess Club are in either the Math Club, the Science Club or both clubs.
• Organizing the information helps us see the relationships between the students and clubs. • All members of the Chess Club are in either the Math Club, the Science Club or both clubs. • Farrah and Gabby could play chess during Math Club.
• Organizing the information helps us see the relationships between the students and clubs. • All members of the Chess Club are in either the Math Club, the Science Club or both clubs. • Farrah and Gabby could play chess during Math Club. • Dante and Ed are in both the Math Club and Science Club.
• Organizing the information helps us see the relationships between the students and clubs. • All members of the Chess Club are in either the Math Club, the Science Club or both clubs. • Farrah and Gabby could play chess during Math Club. • Dante and Ed are in both the Math Club and Science Club. • These are just a few observations from the Venn Diagram.
Math Club (M) as a Set
Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed}
Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed} What do you notice about the set M?
Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed} What do you notice about the set M? • The set is enclosed in curly brackets.
Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed} What do you notice about the set M? • The set is enclosed in curly brackets. • Each name is separated by a comma.
Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed} What do you notice about the set M? • The set is enclosed in curly brackets. • Each name is separated by a comma. • The word “and” is not written in the set.
Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed} What do you notice about the set M? • The set is enclosed in curly brackets. • Each name is separated by a comma. • The word “and” is not written in the set. • Each “piece” in a set is called an element.
You try...
You try... • Write the members of the Science Club (S) as a set.
You try... • Write the members of the Science Club (S) as a set. S = {Dante, Ed, Farrah, Gabby, Hali, Ian}
You try... • Write the members of the Science Club (S) as a set. S = {Dante, Ed, Farrah, Gabby, Hali, Ian} • Write the members of the Chess Club (C) as a set.
You try... • Write the members of the Science Club (S) as a set. S = {Dante, Ed, Farrah, Gabby, Hali, Ian} • Write the members of the Chess Club (C) as a set. C = {Adam, Ed, Farrah, Gabby}
Driving down the road...
Driving down the road... • Imagine driving down Main Street and stopping at a traffic light. Green Street crosses Main Street at the traffic light. What is the piece of the road that belongs to Main Street and Green Street called?
Driving down the road... • Imagine driving down Main Street and stopping at a traffic light. Green Street crosses Main Street at the traffic light. What is the piece of the road that belongs to Main Street and Green Street called? • Intersection!
Driving down the road... • Imagine driving down Main Street and stopping at a traffic light. Green Street crosses Main Street at the traffic light. What is the piece of the road that belongs to Main Street and Green Street called? • Intersection! • In math, intersection has the same meaning. A element that belongs to more than one set Intersects the sets.
Math & Chess Clubs
Math & Chess Clubs • Who is in the Math Club and the Chess Club?
Math & Chess Clubs • Who is in the Math Club and the Chess Club? • Adam and Ed
Math & Chess Clubs • Who is in the Math Club and the Chess Club? • Adam and Ed • They are the Intersection of the Math Club & Chess Club sets.
Math & Chess Clubs • Who is in the Math Club and the Chess Club? • Adam and Ed • They are the Intersection of the Math Club & Chess Club sets. • The symbol to indicate intersection is ∩.
Math & Chess Clubs • Who is in the Math Club and the Chess Club? • Adam and Ed • They are the Intersection of the Math Club & Chess Club sets. • The symbol to indicate intersection is ∩. • M ∩ C = { Adam, Ed}
You try...
You try... • M∩S=
You try... • M∩S= {Dante, Ed}
You try... • M∩S= {Dante, Ed} • S∩C=
You try... • M∩S= {Dante, Ed} • S∩C= {Ed, Farrah, Gabby}
You try... • M∩S= {Dante, Ed} • S∩C= {Ed, Farrah, Gabby} • M∩S∩C=
You try... • M∩S= {Dante, Ed} • S∩C= {Ed, Farrah, Gabby} • M∩S∩C= {Ed}
Labor Unions...
Labor Unions... • Labor Unions address the rights of all workers. If you work at McDonald’s or are a nurse at a hospital, the Labor Union represents your rights.
Labor Unions... • Labor Unions address the rights of all workers. If you work at McDonald’s or are a nurse at a hospital, the Labor Union represents your rights. • In Math, a union has the same idea. It brings together the members of all sets.
Labor Unions... • Labor Unions address the rights of all workers. If you work at McDonald’s or are a nurse at a hospital, the Labor Union represents your rights. • In Math, a union has the same idea. It brings together the members of all sets. • The symbol for Union is ∪. Notice it looks like the letter U.
Back to our clubs...
Back to our clubs... • The Union of the Math Club and Chess Clubs is everyone in both clubs.
Back to our clubs... • The Union of the Math Club and Chess Clubs is everyone in both clubs. • M = {Adam, Barb, Ciera, Dante, Ed}
Back to our clubs... • The Union of the Math Club and Chess Clubs is everyone in both clubs. • M = {Adam, Barb, Ciera, Dante, Ed} • S = {Dante, Ed, Farrah, Gabby, Hali, Ian}
Back to our clubs... • The Union of the Math Club and Chess Clubs is everyone in both clubs. • M = {Adam, Barb, Ciera, Dante, Ed} • S = {Dante, Ed, Farrah, Gabby, Hali, Ian} M ∪ C = {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby}
Back to our clubs... • The Union of the Math Club and Chess Clubs is everyone in both clubs. • M = {Adam, Barb, Ciera, Dante, Ed} • S = {Dante, Ed, Farrah, Gabby, Hali, Ian} M ∪ C = {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby} • Notice that Dante and Ed are only listed once even though they are in both clubs.
You try...
You try... • M∪S=
You try... • M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian}
You try... • M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • S∪C=
You try... • M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • S∪C= {Dante, Ed, Farrah, Gabby, Hali, Ian, Adam}
You try... • M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • S∪C= {Dante, Ed, Farrah, Gabby, Hali, Ian, Adam} • M∪S∪C=
You try... • M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • S∪C= {Dante, Ed, Farrah, Gabby, Hali, Ian, Adam} • M∪S∪C= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian}
What about the outsiders?
What about the outsiders? • Who is not in the Math Club?
What about the outsiders? • Who is not in the Math Club? • Hali, Ian, Farrah, and Gabby
What about the outsiders? • Who is not in the Math Club? • Hali, Ian, Farrah, and Gabby • Not is represented by the symbol ~.
What about the outsiders? • Who is not in the Math Club? • Hali, Ian, Farrah, and Gabby • Not is represented by the symbol ~. • Not is also called the Complement.
What about the outsiders? • Who is not in the Math Club? • Hali, Ian, Farrah, and Gabby • Not is represented by the symbol ~. • Not is also called the Complement. • ~M = {Hali, Ian, Farrah, Gabby}
~(M ∪ S)
~(M ∪ S) • Order of Operations starts with parenthesis. This also applies to sets.
~(M ∪ S) • Order of Operations starts with parenthesis. This also applies to sets. • First, find the union of M and S.
~(M ∪ S) • Order of Operations starts with parenthesis. This also applies to sets. • First, find the union of M and S. • M ∪ S = {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian}
