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Mathematical Proofs: A Transition to Advanced Mathematics

Gary Chartrand, Albert D. Polimeni, Ping Zhang

Chapter 1

Sets - all with Video Answers

Educators

Chapter Questions

03:06

Problem 1

Which of the following are sets?
(a) 1,2,3
(b) $\{1,2\}, 3$
(c) $\{\{1\}, 2\}, 3$
(d) $\{1,\{2\}, 3\}$
(e) $\{1,2, a, b\}$.

Sarah Gift
Sarah Gift
Numerade Educator
05:07

Problem 2

Let $S=\{-2,-1,0,1,2,3\}$. Describe each of the following sets as $\{x \in S: p(x)\}$, where $p(x)$ is some condition on $x$.
(a) $A=\{1,2,3\}$
(b) $B=\{0,1,2,3\}$
(c) $C=\{-2,-1\}$
(d) $D=\{-2,2,3\}$

Abdul Vahid M
Abdul Vahid M
Numerade Educator

Problem 3

Determine the cardinality of each of the following sets:
(a) $A=\{1,2,3,4,5\}$
(b) $B=\{0,2,4, \ldots, 20\}$
(c) $C=\{25,26,27, \ldots, 75\}$
(d) $D=\{\{1,2\},\{1,2,3,4\}\}$
(e) $E=\{\emptyset\}$
(f) $F=\{2,\{2,3,4\}\}$

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02:28

Problem 4

Write each of the following sets by listing its elements within braces.
(a) $A=\{n \in \mathbf{Z}:-4<n \leq 4\}$
(b) $B=\left\{n \in \mathbf{Z}: n^2<5\right\}$
(c) $C=\left\{n \in \mathbf{N}: n^3<100\right\}$
(d) $D=\left\{x \in \mathbf{R}: x^2-x=0\right\}$
(e) $E=\left\{x \in \mathbf{R}: x^2+1=0\right\}$

William Semus
William Semus
Numerade Educator

Problem 5

Write each of the following sets in the form $\{x \in \mathbf{Z}: p(x)\}$, where $p(x)$ is a property concerning $x$.
(a) $A=\{-1,-2,-3, \ldots\}$
(b) $B=\{-3,-2, \ldots, 3\}$
(c) $C=\{-2,-1,1,2\}$

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Problem 6

The set $E=\{2 x: x \in \mathbf{Z}\}$ can be described by listing its elements, namely $E=\{\ldots,-4,-2,0,2,4, \ldots\}$. List the elements of the following sets in a similar manner.
(a) $A=\{2 x+1: x \in \mathbf{Z}\}$
(b) $B=\{4 n: n \in \mathbf{Z}\}$
(c) $C=\{3 q+1: q \in \mathbf{Z}\}$

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Problem 7

The set $E=\{\ldots,-4,-2,0,2,4, \ldots\}$ of even integers can be described by means of a defining condition by $E=\{y=2 x: x \in \mathbf{Z}\}=\{2 x: x \in \mathbf{Z}\}$. Describe the following sets in a similar manner.
(a) $A=\{\ldots,-4,-1,2,5,8, \ldots\}$
(b) $B=\{\ldots,-10,-5,0,5,10, \ldots\}$
(c) $C=\{1,8,27,64,125, \ldots\}$

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02:28

Problem 8

Let $A=\{n \in \mathbf{Z}: 2 \leq|n|<4\}, B=\{x \in \mathbf{Q}: 2<x \leq 4\}$,
$C=\left\{x \in \mathbf{R}: x^2-(2+\sqrt{2}) x+2 \sqrt{2}=0\right\}$ and $D=\left\{x \in \mathbf{Q}: x^2-(2+\sqrt{2}) x+2 \sqrt{2}=0\right\}$.
(a) Describe the set $A$ by listing its elements.
(b) Give an example of three elements that belong to $B$ but do not belong to $A$.
(c) Describe the set $C$ by listing its elements.
(d) Describe the set $D$ in another manner.
(e) Determine the cardinality of each of the sets $A, C$ and $D$.

