The Basic Concepts of Set Theory Chapter 2 The Basic Concepts of Set Theory 2012 Pearson Education, Inc.

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 2012 Pearson Education, Inc.

Surveys and Cardinal Numbers Section 2-4 Surveys and Cardinal Numbers 2012 Pearson Education, Inc.

Surveys and Cardinal Numbers Cardinal Number Formula 2012 Pearson Education, Inc.

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. 2012 Pearson Education, Inc.

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? 2012 Pearson Education, Inc.

Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. H F B 20 Start with like all 3 2012 Pearson Education, Inc.

Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. H F B 20 5 8 Subtract to get 2012 Pearson Education, Inc.

Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. H F B 20 5 25 8 45 7 Subtract to get 2012 Pearson Education, Inc.

Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. H F B 20 5 25 8 45 10 7 Subtract total shown from 140 to get 2012 Pearson Education, Inc.

Analyzing a Survey Solution (from the Venn diagram) a) 45 like only football 10 do not like any sports 2012 Pearson Education, Inc.

Cardinal Number Formula For any two sets A and B, 2012 Pearson Education, Inc.

Example: Applying the Cardinal Number Formula Find n(A) if Solution 2012 Pearson Education, Inc.

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. 2012 Pearson Education, Inc.

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 2012 Pearson Education, Inc.

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) 2012 Pearson Education, Inc.

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. 2012 Pearson Education, Inc.