~(M ∪ S) • Order of Operations starts with parenthesis. This also applies to sets. • First, find the union of M and S. • M ∪ S = {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • Next, find the complement of the union. In-other-words, everyone who is not in M ∪ S
~(M ∪ S) • Order of Operations starts with parenthesis. This also applies to sets. • First, find the union of M and S. • M ∪ S = {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • Next, find the complement of the union. In-other-words, everyone who is not in M ∪ S • ~(M ∪ S) = { }
~(M ∪ S) • Order of Operations starts with parenthesis. This also applies to sets. • First, find the union of M and S. • M ∪ S = {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • Next, find the complement of the union. In-other-words, everyone who is not in M ∪ S • ~(M ∪ S) = { } • Everyone is in either the Math Club or Science Club so we have an empty set. The symbol ∅ or { } represents an empty set.
Try these...
Try these... • ~S=
Try these... • ~S= • {Farrah, Hali, Ian}
Try these... • ~S= • {Farrah, Hali, Ian} • ~ (S ∪ C) =
Try these... • ~S= • {Farrah, Hali, Ian} • ~ (S ∪ C) = • {Barb, Ciera}
Try these... • ~S= • {Farrah, Hali, Ian} • ~ (S ∪ C) = • {Barb, Ciera} • ~ (S ∪ C ∪ M) =
Try these... • ~S= • {Farrah, Hali, Ian} • ~ (S ∪ C) = • {Barb, Ciera} • ~ (S ∪ C ∪ M) = • { } or ∅
How can I use a Venn Diagram? • Visit this Interactive site by Shodor Education Foundation to see how Set operations and a Venn Diagram are applied to an internet search to find useful resources.
Finite versus Infinite
Finite versus Infinite • So far we have dealt with Finite sets.
Finite versus Infinite • So far we have dealt with Finite sets. • What does finite mean?
Finite versus Infinite • So far we have dealt with Finite sets. • What does finite mean? • Finite means there is a limit. In-other-words, we can count the number of items in a finite set.
Finite versus Infinite • So far we have dealt with Finite sets. • What does finite mean? • Finite means there is a limit. In-other-words, we can count the number of items in a finite set. • What do you think infinite means?
Finite versus Infinite • So far we have dealt with Finite sets. • What does finite mean? • Finite means there is a limit. In-other-words, we can count the number of items in a finite set. • What do you think infinite means? • Infinite means there is no limit. It goes on and on and on and on.... Well, you get the idea.
Number Line -5 -4 -3 -2 -1 0 1 2 3 4 5
Number Line -5 -4 -3 -2 -1 0 1 2 3 4 5 • Recall a number is a graphical representation of all numbers.
Number Line -5 -4 -3 -2 -1 0 1 2 3 4 5 • Recall a number is a graphical representation of all numbers. • What term can you apply to the end numbers of a number line?
Number Line -5 -4 -3 -2 -1 0 1 2 3 4 5 • Recall a number is a graphical representation of all numbers. • What term can you apply to the end numbers of a number line? • Infinite because you can never get to the end. It goes on and on.
Number Line -5 -4 -3 -2 -1 0 1 2 3 4 5 • Recall a number is a graphical representation of all numbers. • What term can you apply to the end numbers of a number line? • Infinite because you can never get to the end. It goes on and on. • The negative numbers extend to the left forever getting smaller and smaller.
Number Line -5 -4 -3 -2 -1 0 1 2 3 4 5 • Recall a number is a graphical representation of all numbers. • What term can you apply to the end numbers of a number line? • Infinite because you can never get to the end. It goes on and on. • The negative numbers extend to the left forever getting smaller and smaller. • The positive numbers extend to the right forever getting larger and larger.
Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5
Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • A closed circle is used to indicate a solution on the number line.
Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • A closed circle is used to indicate a solution on the number line. • So x = 2 is at the red circle.
Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • A closed circle is used to indicate a solution on the number line. • So x = 2 is at the red circle. • Shading is used to indicate additional solutions.
Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • A closed circle is used to indicate a solution on the number line. • So x = 2 is at the red circle. • Shading is used to indicate additional solutions. • Such as x ≤ 2 is the red circle and red arrow to the left.
Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • A closed circle is used to indicate a solution on the number line. • So x = 2 is at the red circle. • Shading is used to indicate additional solutions. • Such as x ≤ 2 is the red circle and red arrow to the left. • The arrow at the end indicates the solution goes on forever.
Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5
Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • An open circle is used to indicate the number is not a solution on the number line.
Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • An open circle is used to indicate the number is not a solution on the number line. • Here is the the graph of x < 2.
Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • An open circle is used to indicate the number is not a solution on the number line. • Here is the the graph of x < 2. • The open red circle means x = 2 is not a solution but arrow to the left indicate all the number to the left of 2 are solutions.
Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • An open circle is used to indicate the number is not a solution on the number line. • Here is the the graph of x < 2. • The open red circle means x = 2 is not a solution but arrow to the left indicate all the number to the left of 2 are solutions. • Again, the arrow indicates the solution goes on forever in the negative direction.
Writing Solution Sets - Part I -5 -4 -3 -2 -1 0 1 2 3 4 5
Writing Solution Sets - Part I -5 -4 -3 -2 -1 0 1 2 3 4 5 • The solution x ≤ 2 can be written several ways.
Writing Solution Sets - Part I -5 -4 -3 -2 -1 0 1 2 3 4 5 • The solution x ≤ 2 can be written several ways. • Recall a set is enclosed in curly brackets, { }.
Writing Solution Sets - Part I -5 -4 -3 -2 -1 0 1 2 3 4 5 • The solution x ≤ 2 can be written several ways. • Recall a set is enclosed in curly brackets, { }. • One way is {x ≤ 2}. Because our solution is a set, this is the technical way to write the set.
Writing Solution Sets - Part I -5 -4 -3 -2 -1 0 1 2 3 4 5 • The solution x ≤ 2 can be written several ways. • Recall a set is enclosed in curly brackets, { }. • One way is {x ≤ 2}. Because our solution is a set, this is the technical way to write the set. • It is understood that x ≤ 2 is a set so often the { } are not included. So x ≤ 2 is acceptable.
Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5
Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex.
Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation.
Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation:
Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2}
Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2} • This reads as
Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2} • This reads as the set of all x such that x is less than or equal to 2.
Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2} • This reads as the set of all x such that x is less than or equal to 2.
Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2} • This reads as the set of all x such that x is less than or equal to 2.
Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2} • This reads as the set of all x such that x is less than or equal to 2.
Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2} • This reads as the set of all x such that x is less than or equal to 2.

