INSTRUCTOR’S
SOLUTIONS MANUAL
A NALYSIS WITH AN
I NTRODUCTION TO PROOF
M
SIXTH EDITION
EL
AN
IE
Steven R. Lay and Richard G. Ligo
??
??
All Chapters Included
All Answers Included
, This manual is intended to accompany the 6th edition of Analysis with an Introduction to Proof
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
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
are the exercises that are used in some way in subsequent sections. There is a table on page 3 that
M
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
EL
= the set of rational numbers
R = the set of real numbers
= “for every”
= “there exists”
AN
∋ = “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
attention.
IE
Richard G. Ligo
??
??
iuytre
, All Chapters Included
All Answers Included
Table of Starred Exercises
Note: The prefix P indicates a practice problem, the prefix E indicates an example, the prefix T refers to a
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
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
M
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 5.1.21 5.2.17
EL
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
3.1.30b 3.3.13, E4.1.12, 4.3.15 5.2.13 T5.3.6, T6.1.7, 7.1.14
AN
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
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
IE
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
??
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
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
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
iuytre
, iuytrew
Section 1.1 • Logical Connectives 4
Analysis
with an Introduction to Proof
6th Edition
by Steven R. Lay and Richard G. Ligo
Chapter 1 – Logic and Proof
M
Solutions to Exercises
EL
Section 1.1 – Logical Connectives
1. (a) False: A statement may be false.
(b) False: A statement cannot be both true and false.
AN
(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.
2. (a) False: p is the antecedent.
(b) True: Practice 1.1.6(a).
IE
(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.
??
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
A NALYSIS WITH AN
I NTRODUCTION TO PROOF
M
SIXTH EDITION
EL
AN
IE
Steven R. Lay and Richard G. Ligo
??
??
All Chapters Included
All Answers Included
, This manual is intended to accompany the 6th edition of Analysis with an Introduction to Proof
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
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
are the exercises that are used in some way in subsequent sections. There is a table on page 3 that
M
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
EL
= the set of rational numbers
R = the set of real numbers
= “for every”
= “there exists”
AN
∋ = “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
attention.
IE
Richard G. Ligo
??
??
iuytre
, All Chapters Included
All Answers Included
Table of Starred Exercises
Note: The prefix P indicates a practice problem, the prefix E indicates an example, the prefix T refers to a
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
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
M
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 5.1.21 5.2.17
EL
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
3.1.30b 3.3.13, E4.1.12, 4.3.15 5.2.13 T5.3.6, T6.1.7, 7.1.14
AN
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
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
IE
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
??
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
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
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
iuytre
, iuytrew
Section 1.1 • Logical Connectives 4
Analysis
with an Introduction to Proof
6th Edition
by Steven R. Lay and Richard G. Ligo
Chapter 1 – Logic and Proof
M
Solutions to Exercises
EL
Section 1.1 – Logical Connectives
1. (a) False: A statement may be false.
(b) False: A statement cannot be both true and false.
AN
(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.
2. (a) False: p is the antecedent.
(b) True: Practice 1.1.6(a).
IE
(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.
??
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
