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Thinking Mathematically Chapter 2 Set Theory 2.1 Basic Set Concepts 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. 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}. 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 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 } The Empty Set The empty set, also called the null set, is the set that contains no elements. The empty set is represented by { } or Ø 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} 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” 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} 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) = ? 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? 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. 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? Thinking Mathematically Chapter 2 Set Theory 2.3 Venn Diagrams and Set Operations [we’ll come back to 2.2] 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 Venn Diagrams “Disjoint” sets have no elements in common. U A U A All elements of B are also elements of A. The sets A and B have some common elements. B B U A B 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 }. Complement of a Set U A A’ 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 Definition of Intersection of Sets The intersection of sets A and B, written AB, 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} U A B 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} U A B The Empty Set in Intersection and Union For any set A: 1. A ∩ = 2. A = A 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} • AB=? • AB=? • A=? • A∩ =? 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? Thinking Mathematically Chapter 2 Set Theory 2.2 Subsets Definition of a Subset of a Set Set B is a subset of set A, expressed as BA if every element in set B is also an element in set A. U A B Every set is a subset of itself: A A 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? The Empty Set as a Subset 1. For any set B, B. 2. For any set B other than the empty set, B. 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) Thinking Mathematically Chapter 2 Set Theory 2.4 Set Operations and Venn Diagrams With Three Sets Example: Operations with three sets Exercise Set 2.4 #3, 15 • U = {1, 2, 3, 4, 5, 6, 7} • U = {a, b, c, d, e, f, g, h} A = {1, 3, 5, 7} B = {1, 2, 3} C = {2, 3, 4, 5, 6} (A B) ∩ (A C) A = {a, g, h} B = {b, h, h} C = {b, c, d, e, f} (A B) ∩ (A C) Example – Venn Diagrams Exercise Set 2.4 #35, 37 U A B 4,5 1, 2, 3 7, 8 10, 11 6 9 12 13 AB=? C (A B)’ = ? Example – Venn Diagrams Exercise Set 2.4 #27, 29 U A I II B III IV V VI VII C AC=? A∩B=? 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. U B A U A B U A B 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. U U A B B A U A B Examples: DeMorgan’s Laws U = {1, 2, 3, 4, 5, 6, 7} A = {1, 3, 5, 7} B = {1, 2, 3} • (A ∩ B) ' = ? • A'UB'=? Thinking Mathematically Chapter 2 Set Theory