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

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

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).

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

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

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.

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

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

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?

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

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

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

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

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?

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

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

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

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

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,