1. Set Theory Symbols and Terminology Set – A collection of objects

2. Set Theory Symbols and Terminology Element – An object in a set

3. Set Theory Symbols and Terminology Empty (Null) Set – A set that contains no elements

4. Set Theory Symbols and Terminology Cardinal Number (Cardinality) – The number of elements in a set

5. Set Theory Symbols and Terminology Finite Set – A set that contains a limited number of elements

6. Set Theory Symbols and Terminology Infinite Set – A set that contains an unlimited number of elements

7. Set Theory Symbols and Terminology There are three ways to describe a set Word Description Listing Set Builder Notation

8. Set Theory Symbols and Terminology The following example shows the three ways we can describe the same set.

9. Set Theory Symbols and Terminology Word Description “The set of even counting numbers less than 10”

10. Set Theory Symbols and Terminology Listing E = {2, 4, 6, 8 }

11. Set Theory Symbols and Terminology Set Builder Notation E = {x | x is an even counting number that is less than 10 }

12. Set Theory Symbols and Terminology Cardinal Numbers n(E) means “the number of elements in set E” In this particular case n(E) = 4

13. Example 1) Suppose A is the set of all lower case letters of the alphabet. We could write out set A as follows: A = {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r s,t,u,v,w,x,y,z} We can shorten this notation if we clearly show a pattern, as in the following:

14. A = {a,b,c,d,e…,w,x,y,z} Try writing the following three sets by listing the elements.

15. 1) The set of counting numbers between six and thirteen.

16. B = {7, 8, 9, 10, 11, 12 }

17. 2) C = {5, 6, 7, …, 13}

18. C = {5, 6, 7, 8, 9, 10, 11, 12, 13}

19. 3) D = {x | x is a counting number between 6 and 7}

20. D = { } or Ø

21. Homework Page 54 # 1 - 10

22. Warm Up Find the cardinal number of each set K = {2, 4, 8, 16} M = { 0 } R = {4, 5, …, 12, 13} P = Ø

23. Warm Up Find the cardinal number of each set K = {2, 4, 8, 16} n(K)=4 M = { 0 } n(M)=1 R = {4, 5, …, 12, 13} n(R)=10 P = Ø n(P)=0

24. Pg 54 #1-10 Answers 1. C5. B 2. G6. D 3. E7. H 4. A8. F 9. A = {1, 2, 3, 4, 5, 6} 10. B = {9, 10, 11, 12, 13, 14, 15, 16, 17}

25. Set Theory Symbols and Terminology Empty Set – Example) P = {x | x is a positive number <0} Therefore P = { } or P = Ø but P ≠ {Ø}

26. Set Theory Symbols and Terminology Infinite Set – Example) R = {y | y is an odd whole number} Therefore R = {1, 3, 5, 7, …}

27. Set Theory Symbols and Terminology Finite Set – Example) F = {z | z is a factor of 30} Therefore F = {1, 2, 3, 5, 6, 10, 15, 30}

28. Classwork Page 54 & 55 #11 – 49 odd

29. 11. The set of all whole numbers not greater than 4 can be expressed by listing as A ={0, 1,2,3, 4}. 13. In the set {6, 7,8...., 14}, the ellipsis (three dots) indicates a continuation of the pattern. A complete listing oft his set is B ={6,7,8,9,10, 11, 12, 13,14}.

30. Page 54 & 55 #11 – 49 odd 15. The set { -15, -13, 11,..., -1} contains all integers from -15 to -1 inclusive. Each member is two larger than its predecessor. A complete listing of this set is C ={- 15, -13, -11, -9, -7, -5, -3, -1}. 17. The set {2, 4, 8,..., 256} contains all powers of two from 2 to 256 inclusive. A complete listing of this set D={2, 4,8,16,32,64,128, 256}.

31. Page 54 & 55 #11 – 49 odd 19. A complete listing of the set {x x is an even whole number less than 11 } is E={0, 2, 4, 6, 8, 10}. Remember that 0 is the first whole number. 21. The set of all counting numbers greater than 20 is represented by the listing F={21, 22, 23,... }.

32. Page 54 & 55 #11 – 49 odd 23. The set of Great Lakes is represented by G={Lake Erie, Lake Huron, Lake Michigan, Lake Ontario, Lake Superior}. 25. The set {x | x is a positive multiple of 5} is represented by the listing H={5, 10,15,20,.,. }.

