Surveys and Cardinal Numbers
MATH 100 Survey of Mathematical Ideas
J. Robert Buchanan
Department of Mathematics
Fall 2014
J. Robert Buchanan
Surveys and Cardinal Numbers
Surveys
Today we will learn how to determine the cardinal numbers of
subsets from information obtained by surveys.
There are three ways we can approach such problems:
1
Venn diagrams,
2
Cardinal number formulas,
3
Tables.
J. Robert Buchanan
Surveys and Cardinal Numbers
Venn Diagrams (1 of 4)
Paula Story is a fan of the music of Paul Simon and Art
Garfunkel. In her collection of 22 compact disks, she has the
following:
5 on which Simon and Garfunkel sing
8 on which Simon sings
7 on which Garfunkel sings
12 on which neither Simon nor Garfunkel sings.
J. Robert Buchanan
Surveys and Cardinal Numbers
Venn Diagrams (1 of 4)
Paula Story is a fan of the music of Paul Simon and Art
Garfunkel. In her collection of 22 compact disks, she has the
following:
5 on which Simon and Garfunkel sing
8 on which Simon sings
7 on which Garfunkel sings
12 on which neither Simon nor Garfunkel sings.
1
How many of her compact disks feature only Paul Simon?
2
How many of her compact disks feature only Art
Garfunkel?
3
How many feature at least one of these two artists?
J. Robert Buchanan
Surveys and Cardinal Numbers
Venn Diagrams (2 of 4)
We can start with a Venn diagram with a universal set of all of
Paula Story’s CDs, a set representing CDs on which Simon
sings, and a set on which Garfunkel sings.
U
Simon
J. Robert Buchanan
Garfunkel
Surveys and Cardinal Numbers
Venn Diagrams (2 of 4)
We can start with a Venn diagram with a universal set of all of
Paula Story’s CDs, a set representing CDs on which Simon
sings, and a set on which Garfunkel sings.
U
Simon
Garfunkel
Now we must determine the cardinality of each subset. First,
on how many CDs do both Simon and Garfunkel sing, and on
how many CDs do neither sing?
J. Robert Buchanan
Surveys and Cardinal Numbers
Venn Diagrams (3 of 4)
U
12
Simon
J. Robert Buchanan
5
Garfunkel
Surveys and Cardinal Numbers
Venn Diagrams (3 of 4)
U
12
Simon
5
Garfunkel
Keeping in mind that Simon sings on 8 CDs, how many CDs
feature Simon only?
J. Robert Buchanan
Surveys and Cardinal Numbers
Venn Diagrams (3 of 4)
U
12
3
Simon
J. Robert Buchanan
5
Garfunkel
Surveys and Cardinal Numbers
Venn Diagrams (3 of 4)
U
12
3
Simon
5
Garfunkel
Keeping in mind that Garfunkel sings on 7 CDs, how many CDs
feature Garfunkel only?
J. Robert Buchanan
Surveys and Cardinal Numbers
Venn Diagrams (4 of 4)
U
12
3
Simon
J. Robert Buchanan
2
5
Garfunkel
Surveys and Cardinal Numbers
Venn Diagrams (4 of 4)
U
12
3
Simon
2
5
Garfunkel
Use your i>clicker2 to answer the following questions.
How many of her compact disks feature only Paul Simon?
J. Robert Buchanan
Surveys and Cardinal Numbers
Venn Diagrams (4 of 4)
U
12
3
Simon
2
5
Garfunkel
Use your i>clicker2 to answer the following questions.
How many of her compact disks feature only Paul Simon?
How many of her compact disks feature only Art
Garfunkel?
J. Robert Buchanan
Surveys and Cardinal Numbers
Venn Diagrams (4 of 4)
U
12
3
Simon
2
5
Garfunkel
Use your i>clicker2 to answer the following questions.
How many of her compact disks feature only Paul Simon?
How many of her compact disks feature only Art
Garfunkel?
How many feature at least one of these two artists?
J. Robert Buchanan
Surveys and Cardinal Numbers
Cardinal Number Formula
Definition
For any two sets A and B,
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
J. Robert Buchanan
Surveys and Cardinal Numbers
Cardinal Number Formula
Definition
For any two sets A and B,
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
Example
Find the value of n(A ∩ B) if n(A) = 20, n(B) = 14, and
n(A ∪ B) = 30.
J. Robert Buchanan
Surveys and Cardinal Numbers
Cardinal Number Formula
Definition
For any two sets A and B,
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
Example
Find the value of n(A ∩ B) if n(A) = 20, n(B) = 14, and
n(A ∪ B) = 30.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
30 = 20 + 14 − n(A ∩ B)
n(A ∩ B) = 4
J. Robert Buchanan
Surveys and Cardinal Numbers
Example
Carol Britz who sells college textbooks, interviewed freshmen
on a college campus to find out the main goals of today’s
students.
W : the set of students who want to be wealthy
F : the set of students who want to raise a family
E: the set of students who want to become experts in
their fields
The findings of the survey included the following cardinalities:
n(W ) = 160
n(E ∩ F ) = 90
n(F ) = 140
n(W ∩ F ∩ E) = 80
n(E) = 130
n(E 0 ) = 95
n(W ∩ F ) = 95
n[(W ∪ F ∪ E)0 ] = 10
Find the total number of students surveyed.
