Chapter 2 The Basic Concepts of Set Theory © 2008 Pearson Addison-Wesley. All rights reserved

Chapter 2: The Basic Concepts of Set Theory 2.1 Symbols and Terminology 2.2 Venn Diagrams and Subsets 2.3 Set Operations and Cartesian Products 2.4 Surveys and Cardinal Numbers 2.5 Infinite Sets and Their Cardinalities © 2008 Pearson Addison-Wesley. All rights reserved

Section 2-4 Chapter 1 Surveys and Cardinal Numbers © 2008 Pearson Addison-Wesley. All rights reserved

Surveys and Cardinal Numbers Cardinal Number Formula © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Surveys Problems involving sets of people (or other objects) sometimes require analyzing known information about certain subsets to obtain cardinal numbers of other subsets. The “known information” is often obtained by administering a survey. © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing a Survey Suppose that a group of 140 people were questioned about particular sports that they watch regularly and the following information was produced. 93 like football 40 like football and baseball 70 like baseball 25 like baseball and hockey 40 like hockey 28 like football and hockey 20 like all three a) How many people like only football? b) How many people don’t like any of the sports? © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. B F Start with like all 3 20 H © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. B F 20 Subtract to get 20 8 5 H © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. B F 20 25 Subtract to get 45 20 8 5 7 H © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. B F 20 25 Subtract total shown from 140 to get 45 20 8 5 7 10 H © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Analyzing a Survey Solution (from the Venn diagram) a) 45 like only football 10 do not like any sports © 2008 Pearson Addison-Wesley. All rights reserved

Cardinal Number Formula For any two sets A and B, © 2008 Pearson Addison-Wesley. All rights reserved

Example: Applying the Cardinal Number Formula Find n(A) if Solution © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing Data in a Table On a given day, breakfast patrons were categorized according to age and preferred beverage. The results are summarized on the next slide. There will be questions to follow. © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing Data in a Table Coffee (C) Juice (J) Tea (T) Totals 18-25 (Y) 15 22 18 55 26-33 (M) 30 25 77 Over 33 (O) 45 24 91 90 69 64 223 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing Data in a Table (C) (J) (T) Totals (Y) 15 22 18 55 (M) 30 25 77 (O) 45 24 91 90 69 64 223 Using the letters in the table, find the number of people in each of the following sets. a) b) © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing Data in a Table (C) (J) (T) Totals (Y) 15 22 18 55 (M) 30 25 77 (O) 45 24 91 90 69 64 223 in both Y and C = 15. b) not in O (so Y + M) + those not already counted that are in T = 55 + 77 + 24 = 156. © 2008 Pearson Addison-Wesley. All rights reserved