More Related Content

PDF
Venn diagrams
Lori Rapp
 
PPTX
Junior jets version 6
dmkenney
 
PPTX
Tangent lines PowerPoint
zackomack
 
PPT
Lecture 07
Faisal Mehmood
 
PDF
Unit 1 sets lecture notes
josephdethrow
 
PDF
Pre-Cal 30S December 16, 2008
Darren Kuropatwa
 
PPT
03 more on sets and venn
ethelremitio
 
PPT
Project Town Naku Sa Math
Bret Monsanto
 
Venn diagrams
Lori Rapp
 
Junior jets version 6
dmkenney
 
Tangent lines PowerPoint
zackomack
 
Lecture 07
Faisal Mehmood
 
Unit 1 sets lecture notes
josephdethrow
 
Pre-Cal 30S December 16, 2008
Darren Kuropatwa
 
03 more on sets and venn
ethelremitio
 
Project Town Naku Sa Math
Bret Monsanto
 

Viewers also liked (19)

PPTX
Class 5 - Set Theory and Venn Diagrams
Stephen Parsons
 
PPTX
Sets 3
joycyarmstrong123
 
PPTX
Diagram venn
Pamujiarso Wibowo
 
PDF
Circles and Tangent Lines
Leo Crisologo
 
PPTX
Sets
Juan Apolinario Reyes
 
POTX
Powerpoint slide venn diagram
EdwinaBoggs
 
PPTX
Circle theorem powerpoint updated
Eleanor Gotrel
 
PPT
Funny Venn Diagrams
dirigible
 
PPT
CST 504 Venn Diagrams
Neil MacIntosh
 
PDF
Curves And Surfaces Representation And Application
Diaa ElKott
 
PPTX
Three Circle Venn Diagrams
Passy World
 
PPTX
Venn Diagrams Templates by Creately
Creately
 
PDF
Nakisha wheatley a beginner´s guide to become a better chess player+
Izzquierdo
 
PPT
10.1 tangents to circles
amriawalludin2
 
PPTX
Venn Diagram
khyliesweet
 
PPTX
Sets venn diagrams
zahoormirza194
 
PPTX
Circle theorem powerpoint
ebayliss
 
PPT
Mensuration
ARJUN RASTOGI
 
PPT
Set Theory
TCYonline No.1 testing platform(Top Careers and You)
 
