Set Theory PDF

Chapter 1 Math 3201 1 Chapter 1: Set Theory:  Organizing information into sets and subsets  Graphically illustrating the relationships between sets and subsets using Venn diagrams  Solving problems by using sets, subse...

Chapter 1 Math 3201 1 Chapter 1: Set Theory:  Organizing information into sets and subsets  Graphically illustrating the relationships between sets and subsets using Venn diagrams  Solving problems by using sets, subsets and Venn diagrams. Section 1.1: Types of Sets and Set Notation:  Set: a collection of objects. For example, the set of whole numbers is W = {0, 1, 2, 3, …}  Element: an object in a set. For example, 2 is an element of W  Universal Set: all the elements for the sample. For Example, the universal set of digits is D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}  The number of elements in a set: for example, n(D) = 10  Subset: a set whose elements all belong to another set. For Example, set Q ,the set of odd digits Q = {1, 3, 5, 7, 9} is a subset of set D. Using set notation: 𝑄 ⊂ 𝐷 (Q is a subset of D)  Complement: All the elements of the universal set that do not belong to a subset of it. For example, 𝑄′ = {0, 2, 4, 6, 8} is the complement of Q. Notation used is the prime symbol, 𝑄′ or not 𝑄  Empty Set: a set with no elements. For example the set of odd numbers that are divisible by 2 is the empty set. Notation used: { } or Ø Chapter 1 Math 3201 2  Disjoint Sets: two or more sets having no elements in common. For example, the set of even numbers and odd numbers are disjoint.  Finite Set: a set with a countable number of elements. For example the set E ={2, 4, 6, 8}  Infinite Set: a set with an infinite number of elements. For example the set of natural numbers, N = {1, 2, 3, …} Notation introduced so far:  Sets are defined using brackets. For example to define a universal set with the numbers 1,2 and 3, list its elements: U = {1, 2, 3}  To define the set A that has the numbers 1 and 2 as elements: A = {1, 2}  All elements of A are also elements of U, so A is a subset of U: 𝐴 ⊂ 𝑈  The set 𝐴′ , the complement of A, can be defined as: 𝐴′ = {3}  To define set B, a subset of U that contains the number 4: B = { } or B = Ø 𝐵⊂𝑈 Set Notation You can represent a set by o Listing the elements: ex. A = {1,2,3,4,5} o Using words: A represents all integers from 1 to 5 o Using set notation: 𝐴 = {𝑥|1 ≤ 𝑥 ≤ 5, 𝑥 ∈ 𝐼} Example: Represent using set notation in two different ways: Multiples of 2 from 1 to 20: Chapter 1 Math 3201 3 Venn Diagram can be used to show how sets and subsets are related. Example 1: U U= A B 2 4 A= 5 7 1 6 9 3 8 B= n(A) = n(B) = Example 2: Set A = {multiples of 4} Set B = {multiples of 8} Is 𝐴 ⊂ 𝐵 or is 𝐵 ⊂ 𝐴 ? Chapter 1 Math 3201 4 Practice Questions text page 15-18 Chapter 1 Math 3201 5 ANSWERS: Chapter 1 Math 3201 6 Section 1.2: Exploring Relationships between Sets: Example 1: Given the universal set S, S= {4, 5, 6, 8, 9, 11, 15, 17, 20, 24, 30, 32} (A)place the numbers in the appropriate S regions of the Venn Diagram A = {multiples of 2} A B B = {multiples of 3} (B) Why is there an overlap? (C) Identify the elements that are in both A and B notation: 𝐴 ∩ 𝐵 (D)Identify all the elements that are in A or B notation: 𝐴 ∪ 𝐵 (E) Identify the elements that are not in A or B notation: (𝐴 ∪ 𝐵)′ (F) Identify the elements of A that are not part of B notation: 𝐴\𝐵  Sets that are not disjoint share common elements  Each area of a Venn diagram represents something different Elements in set A and set B (A∩B) U Elements in set A Elements in set B but not in set B but not in set A (A\B) (B\A) A B Elements in set U but not in set A or set B (AUB)′ Chapter 1 Math 3201 7 Explore: In a Newfoundland school, there are 65 Grade 12 students. Of these students, 23 play volleyball, and 26 play basketball. There are 31 students who do not play either sport. The following Venn diagram represents the sets of students. S(allGrade12 students) V(volleyball) B(basketball) 1. Consider the set of students who play volleyball and the set of students who play basketball. Are these two sets disjoint? Explain how you know. 2. Use the Venn Diagram to help determine: a. The number of students who play volleyball only. b. The number of students who play basketball only. c. The number of students who play both volleyball and basketball. 