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Thinking Mathematically

Chapter 2 Set Theory 2.1 Basic Set Concepts. Thinking Mathematically. Basic Set Concepts. A set is a collection of objects. Each object is called an element of the set. A set must be well defined : Its contents can be clearly determined

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Thinking Mathematically

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  1. Chapter 2 Set Theory 2.1 Basic Set Concepts Thinking Mathematically

  2. Basic Set Concepts • A set is a collection of objects. Each object is called an element of the set. • A set must be well defined: • Its contents can be clearly determined • Its clear if an object is or is not a member of the set.

  3. Representing Sets Word Description: Describe the set in your own words, but be specific so the elements are clearly defined. Roster Method: List each element, separated by commas, in braces. Set-Builder Notation: {x | x is … word description}.

  4. The Set of Natural Numbers N = {1,2,3,4,5,…} • This is an example of a set • We will be talking a lot more about sets of numbers in Chapter 5

  5. Examples: Representing Sets Exercise Set 2.1 #3, 5, 13, 15, 25 • Well defined sets (T/F): • The five worst U.S. presidents • The natural numbers greater than one million • Write a description for the set {6, 7, 8, 9, …, 20} • Express this set using the roster method: The set of four seasons in a year. {x | x N and x > 5 }

  6. The empty set, also called the null set, is the set that contains no elements. The empty set is represented by { } or Ø The Empty Set

  7. Examples: Empty Sets Exercise Set 2.1 #35, 37, 41, 45 Which sets are empty • {x | x is a women who served as U.S. president before 2000} • {x | x is the number of women who served as U.S. president before 2000} • {x | x <2 and x > 5} • {x | x is a number less that 2 or greater than 5}

  8. The Notation  and  The symbol  is used to indicate that an object is an element of a set. The symbol  is used to replace the words “is an element of” The symbol  is used to indicate that an object is not an element of a set. The symbol  is used to replace the words “is not an element of”

  9. Example: Set elements Exercise Set 2.1 #51, 59, 63 (T/F) • 5  { 2, 4, 6, …, 20} • 13 {x | x  N and x < 13 } • {3} {3, 4}

  10. Definition of a Set’s Cardinal Number The cardinal number of set A, represented by n(A), is the number of distinct elements in set A. The symbol n(A) is read “n of A”. • Repeated elements are not counted. Exercise Set 2.1 #71 C = {x | x is a day of the week that begins with the letter A} n( C) = ?

  11. Definition of a Finite Set Set A is a finite set if n(A) = 0 or n(A) is a natural number. A set that is not finite is called an infinite set. Exercise Set 2.1 #91 {x | x  N and x >= 100} Finite or infinite?

  12. Definition of Equality of Sets Set A is equal to set B means that set A and set B contain exactly the same elements, regardless of order or possible repetition of elements. We symbolize the equality of sets A and B using the statement A = B.

  13. Definition of Equivalent Sets Set A is equivalent to set B means that set A and set B contain the same number of elements. For equivalent sets, n(A) = n(B). Exercise Set 2.1 #85 A = { 1, 1, 1, 2, 2, 3, 4} B = {4, 3, 2, 1} Are these sets equal? Are these sets equivalent?

  14. Chapter 2 Set Theory 2.3 Venn Diagrams and Set Operations [we’ll come back to 2.2] Thinking Mathematically

  15. Definition of a Universal Set A universal set, symbolized by U, is a set that contains all of the elements being considered in a given discussion or problem. Exercise Set 2.3 #3 A = {Pepsi, Sprite} B = {Coca Cola, Seven-Up} Describe a universal set that includes all elements in sets A and B

  16. U A B U A B U B A Venn Diagrams “Disjoint” sets have no elements in common. All elements of B are also elements of A. The sets A and B have some common elements.

  17. Definition of the Complement of a Set The complement of set A, symbolized by A´, is the set of all elements in the universal set that are not in A. This idea can be expressed in set-builder notation as follows: A´ = {x | x  U and x  A }.

