Question 1

Complete the blank with either $\in$ or $\notin$ to make the statement true.
-$2... \ldots \{ 11,10,9,8 \}$

Accepted Answer

A)  $\in$ 
B)  $\notin$ 
A)  $\in$ 
B)  $\notin$

Question 2

Let A and B be sets with cardinal numbers, n(A)= a and n(B)= b, respectively. Decide whether the statement is true or false.
-$n ( A \cup B ) = n ( A ) + n ( B ) - n ( A \cap B )$

Accepted Answer

A) True 
 B)False

Question 3

Find the cardinal number of the indicated set. Use the cardinal number formula.
-If $n ( B ) = 60 , n ( A \cap B ) = 11$, and $n ( A \cup B ) = 105$, find $n ( A )$.

Accepted Answer

A)  54 
B)  56 
C)  45 
D)  58 
A)  54 
B)  56 
C)  45 
D)  58

Question 4

Identify the set as finite or infinite.
-$\left\{ 1 , \frac { 1 } { 4 } , \frac { 1 } { 16 } , \frac { 1 } { 64 } , \ldots \right\}$

Accepted Answer

A)  Finite 
B)  Infinite 
A)  Finite 
B)  Infinite

Question 5

Find n(A)for the set.
-$A = \{ x \mid x$ is a month in the year $\}$

Accepted Answer

A)  $n ( A ) = 1$ 
B)  $\mathrm { n } ( \mathrm { A } ) = 24$ 
C)  $n ( A ) = 12$ 
D)  $n ( A ) = 52$ 
A)  $n ( A ) = 1$ 
B)  $\mathrm { n } ( \mathrm { A } ) = 24$ 
C)  $n ( A ) = 12$ 
D)  $n ( A ) = 52$

Question 6

Draw an appropriate Venn diagram and use the given information to fill in the number of elements in each region.
-$$\mathrm { n } ( \mathrm { A } ) = 35 , \mathrm { n } \left( \mathrm { A } \cap \mathrm { B } ^ { \prime } \right) = 24 , \mathrm { n } ( \mathrm { A } \cap \mathrm { C } ) = 15 , \mathrm { n } ( \mathrm { B } \cap \mathrm { C } ) = 14 , \mathrm { n } \left( \mathrm { A } ^ { \prime } \cap \mathrm { B } ^ { \prime } \cap \mathrm { C } ^ { \prime } \right) = 10 , \mathrm { n } ( \mathrm { A } \cap \mathrm { B } \cap \mathrm { C } ) = 6 \text {, }$$
$\mathrm { n } ( \mathrm { B } \cup \mathrm { C } ) = 52 , \mathrm { n } \left( \mathrm { B } \cap \mathrm { C } ^ { \prime } \right) = 18$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 7

List the elements in the set.
-The set of all positive integer powers of 3.

Accepted Answer

A) {1, 3, 9, 27, 81, 243, ...} 
B) {3, 6, 9, 12, 15, ...} 
C) {1, 8, 27, 64, 125, ...} 
D) {3, 9, 27, 81, 243, ...} 
A) {1, 3, 9, 27, 81, 243, ...} 
B) {3, 6, 9, 12, 15, ...} 
C) {1, 8, 27, 64, 125, ...} 
D) {3, 9, 27, 81, 243, ...}

Question 8

Find the number of subsets of the set.
-{0}

Accepted Answer

A) 2 
B) 0 
C) 1 
D) 4 
A) 2 
B) 0 
C) 1 
D) 4

Question 9

Determine whether the statement is true or false. 
Let $A = \{ 1,3,5,7 \}$
$\begin{array} { l } 
B = \{ 5,6,7,8 \} \
C = \{ 5,8 \} \
D = \{ 2,5,8 \} \
U = \{ 1,2,3,4,5,6,7,8 \}
\end{array}$
-$A \neq \{ 7,5,3,1 \}$

Accepted Answer

A) True 
 B)False

Question 10

Decide whether $\subset$ ,$\subseteq$ , both, or neither can be placed in the blank to make a true statement.
-$\{ 9,10,11 \}$ _ $\{ 8,9,10,11 \}$

Accepted Answer

A)  $\subseteq$ 
B)  Neither 
C)  Both $\subset$ and $\subseteq$ 
D)  $\subset$ 
A)  $\subseteq$ 
B)  Neither 
C)  Both $\subset$ and $\subseteq$ 
D)  $\subset$

Question 11

Determine whether or not the set is well defined.
-$\{ x \mid x$ is a stock on the AmEx today $\}$

Accepted Answer

A)  Well defined 
B)  Not well defined 
A)  Well defined 
B)  Not well defined

Question 12

Find the indicated cardinal number.
-Find $n ( D \times B )$ given that $B = \{ 1,3 \}$ and $D = \{ 7,8,9,10 \}$.

