Transcript File - Math is Magical

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
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2-4-1
Chapter 1
Section 2-4
Surveys and Cardinal Numbers
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2-4-2
Surveys and Cardinal Numbers
• Surveys
• Cardinal Number Formula
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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
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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2-4-5
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
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2-4-6
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
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2-4-7
Example: Analyzing a Survey
Construct a Venn diagram. Let F = football,
B = baseball, and H = hockey.
F
20
45
8
20
B
25
Subtract to get
5
7
H
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2-4-8
Example: Analyzing a Survey
Construct a Venn diagram. Let F = football,
B = baseball, and H = hockey.
F
20
45
8
20
7
H
B
25
Subtract total
shown from 140
to get
5
10
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2-4-9
Analyzing a Survey
Solution
(from the Venn diagram)
a) 45 like only football
b) 10 do not like any sports
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2-4-10
Cardinal Number Formula
For any two sets A and B,
n  A B   n( A)  n( B)  n( A B).
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2-4-11
Example: Applying the Cardinal
Number Formula
Find n(A) if
n  A B   78, n  A B  =21, and n( B)  36.
Solution
n( A)  n  A B   n( B)  n  A B 
 78  36  21
 63
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2-4-12
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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2-4-13
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
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2-4-14
Example: Analyzing Data in a Table
(C)
(J)
(T)
Totals
(Y)
15
22
18
55
(M)
30
25
22
77
(O)
45
22
24
91
Totals
90
69
64
223
Using the letters in the table, find the number of
people in each of the following sets.
a) Y
C
b) O T
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2-4-15
Example: Analyzing Data in a Table
(Y)
(M)
(O)
Totals
a) Y
(C)
(J)
(T)
Totals
15
30
45
90
22
25
22
69
18
22
24
64
55
77
91
223
C : in both Y and C = 15.
b) O T : not in O (so Y + M) + those not already
counted that are in T = 55 + 77 + 24 = 156.
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2-4-16