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Survey of Mathematical Ideas Math 100 Chapter 2 John Rosson Thursday February 1, 2007

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Basic Concepts of Set Theory 1.Symbols and Terminology 2.Venn Diagrams and Subsets 3.Set Operations and Cartesian Products 4.Cardinal Numbers and Surveys 5.Infinite Sets and Their Cardinalities

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Cardinal Numbers and Surveys Problem: Given partial information about the cardinality of some sets, determine the remaining cardinality information. Example: Suppose we are given the following information about sets A and B. Determine the missing information: n(U), n(A B), n(B).

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Cardinal Numbers and Surveys Look at the problem as filling in a Venn diagram: A B U

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Cardinal Numbers and Surveys Consider the 4 disjoint “pieces” of the diagram. If we knew the cardinalities of these pieces we could calculate everything else. In general, for two sets we need at least 4 pieces of information to solve the problem. A-B B-A U-(A B) ABAB

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Cardinal Number Formula The main tool we will use to solve these types of problems is the: Theorem: (The Cardinal Number Formula) For any two sets A and B, There are four terms in this equation: n(A B), n(A), n(B) and n(A B). So, if we know three of the terms we can solve for the fourth.

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Example (cont.) Returning to our example. Remember, so, using the Cardinal number formula,

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Example (cont.) We can now complete the picture. A-B B-A U-(A B) ABAB 5 6 7 12 From the picture it is easy to see that: B

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Surveys One application of the previous type of problem is in answering questions about surveys. The following example is problem 20, p. 81. Julianne Peterson, a sports psychologist, was planning a study of viewer response to certain aspects of the movies The Natural, Field of Dreams and The Rookie. Upon surveying a group of 55 students, she determined the following: 17 had seen The Natural 17 had seen Field of Dreams 23 had seen The Rookie 6 had seen The Natural and Field of Dreams 8 had seen The Natural and The Rookie 10 had seen Field of Dreams and The Rookie 2 had seen all three movies How many students had seen: 1.Exactly two of these movies? 2.Exactly one of these movies? 3.None of these movies 4.Only The Natural?

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Example (cont.) First define your sets. A=“students who saw The Natural” B=“students who saw Field of Dreams” C=“students who saw The Rookie” Next, translate the survey information into set theory. Upon surveying a group of 55 students, she determined the following: 17 had seen The Natural 17 had seen Field of Dreams 23 had seen The Rookie 6 had seen The Natural and Field of Dreams 8 had seen The Natural and The Rookie 10 had seen Field of Dreams and The Rookie 2 had seen all three movies

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Example (cont.) Draw the 3 set Venn diagram. A B C U

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Example (cont.) Notice that we get 8 disjoint pieces. A-(B C) B-(A C) C-(A B) U-(A B C) (A B)-C (A C)-B(B C)-A ABCABC 2

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Example (cont.) Its easy to finish the picture. A-(B C) B-(A C) C-(A B) U-(A B C) (A B)-C (A C)-B(B C)-A ABCABC 2 6 4 8

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Example (cont.) Finally, we now have a cardinality for each piece. We can now use the completed diagram to answer the questions. A-(B C) B-(A C) C-(A B) U-(A B C) (A B)-C (A C)-B(B C)-A ABCABC 2 6 4 8 5 3 7 20 How many students had seen: 1.Exactly two of these movies? 2.Exactly one of these movies? 3.None of these movies 4.Only The Natural?

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Example (cont.) A-(B C) B-(A C) C-(A B) U-(A B C) (A B)-C (A C)-B(B C)-A ABCABC 2 6 4 8 5 3 7 20 Exactly two of these movies? 4+8+6=18

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Example (cont.) A-(B C) B-(A C) C-(A C) U-(A B C) (A B)-C (A C)-B(B C)-A ABCABC 2 6 4 8 5 3 7 20 Exactly one of these movies? 5+7+3=15

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Example (cont.) A-(B C) B-(A C) C-(A C) U-(A B C) (A B)-C (A C)-B(B C)-A ABCABC 2 6 4 8 5 3 7 20 None of these movies? 20

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Example (cont.) A-(B C) B-(A C) C-(A C) U-(A B C) (A B)-C (A C)-B(B C)-A ABCABC 2 6 4 8 5 3 7 20 Only The Natural? 5

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Assignments 2.5, 3.1, 3.2 Read Section 2.5 Due February 6 Exercises p. 88 1-6, 7, 9, 11, 13, 14, 15, 24, 29, 32, 37, 38, 39, 40, 43. Read Section 3.1 Due February 8 Exercises p. 99 1-9, 39-47, 49-53, 57-74 Read Section 3.2 Due February 13 Exercises p. 111 1-20, 21-25, 45-55.

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