William Semus
William Semus
Numerade Educator
01:42

Problem 9

For $A=\{2,3,5,7,8,10,13\}$, let
$B=\{x \in A: x=y+z$, where $y, z \in A\}$ and $C=\{r \in B: r+s \in B$ for some $s \in B\}$.
Determine $C$.

Aman Gupta
Aman Gupta
Numerade Educator
03:04

Problem 10

Give examples of three sets $A, B$ and $C$ such that
(a) $A \subseteq B \subset C$
(b) $A \in B, B \in C$ and $A \notin C$
(c) $A \in B$ and $A \subset C$.

Mengchun Cai
Mengchun Cai
Numerade Educator
01:39

Problem 11

Let $(a, b)$ be an open interval of real numbers and let $c \in(a, b)$. Describe an open interval $I$ centered at $c$ such that $I \subseteq(a, b)$.

Lucas Finney
Lucas Finney
Numerade Educator
03:02

Problem 12

Which of the following sets are equal?
$$
\begin{array}{ll}
A=\{n \in \mathbf{Z}:|n|<2\} & D=\left\{n \in \mathbf{Z}: n^2 \leq 1\right\} \\
B=\left\{n \in \mathbf{Z}: n^3=n\right\} & E=\{-1,0,1\} . \\
C=\left\{n \in \mathbf{Z}: n^2 \leq n\right\} &
\end{array}
$$

Ayushi Sambyal
Ayushi Sambyal
Numerade Educator
01:03

Problem 13

For a universal set $U=\{1,2, \ldots, 8\}$ and two sets $A=\{1,3,4,7\}$ and $B=\{4,5,8\}$, draw a Venn diagram that represents these sets.

Nick Johnson
Nick Johnson
Numerade Educator
01:06

Problem 14

Find $\mathcal{P}(A)$ and $|\mathcal{P}(A)|$ for
(a) $A=\{1,2\}$.
(b) $A=\{\emptyset, 1,\{a\}\}$.

Sanchit Jain
Sanchit Jain
Numerade Educator
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Problem 15

Find $\mathcal{P}(A)$ for $A=\{0,\{0\}\}$.

Claire Rochford
Claire Rochford
Numerade Educator

Problem 16

Find $\mathcal{P}(\mathcal{P}(\{1\}))$ and its cardinality.

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Problem 17

Find $\mathcal{P}(A)$ and $|\mathcal{P}(A)|$ for $A=\{0, \emptyset,\{\emptyset\}\}$.

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01:06

Problem 18

For $A=\{x: x=0$ or $x \in \mathcal{P}(\{0\})\}$, determine $\mathcal{P}(A)$.

Sanchit Jain
Sanchit Jain
Numerade Educator
View

Problem 19

Give an example of a set $S$ such that
(a) $S \subseteq \mathcal{P}(\mathbf{N})$
(b) $S \in \mathcal{P}(\mathbf{N})$
(c) $S \subseteq \mathcal{P}(\mathbf{N})$ and $|S|=5$
(d) $S \in \mathcal{P}(\mathbf{N})$ and $|S|=5$

Nick Johnson
Nick Johnson
Numerade Educator
01:36

Problem 20

Determine whether the following statements are true or false.
(a) If $\{1\} \in \mathcal{P}(A)$, then $1 \in A$ but $\{1\} \notin A$.
(b) If $A, B$ and $C$ are sets such that $A \subset \mathcal{P}(B) \subset C$ and $|A|=2$, then $|C|$ can be 5 but $|C|$ cannot be 4 .
(c) If a set $B$ has one more element than a set $A$, then $\mathcal{P}(B)$ has at least two more elements than $\mathcal{P}(A)$.
(d) If four sets $A, B, C$ and $D$ are subsets of $\{1,2,3\}$ such that $|A|=|B|=|C|=|D|=2$, then at least two of these sets are equal.

Doruk Isik
Doruk Isik
Numerade Educator
02:20

Problem 21

Three subsets $A, B$ and $C$ of $\{1,2,3,4,5\}$ have the same cardinality. Furthermore,
(a) 1 belongs to $A$ and $B$ but not to $C$.
(b) 2 belongs to $A$ and $C$ but not to $B$.
(c) 3 belongs to $A$ and exactly one of $B$ and $C$.
(d) 4 belongs to an even number of $A, B$ and $C$.
(e) 5 belongs to an odd number of $A, B$ and $C$.
(f) The sums of the elements in two of the sets $A, B$ and $C$ differ by 1 .
What is $B$ ?

