section 2-4 surveys and cardinal numbers slide 2-4-1
TRANSCRIPT
SECTION 2-4
• Surveys and Cardinal Numbers
Slide 2-4-1
SURVEYS AND CARDINAL NUMBERS
• Surveys• Cardinal Number Formula
Slide 2-4-2
SURVEYS
Slide 2-4-3
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.
EXAMPLE: ANALYZING A SURVEY
Slide 2-4-4
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?
EXAMPLE: ANALYZING A SURVEY
Slide 2-4-5
Construct a Venn diagram. Let F = football, B = baseball, and H = hockey.
H
F B
20Start with like all 3
EXAMPLE: ANALYZING A SURVEY
Slide 2-4-6
Construct a Venn diagram. Let F = football, B = baseball, and H = hockey.
H
F B
20
20
58
Subtract to get
EXAMPLE: ANALYZING A SURVEY
Slide 2-4-7
Construct a Venn diagram. Let F = football, B = baseball, and H = hockey.
H
F B
20
20
5
25
8
45
7
Subtract to get
EXAMPLE: ANALYZING A SURVEY
Slide 2-4-8
Construct a Venn diagram. Let F = football, B = baseball, and H = hockey.
H
F B
20
20
5
25
8
45
107
Subtract total shown from 140 to get
ANALYZING A SURVEY
Slide 2-4-9
Solution(from the Venn diagram)
a) 45 like only football
b) 10 do not like any sports
CARDINAL NUMBER FORMULA
For any two sets A and B,
Slide 2-4-10
( ) ( ) ( ).n A B n A n B n A B
EXAMPLE: APPLYING THE CARDINAL NUMBER FORMULA
Find n(A) if
Slide 2-4-11
78, =21, and ( ) 36.n A B n A B n B
Solution ( ) ( )
78 36 21
63
n A n A B n B n A B
EXAMPLE: ANALYZING DATA IN A TABLE
Slide 2-4-12
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.
EXAMPLE: ANALYZING DATA IN A TABLE
Slide 2-4-13
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
EXAMPLE: ANALYZING DATA IN A TABLE
Slide 2-4-14
(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) b)Y C O T
EXAMPLE: ANALYZING DATA IN A TABLE
Slide 2-4-15
(C) (J) (T) Totals
(Y) 15 22 18 55
(M) 30 25 22 77
(O) 45 22 24 91Totals 90 69 64 223
a) in both Y and C:Y C
:O Tb) not in O (so Y + M) + those not already counted that are in T
= 15.
= 55 + 77 + 24 = 156.