Class 5 - Set Theory and Venn Diagrams
Stephen Parsons
 
Sets 3
joycyarmstrong123
 
Diagram venn
Pamujiarso Wibowo
 
Circles and Tangent Lines
Leo Crisologo
 
Sets
Juan Apolinario Reyes
 
Powerpoint slide venn diagram
EdwinaBoggs
 
Circle theorem powerpoint updated
Eleanor Gotrel
 
Funny Venn Diagrams
dirigible
 
CST 504 Venn Diagrams
Neil MacIntosh
 
Curves And Surfaces Representation And Application
Diaa ElKott
 
Three Circle Venn Diagrams
Passy World
 
Venn Diagrams Templates by Creately
Creately
 
Nakisha wheatley a beginner´s guide to become a better chess player+
Izzquierdo
 
10.1 tangents to circles
amriawalludin2
 
Venn Diagram
khyliesweet
 
Sets venn diagrams
zahoormirza194
 
Circle theorem powerpoint
ebayliss
 
Mensuration
ARJUN RASTOGI
 
Set Theory
TCYonline No.1 testing platform(Top Careers and You)
 
Ad

Similar to Sets Notes (20)

PPT
Theory of the sets and Venn diagram-grade 9 math
ivanoberknezev
 
PPTX
Q2 LESSON 7.pptx sets, subsets and intersection
limfamelgo1
 
PPTX
SET THEORY
Aluko Sayo Enoch
 
PDF
Tg 9780195979701
Hafiz Akhtar
 
DOCX
The importance of math
Tayyaba Syed
 
PDF
Sets.pdf
EllenGrace9
 
PPT
Sets powerpoint.ppt
giovanniealvarez1
 
PPTX
Sets Class XI Chapter 1
Nishant Narvekar
 
PPTX
Module week 1 Q1
Rommel Limbauan
 
PPT
Exploring Venn Diagrams in Discrete Mathematics
Dr Chetan Bawankar
 
PDF
Language of Sets
Arlene Leron
 
PPTX
Q2W7-Lesson 1-4- Sets powerpoint presentation
rinatugade30
 
PPTX
SET AND ITS OPERATIONS
RohithV15
 
PPTX
CLASS- 11 Mathematics PPT 2024-25 1 SETS.pptx
sahil388378
 
PPTX
MAAM LENG PPT YAWAAAAA.pptx444156+78196879678697
dagudogalbert5
 
PPTX
Sets.pptx
EllenGrace9
 
PPT
12671414.ppt
franciskishushu
 
PPT
union of ...................SETS-ppt.ppt
MichaelCorpuz11
 
PPTX
#17 Boba PPT Template (1).pptx
KimSeungyoon2
 
PPT
Applications of set theory
Tarun Gehlot
 
Theory of the sets and Venn diagram-grade 9 math
ivanoberknezev
 
Q2 LESSON 7.pptx sets, subsets and intersection
limfamelgo1
 
SET THEORY
Aluko Sayo Enoch
 
Tg 9780195979701
Hafiz Akhtar
 
The importance of math
Tayyaba Syed
 
Sets.pdf
EllenGrace9
 
Sets powerpoint.ppt
giovanniealvarez1
 
Sets Class XI Chapter 1
Nishant Narvekar
 
Module week 1 Q1
Rommel Limbauan
 
Exploring Venn Diagrams in Discrete Mathematics
Dr Chetan Bawankar
 
Language of Sets
Arlene Leron
 
Q2W7-Lesson 1-4- Sets powerpoint presentation
rinatugade30
 
SET AND ITS OPERATIONS
RohithV15
 
CLASS- 11 Mathematics PPT 2024-25 1 SETS.pptx
sahil388378
 
MAAM LENG PPT YAWAAAAA.pptx444156+78196879678697
dagudogalbert5
 
Sets.pptx
EllenGrace9
 
12671414.ppt
franciskishushu
 
union of ...................SETS-ppt.ppt
MichaelCorpuz11
 
#17 Boba PPT Template (1).pptx
KimSeungyoon2
 
Applications of set theory
Tarun Gehlot
 
Ad

More from Lori Rapp (20)

PDF
Piecewise functions
Lori Rapp
 
PDF
Normal curve
Lori Rapp
 
PPT
Circles notes
Lori Rapp
 
PPT
Quadrilateral notes
Lori Rapp
 
KEY
Remainder & Factor Theorems
Lori Rapp
 
KEY
Multiplying polynomials - part 1
Lori Rapp
 
KEY
Develop the Area of a Circle Formula
Lori Rapp
 
KEY
Unit 4 hw 8 - pointslope, parallel & perp
Lori Rapp
 
KEY
Absolute Value Inequalities Notes
Lori Rapp
 
KEY
Compound Inequalities Notes
Lori Rapp
 
KEY
Solving Inequalities Notes
Lori Rapp
 
KEY
Solving quadratic equations part 1
Lori Rapp
 
KEY
Introduction to Equations Notes
Lori Rapp
 
KEY
Associative property
Lori Rapp
 
PDF
Real numbers
Lori Rapp
 
KEY
Unit 4 hw 7 - direct variation & linear equation give 2 points
Lori Rapp
 
KEY
Absolute Value Equations
Lori Rapp
 
KEY
Unit 3 hw 7 - literal equations
Lori Rapp
 
KEY
Unit 3 hw 4 - solving equations variable both sides
Lori Rapp
 
KEY
Unit 3 hw 2 - solving 1 step equations
Lori Rapp
 
Piecewise functions
Lori Rapp
 
Normal curve
Lori Rapp
 
Circles notes
Lori Rapp
 
Quadrilateral notes
Lori Rapp
 
Remainder & Factor Theorems
Lori Rapp
 
Multiplying polynomials - part 1
Lori Rapp
 
Develop the Area of a Circle Formula
Lori Rapp
 
Unit 4 hw 8 - pointslope, parallel & perp
Lori Rapp
 
Absolute Value Inequalities Notes
Lori Rapp
 
Compound Inequalities Notes
Lori Rapp
 
Solving Inequalities Notes
Lori Rapp
 
Solving quadratic equations part 1
Lori Rapp
 
Introduction to Equations Notes
Lori Rapp
 
Associative property
Lori Rapp
 
Real numbers
Lori Rapp
 
Unit 4 hw 7 - direct variation & linear equation give 2 points
Lori Rapp
 
Absolute Value Equations
Lori Rapp
 
Unit 3 hw 7 - literal equations
Lori Rapp
 
Unit 3 hw 4 - solving equations variable both sides
Lori Rapp
 
Unit 3 hw 2 - solving 1 step equations
Lori Rapp
 

Recently uploaded (20)

