INSTRUCTOR’S
SOLUTIONS MANUAL
ST
U
A NALYSIS WITH AN
I NTRODUCTION TO PROOF
VI
Sixth EDITION
A _A
PP
Steven R. Lay and Richard G. Ligo
RO
VE
D
?
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
, 2
This manual is intended to accompany the 6th edition of Analysis with an Introduction to Proof
ST
by Steven R. Lay and Richard G. Ligo (Pearson, 2023). It contains solutions to nearly every exercise
in the text. Those exercises that have hints (or answers) in the back of the book are numbered in bold
print, and the hints are included here for reference. While many of the proofs have been given in full
detail, some of the more routine proofs are only outlines. For some of the problems, other approaches
U
may be equally acceptable. This is particularly true for those problems requesting a counterexample.
We have not tried to be exhaustive in discussing each exercise, but rather to be suggestive.
Let us remind you that the starred exercises are not necessarily the more difficult ones. They
VI
are the exercises that are used in some way in subsequent sections. There is a table on page 3 that
indicates where starred exercises are used later. The following notations are used throughout this
manual:
ℕ = the set of natural numbers {1, 2, 3, 4, …}
ℚ
A
= the set of rational numbers
R = the set of real numbers
∀ = “for every”
_A
∃ = “there exists”
∋ = “such that”
We have tried to be accurate in the preparation of this manual, but unfortunately, some mistake will
inadvertently slip by. I would appreciate having any errors in this manual or the text brought to my
PP
attention.
Richard G. Ligo
RO
VE
D
?
Copyright © 2024 by Pearson Education, Inc. or its affiliates
, 3
Table of Starred Exercises
Note: The prefix P indicates a practice problem, the prefix E indicates an example, the prefix T refers to a
ST
theorem or corollary, and the absence of a prefix before a number indicates an exercise.
Starred Starred
Exercise Later Use Exercise Later use
2.1.26 T3.4.11 4.3.14 4.4.5
U
2.2.10 2.4.26 4.4.10 8.2.15
2.3.32 2.5.3 4.4.16 8.3.10
3.1.3 E7.1.8 4.4.17 T8.3.4
VI
3.1.4 7.1.7 5.1.16 6.2.8
3.1.6 E8.1.2 5.1.18 T6.2.12
3.1.7 4.3.10, 4.3.15, E8.1.10, T9.2.11 5.1.20 5.2.14, 5.3.17
3.1.8 P8.1.5
A
5.1.21 5.2.17
3.1.24 4.1.7f, E5.3.8 5.2.10 T7.2.8
3.1.27 3.3.16 5.2.11 7.2.9b
_A
3.1.30b 3.3.13, E4.1.12, 4.3.15 5.2.13 T5.3.6, T6.1.7, 7.1.14
3.2.6a 4.1.9a, T4.2.1, 6.2.23, 7.2.16, T9.2.11 5.2.16 9.2.15
3.2.6b T6.3.8 5.3.14b T6.2.11, T6.2.13
3.2.6c T4.1.16 6.1.6 6.2.14, 6.2.19
PP
3.2.7 T8.2.10 6.1.8 7.3.15
3.3.7 T7.2.4, 7.2.3 6.1.18b 6.4.10
3.3.14 7.1.15, T7.2.4 6.2.8 T7.2.1
3.4.15 3.5.13, T4.3.12 6.3.15d 9.3.16
3.4.21 3.5.8 7.1.13 P7.2.5
RO
3.5.9 9.2.15 7.1.14 7.2.5
3.6.13 5.5.9 7.1.17 7.2.17
4.1.6b E4.2.2 7.2.9a P7.3.7
4.1.7f T4.2.7, 4.3.10, E8.1.10 7.2.11 T8.2.6
4.1.9a 5.2.10, 9.2.17 7.2.15 7.3.22
VE
4.1.11 E4.3.4 7.2.20 E7.3.9
4.1.12 5.1.17 8.1.7 E8.2.11
4.1.13 5.1.15 8.1.8 8.2.14
4.1.15b 4.4.11, 4.4.18, 5.3.13 8.1.13a 9.3.8
4.1.16 5.1.17 8.2.12 9.2.7, 9.2.8
D
4.2.19 E6.4.6 8.2.15 T8.3.6
4.2.20 5.1.16, T9.1.10 9.1.15a 9.2.9
?
