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Finite Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan

Chapter 6

Sets and counting - all with Video Answers

Educators

Section 1

Sets and Set operation

00:58

Problem 1

write the set in set-builder notation.
The set of gold medalists in the 2014 Winter Olympic Games

AG
Ankit Gupta
Numerade Educator
00:27

Problem 2

Write the set in set-builder notation.
The set of football teams in the NEL

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
00:58

Problem 3

Write the set in set-builder notation.
$\{3,4,5,6,7\}$

AG
Ankit Gupta
Numerade Educator
00:58

Problem 4

Write the set in set-builder notation.
$1,3,5,7,9,11, \ldots, 39\}$

AG
Ankit Gupta
Numerade Educator
01:17

Problem 5

List the elements of the set in roster notation.
$$
\{x \mid x \text { is a digit in the number } 352,646\}
$$

Vysakh M
Vysakh M
Numerade Educator
01:17

Problem 6

List the elements of the set in roster notation.
$$
\{x \mid x \text { is a letter in the word HIPPOPOTAMUS\} }
$$

Vysakh M
Vysakh M
Numerade Educator
01:17

Problem 7

List the elements of the set in roster notation.
$$
\{x \mid 2-x=4 \text { and } x \text { is an integer }\}
$$

Vysakh M
Vysakh M
Numerade Educator
01:17

Problem 8

List the elements of the set in roster notation.
$$
\{x \mid 2-x=4 \text { and } x \text { is a fraction\} }
$$

Vysakh M
Vysakh M
Numerade Educator
00:12

Problem 9

State whether the statements are true or false.
a. $\{a, b, c\}=\{c, a, b\}$
b. $A \in A$

Rebecca Wang
Rebecca Wang
Numerade Educator
01:02

Problem 10

State whether the statements are true or false.
a. $\varnothing \in A$
b. $A \subset A$

Destin Priester
Destin Priester
Numerade Educator
00:41

Problem 11

State whether the statements are true or false.
a. $0 \in \varnothing$
b. $0=Q$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
00:41

Problem 12

State whether the statements are true or false.
a. $\{\varnothing\}=\varnothing$
b. $\{a, b\} \in\{a, b, c\}$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
01:02

Problem 13

State whether the statements are true or false.
$\{$ Chevrolet, Cadillac, Buick $\} \subset\{x \mid x$ is a division of General Motors\}

Destin Priester
Destin Priester
Numerade Educator
00:41

Problem 14

State whether the statements are true or false.
$\{x \mid x$ is a silver medalist in the 2014 Winter Olympic Games $\}=\varnothing$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
00:18

Problem 15

Let $A=\{1,2,3,4,5\}$. Determine whether the statements are true or false.
a. $2 \in A$
b. $A \subseteq\{2,4,6\}$

Amy Jiang
Amy Jiang
Numerade Educator
00:31

Problem 16

Let $A=\{1,2,3,4,5\}$. Determine whether the statements are true or false.
a. $0 \in A$
b. $\{1,3,5\} \in A$

James Kiss
James Kiss
Numerade Educator
01:05

Problem 17

Let $A=\{1,2,3\}$. Which of the following sets are equal to $A$ ?
a. $\{2,1,3\}$
b. $\{3,2,1\}$
c. $\{0,1,2,3\}$

Manisha Sarker
Manisha Sarker
Numerade Educator
00:22

Problem 18

Let $A=\{a, e, l, t, r\} .$ Which of the following sets are equal to $A$ ?
a. $\{x \mid x$ is a letter of the word later\}
b. $\{x \mid x$ is a letter of the word latter\}
c. $\{x \mid x$ is a letter of the word relate $\}$

Elizabeth Xu
Elizabeth Xu
Numerade Educator
01:16

Problem 19

List all subsets of the following sets:
a. $\{1,2\}$
b. $\{1,2,3\}$
c. $\{1,2,3,4\}$

Heather Zimmers
Heather Zimmers
Numerade Educator
00:23

Problem 20

List all subsets of the set $A=\{$ IBM, U.S. Steel. Union Carbide, Boeing\}. Which of these are proper subsets of $A$ ?

