1. SET THEORY Chapter 2

2. DAY 1

3. Set – collection School of fish Gaggle of geese Pride of lions Pod of whales Herd of elephants

4. Set – usually named with a capital letter. Well defined A is the set of the first three lower case letters of the English alphabet.

5. Elements of the set A is the set of the first three lower case letters of the English alphabet. a, b, and c are elements of set A

6. Natural Numbers (Counting Numbers) N = {1, 2, 3,... }

7. Three ways of defining a set: List A = {1,2,3} Description A is the set of the first three counting numbers. Set Builder Notation

8. Universe Empty set

9. Example The set of natural numbers greater than 12 and less than 17.

10. Example {x | x = 2n and n = 1, 2, 3, 4, 5}

11. Example {3, 6, 9, 12,... }

12. Example The set of the first 10 odd natural numbers.

13. Venn Diagrams

14. Set A

15. Complement of A

16. A intersect B

17. A Union B

18. Disjoint Sets

19. Subsets A is a subset of B if every element of A is also an element of B.

20. List all the subsets of {a,b,c} { } {a} {b} {c} {a,b} {a,c} {b,c} {a,b,c}

21. List all the subsets of {a,b,c} Proper Subsets: { }, {a}, {b}, {c}, {a,b}, {a,c}, {b,c} THE Improper Subset: {a,b,c}

22. Subset Notation Let A = {a,b,c} {a} A “The set of a is a subset of A.” (think: The set of a is a proper subset OR IS EQUAL TO A.) {a} A “The set of a is a proper subset of A.”

23. True or False? A = {b,c,f,g} {b,f} A

24. True or False? A = {b,c,f,g} {b,f} A True

25. True or False? A = {b,c,f,g} {b,d} A

26. True or False? A = {b,c,f,g} {b,d} A False Because d A

27. True or False? A = {b,c,f,g} {b,c,f,g} A

28. True or False? A = {b,c,f,g} {b,c,f,g} ATrue {b,c,f,g} A

29. True or False? A = {b,c,f,g} {b,c,f,g} ATrue {b,c,f,g} AFalse Because {b,c,f,g} = A

30. U = {p,q,r,s,t,u,v,w,x,y,z} A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

46. DAY 2

47. Homework Questions Page 83

50. Three types of numbers. Nominal Ordinal Cardinal “The student with ticket 50768-973 has just won second prize – four tickets to the big game this Saturday.”

51. Three types of numbers. Nominal – name or label – for identification Ordinal – tells what order it comes in relation to the rest. Cardinal – Answers the question “how many?” “The student with ticket 50768-973 has just won second prize – four tickets to the big game this Saturday.”

52. Cardinality of the Set If a cardinal number answers the question “how many?” then the cardinality of a set will tell us how many elements are in the set. The notation for “the cardinality of set A” (or the number of elements in A) is n(A)

53. Equal Sets Two sets are equal if the have the exact same elements. Example: A = {a,b,c} and B = {c,a,b} then A = B

54. Consider A = {a,b,c} and C = {x,y,z} They are not equal because they do not have the same exact elements. What characteristic do they share?

55. Equivalent Sets A and C have the same number of elements. Their cardinality is the same. n(A) = 3 and n(C) = 3 n(A) = n(C) A and C are equivalent sets.

56. If two sets are equivalent, you can set up a one- to-one correspondence between them. (That is, you can match them up in pairs.)

57. There are actually 6 different one-to-one correspondences you can set up between these two sets. (6 ways that you can make pairs.) A = {a,b,c} and C = {x,y,z} (make an orderly list)

58. 6 different one-to-one correspondences: A = {a,b,c} and C = {x,y,z} a – x b – y c – z

59. 6 different one-to-one correspondences: A = {a,b,c} and C = {x,y,z} a – xa - x b – yb - z c – zc - y

60. 6 different one-to-one correspondences: A = {a,b,c} and C = {x,y,z} a – xa – xa - y b – yb – zb - x c – zc – yc - z

61. 6 different one-to-one correspondences: A = {a,b,c} and C = {x,y,z} a – xa – xa – ya - y b – yb – zb – xb - z c – zc – yc – zc - x