33. Page 54 & 55 #11 – 49 odd 27. The set {x|x is the reciprocal of a natural number} is represented by the listing I={1, 1/2, 1/3, 1/4, 1/5,... }. 29. The set of all rational numbers may be represented using set-builder notation as J={x|x is a rational number}.

34. Page 54 & 55 #11 – 49 odd 31. The set {1, 3,5,..., 75} may be represented using set- builder notation as K={x|x is an odd natural number less than 76}. 33. The set {2, 4, 6,..., 32} is finite since the cardinal number associated with this set is a whole number.

35. Page 54 & 55 #11 – 49 odd 35. The set {112, 2/3, 3/4,... } is infinite since there is no last element, and we would be unable to count all of the elements. 37. The set {x|x is a natural number greater than 50} is infinite since there is no last element, and therefore its cardinal number is not a whole number.

36. Page 54 & 55 #11 – 49 odd 39. The set {x|x is a rational number} is infinite since there is no last element, and therefore its cardinal number is not a whole number; 41. For any set A, n(A) represents the cardinal number of the set, that is, the number of elements in the set. The set A = {0, 1, 2, 3, 4, 5, 6, 7} contains 8 elements. Thus, n(A) = 8.

37. Page 54 & 55 #11 – 49 odd 43. The set A = {2, 4, 6,..., l000} contains 500 elements. Thus, n(A) = 500. 45. The set A = {a, b, c,,.., z} has 26 elements (letters of the alphabet). Thus n(A) = 26.

38. Page 54 & 55 #11 – 49 odd 47. The set A = the set of integers between -20 and 20 has 39 members. The set can be indicated as {- 19, -18,...,18, 19}, or 19 negative integers, 19 positive integers, and 0. Thus, n(A) = 39. 49. The set A = { 1/3, 2/4, 3/5, 4/6,..., 27/29, 28/30} has 28 elements. Thus, n(A) = 28.

39. Equal and Equivalent Sets Equal Sets – Two sets are equal if they contain the EXACT same elements. A={1,4,9,16,25} B={1,9,4,25,16}

40. Equal and Equivalent Sets Equivalent Sets – Two sets are equivalent if they contain the same NUMBER of elements. A={1,3,5,7,9} B={1,2,4,8,16}

41. Well Defined and Not Well Defined Sets

44. On your own, come up with one example of a well defined set and one example of a not well defined set. Place your sets in the appropriate section of the board.

45. Elements of Sets

47. Homework Do page 55 #53 – 84 QUIZ tomorrow on pages 54-55 #1 - 84

48. Homework Answers

49. Tuesday Oct 5 Quiz Today After Quiz, do page 56 Question #92. Finish it for homework and be prepared to turn it in.

50. Wednesday Oct 6 Venn Diagrams and Subsets Consider the set of counting numbers less than or equal to 20. U={x|x is a counting number less than 20} U={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} Use the following Venn Diagram to divide the numbers into groups of even numbers and groups of multiples of three.

51. Even Numbers Multiples of 3 U Counting Numbers ≤ 20

52. Even Numbers Multiples of 3 1 2 3 4 6 10 8 12 9 15 5 7 11 13 17 19 14 1618 20 U Counting Numbers ≤ 20

53. Wednesday Oct 6 Venn Diagrams and Subsets Universal Set – The set of all objects under discussion. For our example, the universal set is the set of all counting numbers less than or equal to 20. The universal set is always denoted by the letter U

54. Wednesday Oct 6 Venn Diagrams and Subsets Let’s let A represent the set of all even numbers less than or equal to 20 and B will represent the set of all multiples of 3 that are less than or equal to 20 A = {2,4,6,8,10,12,14,16,18,20} B = {3,6,9,12,15,18}

55. Even Numbers Multiples of 3 1 2 3 4 6 10 8 12 9 15 5 7 11 13 17 19 14 1618 20 U Counting Numbers ≤ 20 A B

56. Wednesday Oct 6 Venn Diagrams and Subsets Complement of a Set – The complement of a set is the set of all elements of the universal set that are NOT elements of the set in question. In our example the complement of A, written A´, is A´={1, 3, 5, 7, 9, 11, 13, 15, 17, 19}

57. Even Numbers Multiples of 3 1 3 9 15 5 7 11 13 17 19 U Counting Numbers ≤ 20 A B 2 4 6 10 8 12 14 1618 20

58. Wednesday Oct 6 Venn Diagrams and Subsets Subset of a Set – The subset of a set is the set where ALL elements of one set are also elements another set. Using our example, A is a subset of U. B is also a subset of U