PPTX
Renaissance Architecture: A Journey from Faith to Humanism
DrMonicaSharma
 
PDF
Saundersa Comprehensive Review for the NCLEX-RN Examination.pdf
sreejithcareers
 
PDF
Chapter 2 Heredity, Prenatal Development, and Birth.pdf
MarjorieLopezTiu
 
PPTX
Institutional Correction lecture only . . .
limvtheghins
 
PPTX
IMMUNITY IMMUNITY refers to protection against infection, and the immune syst...
shilpa939953
 
PDF
Physiotherapy_for_Respiratory_and_Cardiac_Problems WEBBER.pdf
DrMONISHAphysio
 
PDF
102 student loan defaulters named and shamed – Is someone you know on the list?
Kweku Zurek
 
PDF
01-Introduction-to-Information-Management.pdf
PrinceIbarra
 
PPTX
Cell Types and Its function , kingdom of life
ClaireMangundayao1
 
PDF
FourierSeries-QuestionsWithAnswers(Part-A).pdf
Raji Murugan
 
PPTX
master seminar digital applications in india
ananyaray35
 
PPTX
Final Presentation General Medicine 03-08-2024.pptx
ZaheerAhmad228692
 
PDF
Abdominal Access Techniques with Prof. Dr. R K Mishra
mohitsuren827
 
PDF
Basic Mud Logging Guide for educational purpose
wdandworks
 
PDF
grade 11-chemistry_fetena_net_5883.pdf teacher guide for all student
banteamlakmebratu738
 
PDF
Complications of Minimal Access Surgery at WLH
mohitsuren827
 
PDF
O5-L3 Freight Transport Ops (International) V1.pdf
labyankof
 
PPTX
PPH.pptx obstetrics and gynecology in nursing
SrideviDevaraj5
 
PPTX
PPT- ENG7_QUARTER1_LESSON1_WEEK1. IMAGERY -DESCRIPTIONS pptx.pptx
KMDee
 
PDF
RMMM.pdf make it easy to upload and study
sajedul2
 
Renaissance Architecture: A Journey from Faith to Humanism
DrMonicaSharma
 
Saundersa Comprehensive Review for the NCLEX-RN Examination.pdf
sreejithcareers
 
Chapter 2 Heredity, Prenatal Development, and Birth.pdf
MarjorieLopezTiu
 
Institutional Correction lecture only . . .
limvtheghins
 
IMMUNITY IMMUNITY refers to protection against infection, and the immune syst...
shilpa939953
 
Physiotherapy_for_Respiratory_and_Cardiac_Problems WEBBER.pdf
DrMONISHAphysio
 
102 student loan defaulters named and shamed – Is someone you know on the list?
Kweku Zurek
 
01-Introduction-to-Information-Management.pdf
PrinceIbarra
 
Cell Types and Its function , kingdom of life
ClaireMangundayao1
 
FourierSeries-QuestionsWithAnswers(Part-A).pdf
Raji Murugan
 
master seminar digital applications in india
ananyaray35
 
Final Presentation General Medicine 03-08-2024.pptx
ZaheerAhmad228692
 
Abdominal Access Techniques with Prof. Dr. R K Mishra
mohitsuren827
 
Basic Mud Logging Guide for educational purpose
wdandworks
 
grade 11-chemistry_fetena_net_5883.pdf teacher guide for all student
banteamlakmebratu738
 
