Chapter 2

Basic Structures: Sets, Functions, Sequences, Sums,and Matrices

Educators

DI

Problem 1

List the members of these sets.
a) $\left\{x | x \text { is a real number such that } x^{2}=1\right\}$
b) $\{x | x \text { is a positive integer less than } 12\}$
c) $\{x | x \text { is the square of an integer and } x<100\}$
d) $\left\{x | x \text { is an integer such that } x^{2}=2\right\}$

Clarissa N.
Numerade Educator

Problem 2

Use set builder notation to give a description of each of these sets.
a) $\{0,3,6,9,12\}$
b) $\{-3,-2,-1,0,1,2,3\}$
c) $\{m, n, o, p\}$

DI
Doruk I.
Numerade Educator

Problem 3

Which of the intervals $(0,5),(0,5],[0,5),[0,5],(1,4]$ $[2,3],(2,3)$ contains
$$
\begin{array}{lll}{\text { a) } 0 ?} & {\text { b) } 1 ?} \\ {\text { c) } 2 ?} & {\text { d) } 3 ?} \\ {\text { e) } 4 ?} & {\text { f) } 5 ?}\end{array}
$$

Clarissa N.
Numerade Educator

Problem 4

For each of these intervals, list all its clements or explain why it is empty.
$$
\begin{array}{ll}{\text { a) }[a, a]} & {\text { b) }[a, a)} \\ {\text { c) }(a, a]} & {\text { d) }(a, a)} \\ {\text { e) }(a, b), \text { where } a > b} & {\text { f) }[a, b], \text { where } a > b}\end{array}
$$

DI
Doruk I.
Numerade Educator

Problem 5

For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other.
a) the set of airline flights from New York to New Delhi, the set of nonstop airline flights from New York to
New Delhi
b) the set of people who speak English, the set of people who speak Chinese
c) the set of flying squirrels, the set of living creatures that can fly

Clarissa N.
Numerade Educator

Problem 6

For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other.
a) the set of people who speak English, the set of people who speak English with an Australian accent
b) the set of fruits, the set of citralian accent
c) the set of fruits, the set of citralian accent
c) the set of students studying discrete mathematics, the set of students studying data structures

DI
Doruk I.
Numerade Educator

Problem 7

Determine whether each of these pairs of sets are equal.
$$
\begin{array}{l}{\text { a) }\{1,3,3,3,5,5,5,5,5\},\{5,3,1\}} \\ {\text { b) }\{\{1\}\},\{1,\{1\}\}} & {\text { c) } \emptyset,\{\emptyset\}}\end{array}
$$

Clarissa N.
Numerade Educator

Problem 8

Suppose that $A=\{2,4,6\}, B=\{2,6\}, C=\{4,6\},$ and $D=\{4,6,8\} .$ Determine which of these sets are subsets of which other of these sets.

DI
Doruk I.
Numerade Educator

Problem 9

For each of the following sets, determine whether 2 is an element of that set.
$$
\begin{array}{l}{\text { a) }\{x \in \mathbf{R} | x \text { is an integer greater than } 1\}} \\ {\text { b) }\{x \in \mathbf{R} | x \text { is the square of an integer }\}} \\ {\text { c) }\{2,\{2\}\}} \\ {\text { e) }\{\{2\},\{2,\{2\}\}\}} & {\text { f) }\{\{2\},\{\{2\}\}\}}\end{array}
$$

Clarissa N.
Numerade Educator

Problem 10

For each of the sets in Exercise 9 , determine whether $\{2\}$ is an element of that set.

DI
Doruk I.
Numerade Educator

Problem 11

Determine whether each of these statements is true or false.
$$
\begin{array}{ll}{\text { a) } 0 \in \emptyset} & {\text { b) } \emptyset \in\{0\}} \\ {\text { c) }\{0\} \subset \emptyset} & {\text { d) } \subset\{0\}} \\ {\text { e) }\{0\} \in\{0\}} & {\text { f) }\{0\} \subset\{0\}} \\ {\text { g) }\{\emptyset\} \subseteq\{\emptyset\}}\end{array}
$$

Clarissa N.
Numerade Educator

Problem 12

Determine whether these statements are true or false.
a) $\quad \emptyset \in\{\emptyset\}$
b) $\quad g \in\{\emptyset,\{\emptyset\}\}$
c) $\{\mathfrak{D}\} \in\{\mathfrak{O}\}$
d) $\{\emptyset\} \in\{\{\emptyset\}\}$
e) $\{\mathfrak{Q}\} \subset\{\emptyset,\{\mathfrak{D}\}\}$
f) $\{\{\emptyset\}\} \subset\{\emptyset,\{\emptyset\}\}$
g) $\{\{\varnothing\}\} \subset\{\{\emptyset\},\{\emptyset\}\}$

DI
Doruk I.
Numerade Educator

Problem 13

Determine whether each of these statements is true or false.
$$
\begin{array}{lll}{\text { a) }} & {x \in\{x\}} & {\text { b) }\{x\} \subseteq\{x\}} & {\text { c) }\{x\} \in\{x\}} \\ {\text { d) }} & {\{x\} \in\{\{x\}\}} & {\text { e) } \emptyset \subseteq\{x\}} & {\text { f) } \emptyset \in\{x\}}\end{array}
$$

Clarissa N.
Numerade Educator

Problem 14

Use a Venn diagram to illustrate the subset of odd integers in the set of all positive integers not exceeding 10 .

