Discrete Mathematics and its Applications

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) $\left\{x | x \text { is an integer such that } x^{2}=2\right\}$

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) $\{m, n, o, p\}$

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

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

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

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

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

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.

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

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

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

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) $\{\{\varnothing\}\} \subset\{\{\emptyset\},\{\emptyset\}\}$

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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

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

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

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

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

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

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?

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

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

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

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

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

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

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

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

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

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

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 $b=d . ]$

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.

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