Ashley Volpe
Ashley Volpe
Numerade Educator
01:10

Problem 21

Find the smallest possible set (i.e., the set with the least number of elements) that contains the given sets as subsets.
$\{1,2\},\{1,3,4\},\{4,6,8,10\}$

Aman Gupta
Aman Gupta
Numerade Educator
01:10

Problem 22

Find the smallest possible set (i.e., the set with the least number of elements) that contains the given sets as subsets.
$\{1,2,4\},\{a, b\}$

Aman Gupta
Aman Gupta
Numerade Educator
01:10

Problem 24

Find the smallest possible set (i.e., the set with the least number of elements) that contains the given sets as subsets.
$$
\begin{array}{l}
\text { \{GM, Ford, Chrysler }\} \text { , \{Daimler-Benz, Volkswagen\}, }\\
\text { \{Toyota, Nissan\} }
\end{array}
$$

Aman Gupta
Aman Gupta
Numerade Educator
00:47

Problem 25

Use Venn diagrams to represent the following relationships:
a. $A \subset B$ and $B \subset C$
b. $A \subset U$ and $B \subset U$, where $A$ and $B$ have no elements in common
c. The sets $A, B$, and $C$ are equal.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:18

Problem 26

Let $U$ denote the set of all students who applied for admission to the freshman class at Faber College for the upcoming academic year, and let
$A=\{x \in U \mid x$ is a successful applicant $\}$ $B=\{x \in U \mid x$ is a female student who enrolled in the freshman class \}
$C=\{x \in U \mid x$ is a male student who enrolled in the freshman class\}
a. Use Venn diagrams to represent the sets $U, A, B$, and $C$.
b. Determine whether the following statements are true or false.
i. $A \subseteq B$
ii. $B \subset A$
iii. $C \subset B$

Amy Jiang
Amy Jiang
Numerade Educator
01:38

Problem 27

Write an expression describing the shaded portion(s) of the Venn diagram.

Nick Johnson
Nick Johnson
Numerade Educator
01:38

Problem 28

Write an expression describing the shaded portion(s) of the Venn diagram.

Nick Johnson
Nick Johnson
Numerade Educator
00:55

Problem 29

Shade the portion of the accompanying fiqure that represents each set.
a. $A \cap B^{c}$
b. $A^{c} \cap B$

Amy Jiang
Amy Jiang
Numerade Educator
01:02

Problem 30

Shade the portion of the accompanying fiqure that represents each set.
a. $A^{c} \cap B^{c}$
b. $(A \cup B)^{c}$

Amy Jiang
Amy Jiang
Numerade Educator
01:54

Problem 31

Shade the portion of the accompanying figure that represents each set.
a. $A \cup B \cup C$
b. $A \cap B \cap C$

Maxime Rossetti
Maxime Rossetti
Numerade Educator
01:54

Problem 32

Shade the portion of the accompanying figure that represents each set.
a. $A \cap B \cap C$
b. $A^{c} \cap B \cap C$

Maxime Rossetti
Maxime Rossetti
Numerade Educator
01:10

Problem 33

Shade the portion of the accompanying figure that represents each set.
a. $A^{c} \cap B^{c} \cap C^{2}$
b. $(A \cup B)^{c} \cap C$

Amy Jiang
Amy Jiang
Numerade Educator
01:10

Problem 34

Shade the portion of the accompanying figure that represents each set.
a. $A \cup(B \cap C)^{c}$
b. $(A \cup B \cup C)^{c}$

Amy Jiang
Amy Jiang
Numerade Educator
01:07

Problem 35

Let $U=\{1,2,3,4,5,6,7,8,9,10\}$, $A=\{1,3,5,7,9\}, B=\{2,4,6,8,10\}$, and $C=\{1,2,4,5,8,9\} .$
List the elements of each set.
a. $A^{c}$
b. $B \cup C$
c. $C \cup C^{c}$

Manisha Sarker
Manisha Sarker
Numerade Educator
01:23

Problem 36

Let $U=\{1,2,3,4,5,6,7,8,9,10\}$, $A=\{1,3,5,7,9\}, B=\{2,4,6,8,10\}$, and $C=\{1,2,4,5,8,9\} .$
List the elements of each set.
a. $C \cap C$
b. $(A \cap C)^{c}$
c. $A \cup(B \cap C)$