62. 6 different one-to-one correspondences: A = {a,b,c} and C = {x,y,z} a – xa – xa – ya – y b – yb – zb – xb - z c – zc – yc – zc – x a – z b – x c – y

63. 6 different one-to-one correspondences: A = {a,b,c} and C = {x,y,z} a – xa – xa – ya – y b – yb – zb – xb - z c – zc – yc – zc – x a – za - z b – xb - y c – yc - x

64. A = {x|x is a moon of Mars} B = {x|x is a former U.S. president whose last name is Adams} C = {x|x is one of the Bronte sisters of nineteenth- century literary fame} D = {x|x is a satellite of the fourth-closest planet to the sun} Which of these sets are equal and which are equivalent? What do we need to know about each set to answer this question?

65. A = {x|x is a moon of Mars} A = {Deimos, Phobos} n(A) =

66. A = {Deimos, Phobos} n(A) = 2 B = {x|x is a former U.S. president whose last name is Adams} B = {John Adams, John Quincy Adams} n(B) =

67. A = {Deimos, Phobos} n(A) = 2 B = {John Adams, John Quincy Adams} n(B) = 2 C = {x|x is one of the Bronte sisters of nineteenth-century literary fame} C = {Anne, Charlotte, Emily} n(C) =

68. A = {Deimos, Phobos} n(A) = 2 B = {John Adams, John Quincy Adams} n(B) = 2 C = {Anne, Charlotte, Emily} n(C) = 3 D = {x|x is a satellite of the fourth-closest planet to the sun} D = {Deimos, Phobos} n(D) =

69. A = {Deimos, Phobos} n(A) = 2 B = {John Adams, John Quincy Adams} n(B) = 2 C = {Anne, Charlotte, Emily} n(C) = 3 D = {Deimos, Phobos} n(D) =2

70. Finite/Infinite Whole numbers? Real numbers between 0 and 1? Factors of 20? Multiples of 20? Number of grains of sand on the earth?

71. Example 2.9 Page 94

72. n(U) = 60 n(S) = 24 n(E) = 22 n(H) = 17 5 both S and E 4 both S and H 3 both E and H 2 all three

73. Attribute Lab Three attributes considered are Size Color Shape

76. A and B – You must get through the first door AND the second door. (more restrictive) A or B – You may go in the first door OR the second door. (more generous)

78. Day 3

79. Homework Questions Page 97

82. Binary Operations Addition Subtraction Multiplication Division

83. __________ + __________ = __________

84. Addend + Addend = Sum __________ - __________ = __________

85. Addend + Addend = Sum Minuend – Subtrahend = Difference __________ X __________ = __________

86. Addend + Addend = Sum Minuend – Subtrahend = Difference Factor X Factor = Product __________ __________ = __________

87. Addend + Addend = Sum Minuend – Subtrahend = Difference Factor X Factor = Product Dividend Divisor = Quotient

88. Properties Pages 104 and 120 Closure

89. Counting Numbers = {1, 2, 3,... } Whole Numbers = {0, 1, 2, 3,... }

90. Closure Examples Is the set of Whole Numbers closed with respect to Addition? Subtraction? Multiplication? Division?

91. Closure Examples Is the set of Even Counting Numbers closed with respect to Addition? Subtraction? Multiplication? Division?

92. Closure Examples Is {0, 1} closed with respect to Addition? Subtraction? Multiplication? Division?

93. Properties Pages 104 and 120 Closure Commutative Associative Identity Element for Addition Identity Element for Multiplication Multiplication-by-Zero Property Distributive Property of Multiplication over Addition

94. Examples 2 + (3 + 4) = 5 + 4

95. Examples 2 + (3 + 4) = 5 + 4Associative 2 + (3 + 4) = 7 + 2

96. Examples 2 + (3 + 4) = 5 + 4Associative 2 + (3 + 4) = 7 + 2Commutative 2(3 + 4) = 6 + 8

97. Examples 2 + (3 + 4) = 5 + 4Associative 2 + (3 + 4) = 7 + 2Commutative 2(3 + 4) = 6 + 8Distributive 2(3 + 4) = (7)2

98. Examples 2 + (3 + 4) = 5 + 4Associative 2 + (3 + 4) = 7 + 2Commutative 2(3 + 4) = 6 + 8Distributive 2(3 + 4) = (7)2Commutative

99. Conceptual Models Addition –Set Model

100. Conceptual Models Addition Subtraction (page 108) –Take-away –Missing Addend –Comparison –Number-line

101. Take-away - Missing Addend Comparison - Number-line Identify which model would illustrate the problem best. Mary got 43 pieces of candy. Karen got 36 pieces. How many more pieces does Mary have than Karen?