Aman Gupta
Aman Gupta
Numerade Educator

Problem 22

Let $U=\{1,3, \ldots, 15\}$ be the universal set, $A=\{1,5,9,13\}$, and $B=\{3,9,15\}$. Determine the following:
(a) $A \cup B$
(b) $A \cap B$
(c) $A-B$
(d) $B-A$
(e) $\bar{A}$
(f) $A \cap \bar{B}$.

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02:06

Problem 23

Give examples of two sets $A$ and $B$ such that $|A-B|=|A \cap B|=|B-A|=3$. Draw the accompanying Venn diagram.

William Semus
William Semus
Numerade Educator

Problem 24

Give examples of three sets $A, B$ and $C$ such that $B \neq C$ but $B-A=C-A$.

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03:04

Problem 25

Give examples of three sets $A, B$ and $C$ such that
(a) $A \in B, A \subseteq C$ and $B \nsubseteq C$
(b) $B \in A, B \subset C$ and $A \cap C \neq \emptyset$
(c) $A \in B, B \subseteq C$ and $A \nsubseteq C$.

Mengchun Cai
Mengchun Cai
Numerade Educator
03:08

Problem 26

Let $U$ be a universal set and let $A$ and $B$ be two subsets of $U$. Draw a Venn diagram for each of the following sets.
(a) $\overline{A \cup B}$
(b) $\bar{A} \cap \bar{B}$
(c) $\overline{A \cap B}$
(d) $\bar{A} \cup \bar{B}$.
What can you say about parts (a) and (b)? parts (c) and (d)?

Kari Hasz
Kari Hasz
Numerade Educator
00:25

Problem 27

Give an example of a universal set $U$, two sets $A$ and $B$ and accompanying Venn diagram such that $|A \cap B|=|A-B|=|B-A|=|\overline{A \cup B}|=2$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
04:22

Problem 28

Let $A, B$ and $C$ be nonempty subsets of a universal set $U$. Draw a Venn diagram for each of the following set operations.
(a) $(C-B) \cup A$
(b) $C \cap(A-B)$.

Aman Gupta
Aman Gupta
Numerade Educator
03:04

Problem 29

Let $A=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\}\}$.
(a) Determine which of the following are elements of $A: \emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}$.
(b) Determine $|A|$.
(c) Determine which of the following are subsets of $A: \emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}$. For (d)-(i), determine the indicated sets.
(d) $\emptyset \cap A$
(e) $\{\emptyset\} \cap A$
(f) $\{\emptyset,\{\emptyset\}\} \cap A$
(g) $\emptyset \cup A$
(h) $\{\emptyset\} \cup A$
(i) $\{\emptyset,\{\emptyset\}\} \cup A$.

Rahul Kumar
Rahul Kumar
Numerade Educator
03:43

Problem 30

Let $A=\{x \in \mathbf{R}:|x-1| \leq 2\}, B=\{x \in \mathbf{R}:|x| \geq 1\}$ and $C=\{x \in \mathbf{R}:|x+2| \leq 3\}$.
(a) Express $A, B$ and $C$ using interval notation.
(b) Determine each of the following sets using interval notation: $A \cup B, A \cap B, B \cap C, B-C$.

Aayush Gupta
Aayush Gupta
Numerade Educator
01:08

Problem 31

Give an example of four different sets $A, B, C$ and $D$ such that (1) $A \cup B=\{1,2\}$ and $C \cap D=\{2,3\}$ and (2) if $B$ and $C$ are interchanged and $\cup$ and $\cap$ are interchanged, then we get the same result.

Dheeraj
Dheeraj
Numerade Educator
01:08

Problem 32

Give an example of four different subsets $A, B, C$ and $D$ of $\{1,2,3,4\}$ such that all intersections of two subsets are different.

Dheeraj
Dheeraj
Numerade Educator
02:06

Problem 33

Give an example of two nonempty sets $A$ and $B$ such that $\{A \cup B, A \cap B, A-B, B-A\}$ is the power set of some set.