Complications of Minimal Access Surgery at WLH
mohitsuren827
 
O5-L3 Freight Transport Ops (International) V1.pdf
labyankof
 
PPH.pptx obstetrics and gynecology in nursing
SrideviDevaraj5
 
PPT- ENG7_QUARTER1_LESSON1_WEEK1. IMAGERY -DESCRIPTIONS pptx.pptx
KMDee
 
RMMM.pdf make it easy to upload and study
sajedul2
 

Sets Notes

  • 1. Venn Diagrams & Sets
  • 2. • Adam, Barb, Ciera, Dante, and Ed are members of the Math Club.
  • 3. • Adam, Barb, Ciera, Dante, and Ed are members of the Math Club. • Dante, Ed, Farrah, Gabby, Hali, and Ian are members of the Science Club.
  • 4. • Adam, Barb, Ciera, Dante, and Ed are members of the Math Club. • Dante, Ed, Farrah, Gabby, Hali, and Ian are members of the Science Club. • Adam, Ed, Farrah, and Gabby are members of the Chess Club.
  • 5. • Adam, Barb, Ciera, Dante, and Ed are members of the Math Club. • Dante, Ed, Farrah, Gabby, Hali, and Ian are members of the Science Club. • Adam, Ed, Farrah, and Gabby are members of the Chess Club. • Your task is to organize this data on paper in any manner you choose (such as a diagram, table, etc.). Then you will answer questions about the data.
  • 6. Possible Table Organization Adam Barb Ciera Dante Ed Farrah Gabby Hali Ian Math X X X X X Science X X X X X X Chess X X X X
  • 7. Possible Diagram Organization Sc l ub ien h C ce Mat Cl Barb ub Dante Hali Ciera Ian Ed Adam Farrah Gabby Chess Club
  • 9. Venn Diagram • The last data organization is called a Venn Diagram.
  • 10. Venn Diagram • The last data organization is called a Venn Diagram. • Useful for organizing 2 or more sets.
  • 11. Venn Diagram • The last data organization is called a Venn Diagram. • Useful for organizing 2 or more sets. • Can you determine the sets we used based on the Venn Diagram?
  • 12. Venn Diagram • The last data organization is called a Venn Diagram. • Useful for organizing 2 or more sets. • Can you determine the sets we used based on the Venn Diagram? • The sets we used are Math Club, Science Club, and Chess Club.
  • 13. Use the Venn Diagram to answer...
  • 14. Use the Venn Diagram to answer... • What observation can you make about each student?
  • 15. Use the Venn Diagram to answer... • What observation can you make about each student? • What observation can you make about each club?
  • 16. Use the Venn Diagram to answer... • What observation can you make about each student? • What observation can you make about each club? • What are the benefits to visualizing the relationship between the clubs and students?
  • 17. • Organizing the information helps us see the relationships between the students and clubs.
  • 18. • Organizing the information helps us see the relationships between the students and clubs. • All members of the Chess Club are in either the Math Club, the Science Club or both clubs.
  • 19. • Organizing the information helps us see the relationships between the students and clubs. • All members of the Chess Club are in either the Math Club, the Science Club or both clubs. • Farrah and Gabby could play chess during Math Club.
  • 20. • Organizing the information helps us see the relationships between the students and clubs. • All members of the Chess Club are in either the Math Club, the Science Club or both clubs. • Farrah and Gabby could play chess during Math Club. • Dante and Ed are in both the Math Club and Science Club.
  • 21. • Organizing the information helps us see the relationships between the students and clubs. • All members of the Chess Club are in either the Math Club, the Science Club or both clubs. • Farrah and Gabby could play chess during Math Club. • Dante and Ed are in both the Math Club and Science Club. • These are just a few observations from the Venn Diagram.
  • 22. Math Club (M) as a Set
  • 23. Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed}
  • 24. Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed} What do you notice about the set M?
  • 25. Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed} What do you notice about the set M? • The set is enclosed in curly brackets.
  • 26. Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed} What do you notice about the set M? • The set is enclosed in curly brackets. • Each name is separated by a comma.
  • 27. Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed} What do you notice about the set M? • The set is enclosed in curly brackets. • Each name is separated by a comma. • The word “and” is not written in the set.
  • 28. Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed} What do you notice about the set M? • The set is enclosed in curly brackets. • Each name is separated by a comma. • The word “and” is not written in the set. • Each “piece” in a set is called an element.
  • 30. You try... • Write the members of the Science Club (S) as a set.
  • 31. You try... • Write the members of the Science Club (S) as a set. S = {Dante, Ed, Farrah, Gabby, Hali, Ian}
  • 32. You try... • Write the members of the Science Club (S) as a set. S = {Dante, Ed, Farrah, Gabby, Hali, Ian} • Write the members of the Chess Club (C) as a set.
  • 33. You try... • Write the members of the Science Club (S) as a set. S = {Dante, Ed, Farrah, Gabby, Hali, Ian} • Write the members of the Chess Club (C) as a set. C = {Adam, Ed, Farrah, Gabby}
  • 34. Driving down the road...
  • 35. Driving down the road... • Imagine driving down Main Street and stopping at a traffic light. Green Street crosses Main Street at the traffic light. What is the piece of the road that belongs to Main Street and Green Street called?
  • 36. Driving down the road... • Imagine driving down Main Street and stopping at a traffic light. Green Street crosses Main Street at the traffic light. What is the piece of the road that belongs to Main Street and Green Street called? • Intersection!
  • 37. Driving down the road... • Imagine driving down Main Street and stopping at a traffic light. Green Street crosses Main Street at the traffic light. What is the piece of the road that belongs to Main Street and Green Street called? • Intersection! • In math, intersection has the same meaning. A element that belongs to more than one set Intersects the sets.
  • 38. Math & Chess Clubs
  • 39. Math & Chess Clubs • Who is in the Math Club and the Chess Club?
  • 40. Math & Chess Clubs • Who is in the Math Club and the Chess Club? • Adam and Ed