102. Take-away - Missing Addend Comparison - Number-line Identify which model would illustrate the problem best. Mary gave 20 pieces of her 43 pieces of candy to her brother. How many pieces does she have left?

103. Take-away - Missing Addend Comparison - Number-line Identify which model would illustrate the problem best. Karen’s older brother collected 53 pieces. How many more pieces would Karen need to have as many as her brother?

104. Take-away - Missing Addend Comparison - Number-line Identify which model would illustrate the problem best. Ken left home and walked 10 blocks east. The last 4 blocks were after crossing Main Street. How far is Main Street from Ken’s house?

105. Conceptual Models Addition Subtraction Multiplication (page 115) –Repeated Addition –Number-line –Rectangular Array –Multiplication Tree

106. Multiplication Tree Melissa has 4 flags colored red, yellow, green and blue. How many ways can she display them on a flagpole?

109. Conceptual Models Addition Subtraction Multiplication –Repeated Addition –Number-line –Rectangular Array –Multiplication Tree –Cartesian Product

110. Cartesian Product The Cartesian Product of A and B is a set of ordered pairs written A X B, and read “A cross B.” A X B = {(a,b) | a A and b B}

111. Cartesian Product A X B = {(a,b) | a A and b B} Example: A = {5, 6, 7}B = {6, 8} A X B = {(

112. Cartesian Product A X B = {(a,b) | a A and b B} Example: A = {5, 6, 7}B = {6, 8} A X B = {(5,6), (5,8), (6,6), (6,8), (7,6), (7,8)}

113. Cartesian Product Example: A = {5, 6, 7}B = {6, 8} A X B = {(5,6), (5,8), (6,6), (6,8), (7,6), (7,8)} NOTE: n(A) = 3, n(B) = 2 and n(AXB) = 6

114. How many different things can you order at the yogurt shop if you must choose from a waffle cone or a sugar cone and either vanilla, chocolate, mint, or raspberry yogurt? C = {w, s}, Y = {v, c, m, r}

115. Cartesian Product C = {w, s}, Y = {v, c, m, r} C X Y = {(w, v), (w, c), (w, m), (w, r), (s, v), (s, c), (s, m), (s, r)} n(C X Y) = 8

116. Conceptual Models Addition Subtraction Multiplication Division (Page 121) –Repeated Subtraction –Sharing –Missing Factor

117. Division Example Describe how you would divide 78 by 13 using counters and each of the following models. –Repeated Subtraction –Sharing –Missing Factor

118. Family of Facts 20 4 = 55 X 4 = 20 and 20 5 = 44 X 5 = 20

119. Family of Facts 0 ÷ 4 = 0 and 0 X 4 = 0 4 ÷ 0 = ?? and ?? X 0 = 4

120. Division by Zero is Undefined.

121. Extra Practice Worksheet

122. DAY 4

123. Homework Pages 111 and 130

126. Worksheet Answers

128. Math and Music The Magical Connection! Scholastic Parent and Child Magazine Spelling Phone Numbers School House Rock

129. “Skip to My Lou” Chorus:Times facts, they’re a breeze; Learn a few, then work on speed. Times facts, you’ll be surprised By just how fast you can memorize.

130. 3 time 7 is 21 Now, at last we’ve all begun. 4 times 7 is 28 Let’s sing what we appreciate. (Chorus) 5 times 7 is 35. Yes, by gosh, we’re still alive. 6 times 7 is 42. I forgot what we’re supposed to do. (Chorus)

131. Print Review for Test

132. Venn Diagram Lab