SECTION 2-4
• Surveys and Cardinal Numbers
Slide 2-4-1
SURVEYS AND CARDINAL NUMBERS
• Surveys
• Cardinal Number Formula
Slide 2-4-2
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.
Slide 2-4-3
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
70 like baseball
40 like hockey
40 like football and baseball
25 like baseball and 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?
Slide 2-4-4
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
Slide 2-4-5
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
Slide 2-4-6
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
Slide 2-4-7
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
Slide 2-4-8
ANALYZING A SURVEY
Solution
(from the Venn diagram)
a) 45 like only football
b) 10 do not like any sports
Slide 2-4-9
CARDINAL NUMBER FORMULA
For any two sets A and B,
n  A B   n( A)  n( B)  n( A B).
Slide 2-4-10
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
Slide 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.
Slide 2-4-12
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
22
77
Over 33
(O)
45
22
24
91
Totals
90
69
64
223
Slide 2-4-13
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
Slide 2-4-14
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.
Slide 2-4-15