  18. Complement of a Set U A A’

  19. Example: Set Complement Exercise Set 2.3 #11 U = {1, 2, 3,…, 20} A = {1, 2, 3, 4, 5} B = {6, 7, 8, 9} C = {1, 3, 5, …, 19} D = {2, 4, 6, …, 20} C´ = ?

  20. U B A Definition of Intersection of Sets The intersection of sets A and B, written AB, is the set of elements common to both set A and set B. This definition can be expressed in set builder notation as follows: A  B = { x | x  A AND x  B}

  21. U B A Definition of the Union of Sets The union of sets A and B, written A  B, is the set of elements that are members of set A or of set B or of both sets. This definition can be expressed in set-builder notation as follows: A  B = {x | x  A OR x  B}

  22. The Empty Set in Intersection and Union For any set A: 1. A ∩  =  2. A  = A

  23. Examples: Union / Intersection Exercise Set 2.3 #17, 19, 33, 35 U = {1, 2, 3, 4, 5, 6, 7} A = {1, 3, 5, 7} B = {1, 2, 3} C = {2, 3, 4, 5, 6} • A  B = ? • A  B = ? • A  = ? • A ∩  = ?

  24. Cardinal Number of the Union of Two Sets n(A U B) = n(A) + n(B) – n(A ∩B) Exercise Set 2.3 #93 • Set A 17 elements • Set B 20 elements • There are 6 elements common to the two sets • How many elements in the union?

  25. Chapter 2 Set Theory 2.2 Subsets Thinking Mathematically

  26. Set B is a subset of set A, expressed as B  A if every element in set B is also an element in set A. U A B Definition of a Subset of a Set Every set is a subset of itself: A A

  27. Definition of a Proper Subset of a Set Set B is a proper subset of set A, expressed as B  A, if set B is a subset of set A and sets A and B are not equal ( A  B ). What is an improper subset?

  28. The Empty Set as a Subset • For any set B,  B. • For any set B other than the empty set,  B.

  29. Example: Subsets • Exercise Set 2.2 #3, 45, 43, 47 • {-3, 0, 3} ____ {-3, -1, 1, 3} • (, , both, neither) • {Ralph}  {Ralph, Alice, Trixie, Norton} (T/F) • Ralph  {Ralph, Alice, Trixie, Norton} (T/F) •  {Archie, Edith, Mike, Gloria} (T/F)

  30. Chapter 2 Set Theory 2.4 Set Operations and Venn Diagrams With Three Sets Thinking Mathematically

  31. Example: Operations with three setsExercise Set 2.4 #3, 15 • U = {a, b, c, d, e, f, g, h} A = {a, g, h} B = {b, h, h} C = {b, c, d, e, f} (A  B) ∩ (A  C) • U = {1, 2, 3, 4, 5, 6, 7} A = {1, 3, 5, 7} B = {1, 2, 3} C = {2, 3, 4, 5, 6} (A  B) ∩ (A  C)

  32. Example – Venn Diagrams • Exercise Set 2.4 #35, 37 U A B 4,5 10, 11 1, 2, 3 6 7, 8 9 12 C 13 (A  B)’ = ? A  B = ?

  33. Example – Venn Diagrams • Exercise Set 2.4 #27, 29 U A B II III I V IV VI VII C A  C = ? A ∩ B = ?

  34. U B A U U B A B A De Morgan’s Laws(using Venn Diagrams as a proof) • (A U B)' = A' ∩ B': The complement of the union of two sets is the intersection of the complement of those sets.

  35. U U B A B A U B A De Morgan’s Laws • (A ∩ B)' = A' U B': The complement of the intersection of two sets is the union of the complement of those sets.

  36. Examples: DeMorgan’s Laws U = {1, 2, 3, 4, 5, 6, 7} A = {1, 3, 5, 7} B = {1, 2, 3} • (A ∩ B) ' = ? • A 'U B ' = ?

  37. Chapter 2 Set Theory Thinking Mathematically

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