Manisha Sarker
Manisha Sarker
Numerade Educator
01:23

Problem 37

Let $U=\{1,2,3,4,5,6,7,8,9,10\}$, $A=\{1,3,5,7,9\}, B=\{2,4,6,8,10\}$, and $C=\{1,2,4,5,8,9\} .$
List the elements of each set.
a. $(A \cap B) \cup C$
b. $(A \cup B \cup C)^{c}$
c. $(A \cap B \cap C)^{c}$

Manisha Sarker
Manisha Sarker
Numerade Educator
01:23

Problem 38

Let $U=\{1,2,3,4,5,6,7,8,9,10\}$, $A=\{1,3,5,7,9\}, B=\{2,4,6,8,10\}$, and $C=\{1,2,4,5,8,9\} .$
List the elements of each set.
a. $A^{c} \cap\left(B \cap C^{2}\right)$
b. $\left(A \cup B^{\star}\right) \cup\left(B \cap C^{c}\right)$
c. $(A \cup B)^{c} \cap C^{2}$

Manisha Sarker
Manisha Sarker
Numerade Educator
View

Problem 39

Determine whether the pairs of sets are disjoint.
a. $\{1,2,3,4\},\{4,5,6,7\}$
b. $\{a, c, e, g\},\{b, d, f\}$

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 40

Determine whether the pairs of sets are disjoint.
a. $\varnothing,\{1,3,5\}$
b. $\{0,1,3,4\},\{0,2,5,7\}$

Nick Johnson
Nick Johnson
Numerade Educator
01:14

Problem 41

Describe each set in words.
a. $T^{x}$
b. $C^{e}$

BH
Bridget Hintz
Numerade Educator
09:40

Problem 42

Describe each set in words.
a. $T \cup C$
b. $T \cap C$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
00:18

Problem 43

Describe each set in words.
a. $T \cap C^{c}$
b. $T^{c} \cap C$

Darian Kaulahao
Darian Kaulahao
Numerade Educator
09:40

Problem 44

Describe each set in words.
a. $T^{c} \cap C^{c}$
b. $(T \cup C)^{c}$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:14

Problem 45

Describe each set in words.
a. $D^{2}$
b. $\bar{N}^{2}$

BH
Bridget Hintz
Numerade Educator
09:40

Problem 46

Describe each set in words.
a. $N \cup D$
b. $N \cap M$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
00:18

Problem 47

Describe each set in words.
a. $D \cap M$
b. $D \cap A$

Elizabeth Xu
Elizabeth Xu
Numerade Educator
09:40

Problem 48

Describe each set in words.
a. $N \cap F$
b. $(D \cup N)^{c}$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
02:44

Problem 49

Let $U$ denote the set of all senators in Congress, and let
$$
\begin{array}{l}
D=\{x \in U \mid x \text { is a Democrat }\} \\
R=\{x \in U \mid x \text { is a Republican }\} \\
F=\{x \in U \mid x \text { is a female }\} \\
L=\{x \in U \mid x \text { is a lawyer }\}
\end{array}
$$
a. The set of all Democrats who are female
b. The set of all Republicans who are male and are not lawyers

Mitchell Cutler
Mitchell Cutler
Numerade Educator
02:44

Problem 50

Let $U$ denote the set of all senators in Congress, and let
$$
\begin{array}{l}
D=\{x \in U \mid x \text { is a Democrat }\} \\
R=\{x \in U \mid x \text { is a Republican }\} \\
F=\{x \in U \mid x \text { is a female }\} \\
L=\{x \in U \mid x \text { is a lawyer }\}
\end{array}
$$
a. The set of all Democrats who are female or are lawyers
b. The set of all senators who are not Democrats or are Lawyers

Mitchell Cutler
Mitchell Cutler
Numerade Educator
01:36

Problem 51

a. The set of students who have not had a course in economics
b. The set of students who have had courses in accounting and economics
c. The set of students who have had courses in accounting and economics but not marketing

Aman Gupta
Aman Gupta
Numerade Educator
04:48

Problem 52

Let $U$ denote the set of all senators in Congress, and let
$$
\begin{array}{l}
D=\{x \in U \mid x \text { is a Democrat }\} \\
R=\{x \in U \mid x \text { is a Republican }\} \\
F=\{x \in U \mid x \text { is a female }\} \\
L=\{x \in U \mid x \text { is a lawyer }\}
\end{array}
$$
a. The set of students who have had courses in economics but not courses in accounting or marketing
b. The set of students who have had at least one of the three courses
c. The set of students who have had all three courses