William Semus
William Semus
Numerade Educator

Problem 34

Give an example of two subsets $A$ and $B$ of $\{1,2,3\}$ such that all of the following sets are different: $A \cup B$, $A \cup \bar{B}, \bar{A} \cup B, \bar{A} \cup \bar{B}, A \cap B, A \cap \bar{B}, \bar{A} \cap B, \bar{A} \cap \bar{B}$.

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Problem 35

Give examples of a universal set $U$ and sets $A, B$ and $C$ such that each of the following sets contains exactly one element: $A \cap B \cap C,(A \cap B)-C,(A \cap C)-B,(B \cap C)-A, A-(B \cup C), B-(A \cup C)$, $C-(A \cup B), \overline{A \cup B \cup C}$. Draw the accompanying Venn diagram.

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Problem 36

For a real number $r$, define $S_r$ to be the interval $[r-1, r+2]$. Let $A=\{1,3,4\}$. Determine $\bigcup_{\alpha \in A} S_\alpha$ and $\bigcap_{\alpha \in A} S_\alpha$.

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02:38

Problem 37

Let $A=\{1,2,5\}, B=\{0,2,4\}, C=\{2,3,4\}$ and $S=\{A, B, C\}$. Determine $\bigcup_{X \in \mathcal{S}} X$ and $\bigcap_{X \in \mathcal{S}} X$.

Sarah Gift
Sarah Gift
Numerade Educator

Problem 38

For a real number $r$, define $A_r=\left\{r^2\right\}, B_r$ as the closed interval $[r-1, r+1]$ and $C_r$ as the interval $(r, \infty)$. For $S=\{1,2,4\}$, determine
(a) $\bigcup_{\alpha \in S} A_\alpha$ and $\bigcap_{\alpha \in S} A_\alpha$
(b) $\bigcup_{\alpha \in S} B_\alpha$ and $\bigcap_{\alpha \in S} B_\alpha$
(c) $\bigcup_{\alpha \in S} C_\alpha$ and $\bigcap_{\alpha \in S} C_\alpha$.

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Problem 39

Let $A=\{a, b, \ldots, z\}$ be the set consisting of the letters of the alphabet. For $\alpha \in A$, let $A_\alpha$ consist of $\alpha$ and the two letters that follow it, where $A_y=\{y, z, a\}$ and $A_z=\{z, a, b\}$. Find a set $S \subseteq A$ of smallest cardinality such that $\bigcup_{\alpha \in S} A_\alpha=A$. Explain why your set $S$ has the required properties.

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07:06

Problem 40

For $i \in \mathbf{Z}$, let $A_i=\{i-1, i+1\}$. Determine the following:
(a) $\bigcup_{i=1}^5 A_{2 i}$
(b) $\bigcup_{i=1}^5\left(A_i \cap A_{i+1}\right)$
(c) $\bigcup_{i=1}^5\left(A_{2 i-1} \cap A_{2 i+1}\right)$.

Mengchun Cai
Mengchun Cai
Numerade Educator
07:06

Problem 41

For each of the following, find an indexed collection $\left\{A_n\right\}_{n \in \mathbf{N}}$ of distinct sets (that is, no two sets are equal) satisfying the given conditions.
(a) $\cap_{n=1}^{\infty} A_n=\{0\}$ and $\bigcup_{n=1}^{\infty} A_n=[0,1]$
(b) $\bigcap_{n=1}^{\infty} A_n=\{-1,0,1\}$ and $\bigcup_{n=1}^{\infty} A_n=\mathbf{Z}$.

Mengchun Cai
Mengchun Cai
Numerade Educator

Problem 42

For each of the following collections of sets, define a set $A_n$ for each $n \in \mathbf{N}$ such that the indexed collection $\left\{A_n\right\}_{n \in \mathrm{N}}$ is precisely the given collection of sets. Then find both the union and intersection of the indexed collection of sets.
(a) $\{[1,2+1),[1,2+1 / 2),[1,2+1 / 3), \ldots\}$
(b) $\{(-1,2),(-3 / 2,4),(-5 / 3,6),(-7 / 4,8), \ldots\}$.