  • 41. Math & Chess Clubs • Who is in the Math Club and the Chess Club? • Adam and Ed • They are the Intersection of the Math Club & Chess Club sets.
  • 42. Math & Chess Clubs • Who is in the Math Club and the Chess Club? • Adam and Ed • They are the Intersection of the Math Club & Chess Club sets. • The symbol to indicate intersection is ∩.
  • 43. Math & Chess Clubs • Who is in the Math Club and the Chess Club? • Adam and Ed • They are the Intersection of the Math Club & Chess Club sets. • The symbol to indicate intersection is ∩. • M ∩ C = { Adam, Ed}
  • 46. You try... • M∩S= {Dante, Ed}
  • 47. You try... • M∩S= {Dante, Ed} • S∩C=
  • 48. You try... • M∩S= {Dante, Ed} • S∩C= {Ed, Farrah, Gabby}
  • 49. You try... • M∩S= {Dante, Ed} • S∩C= {Ed, Farrah, Gabby} • M∩S∩C=
  • 50. You try... • M∩S= {Dante, Ed} • S∩C= {Ed, Farrah, Gabby} • M∩S∩C= {Ed}
  • 52. Labor Unions... • Labor Unions address the rights of all workers. If you work at McDonald’s or are a nurse at a hospital, the Labor Union represents your rights.
  • 53. Labor Unions... • Labor Unions address the rights of all workers. If you work at McDonald’s or are a nurse at a hospital, the Labor Union represents your rights. • In Math, a union has the same idea. It brings together the members of all sets.
  • 54. Labor Unions... • Labor Unions address the rights of all workers. If you work at McDonald’s or are a nurse at a hospital, the Labor Union represents your rights. • In Math, a union has the same idea. It brings together the members of all sets. • The symbol for Union is ∪. Notice it looks like the letter U.
  • 55. Back to our clubs...
  • 56. Back to our clubs... • The Union of the Math Club and Chess Clubs is everyone in both clubs.
  • 57. Back to our clubs... • The Union of the Math Club and Chess Clubs is everyone in both clubs. • M = {Adam, Barb, Ciera, Dante, Ed}
  • 58. Back to our clubs... • The Union of the Math Club and Chess Clubs is everyone in both clubs. • M = {Adam, Barb, Ciera, Dante, Ed} • S = {Dante, Ed, Farrah, Gabby, Hali, Ian}
  • 59. Back to our clubs... • The Union of the Math Club and Chess Clubs is everyone in both clubs. • M = {Adam, Barb, Ciera, Dante, Ed} • S = {Dante, Ed, Farrah, Gabby, Hali, Ian} M ∪ C = {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby}
  • 60. Back to our clubs... • The Union of the Math Club and Chess Clubs is everyone in both clubs. • M = {Adam, Barb, Ciera, Dante, Ed} • S = {Dante, Ed, Farrah, Gabby, Hali, Ian} M ∪ C = {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby} • Notice that Dante and Ed are only listed once even though they are in both clubs.
  • 63. You try... • M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian}
  • 64. You try... • M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • S∪C=
  • 65. You try... • M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • S∪C= {Dante, Ed, Farrah, Gabby, Hali, Ian, Adam}
  • 66. You try... • M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • S∪C= {Dante, Ed, Farrah, Gabby, Hali, Ian, Adam} • M∪S∪C=
  • 67. You try... • M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • S∪C= {Dante, Ed, Farrah, Gabby, Hali, Ian, Adam} • M∪S∪C= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian}
  • 68. What about the outsiders?
  • 69. What about the outsiders? • Who is not in the Math Club?
  • 70. What about the outsiders? • Who is not in the Math Club? • Hali, Ian, Farrah, and Gabby
  • 71. What about the outsiders? • Who is not in the Math Club? • Hali, Ian, Farrah, and Gabby • Not is represented by the symbol ~.
  • 72. What about the outsiders? • Who is not in the Math Club? • Hali, Ian, Farrah, and Gabby • Not is represented by the symbol ~. • Not is also called the Complement.
  • 73. What about the outsiders? • Who is not in the Math Club? • Hali, Ian, Farrah, and Gabby • Not is represented by the symbol ~. • Not is also called the Complement. • ~M = {Hali, Ian, Farrah, Gabby}
  • 75. ~(M ∪ S) • Order of Operations starts with parenthesis. This also applies to sets.
  • 76. ~(M ∪ S) • Order of Operations starts with parenthesis. This also applies to sets. • First, find the union of M and S.
  • 77. ~(M ∪ S) • Order of Operations starts with parenthesis. This also applies to sets. • First, find the union of M and S. • M ∪ S = {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian}
  • 78. ~(M ∪ S) • Order of Operations starts with parenthesis. This also applies to sets. • First, find the union of M and S. • M ∪ S = {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • Next, find the complement of the union. In-other-words, everyone who is not in M ∪ S
  • 79. ~(M ∪ S) • Order of Operations starts with parenthesis. This also applies to sets. • First, find the union of M and S. • M ∪ S = {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • Next, find the complement of the union. In-other-words, everyone who is not in M ∪ S • ~(M ∪ S) = { }
  • 80. ~(M ∪ S) • Order of Operations starts with parenthesis. This also applies to sets. • First, find the union of M and S. • M ∪ S = {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian} • Next, find the complement of the union. In-other-words, everyone who is not in M ∪ S • ~(M ∪ S) = { } • Everyone is in either the Math Club or Science Club so we have an empty set. The symbol ∅ or { } represents an empty set.
  • 83. Try these... • ~S= • {Farrah, Hali, Ian}
  • 84. Try these... • ~S= • {Farrah, Hali, Ian} • ~ (S ∪ C) =
  • 85. Try these... • ~S= • {Farrah, Hali, Ian} • ~ (S ∪ C) = • {Barb, Ciera}
  • 86. Try these... • ~S= • {Farrah, Hali, Ian} • ~ (S ∪ C) = • {Barb, Ciera} • ~ (S ∪ C ∪ M) =
  • 87. Try these... • ~S= • {Farrah, Hali, Ian} • ~ (S ∪ C) = • {Barb, Ciera} • ~ (S ∪ C ∪ M) = • { } or ∅
  • 88. How can I use a Venn Diagram? • Visit this Interactive site by Shodor Education Foundation to see how Set operations and a Venn Diagram are applied to an internet search to find useful resources.
  • 90. Finite versus Infinite • So far we have dealt with Finite sets.
  • 91. Finite versus Infinite • So far we have dealt with Finite sets. • What does finite mean?
  • 92. Finite versus Infinite • So far we have dealt with Finite sets. • What does finite mean? • Finite means there is a limit. In-other-words, we can count the number of items in a finite set.
  • 93. Finite versus Infinite • So far we have dealt with Finite sets. • What does finite mean? • Finite means there is a limit. In-other-words, we can count the number of items in a finite set. • What do you think infinite means?