DI
Doruk I.
Numerade Educator

Problem 15

Use a Venn diagram to illustrate the set of all months of the year whose names do not contain the letter $R$ in the set of all months of the year.

Clarissa N.
Numerade Educator

Problem 16

Use a Venn diagram to illustrate the relationship $A \subseteq B$ and $B \subseteq C .$

DI
Doruk I.
Numerade Educator

Problem 17

Use a Venn diagram to illustrate the relationships $A \subset B$ and $B \subset C .$

Clarissa N.
Numerade Educator

Problem 18

Use a Venn diagram to illustrate the relationships $A \subset B$ and $A \subset C .$

DI
Doruk I.
Numerade Educator

Problem 19

Suppose that $A, B,$ and $C$ are sets such that $A \subseteq B$ and $B \subset C$ . Show that $A \subset C$ .

Clarissa N.
Numerade Educator

Problem 20

Find two sets $A$ and $B$ such that $A \in B$ and $A \subseteq B$ .

DI
Doruk I.
Numerade Educator

Problem 21

What is the cardinality of each of these sets?
$$
\begin{array}{ll}{\text { a) }\{a\}} & {\text { b) }\{\{a\}\}} \\ {\text { c) }\{a,\{a\}\}} & {\text { d) }\{a,\{a\},\{a,\{a\}\}\}}\end{array}
$$

Clarissa N.
Numerade Educator

Problem 22

What is the cardinality of each of these sets?
$$
\begin{array}{ll}{\text { a) } \emptyset} & {\text { b) }\{\emptyset\}} \\ {\text { c) }\{\emptyset,\{\emptyset\}\}} & {\text { d) }\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}}\end{array}
$$

DI
Doruk I.
Numerade Educator

Problem 23

Find the power set of each of these sets, where $a$ and $b$ are distinct elements.
$$
\begin{array}{llll}{\text { a) }\{a\}} & {\text { b) }\{a, b\}} & {\text { c) }\{\varnothing,\{\Phi\}\}}\end{array}
$$

Clarissa N.
Numerade Educator

Problem 24

Can you conclude that $A=B$ if $A$ and $B$ are two sets with the same power set?

DI
Doruk I.
Numerade Educator

Problem 25

How many elements does each of these sets have where $a$ and $b$ are distinct elements?
$$
\begin{array}{l}{\text { a) } \mathcal{P}(\{a, b,\{a, b\}\})} \\ {\text { b) } \mathcal{P}(\{\Phi, a,\{a\},\{\{a\}\}\})} \\ {\text { c) } \mathcal{P}(\mathcal{P}(\emptyset))}\end{array}
$$

Clarissa N.
Numerade Educator

Problem 26

Determine whether each of these sets is the power set of a set, where $a$ and $b$ are distinct elements.
$$
\begin{array}{ll}{\text { a) } \emptyset} & {\text { b) }\{\emptyset,\{a\}\}} \\ {\text { c) }\{\emptyset,\{a\},\{\emptyset, a\}\}} & {\text { d) }\{\emptyset,\{a\},\{b\},\{a, b\}\}}\end{array}
$$

DI
Doruk I.
Numerade Educator

Problem 27

Prove that $\mathcal{P}(A) \subseteq \mathcal{P}(B)$ if and only if $A \subseteq B$

Clarissa N.
Numerade Educator

Problem 28

Show that if $A \subseteq C$ and $B \subseteq D,$ then $A \times B \subseteq C \times D$

DI
Doruk I.
Numerade Educator

Problem 29

Let $A=\{a, b, c, d\}$ and $B=\{y, z\} .$ Find
$$
\begin{array}{lll}{\text { a) } A \times B} & {\text { b) } B \times A}\end{array}
$$

Clarissa N.
Numerade Educator

Problem 30

What is the Cartesian product $A \times B,$ where $A$ is the set of courses offered by the mathematics department at a university and $B$ is the set of mathematics professors at this university? Give an example of how this Cartesian product can be used.

DI
Doruk I.
Numerade Educator

Problem 31

What is the Cartesian product $A \times B \times C,$ where $A$ is the set of all airlines and $B$ and $C$ are both the set of all cities in the United States? Give an example of how this Cartesian product can be used.

Clarissa N.
Numerade Educator

Problem 32

Suppose that $A \times B=\emptyset,$ where $A$ and $B$ are sets. What can you conclude?