3. Describe how you solved the problem. Chapter 1 Math 3201 8 Practice Questions P.20 – 21 Chapter 1 Math 3201 9 Answers:  Chapter 1 Math 3201 10 Section 1.3: Intersection and Union or Two Sets: Venn Diagram/Definitions U = set of all integers from -3 to +3 A = set of non-negative integers B = set of integers divisible by 2 Set Notation Meaning Venn Diagram Answer 𝐴∪𝐵 Any element that is A B {-2, 0, 1, 2, 3} in either of the sets A union B 13 02 -2 Any element that is A OR B in at least one of -1, -3 the sets 𝐴∩𝐵 Only elements that A B {0, 2} are in both A and B A intersection B 13 02 -2 Represented by the A AND B overlap region for -1, -3 non-disjoint sets A B A\ B Elements found in {1, 3} set A but excluding 13 02 -2 A minus B the ones that are also in set B -1, -3 𝐴′ A B All elements in the {-3, -2, -1} A complement universal set 13 02 -2 outside of A Not A -1, -3 A B (𝐴 ∪ 𝐵)′ Elements outside {-3, -1} A and B 13 02 -2 Not(A union B) -1, -3 A B (𝐴 ∩ 𝐵)′ Elements outside of {-3, -2, -1, 1,3} the overlap of A 13 02 -2 Not(A intersect B) and B -1, -3 Chapter 1 Math 3201 11 Venn Diagrams can help develop formulas to determine the number of elements in certain sets. Example 1: What formula can be used to determine the 𝑛(𝐴\𝐵) U A B 2 4 5 7 1 6 9 3 8 There is more than one formula that can be used… Just as long as it makes sense! 𝑛(𝐴\𝐵) = 𝑛(𝐴) − 𝑛(𝐴 ∩ 𝐵) Or 𝑛(𝐴\𝐵) = 𝑛(𝐴 ∪ 𝐵) − 𝑛(𝐵) Example 2: Given the following sets: Set A = {2, 3, 6, 8, 9} Set B = {4, 5, 6, 7, 9} (A)What elements are in 𝐴 ∪ 𝐵 ? (B) What is the (𝐴 ∪ 𝐵) ? (C) The 𝑛(𝐴) = 5 and the 𝑛(𝐵) = 5, should the 𝑛(𝐴 ∪ 𝐵) = 10 ? (D) How can you compensate for this over-counting? Chapter 1 Math 3201 12 The Principle of Inclusion and Exclusion  If two sets, A and B, contain common elements, to calculate the number of elements in A or B, 𝑛(𝐴 ∪ 𝐵), you must subtract the elements in the intersection so that they are not counted twice. 𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 ∩ 𝐵)  If two sets, A and B, are disjoint, they do not have any intersection. 𝑛(𝐴 ∩ 𝐵) = 0 and 𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵)  𝑛(𝐴 ∪ 𝐵) can also be determined using 𝐴\𝐵 𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴\𝐵) + 𝑛(𝐵\𝐴) + 𝑛(𝐴 ∩ 𝐵) Example 3: (Ex.4 page 29) Morgan surveyed the 30 students in her math class about their eating habits.  18 of these students eat breakfast  5 of the 18 also eat a healthy lunch  3 students do not eat breakfast and do not eat a healthy lunch. How many students eat a healthy lunch? Tyler solved the problem, as shown below but made an error. What error did Tyler make? Determine the correct solution. C There are 3 students who don’t belong in either B L region. This means there are 30 – 3 = 27 in B or L 18 5 x 18 + 5 + x = 27 x=4 3 therefore, n(L) = 5 + 4 n(L) = 9 Practice problem: Page 32-34 #1,3,8,10,15,16 Chapter 1 Math 3201 13 Section 1.4: Applications of Set Theory: Working with 3 sets in a Venn Diagram:  Start at center where 3 sets intersect.  Reminders the numbers in all the regions total the number in the universal set.  Careful of wording. “only” means just that region Example 1: U P C There are 36 students who study science. 