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
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.
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.
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.
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}
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.
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}
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