DI
Doruk I.
Numerade Educator

Problem 33

Let $A$ be a set. Show that $\emptyset \times A=A \times \emptyset=\emptyset$

Clarissa N.
Numerade Educator

Problem 34

Let $A=\{a, b, c\}, B=\{x, y\},$ and $C=\{0,1\} .$ Find
$$
\begin{array}{ll}{\text { a) } A \times B \times C} & {\text { b) } C \times B \times A} \\ {\text { c) } C \times A \times B .} & {\text { d) } B \times B \times B}\end{array}
$$

DI
Doruk I.
Numerade Educator

Problem 35

Find $A^{2}$ if
$$
A=\{0,1,3\}, \quad \text { b) } A=\{1,2, a, b\}
$$

Clarissa N.
Numerade Educator

Problem 36

Find $A^{3}$ if
a) $A=\left\{\begin{array}{ll}{a \}} & {\text { b) } A=\{0, a\}}\end{array}\right.$

DI
Doruk I.
Numerade Educator

Problem 37

How many different elements does $A \times B$ have if $A$ has $m$ elements and $B$ has $n$ elements?

Clarissa N.
Numerade Educator

Problem 38

How many different elements does $A \times B \times C$ have if $A$ has $m$ elements, $B$ has $n$ elements, and $C$ has $p$ elements?

DI
Doruk I.
Numerade Educator

Problem 39

How many different elements does $A^{n}$ have when $A$ has $m$ elements and $n$ is a positive integer?

Clarissa N.
Numerade Educator

Problem 40

Show that $A \times B \neq B \times A,$ when $A$ and $B$ are nonempty, unless $A=B .$

DI
Doruk I.
Numerade Educator

Problem 41

Explain why $A \times B \times C$ and $(A \times B) \times C$ are not the same.

Clarissa N.
Numerade Educator

Problem 42

Explain why $(A \times B) \times(C \times D)$ and $A \times(B \times C) \times D$ are not the same.

DI
Doruk I.
Numerade Educator

Problem 43

Prove or disprove that if $A$ and $B$ are sets, then $\mathcal{P}(A \times B)=$ $\mathcal{P}(A) \times \mathcal{P}(B)$

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Problem 44

Prove or disprove that if $A, B,$ and $C$ are nonempty sets and $A \times B=A \times C,$ then $B=C$

DI
Doruk I.
Numerade Educator

Problem 45

Translate each of these quantifications into English and determine its truth value.
$$
\begin{array}{ll}{\text { a) } \forall x \in \mathbf{R}\left(x^{2} \neq-1\right)} & {\text { b) } \exists x \in \mathbf{Z}\left(x^{2}=2\right)} \\ {\text { c) } \forall x \in \mathbf{Z}\left(x^{2}>0\right)} & {\text { d) } \exists x \in \mathbf{R}\left(x^{2}=x\right)}\end{array}
$$

Clarissa N.
Numerade Educator

Problem 46

Translate each of these quantifications into English and determine its truth value.
$$
\begin{array}{ll}{\text { a) } \exists x \in \mathbf{R}\left(x^{3}=-1\right)} & {\text { b) } \exists x \in \mathbf{Z}(x+1>x)} \\ {\text { c) } \forall x \in \mathbf{Z}(x-1 \in \mathbf{Z})} & {\text { d) } \forall x \in \mathbf{Z}\left(x^{2} \in \mathbf{Z}\right)}\end{array}
$$

DI
Doruk I.
Numerade Educator

Problem 47

Find the truth set of each of these predicates where the domain is the set of integers.
$$
\begin{array}{l}{\text { a) } P(x) : x^{2}<3} \\ {\text { c) } R(x) : 2 x+1=0}\end{array} \quad \text { b) } Q(x) : x^{2}>x
$$

Clarissa N.
Numerade Educator

Problem 48

Find the truth set of each of these predicates where the domain is the set of integers.
$$
\begin{array}{l}{\text { a) } P(x) : x^{3} \geq 1} \\ {\text { c) } R(x) : x<x^{2}}\end{array} \quad \text { b) } Q(x) : x^{2}=2
$$

DI
Doruk I.
Numerade Educator

Problem 49

The defining property of an ordered pair is that two ordered pairs are equal if and only if their first elements are equal and their second elements are equal. Surprisingly, instead of taking the ordered pair as a primitive concept, we can construct ordered pairs using basic notions from set theory. Show that if we define the ordered pair $(a, b)$ to be $\{\{a\},\{a, b\}\},$ then $(a, b)=(c, d)$ if and only if $a=c$ and $b=d .[\text { Hint: First show that }\{\{a\},\{a, b\}\}=$ $\{\{c\},\{c, d\}\}$ if and only if $a=c$ and $b=d . ]$

Clarissa N.
Numerade Educator

Problem 50

This exercise presents Russell's paradox. Let $S$ be the set that contains a set $x$ if the set $x$ does not belong to itself, so that $S=\{x | x \notin x\}$
a) Show the assumption that $S$ is a member of $S$ leads to a contradiction.
b) Show the assumption that $S$ is not a member of $S$ leads to a contradiction. By parts ( a and (b) it follows that the set $S$ cannot be defined as it was. This paradox can be avoided by restricting the types of elements that sets can have.

DI
Doruk I.
Numerade Educator

Problem 51

Describe a procedure for listing all the subsets of a finite set.

Clarissa N.
Numerade Educator