1. Chapter 2 The Basic Concepts of Set Theory
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2. 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
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3. Section 2-4 Chapter 1 Surveys and Cardinal Numbers
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4. Surveys and Cardinal Numbers
Cardinal Number Formula
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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.
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6. 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?
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7. 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
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8. 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
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9. 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
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10. 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
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Analyzing a Survey
Solution
(from the Venn diagram)
a) 45 like only football
10 do not like any sports
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12. Cardinal Number Formula
For any two sets A and B,
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13. Example: Applying the Cardinal Number Formula
Find n(A) if
Solution
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14. 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.
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15. 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
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16. 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)
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17. 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
= = 156.
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