  • 94. Finite versus Infinite • So far we have dealt with Finite sets. • What does finite mean? • Finite means there is a limit. In-other-words, we can count the number of items in a finite set. • What do you think infinite means? • Infinite means there is no limit. It goes on and on and on and on.... Well, you get the idea.
  • 95. Number Line -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 96. Number Line -5 -4 -3 -2 -1 0 1 2 3 4 5 • Recall a number is a graphical representation of all numbers.
  • 97. Number Line -5 -4 -3 -2 -1 0 1 2 3 4 5 • Recall a number is a graphical representation of all numbers. • What term can you apply to the end numbers of a number line?
  • 98. Number Line -5 -4 -3 -2 -1 0 1 2 3 4 5 • Recall a number is a graphical representation of all numbers. • What term can you apply to the end numbers of a number line? • Infinite because you can never get to the end. It goes on and on.
  • 99. Number Line -5 -4 -3 -2 -1 0 1 2 3 4 5 • Recall a number is a graphical representation of all numbers. • What term can you apply to the end numbers of a number line? • Infinite because you can never get to the end. It goes on and on. • The negative numbers extend to the left forever getting smaller and smaller.
  • 100. Number Line -5 -4 -3 -2 -1 0 1 2 3 4 5 • Recall a number is a graphical representation of all numbers. • What term can you apply to the end numbers of a number line? • Infinite because you can never get to the end. It goes on and on. • The negative numbers extend to the left forever getting smaller and smaller. • The positive numbers extend to the right forever getting larger and larger.
  • 101. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 102. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • A closed circle is used to indicate a solution on the number line.
  • 103. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • A closed circle is used to indicate a solution on the number line. • So x = 2 is at the red circle.
  • 104. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • A closed circle is used to indicate a solution on the number line. • So x = 2 is at the red circle. • Shading is used to indicate additional solutions.
  • 105. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • A closed circle is used to indicate a solution on the number line. • So x = 2 is at the red circle. • Shading is used to indicate additional solutions. • Such as x ≤ 2 is the red circle and red arrow to the left.
  • 106. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • A closed circle is used to indicate a solution on the number line. • So x = 2 is at the red circle. • Shading is used to indicate additional solutions. • Such as x ≤ 2 is the red circle and red arrow to the left. • The arrow at the end indicates the solution goes on forever.
  • 107. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 108. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • An open circle is used to indicate the number is not a solution on the number line.
  • 109. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • An open circle is used to indicate the number is not a solution on the number line. • Here is the the graph of x < 2.
  • 110. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • An open circle is used to indicate the number is not a solution on the number line. • Here is the the graph of x < 2. • The open red circle means x = 2 is not a solution but arrow to the left indicate all the number to the left of 2 are solutions.
  • 111. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5 • An open circle is used to indicate the number is not a solution on the number line. • Here is the the graph of x < 2. • The open red circle means x = 2 is not a solution but arrow to the left indicate all the number to the left of 2 are solutions. • Again, the arrow indicates the solution goes on forever in the negative direction.
  • 112. Writing Solution Sets - Part I -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 113. Writing Solution Sets - Part I -5 -4 -3 -2 -1 0 1 2 3 4 5 • The solution x ≤ 2 can be written several ways.
  • 114. Writing Solution Sets - Part I -5 -4 -3 -2 -1 0 1 2 3 4 5 • The solution x ≤ 2 can be written several ways. • Recall a set is enclosed in curly brackets, { }.
  • 115. Writing Solution Sets - Part I -5 -4 -3 -2 -1 0 1 2 3 4 5 • The solution x ≤ 2 can be written several ways. • Recall a set is enclosed in curly brackets, { }. • One way is {x ≤ 2}. Because our solution is a set, this is the technical way to write the set.
  • 116. Writing Solution Sets - Part I -5 -4 -3 -2 -1 0 1 2 3 4 5 • The solution x ≤ 2 can be written several ways. • Recall a set is enclosed in curly brackets, { }. • One way is {x ≤ 2}. Because our solution is a set, this is the technical way to write the set. • It is understood that x ≤ 2 is a set so often the { } are not included. So x ≤ 2 is acceptable.
  • 117. Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 118. Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex.
  • 119. Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation.
  • 120. Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation:
  • 121. Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2}
  • 122. Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2} • This reads as
  • 123. Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2} • This reads as the set of all x such that x is less than or equal to 2.
  • 124. Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2} • This reads as the set of all x such that x is less than or equal to 2.
  • 125. Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2} • This reads as the set of all x such that x is less than or equal to 2.
  • 126. Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2} • This reads as the set of all x such that x is less than or equal to 2.
  • 127. Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5 • Set Builder Notation looks a little more complicated but is useful when sets become more complex. • It’s good to recognize Set Builder Notation because you never know when a text book or standardized test will use this notation. • Here is x ≤ 2 in Set Builder Notation: { x | x≤2} • This reads as the set of all x such that x is less than or equal to 2.

Editor's Notes