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

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

2 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

3 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).

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

5 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) ABAB

6 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.

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

8 Example (cont.) We can now complete the picture. A-B B-A U-(A  B) ABAB From the picture it is easy to see that: B

9 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?

10 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

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

12 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 ABCABC 2

13 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 ABCABC

14 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 ABCABC 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?

15 Example (cont.) A-(B  C) B-(A  C) C-(A  B) U-(A  B  C) (A  B)-C (A  C)-B(B  C)-A ABCABC Exactly two of these movies? 4+8+6=18

16 Example (cont.) A-(B  C) B-(A  C) C-(A  C) U-(A  B  C) (A  B)-C (A  C)-B(B  C)-A ABCABC Exactly one of these movies? 5+7+3=15

17 Example (cont.) A-(B  C) B-(A  C) C-(A  C) U-(A  B  C) (A  B)-C (A  C)-B(B  C)-A ABCABC None of these movies? 20

18 Example (cont.) A-(B  C) B-(A  C) C-(A  C) U-(A  B  C) (A  B)-C (A  C)-B(B  C)-A ABCABC Only The Natural? 5

19 Assignments 2.5, 3.1, 3.2 Read Section 2.5 Due February 6 Exercises p , 7, 9, 11, 13, 14, 15, 24, 29, 32, 37, 38, 39, 40, 43. Read Section 3.1 Due February 8 Exercises p , 39-47, 49-53, Read Section 3.2 Due February 13 Exercises p , 21-25,


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