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All rights reserved
Surveys and Cardinal Numbers
© 2008 Pearson Addison-Wesley. All rights reserved
2-4-2
Surveys and Cardinal Numbers
• Surveys
• Cardinal Number Formula
2-4-3
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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2-4-4
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Analyzing a Survey
Example: Out of 100 customers at a pizza parlor,
60 ordered pizzas with both cheese and pepperoni.
80 had cheese on their pizza and 72 had pepperoni.
How many customers ordered cheese with no pepperoni?
A Venn diagram can help organize information to be able to answer questions like this.
2-4-5
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Analyzing a Survey
Construct a Venn diagram. Let C = cheese,
P = pepperoni.
C
P
Start with both
60
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2-4-6
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Example: Analyzing a Survey
Construct a Venn diagram. Let C = cheese,
P = pepperoni.
C
20
60
P
12
Subtract to get
© 2008 Pearson Addison-Wesley. All rights reserved
2-4-7
Example: Analyzing a Survey
Construct a Venn diagram. Let C = cheese,
P = pepperoni.
C
20
60
P
12
Subtract total shown from 100 in U to get
8
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2-4-8
Analyzing a Survey
Solution
(from the Venn diagram)
20 had cheese with no pepperoni.
Other questions we could answer?
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2-4-9
Cardinal Number Formula
For any two sets A and B ,
( U B
) = ( ) + ( ) − ( I B ).
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2-4-10
Example: Applying the
Cardinal Number Formula
Find n ( A ) if
( U B
) = 78, ( I B
) =21, and ( ) = 36.
Solution
( U B
) = ( ) + ( ) − ( I B ).
78 = n ( A ) + 36 – 21
78 = n ( A ) + 15
– 15 – 15
63 = n ( A )
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2-4-11
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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2-4-12
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Example: Analyzing Data in a Table
18-25
( Y )
26-33
( M )
Over 33
( O )
Totals
Coffee
( C )
15
30
45
90
Juice
( J )
22
25
22
69
Tea
( T )
18
22
Totals
55
77
24
64
91
223
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2-4-13
Example: Analyzing Data in a Table
( Y )
( M )
( O )
Totals
( C )
15
30
45
90
( J )
22
25
22
69
( T )
18
22
24
64
Totals
55
77
91
223
Using the letters in the table, find the number of people in each of the following sets.
a) Y I C b) O ′ U T
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2-4-14
Example: Analyzing Data in a Table
( Y )
( M )
( O )
Totals
( C
15
30
45
90
) ( J
22
25
22
69
) ( T )
18
22
24
64
Totals
55
77
91
223 a) Y I C : in both Y and C = 15.
O ′ U T : O (so Y + M ) + those not already counted that are in T = 55 + 77 + 24 = 156.
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