Chapter 2 The Basic Concepts of Set Theory
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 2. 2 2. 3 2. 4 2. 5 Symbols and Terminology Venn Diagrams and Subsets Set Operations and Cartesian Products Surveys and Cardinal Numbers Infinite Sets and Their Cardinalities © 2008 Pearson Addison-Wesley. All rights reserved 2
Chapter 1 Section 2 -4 Surveys and Cardinal Numbers © 2008 Pearson Addison-Wesley. All rights reserved
Surveys and Cardinal Numbers • Surveys • Cardinal Number Formula © 2008 Pearson Addison-Wesley. All rights reserved 4
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 5
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 6
Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. B F 20 Start with like all 3 H © 2008 Pearson Addison-Wesley. All rights reserved 7
Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. F B 20 8 20 Subtract to get 5 H © 2008 Pearson Addison-Wesley. All rights reserved 8
Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. F 45 20 8 20 B 25 Subtract to get 5 7 H © 2008 Pearson Addison-Wesley. All rights reserved 9
Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. F 45 20 8 20 7 H B 25 Subtract total shown from 140 to get 5 10 © 2008 Pearson Addison-Wesley. All rights reserved 10
Analyzing a Survey Solution (from the Venn diagram) a) 45 like only football b) 10 do not like any sports © 2008 Pearson Addison-Wesley. All rights reserved 11
Cardinal Number Formula For any two sets A and B, © 2008 Pearson Addison-Wesley. All rights reserved 12
Example: Applying the Cardinal Number Formula Find n(A) if Solution © 2008 Pearson Addison-Wesley. All rights reserved 13
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 14
Example: Analyzing Data in a Table Coffee (C) Juice (J) Tea (T) Totals 18 -25 (Y) 15 22 18 55 26 -33 (M) Over 33 (O) Totals 30 25 22 77 45 22 24 91 90 69 64 223 © 2008 Pearson Addison-Wesley. All rights reserved 15
Example: Analyzing Data in a Table (J) 22 (T) 18 Totals (Y) (C) 15 (M) 30 25 22 77 (O) 45 90 22 69 24 64 91 223 Totals 55 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 16
Example: Analyzing Data in a Table (Y) (M) (O) Totals a) (C) (J) (T) Totals 15 30 45 90 22 25 22 69 18 22 24 64 55 77 91 223 in both Y and C = 15. 1. b) not in O (so Y + M) + those not already counted that 1. are in T+ 77 + 24 = 156. = 55 © 2008 Pearson Addison-Wesley. All rights reserved 17
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