Matthew Winsor
Matthew Winsor
Numerade Educator
03:29

Problem 53

In a reader survey conducted by USA Today, readers were asked to name the best U.S. city for Italian restaurants. The results follow:
\begin{tabular}{lccccc}
\hline & New City & York & Chicago & Boston & Las & San \\
\hline Respondents $(\%)$ & 55 & 16 & 15 & 7 & 7 \\
\hline
\end{tabular}
Let $A$ denote the set of cities that were voted the best by more than $10 \%$ of the respondents, let $B$ be the set of cities that were voted the best by between $10 \%$ and $20 \%$ of the respondents, and let $C$ denote the set of cities that were voted the best by fewer than $10 \%$ of the respondents. Find the following sets:
a. $A, B$, and $C$
b. $A \cup B$
c. $A \cap B$
d. $A^{c} \cap B$
e. $A \cap B^{\circ}$
f. $(A \cup B)^{c}$
Source: travel.usatoday.com.

Carson Merrill
Carson Merrill
Numerade Educator
01:03

Problem 54

INVENTORY Loss The biggest cause of inventory loss, called shrinkage, is shoplifting, followed closely by employee theft. In a study conducted by the Center for Retail Research, the nine countries with the highest shrinkage rates, measured in the dollar amount lost for every $\$ 100$ in sales, are as follows:
\begin{tabular}{lllllllll}
\hline & \multicolumn{6}{c} { South } \\
Country & India & Russia & Marocco & Africa & Brazil & Mexico & Thailand & Turkey \\
\hline Shrinkage & & & & & & & & \\
Rate (\$) & $2.38$ & $1.74$ & $1.72$ & $1.71$ & $1.69$ & $1.64$ & $1.64$ & $1.63$ \\
\hline
\end{tabular}
Let $A$ denote the set of countries that have a shrinkage rate greater than $\$ 1.65$, let $B$ be the set of countries that have a shrinkage rate between $\$ 1.65$ and $\$ 1.73$, and let $C$ be the set of countries that have a shrinkage rate less than $\$ 1.70 .$ Find the following sets:
a. $A, B$, and $\bar{C}$
b. $A \cap B$
c. $A^{e} \cap B$
d. $A \cap B^{c}$
e. $A^{c} \cup B^{c}$
Source: Center for Retail Research Graphics.

Joseph Liao
Joseph Liao
Numerade Educator
00:10

Problem 55

Refer to the following diagram, where $U$ is the set of all tourists surveyed over a 1-week period in London and where
$$
\begin{array}{l}
A=\{x \in U \mid x \text { has taken the underground [subway }]\} \\
B=\{x \in U \mid x \text { has taken a cab }\} \\
C=\{x \in U \mid x \text { has taken a bus }\}
\end{array}
$$
a. Region
b. Regions 1 and 4 together
c. Regions $4,5,7$, and 8 together

Coach Rye
Coach Rye
Numerade Educator
00:10

Problem 56

Refer to the following diagram, where $U$ is the set of all tourists surveyed over a 1-week period in London and where
$$
\begin{array}{l}
A=\{x \in U \mid x \text { has taken the underground [subway }]\} \\
B=\{x \in U \mid x \text { has taken a cab }\} \\
C=\{x \in U \mid x \text { has taken a bus }\}
\end{array}
$$
a. Region 3
b. Regions 4 and 6 together
c. Regions 5,6, and 7 together

Coach Rye
Coach Rye
Numerade Educator
01:21

Problem 57

Use Venn diagrams to illustrate each statement.
$A \subseteq A \cup B ; B \subseteq A \cup B$

Laurie Huffman
Laurie Huffman
Numerade Educator
02:51

Problem 58

Use Venn diagrams to illustrate each statement.
$A \cap B \subseteq A ; A \cap B \subseteq B$

James Chok
James Chok
Numerade Educator
01:21

Problem 59

Use Venn diagrams to illustrate each statement.
$A \cup(B \cup C)=(A \cup B) \cup C$