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Problem 43

For $r \in \mathbf{R}^{+}$, let $A_r=\{x \in \mathbf{R}:|x|<r\}$. Determine $\bigcup_{r \in \mathbf{R}^{+}} A_r$ and $\bigcap_{r \in \mathbf{R}^{+}} A_r$.

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00:40

Problem 44

Each of the following sets is a subset of $A=\{1,2, \ldots, 10\}$ :
$$
\begin{aligned}
& A_1=\{1,5,7,9,10\}, A_2=\{1,2,3,8,9\}, A_3=\{2,4,6,8,9\}, \\
& A_4=\{2,4,8\}, A_5=\{3,6,7\}, A_6=\{3,8,10\}, A_7=\{4,5,7,9\}, \\
& A_8=\{4,5,10\}, A_9=\{4,6,8\}, A_{10}=\{5,6,10\}, \\
& A_{11}=\{5,8,9\}, A_{12}=\{6,7,10\}, A_{13}=\{6,8,9\}
\end{aligned}
$$
Find a set $I \subseteq\{1,2, \ldots, 13\}$ such that for every two distinct elements $j, k \in I, A_j \cap A_k=\emptyset$ and $\left|\bigcup_{i \in I} A_i\right|$ is maximum.

Kayleah Tsai
Kayleah Tsai
Numerade Educator

Problem 45

For $n \in \mathbf{N}$, let $A_n=\left(-\frac{1}{n}, 2-\frac{1}{n},\right)$. Determine $\bigcup_{n \in \mathbf{N}} A_n$ and $\bigcap_{n \in \mathbf{N}} A_n$.

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Problem 46

Which of the following are partitions of $A=\{a, b, c, d, e, f, g\}$ ? For each collection of subsets that is not a partition of $A$, explain your answer.
(a) $S_1=\{\{a, c, e, g\},\{b, f\},\{d\}\}$
(b) $S_2=\{\{a, b, c, d\},\{e, f\}\}$
(c) $S_3=\{A\}$
(d) $S_4=\{\{a\}, \emptyset,\{b, c, d\},\{e, f, g\}\}$
(e) $S_5=\{\{a, c, d\},\{b, g\},\{e\},\{b, f\}\}$.

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04:06

Problem 47

Which of the following sets are partitions of $A=\{1,2,3,4,5\}$ ?
(a) $S_1=\{\{1,3\},\{2,5\}\}$
(b) $S_2=\{\{1,2\},\{3,4,5\}\}$
(c) $S_3=\{\{1,2\},\{2,3\},\{3,4\},\{4,5\}\}$
(d) $S_4=A$.

Ibrahima Barry
Ibrahima Barry
Numerade Educator

Problem 48

Let $A=\{1,2,3,4,5,6\}$. Give an example of a partition $S$ of $A$ such that $|S|=3$.

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Problem 49

Give an example of a set $A$ with $|A|=4$ and two disjoint partitions $S_1$ and $S_2$ of $A$ with
$$
\left|S_1\right|=\left|S_2\right|=3 \text {. }
$$

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03:49

Problem 50

Give an example of a partition of $\mathbf{N}$ into three subsets.

James Kiss
James Kiss
Numerade Educator

Problem 51

Give an example of a partition of $\mathbf{Q}$ into three subsets.

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Problem 52

Give an example of three sets $A, S_1$ and $S_2$ such that $S_1$ is a partition of $A, S_2$ is a partition of $S_1$ and $\left|S_2\right|<\left|S_1\right|<|A|$.

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Problem 53

Give an example of a partition of $\mathbf{Z}$ into four subsets.

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Problem 54

Let $A=\{1,2, \ldots, 12\}$. Give an example of a partition $S$ of $A$ satisfying the following requirements: (i) $|S|=5$, (ii) there is a subset $T$ of $S$ such that $|T|=4$ and $\left|\cup_{X \in T} X\right|=10$ and (iii) there is no element $B \in S$ such that $|B|=3$.