Copyright © 2024 by Pearson Education, Inc. or its affiliates
, Section 1.1 • Logical Connectives 4
This work is protected by United States copyright laws and is provided solely for
the use of instructors in teaching their courses and assessing student learning.
Dissemination or sale of any part of this work (including on the World Wide Web)
will destroy the integrity of the work and is not permitted. The work and materials
from it should never be made available to students except by instructors using
ST
the accompanying text in their classes. All recipients of this work are expected to
abide by these restrictions and to honor the intended pedagogical purposes and
the needs of other instructors who rely on these materials.
U
VI
Analysis
A
with an Introduction to Proof
6th Edition
_A
by Steven R. Lay and Richard G. Ligo
PP
Chapter 1 – Logic and Proof
Solutions to Exercises
RO
Section 1.1 – Logical Connectives
1. (a) False: A statement may be false.
(b) False: A statement cannot be both true and false.
(c) True: See the comment after Practice 1.1.4.
(d) False: See the comment before Example 1.1.3.
(e) False: If the statement is false, then its negation is true.
VE
2. (a) False: p is the antecedent.
(b) True: Practice 1.1.6(a).
(c) False: See the paragraph before Practice 1.1.5.
(d) False: “p whenever q” is “if q, then p”.
(e) False: The negation of p ⇒ q is p ∧ ~ q.
D
3. Answers in Book: (a) The 3 × 3 identity matrix is not singular.
(b) The function f (x) = sin x is not bounded on R.
?
(c) The function f is not linear or the function g is not linear.
(d) Six is not prime and seven is not odd.
(e) x is in D and f (x) ≥ 5.
Copyright © 2024 by Pearson Education, Inc. or its affiliates
SOLUTIONS MANUAL
ST
U
A NALYSIS WITH AN
I NTRODUCTION TO PROOF
VI
Sixth EDITION
A _A
PP
Steven R. Lay and Richard G. Ligo
RO
VE
D
?
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
, 2
This manual is intended to accompany the 6th edition of Analysis with an Introduction to Proof
ST
by Steven R. Lay and Richard G. Ligo (Pearson, 2023). It contains solutions to nearly every exercise
in the text. Those exercises that have hints (or answers) in the back of the book are numbered in bold
print, and the hints are included here for reference. While many of the proofs have been given in full
detail, some of the more routine proofs are only outlines. For some of the problems, other approaches
U
may be equally acceptable. This is particularly true for those problems requesting a counterexample.
We have not tried to be exhaustive in discussing each exercise, but rather to be suggestive.
Let us remind you that the starred exercises are not necessarily the more difficult ones. They
VI
are the exercises that are used in some way in subsequent sections. There is a table on page 3 that
indicates where starred exercises are used later. The following notations are used throughout this
manual:
ℕ = the set of natural numbers {1, 2, 3, 4, …}
ℚ
A
= the set of rational numbers
R = the set of real numbers
∀ = “for every”
_A
∃ = “there exists”
∋ = “such that”
We have tried to be accurate in the preparation of this manual, but unfortunately, some mistake will
inadvertently slip by. I would appreciate having any errors in this manual or the text brought to my
PP
attention.
Richard G. Ligo
RO
VE
D
?
Copyright © 2024 by Pearson Education, Inc. or its affiliates
, 3
Table of Starred Exercises
Note: The prefix P indicates a practice problem, the prefix E indicates an example, the prefix T refers to a
ST
theorem or corollary, and the absence of a prefix before a number indicates an exercise.