Laurie Huffman
Laurie Huffman
Numerade Educator
02:51

Problem 60

Use Venn diagrams to illustrate each statement.
$A \cap(B \cap C)=(A \cap B) \cap C$

James Chok
James Chok
Numerade Educator
02:51

Problem 61

Use Venn diagrams to illustrate each statement.
$A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$

James Chok
James Chok
Numerade Educator
01:55

Problem 62

Use Venn diagrams to illustrate each statement.
$(A \cup B)^{c}=A^{c} \cap B^{c}$

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator
05:07

Problem 63

In Exercises 63 and 64, let
$$
\begin{array}{l}
U=\{1,2,3,4,5,6,7,8,9,10\} \\
A=\{1,3,5,7,9\} \\
B=\{1,2,4,7,8\} \\
C=\{2,4,6,8\}
\end{array}
$$
a. $A \cup(B \cup C)=(A \cup B) \cup C$
b. $A \cap(B \cap C)=(A \cap B) \cap C$

Abdul Vahid M
Abdul Vahid M
Numerade Educator
05:15

Problem 64

Let
$$
\begin{array}{l}
U=\{1,2,3,4,5,6,7,8,9,10\} \\
A=\{1,3,5,7,9\} \\
B=\{1,2,4,7,8\} \\
C=\{2,4,6,8\}
\end{array}
$$
a. $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$
b. $(A \cup B)^{c}=A^{c} \cap B^{c}$

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator
00:57

Problem 65

Refer to the accompanying figure, and list the points that belong to each set.
a. $A \cup B$
b. $A \cap B$

Amy Jiang
Amy Jiang
Numerade Educator
01:10

Problem 66

Refer to the accompanying figure, and list the points that belong to each set.
a. $A \cap(B \cup C)$
b. $(B \cap C)^{c}$

Amy Jiang
Amy Jiang
Numerade Educator
00:12

Problem 67

Refer to the accompanying figure, and list the points that belong to each set.
a. $(B \cup C)^{c}$
b. $A^{c}$

Amy Jiang
Amy Jiang
Numerade Educator
01:10

Problem 68

Refer to the accompanying figure, and list the points that belong to each set.
a. $(A \cap B) \cap \mathrm{C}$
b. $(A \cup B \cup C)^{2}$

Amy Jiang
Amy Jiang
Numerade Educator
02:10

Problem 69

Suppose $A \subset B$ and $B \subset C$, where $A$ and $B$ are any two sets. What conclusion can be drawn regarding the sets $A$ and $C$ ?

Anthony Ramos
Anthony Ramos
Numerade Educator
04:11

Problem 70

Verify the assertion that two sets $A$ and $B$ are equal if and only if (1) $A \subseteq B$ and (2) $B \subseteq A$.

Anthony Ramos
Anthony Ramos
Numerade Educator
00:53

Problem 71

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
A set is never a subset of itself.

Destin Priester
Destin Priester
Numerade Educator
01:02

Problem 72

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
A proper subset of a set is itself a subset of the set but not necessarily vice versa.

Destin Priester
Destin Priester
Numerade Educator
00:25

Problem 73

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $A \cup B=\varnothing$, then $A=\varnothing$ and $B=\varnothing$.

Tanishq Gupta
Tanishq Gupta
Numerade Educator
00:25

Problem 74

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $A \cap B=\varnothing$, then either $A=\varnothing$ or $B=\varnothing$.

Tanishq Gupta
Tanishq Gupta
Numerade Educator
00:25

Problem 75

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
$\left(A \cup A^{2}\right)^{c}=\varnothing$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
01:27

Problem 76

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $A \subseteq B$, then $A \cap B=A$.

Ahmad Reda
Ahmad Reda
Numerade Educator
01:27

Problem 77

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $A \subseteq B$, then $A \cup B=B$.

Ahmad Reda
Ahmad Reda
Numerade Educator
01:08

Problem 78

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $A \cup B=A$, then $A \subseteq B$

Carson Merrill
Carson Merrill
Numerade Educator
01:11

Problem 79

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $A \subset B$, then $A^{c} \supset B^{c}$.

Carson Merrill
Carson Merrill
Numerade Educator
00:25

Problem 80

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
$4 \cap \varnothing=\varnothing$

Tanishq Gupta
Tanishq Gupta
Numerade Educator