59. Wednesday Oct 6 Venn Diagrams and Subsets U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20} A = {2,4,6,8,10,12,14,16,18,20} U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20} B = {3,6,9,12,15,18}

60. Wednesday Oct 6 Venn Diagrams and Subsets

64. Even Numbers Multiples of 3 1 2 3 4 6 10 8 12 9 15 5 7 11 13 17 19 14 1618 20 U Counting Numbers ≤ 20 A B

65. Wednesday Oct 6 Venn Diagrams and Subsets

67. Wednesday Oct 6 Homework Page 61 #1 – 14

68. Subsets and Proper Subsets List all of the subsets { } or Ø

69. Subsets and Proper Subsets List all of the subsets { } or Ø Ø

70. Subsets and Proper Subsets List all of the subsets {a}

71. Subsets and Proper Subsets List all of the subsets {a} Ø {a}

72. Subsets and Proper Subsets List all of the subsets {a, b}

73. Subsets and Proper Subsets List all of the subsets {a, b} Ø {a} {b} {a, b}

74. Subsets and Proper Subsets List all of the subsets {a, b, c}

75. Subsets and Proper Subsets List all of the subsets {a, b, c } Ø {a}; {b}; {c} {a, b}; {a, c}; {b, c} {a, b, c }

76. Subsets and Proper Subsets List all of the subsets {a, b, c, d}

77. Subsets and Proper Subsets List all of the subsets {a, b, c, d } Ø {a}; {b}; {c}; {d} {a, b}; {a, c}; {a, d}; {b, c}; {b, d}; {c, d} {a, b, c }; {a, b, d }; {a, c, d }; {b, c, d } {a, b, c, d }

78. Subsets and Proper Subsets List all of the proper subsets { } or Ø

79. Subsets and Proper Subsets List all of the proper subsets { } or Ø

80. Subsets and Proper Subsets List all of the proper subsets {a}

81. Subsets and Proper Subsets List all of the proper subsets {a} Ø

82. Subsets and Proper Subsets List all of the proper subsets {a, b}

83. Subsets and Proper Subsets List all of the proper subsets {a, b} Ø {a} {b}

84. Subsets and Proper Subsets List all of the proper subsets {a, b, c}

85. Subsets and Proper Subsets List all of the proper subsets {a, b, c } Ø {a}; {b}; {c} {a, b}; {a, c}; {b, c}

86. Subsets and Proper Subsets List all of the proper subsets {a, b, c, d}

87. Subsets and Proper Subsets List all of the proper subsets {a, b, c, d } Ø {a}; {b}; {c}; {d} {a, b}; {a, c}; {a, d}; {b, c}; {b, d}; {c, d} {a, b, c }; {a, b, d }; {a, c, d }; {b, c, d }

88. Subsets and Proper Subsets

89. Homework Page 61-62 #15-53

91. Page 62 #55 - 60

92. U = {Higher Cost, Lower Cost, Educational, More Time to See Sights, Less Time to See Sights, Cannot Visit Relatives, Can Visit Relatives}

93. Page 62 #55 - 60 U = {Higher Cost, Lower Cost, Educational, More Time to See Sights, Less Time to See Sights, Cannot Visit Relatives, Can Visit Relatives} F = {Higher Cost, Educational, More Time to See Sights, Cannot Visit Relatives} F΄ = {Lower Cost, Less Time to See Sights, Can Visit Relatives}

94. Page 62 #55 - 60 U = {Higher Cost, Lower Cost, Educational, More Time to See Sights, Less Time to See Sights, Cannot Visit Relatives, Can Visit Relatives} D = {Lower Cost, Educational, Less Time to See Sights, Can Visit Relatives} D΄ = {Higher Cost, More Time to See Sights, Cannot Visit Relatives}

95. Page 62 #55 - 60 F = {Higher Cost, Educational, More Time to See Sights, Cannot Visit Relatives} D = {Lower Cost, Educational, Less Time to See Sights, Can Visit Relatives} Both F and D = {Educational}

96. Page 62 #55 - 60 F΄ = {Lower Cost, Less Time to See Sights, Can Visit Relatives} D΄ = {Higher Cost, More Time to See Sights, Cannot Visit Relatives} Both F΄ and D ΄ = Ø

97. Page 62 #55 - 60 F = {Higher Cost, Educational, More Time to See Sights, Cannot Visit Relatives} D΄ = {Higher Cost, More Time to See Sights, Cannot Visit Relatives} Both F and D ΄ = {Higher Cost, More Time to See Sights, Cannot Visit Relatives}