Accepted Answer

A)  12 
B)  16 
C)  7 
D)  8 
A)  12 
B)  16 
C)  7 
D)  8

Question 13

Solve the problem.
-In a survey of 280 people, a travel company asked people about places they plan to visit in the next 5 years. The results were as follows:
48 plan to visit Europe
58 plan to visit Latin America
34 plan to visit Asia
14 plan to visit Europe and Latin America
12 plan to visit Latin America and Asia
11 plan to visit Europe and Asia
4 plan to visit all three
How many people plan to visit exactly two of these places?

Accepted Answer

A) 25 
B) 37 
C) 18 
D) 29 
A) 25 
B) 37 
C) 18 
D) 29

Question 14

List the elements in the set . Let
$\begin{array} { l } 
\mathrm { U } = \{ \mathrm { q } , \mathbf { r } , \mathrm { s } , \mathrm { t } , \mathbf { u } , \mathrm { v } , \mathbf { w } , \mathbf { x } , \mathrm { y } , \mathrm { z } \} \
\mathrm { A } = \{ \mathrm { q } , \mathrm { s } , \mathrm { u } , \mathbf { w } , \mathrm { y } \} \
\mathrm { B } = \{ \mathrm { q } , \mathrm { s } , \mathrm { y } , \mathrm { z } \} \
\mathrm { C } = \{ \mathrm { v } , \mathbf { w } , \mathbf { x } , \mathrm { y } , \mathrm { z } \}
\end{array}$
-$C - A$

Accepted Answer

A)  $\{ v , x , z \}$ 
B)  $\{ q , s , u , v , x , z \}$ 
C)  $\{ w , y \}$ 
D)  $\{ q , s , u \}$ 
A)  $\{ v , x , z \}$ 
B)  $\{ q , s , u , v , x , z \}$ 
C)  $\{ w , y \}$ 
D)  $\{ q , s , u \}$

Question 15

Find n(A)for the set.
-A = {2, 4, 6, 8, 10}

Accepted Answer

A) n(
A) = 2 
B) n(
A) = 4 
C) n(
A) = 5 
D) n(
A) = 10 
A) n(
A) = 2 
B) n(
A) = 4 
C) n(
A) = 5 
D) n(
A) = 10

Question 16

List the elements in the set . Let
$\begin{array} { l } 
\mathrm { U } = \{ \mathrm { q } , \mathbf { r } , \mathrm { s } , \mathrm { t } , \mathbf { u } , \mathrm { v } , \mathbf { w } , \mathbf { x } , \mathrm { y } , \mathrm { z } \} \
\mathrm { A } = \{ \mathrm { q } , \mathrm { s } , \mathrm { u } , \mathbf { w } , \mathrm { y } \} \
\mathrm { B } = \{ \mathrm { q } , \mathrm { s } , \mathrm { y } , \mathrm { z } \} \
\mathrm { C } = \{ \mathrm { v } , \mathbf { w } , \mathbf { x } , \mathrm { y } , \mathrm { z } \}
\end{array}$
-$A \cup C$

Accepted Answer

A)  $\{ q , s , u , v , w , x , y , z \}$ 
B)  $\{ q , s , u , w , y , v , w , x , y , z \}$ 
C)  $\{ \mathrm { w } , \mathrm { y } \}$ 
D)  $\{ q , s , u , v , w , y , z \}$ 
A)  $\{ q , s , u , v , w , x , y , z \}$ 
B)  $\{ q , s , u , w , y , v , w , x , y , z \}$ 
C)  $\{ \mathrm { w } , \mathrm { y } \}$ 
D)  $\{ q , s , u , v , w , y , z \}$

Question 17

The lists below show five agricultural crops in Alabama, Arkansas, and Louisiana. 
$\begin{array} { l l l } 
\underline { \text { Alabama } } & \underline { \text { Arkansas } } & \underline { \text { Louisiana } } \
\hline \text { soybeans (s) } & \text { soybeans (s) } & \text { soybeans (s) } \
\text { peanuts (p) } & \text { rice (r) } & \text { sugarcane (n) } \
\text { corn (c) } & \text { cotton (t) } & \text { rice (r) } \
\text { hay (h) } & \text { hay (h) } & \text { corn (c) } \
\text { wheat (w) } & \text { wheat (w) } & \text { cotton (t) }
\end{array}$
 Let U be the smallest possible universal set that includes all of the crops listed, and let A, K and L be the sets of five crops
in Alabama, Arkansas, and Louisiana, respectively. Find each of the following sets.
-The set of crops common to A, K, and L

Accepted Answer

A) {s} 
B) {c, h, n, p, r, s, t, w} 
C) {n, p, s} 
D) {n, p} 
A) {s} 
B) {c, h, n, p, r, s, t, w} 
C) {n, p, s} 
D) {n, p}

Question 18

Find the indicated cardinal number.
-Find $n ( C \times D )$ given that $C = \{ 4,5,6 \}$ and $D = \{ 7,8,9,10 \}$.

