Section 2'5 Surveys and Cardinal Numbers

1
Section 2.5 Surveys and Cardinal Numbers

• Thinking Mathematically
• Math 1001

2
Objectives

• Use Venn diagrams to visualize a surveys
  results.
• Use the formula for n(AUB).
• Use Survey Results to complete Venn diagrams and
  answer questions about the survey.

3
Example 1a Using a Venn Diagram to Visualize the
Results of a Survey

• How many students are willing to donate blood?
• Regions I and II
• 370 120 490
• 490 students

U
A
B
I
III
II
290
370
120
220
IV
A Set of students willing to donate blood
B Set of students willing to serve breakfast
4
Example 1b Using a Venn Diagram to Visualize the
Results of a Survey

• How many students are willing to serve a free
  breakfast to blood donors?
• Regions II and III
• 120 220 340
• 340 students

U
A
B
I
III
II
290
370
120
220
IV
A Set of students willing to donate blood
B Set of students willing to serve breakfast
5
Example 1c Using a Venn Diagram to Visualize the
Results of a Survey

• How many students are willing to donate blood and
  serve breakfast?
• Region II
• 120 students

U
A
B
I
III
II
290
370
120
220
IV
A Set of students willing to donate blood
B Set of students willing to serve breakfast
6
Example 1d Using a Venn Diagram to Visualize the
Results of a Survey

• How many students are willing to donate blood or
  serve breakfast?
• Regions I, II, III
• 370 120 220 710
• 710 students

U
A
B
I
III
II
290
370
120
220
IV
A Set of students willing to donate blood
B Set of students willing to serve breakfast
7
Example 1e Using a Venn Diagram to Visualize the
Results of a Survey

• How many students are willing to donate blood but
  not serve breakfast?
• Region I
• 370
• 370 students

U
A
B
I
III
II
290
370
120
220
IV
A Set of students willing to donate blood
B Set of students willing to serve breakfast
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Example 1f Using a Venn Diagram to Visualize the
Results of a Survey

• How many students are willing to serve breakfast
  but not donate blood?
• Region III
• 220
• 220 students

U
A
B
I
III
II
290
370
120
220
IV
A Set of students willing to donate blood
B Set of students willing to serve breakfast
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Example 1g Using a Venn Diagram to Visualize the
Results of a Survey

• How many students are neither willing to donate
  blood nor serve breakfast?
• Region IV
• 290
• 290 students

U
A
B
I
III
II
290
370
120
220
IV
A Set of students willing to donate blood
B Set of students willing to serve breakfast
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Example 1h Using a Venn Diagram to Visualize the
Results of a Survey

• How many students were surveyed?
• Regions I, II, III, IV
• 370 120 220 290 1000
• 1000 students

U
A
B
I
III
II
290
370
120
220
IV
A Set of students willing to donate blood
B Set of students willing to serve breakfast
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The Cardinal Number of the Union of Two Sets

• Example 1d How many students are willing to
  donate blood or serve breakfast?
• Regions I, II, III
• 370 120 220 710

n(AUB)
U
A
B
I
III
II
290
370
120
220
IV
n(A) 370 120 490
A Set of students willing to donate blood
n(B) 220 120 340
B Set of students willing to serve breakfast
490 340 830
710 ? 830 because we added 120 twice.
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Formula for the Cardinal Number of the Union of
Two Sets

• n(AUB) n(A) n(B) n(AnB)
• To find the cardinal number in the union of sets
  A and B, add the number of elements in set A and
  B and then subtract the number of elements common
  to both sets.

490 340 120 710
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In a survey of 520 college students

• 275 were taking history
• 300 were taking mathematics
• 175 were taking both history and mathematics.
• Construct the Venn Diagram

Region II 175
U
History
Math
Region I ?
I
III
II
20
100
175
225
275 175 100
IV
Region III ?
300 175 225
Region IV
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Problem 24 Page 81A Survey of 75 college
students Where do you get your news?

• 29 newspapers
• 43 television
• 7 both newspapers and television

• How many got the news from
• Only newspapers?
• Only television?
• Papers or TV?
• How many did not get the news from either papers
  or TV?

22
36
U
Papers
TV
22 7 36 65
I
III
II
75-(22736) 10
29-722
7
43-736
IV
75 (22 7 36) 10
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Problem 26 Page 82 Survey of 180 College mens
participation

• How many participated in only campus sports?
• How many participated in frats and sports but not
  tutoring?
• How many participated in frats or sports but not
  tutoring?
• How many participated in exactly one of these
  activities?
• How many participated in exactly two of these
  activities?
• How many did not participate in any activities?

30
8
43 - 8 - 5 - 7 23

• 43 fraternities
• 52 sports
• 35 tutoring
• 13 frats and sports
• 14 sports and tutors
• 12 frats and tutors
• 5 all three

52 - 8 - 5 - 9 30
35 - 7 - 5 - 9 14
61
23
8
30
U
Frat
67
Sport
14
23
30
8
30
23
5
24
9
8
7
9
7
14
Tutor
VIII
180 - (5789142330) 84
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Section 2.5 Homework

• Page 80
• 1-9, 14-16, 19-20, 24, 26, 30,