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03:49

Problem 55

A set $S$ is partitioned into two subsets $S_1$ and $S_2$. This produces a partition $\mathcal{P}_1$ of $S$ where $\mathcal{P}_1=\left\{S_1, S_2\right\}$ and so $\left|\mathcal{P}_1\right|=2$. One of the sets in $\mathcal{P}_1$ is then partitioned into two subsets, producing a partition $\mathcal{P}_2$ of $S$ with $\left|\mathcal{P}_2\right|=3$. A total of $\left|\mathcal{P}_1\right|$ sets in $\mathcal{P}_2$ are partitioned into two subsets each, producing a partition $\mathcal{P}_3$ of $S$. Next, a total of $\left|\mathcal{P}_2\right|$ sets in $\mathcal{P}_3$ are partitioned into two subsets each, producing a partition $\mathcal{P}_4$ of $S$. This is continued until a partition $\mathcal{P}_6$ of $S$ is produced. What is $\left|\mathcal{P}_6\right|$ ?

James Kiss
James Kiss
Numerade Educator

Problem 56

We mentioned that there are three ways that a collection $\mathcal{S}$ of subsets of a nonempty set $A$ is defined to be a partition of $A$.
Definition 1 The collection $\mathcal{S}$ consists of pairwise disjoint nonempty subsets of $A$ and every element of $A$ belongs to a subset in $\mathcal{S}$.
Definition 2 The collection $\mathcal{S}$ consists of nonempty subsets of $A$ and every element of $A$ belongs to exactly one subset in $\mathcal{S}$.
Definition 3 The collection $\mathcal{S}$ consists of subsets of $A$ satisfying the three properties (1) every subset in $S$ is nonempty, (2) every two subsets of $A$ are equal or disjoint and (3) the union of all subsets in $S$ is $A$.
(a) Show that any collection $\mathcal{S}$ of subsets of $A$ satisfying Definition 1 satisfies Definition 2.
(b) Show that any collection $\mathcal{S}$ of subsets of $A$ satisfying Definition 2 satisfies Definition 3.
(c) Show that any collection $\mathcal{S}$ of subsets of $A$ satisfying Definition 3 satisfies Definition 1.

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01:03

Problem 57

Let $A=\{x, y, z\}$ and $B=\{x, y\}$. Determine $A \times B$.

Manisha Sarker
Manisha Sarker
Numerade Educator
01:06

Problem 58

Let $A=\{1,\{1\},\{\{1\}\}\}$. Determine $A \times A$.

Sanchit Jain
Sanchit Jain
Numerade Educator
01:06

Problem 59

For $A=\{a, b\}$, determine $A \times \mathcal{P}(A)$.

Sanchit Jain
Sanchit Jain
Numerade Educator
01:06

Problem 60

For $A=\{\emptyset,\{\emptyset\}\}$, determine $A \times \mathcal{P}(A)$.

Sanchit Jain
Sanchit Jain
Numerade Educator
01:06

Problem 61

For $A=\{1,2\}$ and $B=\{\emptyset\}$, determine $A \times B$ and $\mathcal{P}(A) \times \mathcal{P}(B)$.

Sanchit Jain
Sanchit Jain
Numerade Educator
02:22

Problem 62

Describe the graph of the circle whose equation is $x^2+y^2=4$ as a subset of $\mathbf{R} \times \mathbf{R}$.

Grace Bajar
Grace Bajar
Numerade Educator
01:52

Problem 63

List the elements of the set $S=\{(x, y) \in \mathbf{Z} \times \mathbf{Z}:|x|+|y|=3\}$. Plot the corresponding points in the Euclidean $x y$-plane.

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
01:06

Problem 64

For $A=\{1,2\}$ and $B=\{1\}$, determine $\mathcal{P}(A \times B)$.

Sanchit Jain
Sanchit Jain
Numerade Educator

Problem 65

For $A=\{x \in \mathbf{R}:|x-1| \leq 2\}$ and $B=\{y \in \mathbf{R}:|y-4| \leq 2\}$, give a geometric description of the points in the $x y$-plane belonging to $A \times B$.

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Problem 66

For $A=\{a \in \mathbf{R}:|a| \leq 1\}$ and $B=\{b \in \mathbf{R}:|b|=1\}$, give a geometric description of the points in the $x y$-plane belonging to $(A \times B) \cup(B \times A)$.