14 study Physics 18 study chemistry 24 study biology 5 study physics and chemistry 8 study physics and biology 10 study biology and chemistry 3 study all three subjects. B Determine the number of students who A) Study physics and biology only B) Study at least two subjects C) Study biology only Example 2: A survey of a machine shop reveals the following information about its employees: 44 can run a lathe S L M 49 can run a milling machine 56 can operate a press punch 27 can run a lathe and milling machine 19 can run a milling machine and operate a press punch 24 can run a lathe and operate a press punch 10 can operate all three machines. 9 cannot operate any of the machines P How many people are employed at the machine shop? Chapter 1 Math 3201 14 Example 3: There are 25 dogs at the dog show. 12 dogs are black, 8 dogs have a short tail, 15 dogs have long hair 1 dog is black with a short tail and long hair D B S 3 dogs are black with short tails but do not have long hair 2 dogs have short tails and long hair but are not black. If all the dogs in the kennel have at least one of the mentioned characteristics, how many dogs are black with long hair but do not have short tails? L Example 4: 28 children have a dog, a cat, or a bird 13 children have a dog, 13 children have a cat, and 13 children have a bird. 4 children have only a dog and a cat 3 children have only a dog and a bird P B C 2 children have only a cat and a bird No child has two of each type of pet. A) How many children have a cat, a dog, and a bird? B) How many children have only 1 pet? D Practice problem: Page 51-54 #2,4,6,9,14 Review Page 56 #1,2,4, Page 58#3-7  Chapter 1 Math 3201 15 Chapter 1 Review Section 1: Multiple Choice. 1. Given A = {1, 3, 6, 8, 9, 12, 15} and B = {2, 7, 14}, which is false? 1. ____ (A) B is the complement of A (B) A  B   (C) A and B are disjoint sets (D) B  A 2. Consider the sets: A = {1, 3, 4, 7, 8, 9} B = {1, 2, 3, 4, 5} C = {1, 3} 2.____ What is A∩B ? A) {1, 3} B) {1, 3, 4} C) {1, 2, 3, 4, 5} D) {1, 2, 3, 4, 5, 7, 8, 9} 3. Describe the shaded region: 3.____ A) B B) B' C) A' D) B\A 4. In a class there are 30 students. 4.____  21 students like Math  16 students like English  6 students don't like Math or English How many students like both Math and English? A) 5 B) 7 C)13 D) 24 5. Consider the sets: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 3, 5, 7, 9} B = {2, 4, 6} What is A' ? 5.____ A) {2, 4, 6} B) {2, 4, 6, 8} C) {1, 3, 5, 7, 9} D) {1, 2, 3, 4, 5, 6, 7, 8, 9} 6. Consider the sets: P = {2, 4, 6, 8, } Q = {𝑂𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 0 𝑎𝑛𝑑 10} 6.____ R = {1, 3, 5, 7} S = {1, 2, 3, 4}. Which of the following is true? A) P ⊂ Q B) S ⊂ P C) Q ⊂ R D) R ⊂ Q  Chapter 1 Math 3201 16 7. A summer camp offers canoeing, rock climbing, A and archery. C The Venn diagram shows the types of activities 3 17 6 the campers like. 5 11 5 Use the diagram to determine 𝑛((𝐴 ∪ 𝐶)\𝑅). 20 7.____ R (A) 26 (B) 47 (C) 42 (D) 67 8. Set M consists of the multiples of 3 from 1 to 30. Which represents set notation? 8._____ (A) M={1, 2, 3, … , 28, 29, 30} (B) M={3𝑥|1 ≤ 𝑥 ≤ 10, 𝑥 ∈ 𝑁} (C) M={3| 1 ≤ 𝑥 ≤ 30, 𝑥 ∈ 𝑁} (D) M={3𝑥|1 ≤ 𝑥 ≤ 30, 𝑥 ∈ 𝑁} Section 2: Constructed Response 9. Carlos surveyed 75 students about their favorite subjects in school. He recorded his results. Favorite Subject Number of Students mathematics 27 science 25 neither mathematics nor science 30 (A) Determine how many students like mathematics and science. (B) Determine how many students like only mathematics or only science. 10. 66 students were surveyed to determine their travel interests. 32 students wanted to go to Spain 27 students wanted to go to New York 44 students wanted to go to Paris 16 students wanted to go to New York and Paris 18 students wanted to go to Paris and Spain 10 students wanted to go to all three destinations How many students wanted to go to New York and Spain but not Paris? Answers: 1d 2b 3d 4c 5b 6d 7a 8b 9A)7 B)3810 10. 3