Accepted Answer

A)  27 
B)  12 
C)  81 
D)  7 
A)  27 
B)  12 
C)  81 
D)  7

Question 19

Write the set in set-builder notation.
-The set of all cars owned by students

Accepted Answer

A)  $\{ x$ is a student with a car $\}$ 
B)  $\{ x \mid x$ is a car owned by a student $\}$ 
C)  $\{ x \mid x$ is a student with a car $\}$ 
D)  $\{ x$ is a car $\}$ 
A)  $\{ x$ is a student with a car $\}$ 
B)  $\{ x \mid x$ is a car owned by a student $\}$ 
C)  $\{ x \mid x$ is a student with a car $\}$ 
D)  $\{ x$ is a car $\}$

Question 20

$\text { Use } \subset \text { or } \not \subset \text { in the blank to make a true statement. }$

-$\{ x \mid x$ is a counting number larger than $5 \} \ldots \{ 7,8,9 , \ldots \}$

Accepted Answer

A)  $\nsubseteq$ 
B)  $\subseteq$ 
A)  $\nsubseteq$ 
B)  $\subseteq$

Complete the blank with either $\in$ or $\notin$ to make the statement true. - $2... \ldots \{ 11,10,9,8 \}$

Let A and B be sets with cardinal numbers, n(A)= a and n(B)= b, respectively. Decide whether the statement is true or false. - $n ( A \cup B ) = n ( A ) + n ( B ) - n ( A \cap B )$

Find the cardinal number of the indicated set. Use the cardinal number formula. -If $n ( B ) = 60 , n ( A \cap B ) = 11$ , and $n ( A \cup B ) = 105$ , find $n ( A )$ .

Identify the set as finite or infinite. - $\left\{ 1 , \frac { 1 } { 4 } , \frac { 1 } { 16 } , \frac { 1 } { 64 } , \ldots \right\}$

Find n(A)for the set. - $A = \{ x \mid x$ is a month in the year $\}$

List the elements in the set. -The set of all positive integer powers of 3.

Find the number of subsets of the set. -{0}

Determine whether the statement is true or false. Let $A = \{ 1,3,5,7 \}$ B=\{5,6,7,8\} C=\{5,8\} D=\{2,5,8\} U=\{1,2,3,4,5,6,7,8\} - $A \neq \{ 7,5,3,1 \}$

Decide whether $\subset$ , $\subseteq$ , both, or neither can be placed in the blank to make a true statement. - $\{ 9,10,11 \}$ _ $\{ 8,9,10,11 \}$

Determine whether or not the set is well defined. - $\{ x \mid x$ is a stock on the AmEx today $\}$

Find the indicated cardinal number. -Find $n ( D \times B )$ given that $B = \{ 1,3 \}$ and $D = \{ 7,8,9,10 \}$ .

List the elements in the set . Let =\{,,,,,,,,,\} =\{,,,,\} =\{,,,\} =\{,,,,\} - $C - A$

Find n(A)for the set. -A = {2, 4, 6, 8, 10}

List the elements in the set . Let =\{,,,,,,,,,\} =\{,,,,\} =\{,,,\} =\{,,,,\} - $A \cup C$

Find the indicated cardinal number. -Find $n ( C \times D )$ given that $C = \{ 4,5,6 \}$ and $D = \{ 7,8,9,10 \}$ .

Write the set in set-builder notation. -The set of all cars owned by students

$\text { Use } \subset \text { or } \not \subset \text { in the blank to make a true statement. }$ - $\{ x \mid x$ is a counting number larger than $5 \} \ldots \{ 7,8,9 , \ldots \}$

The Art of Problem Solving

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Exam 2: The Basic Concepts of Set Theory

Complete the blank with either ∈\in∈ or ∉\notin∈/ to make the statement true. - 2...…{11,10,9,8}2... \ldots \{ 11,10,9,8 \}2...…{11,10,9,8}

Let A and B be sets with cardinal numbers, n(A)= a and n(B)= b, respectively. Decide whether the statement is true or false. - n(A∪B)=n(A)+n(B)−n(A∩B)n ( A \cup B ) = n ( A ) + n ( B ) - n ( A \cap B )n(A∪B)=n(A)+n(B)−n(A∩B)

Find the cardinal number of the indicated set. Use the cardinal number formula. -If n(B)=60,n(A∩B)=11n ( B ) = 60 , n ( A \cap B ) = 11n(B)=60,n(A∩B)=11 , and n(A∪B)=105n ( A \cup B ) = 105n(A∪B)=105 , find n(A)n ( A )n(A) .

Identify the set as finite or infinite. - {1,14,116,164,…}\left\{ 1 , \frac { 1 } { 4 } , \frac { 1 } { 16 } , \frac { 1 } { 64 } , \ldots \right\}{1,41​,161​,641​,…}

Find n(A)for the set. - A={x∣xA = \{ x \mid xA={x∣x is a month in the year }\}}