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01:47

Problem 67

The set $T=\{2 k+1: k \in \mathbf{Z}\}$ can be described as $T=\{\ldots,-3,-1,1,3, \ldots\}$. Describe the following sets in a similar manner.
(a) $A=\{4 k+3: k \in \mathbf{Z}\}$
(b) $B=\{5 k-1: k \in \mathbf{Z}\}$.

Willis James
Willis James
Numerade Educator

Problem 68

Let $S=\{-10,-9, \ldots, 9,10\}$. Describe each of the following sets as $\{x \in S: p(x)\}$, where $p(x)$ is some condition on $x$.
(a) $A=\{-10,-9, \ldots,-1,1, \ldots, 9,10\}$
(b) $B=\{-10,-9, \ldots,-1,0\}$
(c) $C=\{-5,-4, \ldots, 0,1, \ldots, 7\}$
(d) $D=\{-10,-9, \ldots, 4,6,7, \ldots, 10\}$.

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Problem 69

Describe each of the following sets by listing its elements within braces.
(a) $\left\{x \in \mathbf{Z}: x^3-4 x=0\right\}$
(b) $\{x \in \mathbf{R}:|x|=-1\}$
(c) $\{m \in \mathbf{N}: 2<m \leq 5\}$
(d) $\{n \in \mathbf{N}: 0 \leq n \leq 3\}$
(e) $\left\{k \in \mathbf{Q}: k^2-4=0\right\}$
(f) $\left\{k \in \mathbf{Z}: 9 k^2-3=0\right\}$
(g) $\left\{k \in \mathbf{Z}: 1 \leq k^2 \leq 10\right\}$.

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03:42

Problem 70

Determine the cardinality of each of the following sets.
(a) $A=\{1,2,3,\{1,2,3\}, 4,\{4\}\}$
(b) $B=\{x \in \mathbf{R}:|x|=-1\}$
(c) $C=\{m \in \mathbf{N}: 2<m \leq 5\}$
(d) $D=\{n \in \mathbf{N}: n<0\}$
(e) $E=\left\{k \in \mathbf{N}: 1 \leq k^2 \leq 100\right\}$
(f) $F=\left\{k \in \mathbf{Z}: 1 \leq k^2 \leq 100\right\}$.

Clayton Schubring
Clayton Schubring
Numerade Educator
01:38

Problem 71

For $A=\{-1,0,1\}$ and $B=\{x, y\}$, determine $A \times B$.

Vicki Stebbins
Vicki Stebbins
Numerade Educator
02:51

Problem 72

Let $U=\{1,2,3\}$ be the universal set and let $A=\{1,2\}, B=\{2,3\}$ and $C=\{1,3\}$. Determine the following.
(a) $(A \cup B)-(B \cap C)$
(b) $\bar{A}$
(c) $\overline{B \cup C}$
(d) $A \times B$.

Aman Gupta
Aman Gupta
Numerade Educator
02:06

Problem 73

Let $A=\{1,2, \ldots, 10\}$. Give an example of two sets $S$ and $B$ such that $S \subseteq \mathcal{P}(A),|S|=4, B \in S$ and $|B|=2$.

William Semus
William Semus
Numerade Educator

Problem 74

For $A=\{1\}$ and $C=\{1,2\}$, give an example of a set $B$ such that $\mathcal{P}(A) \subset B \subset \mathcal{P}(C)$.

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Problem 75

Give examples of two sets $A$ and $B$ such that $A \cap \mathcal{P}(A) \in B$ and $\mathcal{P}(A) \subseteq A \cup B$.

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03:02

Problem 76

Which of the following sets are equal?
$$
\begin{array}{ll}
A=\{n \in \mathbf{Z}:-4 \leq n \leq 4\} & D=\left\{x \in \mathbf{Z}: x^3=4 x\right\} \\
B=\{x \in \mathbf{N}: 2 x+2=0\} & E=\{-2,0,2\} . \\
C=\{x \in \mathbf{Z}: 3 x-2=0\} &
\end{array}
$$

Ayushi Sambyal
Ayushi Sambyal
Numerade Educator
01:22

Problem 77

Let $A$ and $B$ be subsets of some unknown universal set $U$. Suppose that $\bar{A}=\{3,8,9\}, A-B=\{1,2\}$, $B-A=\{8\}$ and $A \cap B=\{5,7\}$. Determine $U, A$ and $B$.