Starred Starred
Exercise Later Use Exercise Later use
2.1.26 T3.4.11 4.3.14 4.4.5
U
2.2.10 2.4.26 4.4.10 8.2.15
2.3.32 2.5.3 4.4.16 8.3.10
3.1.3 E7.1.8 4.4.17 T8.3.4
VI
3.1.4 7.1.7 5.1.16 6.2.8
3.1.6 E8.1.2 5.1.18 T6.2.12
3.1.7 4.3.10, 4.3.15, E8.1.10, T9.2.11 5.1.20 5.2.14, 5.3.17
3.1.8 P8.1.5
A
5.1.21 5.2.17
3.1.24 4.1.7f, E5.3.8 5.2.10 T7.2.8
3.1.27 3.3.16 5.2.11 7.2.9b
_A
3.1.30b 3.3.13, E4.1.12, 4.3.15 5.2.13 T5.3.6, T6.1.7, 7.1.14
3.2.6a 4.1.9a, T4.2.1, 6.2.23, 7.2.16, T9.2.11 5.2.16 9.2.15
3.2.6b T6.3.8 5.3.14b T6.2.11, T6.2.13
3.2.6c T4.1.16 6.1.6 6.2.14, 6.2.19
PP
3.2.7 T8.2.10 6.1.8 7.3.15
3.3.7 T7.2.4, 7.2.3 6.1.18b 6.4.10
3.3.14 7.1.15, T7.2.4 6.2.8 T7.2.1
3.4.15 3.5.13, T4.3.12 6.3.15d 9.3.16
3.4.21 3.5.8 7.1.13 P7.2.5
RO
3.5.9 9.2.15 7.1.14 7.2.5
3.6.13 5.5.9 7.1.17 7.2.17
4.1.6b E4.2.2 7.2.9a P7.3.7
4.1.7f T4.2.7, 4.3.10, E8.1.10 7.2.11 T8.2.6
4.1.9a 5.2.10, 9.2.17 7.2.15 7.3.22
VE
4.1.11 E4.3.4 7.2.20 E7.3.9
4.1.12 5.1.17 8.1.7 E8.2.11
4.1.13 5.1.15 8.1.8 8.2.14
4.1.15b 4.4.11, 4.4.18, 5.3.13 8.1.13a 9.3.8
4.1.16 5.1.17 8.2.12 9.2.7, 9.2.8
D
4.2.19 E6.4.6 8.2.15 T8.3.6
4.2.20 5.1.16, T9.1.10 9.1.15a 9.2.9
?
Copyright © 2024 by Pearson Education, Inc. or its affiliates
, Section 1.1 • Logical Connectives 4
This work is protected by United States copyright laws and is provided solely for
the use of instructors in teaching their courses and assessing student learning.
Dissemination or sale of any part of this work (including on the World Wide Web)
will destroy the integrity of the work and is not permitted. The work and materials
from it should never be made available to students except by instructors using
ST
the accompanying text in their classes. All recipients of this work are expected to
abide by these restrictions and to honor the intended pedagogical purposes and
the needs of other instructors who rely on these materials.
U
VI
Analysis
A
with an Introduction to Proof
6th Edition
_A
by Steven R. Lay and Richard G. Ligo
PP
Chapter 1 – Logic and Proof
Solutions to Exercises
RO
Section 1.1 – Logical Connectives
1. (a) False: A statement may be false.
(b) False: A statement cannot be both true and false.
(c) True: See the comment after Practice 1.1.4.
(d) False: See the comment before Example 1.1.3.
(e) False: If the statement is false, then its negation is true.
VE
2. (a) False: p is the antecedent.
(b) True: Practice 1.1.6(a).
(c) False: See the paragraph before Practice 1.1.5.
(d) False: “p whenever q” is “if q, then p”.
(e) False: The negation of p ⇒ q is p ∧ ~ q.
D
3. Answers in Book: (a) The 3 × 3 identity matrix is not singular.
(b) The function f (x) = sin x is not bounded on R.
?
(c) The function f is not linear or the function g is not linear.
(d) Six is not prime and seven is not odd.
(e) x is in D and f (x) ≥ 5.
Copyright © 2024 by Pearson Education, Inc. or its affiliates
