Sets Notes

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  • Sets Notes

    1. 1. Venn Diagrams & Sets
    2. 2. • Adam, Barb, Ciera, Dante, and Ed are members of the Math Club.
    3. 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. 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. 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. 6. Possible Table Organization Adam Barb Ciera Dante Ed Farrah Gabby Hali Ian Math X X X X XScience X X X X X XChess X X X X
    7. 7. Possible Diagram Organization Sc l ub ien h C ceMat Cl Barb ub Dante Hali Ciera Ian Ed Adam Farrah Gabby Chess Club
    8. 8. Venn Diagram
    9. 9. Venn Diagram• The last data organization is called a Venn Diagram.
    10. 10. Venn Diagram• The last data organization is called a Venn Diagram.• Useful for organizing 2 or more sets.
    11. 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. 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. 13. Use the Venn Diagram to answer...
    14. 14. Use the Venn Diagram to answer...• What observation can you make about each student?
    15. 15. Use the Venn Diagram to answer...• What observation can you make about each student?• What observation can you make about each club?
    16. 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. 17. • Organizing the information helps us see the relationships between the students and clubs.
    18. 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. 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. 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. 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. 22. Math Club (M) as a Set
    23. 23. Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed}
    24. 24. Math Club (M) as a Set M = {Adam, Barb, Ciera, Dante, Ed}What do you notice about the set M?
    25. 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. 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. 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. 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.
    29. 29. You try...
    30. 30. You try...• Write the members of the Science Club (S) as a set.
    31. 31. You try...• Write the members of the Science Club (S) as a set. S = {Dante, Ed, Farrah, Gabby, Hali, Ian}
    32. 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. 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. 34. Driving down the road...
    35. 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. 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. 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. 38. Math & Chess Clubs
    39. 39. Math & Chess Clubs• Who is in the Math Club and the Chess Club?
    40. 40. Math & Chess Clubs• Who is in the Math Club and the Chess Club? • Adam and Ed
    41. 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. 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. 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}
    44. 44. You try...
    45. 45. You try...• M∩S=
    46. 46. You try...• M∩S= {Dante, Ed}
    47. 47. You try...• M∩S= {Dante, Ed}• S∩C=
    48. 48. You try...• M∩S= {Dante, Ed}• S∩C= {Ed, Farrah, Gabby}
    49. 49. You try...• M∩S= {Dante, Ed}• S∩C= {Ed, Farrah, Gabby}• M∩S∩C=
    50. 50. You try...• M∩S= {Dante, Ed}• S∩C= {Ed, Farrah, Gabby}• M∩S∩C= {Ed}
    51. 51. Labor Unions...
    52. 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. 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. 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. 55. Back to our clubs...
    56. 56. Back to our clubs...• The Union of the Math Club and Chess Clubs is everyone in both clubs.
    57. 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. 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. 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. 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.
    61. 61. You try...
    62. 62. You try...• M∪S=
    63. 63. You try...• M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian}
    64. 64. You try...• M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian}• S∪C=
    65. 65. You try...• M∪S= {Adam, Barb, Ciera, Dante, Ed, Farrah, Gabby, Hali, Ian}• S∪C= {Dante, Ed, Farrah, Gabby, Hali, Ian, Adam}
    66. 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. 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. 68. What about the outsiders?
    69. 69. What about the outsiders?• Who is not in the Math Club?
    70. 70. What about the outsiders?• Who is not in the Math Club? • Hali, Ian, Farrah, and Gabby
    71. 71. What about the outsiders?• Who is not in the Math Club? • Hali, Ian, Farrah, and Gabby• Not is represented by the symbol ~.
    72. 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. 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}
    74. 74. ~(M ∪ S)
    75. 75. ~(M ∪ S)• Order of Operations starts with parenthesis. This also applies to sets.
    76. 76. ~(M ∪ S)• Order of Operations starts with parenthesis. This also applies to sets.• First, find the union of M and S.
    77. 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. 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. 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. 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.
    81. 81. Try these...
    82. 82. Try these...• ~S=
    83. 83. Try these...• ~S= • {Farrah, Hali, Ian}
    84. 84. Try these...• ~S= • {Farrah, Hali, Ian}• ~ (S ∪ C) =
    85. 85. Try these...• ~S= • {Farrah, Hali, Ian}• ~ (S ∪ C) = • {Barb, Ciera}
    86. 86. Try these...• ~S= • {Farrah, Hali, Ian}• ~ (S ∪ C) = • {Barb, Ciera}• ~ (S ∪ C ∪ M) =
    87. 87. Try these...• ~S= • {Farrah, Hali, Ian}• ~ (S ∪ C) = • {Barb, Ciera}• ~ (S ∪ C ∪ M) = • { } or ∅
    88. 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.
    89. 89. Finite versus Infinite
    90. 90. Finite versus Infinite• So far we have dealt with Finite sets.
    91. 91. Finite versus Infinite• So far we have dealt with Finite sets.• What does finite mean?
    92. 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. 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. 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. 95. Number Line-5 -4 -3 -2 -1 0 1 2 3 4 5
    96. 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. 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. 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. 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. 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. 101. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5
    102. 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. 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. 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. 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. 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. 107. Graphing Solutions -5 -4 -3 -2 -1 0 1 2 3 4 5
    108. 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. 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. 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. 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. 112. Writing Solution Sets - Part I -5 -4 -3 -2 -1 0 1 2 3 4 5
    113. 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. 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. 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. 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. 117. Writing Solution Sets - Part II -5 -4 -3 -2 -1 0 1 2 3 4 5
    118. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

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