Aman Gupta
Aman Gupta
Numerade Educator

Problem 78

Let $I$ denote the interval $[0, \infty)$. For each $r \in I$, define
$$
\begin{aligned}
A_r & =\left\{(x, y) \in \mathbf{R} \times \mathbf{R}: x^2+y^2=r^2\right\} \\
B_r & =\left\{(x, y) \in \mathbf{R} \times \mathbf{R}: x^2+y^2 \leq r^2\right\} \\
C_r & =\left\{(x, y) \in \mathbf{R} \times \mathbf{R}: x^2+y^2>r^2\right\} .
\end{aligned}
$$
(a) Determine $\bigcup_{r \in I} A_r$ and $\bigcap_{r \in I} A_r$.
(b) Determine $\bigcup_{r \in I} B_r$ and $\bigcap_{r \in I} B_r$.
(c) Determine $\bigcup_{r \in I} C_r$ and $\bigcap_{r \in I} C_r$.

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00:48

Problem 79

Give an example of four sets $A_1, A_2, A_3, A_4$ such that $\left|A_i \cap A_j\right|=|i-j|$ for every two integers $i$ and $j$ with $1 \leq i<j \leq 4$.

James Kiss
James Kiss
Numerade Educator
06:14

Problem 80

(a) Give an example of two problems suggested by Exercise 1.79 (above).
(b) Solve one of the problems in (a).

Faizanullah Kazmi
Faizanullah Kazmi
Numerade Educator

Problem 81

Let $A=\{1,2,3\}, B=\{1,2,3,4\}$ and $C=\{1,2,3,4,5\}$. For the sets $S$ and $T$ described below, explain whether $|S|<|T|,|S|\rangle|T|$ or $|S|=|T|$.
(a) Let $B$ be the universal set and let $S$ be the set all subsets $X$ of $B$ for which $|X| \neq|\bar{X}|$. Let $T$ be the set of 2-element subsets of $C$.
(b) Let $S$ be the set of all partitions of the set $A$ and let $T$ be the set of 4-element subsets of $C$.
(c) Let $S=\{(b, a): b \in B, a \in A, a+b$ is odd $\}$ and let $T$ be the set of all nonempty proper subsets of $A$.

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04:47

Problem 82

Give an example of a set $A=\{1,2, \ldots, k\}$ for a smallest $k \in \mathbf{N}$ containing subsets $A_1, A_2, A_3$ such that $\left|A_i-A_j\right|=\left|A_j-A_i\right|=|i-j|$ for every two integers $i$ and $j$ with $1 \leq i<j \leq 3$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator

Problem 83

(a) For $A=\{-3,-2, \ldots, 4\}$ and $B=\{1,2, \ldots, 6\}$, determine
$$
S=\left\{(a, b) \in A \times B: a^2+b^2=25\right\} \text {. }
$$
(b) For $C=\{a \in B:(a, b) \in S\}$ and $D=\{b \in A:(a, b) \in S\}$, where $A, B, S$ are the sets in (a), determine $C \times D$.

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01:08

Problem 84

For $A=\{1,2,3\}$, let $B$ be the set of 2-element sets belonging to $\mathcal{P}(A)$ and let $C$ be the set consisting of the sets that are the intersections of two distinct elements of $B$. Determine $D=\mathcal{P}(C)$.

Dheeraj
Dheeraj
Numerade Educator
10:52

Problem 85

For a real number $r$, let $A_r=\{r, r+1\}$. Let $S=\left\{x \in \mathbf{R}: x^2+2 x-1=0\right\}$.
(a) Determine $B=A_s \times A_t$ for the distinct elements $s, t \in S$, where $s<t$.
(b) Let $C=\{a b:(a, b) \in B\}$. Determine the sum of the elements of $C$.

Charles Carter
Charles Carter
Numerade Educator