J. Robert Buchanan
Surveys and Cardinal Numbers
Solution
Number of students surveyed is n(U).
J. Robert Buchanan
Surveys and Cardinal Numbers
Solution
Number of students surveyed is n(U).
n(U) = n(E ∪ E 0 )
J. Robert Buchanan
Surveys and Cardinal Numbers
Solution
Number of students surveyed is n(U).
n(U) = n(E ∪ E 0 )
= n(E) + n(E 0 ) − n(E ∩ E 0 )
J. Robert Buchanan
Surveys and Cardinal Numbers
Solution
Number of students surveyed is n(U).
n(U) = n(E ∪ E 0 )
= n(E) + n(E 0 ) − n(E ∩ E 0 )
= 130 + 95 − n(∅)
= 130 + 95 − 0
= 225
J. Robert Buchanan
Surveys and Cardinal Numbers
Tables
A study of US Army housing trends categorized personnel as
commissioned officers (C), warrant officers (W ), or enlisted
(E), and categorized their living facilities as on-base (B), rented
off-base (R), or owned off-base (O). The survey yielded the
following data.
C
Personnel W
E
Totals
B
12
4
374
390
J. Robert Buchanan
Facilities
R
29
5
71
105
O
54
6
285
345
Totals
95
15
730
840
Surveys and Cardinal Numbers
Tables
A study of US Army housing trends categorized personnel as
commissioned officers (C), warrant officers (W ), or enlisted
(E), and categorized their living facilities as on-base (B), rented
off-base (R), or owned off-base (O). The survey yielded the
following data.
C
Personnel W
E
Totals
B
12
4
374
390
Facilities
R
29
5
71
105
O
54
6
285
345
Totals
95
15
730
840
Use your i>clicker2 to enter the cardinalities of the following
sets:
W ∩O
J. Robert Buchanan
Surveys and Cardinal Numbers
Tables
A study of US Army housing trends categorized personnel as
commissioned officers (C), warrant officers (W ), or enlisted
(E), and categorized their living facilities as on-base (B), rented
off-base (R), or owned off-base (O). The survey yielded the
following data.
C
Personnel W
E
Totals
B
12
4
374
390
Facilities
R
29
5
71
105
O
54
6
285
345
Totals
95
15
730
840
Use your i>clicker2 to enter the cardinalities of the following
sets:
W ∩O
C∪B
J. Robert Buchanan
Surveys and Cardinal Numbers
Tables
A study of US Army housing trends categorized personnel as
commissioned officers (C), warrant officers (W ), or enlisted
(E), and categorized their living facilities as on-base (B), rented
off-base (R), or owned off-base (O). The survey yielded the
following data.
C
Personnel W
E
Totals
B
12
4
374
390
Facilities
R
29
5
71
105
O
54
6
285
345
Totals
95
15
730
840
Use your i>clicker2 to enter the cardinalities of the following
sets:
W ∩O
C∪B
R0 ∪ W 0
J. Robert Buchanan
Surveys and Cardinal Numbers
Tables
A study of US Army housing trends categorized personnel as
commissioned officers (C), warrant officers (W ), or enlisted
(E), and categorized their living facilities as on-base (B), rented
off-base (R), or owned off-base (O). The survey yielded the
following data.
C
Personnel W
E
Totals
B
12
4
374
390
Facilities
R
29
5
71
105
O
54
6
285
345
Totals
95
15
730
840
Use your i>clicker2 to enter the cardinalities of the following
sets:
(C ∩ B) ∪ (E ∩ O)
W ∩O
C∪B
R0 ∪ W 0
J. Robert Buchanan
Surveys and Cardinal Numbers
Tables
A study of US Army housing trends categorized personnel as
commissioned officers (C), warrant officers (W ), or enlisted
(E), and categorized their living facilities as on-base (B), rented
off-base (R), or owned off-base (O). The survey yielded the
following data.
C
Personnel W
E
Totals
B
12
4
374
390
Facilities
R
29
5
71
105
O
54
6
285
345
Totals
95
15
730
840
Use your i>clicker2 to enter the cardinalities of the following
sets:
W ∩O
(C ∩ B) ∪ (E ∩ O)
C∪B
(C ∪ W ) ∩ (B ∪ R)
R0
∪
W0
J. Robert Buchanan
Surveys and Cardinal Numbers
Tables
A study of US Army housing trends categorized personnel as
commissioned officers (C), warrant officers (W ), or enlisted
(E), and categorized their living facilities as on-base (B), rented
off-base (R), or owned off-base (O). The survey yielded the
following data.
C
Personnel W
E
Totals
B
12
4
374
390
Facilities
R
29
5
71
105
O
54
6
285
345
Totals
95
15
730
840
Use your i>clicker2 to enter the cardinalities of the following
sets:
W ∩O
(C ∩ B) ∪ (E ∩ O)
C∪B
(C ∪ W ) ∩ (B ∪ R)
R0
B ∩ (W ∪ R)0
∪
W0
J. Robert Buchanan
Surveys and Cardinal Numbers