INSTRUCTOR’S
SOLUTIONS MANUAL
ANALYSIS WITH AN
INTRODUCTION TO PROOF
M
FIFTH EDITION
EL
All Chapters Included
AN
All Answers Included
IE
Steven Lay
??
??
, qwertyui
This manual is intended to accompany the 5th edition of Analysis with an Introduction to Proof
by Steven R. Lay (Pearson, 2013). 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. I have not tried
to be exhaustive in discussing each exercise, but rather to be suggestive.
Let me 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
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, …}
M
= the set of rational numbers
R = the set of real numbers
= “for every”
EL
= “there exists”
e = “such that”
I have tried to be accurate in the preparation of this manual. Undoubtedly, however, some
AN
mistakes will inadvertently slip by. I would appreciate having any errors in this manual or the text
brought to my attention.
Steven R. Lay
IE
??
??
All Chapters Included
All Answers Included
wertyui
, qwertyui
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 8.2.14
2.3.32 2.5.3 4.4.16 8.3.9
3.1.3 E7.1.7 4.4.17 T8.3.3
M
3.1.4 7.1.7 5.1.14 6.2.8
3.1.6 E8.1.1 5.1.16 T6.2.9
3.1.7 4.3.10, 4.3.15, E8.1.7, T9.2.9 5.1.18 5.2.14, 5.3.15
3.1.8 P8.1.3 5.1.19 5.2.17
EL
3.1.24 4.1.7f, E5.3.7 5.2.10 T7.2.8
3.1.27 3.3.14 5.2.11 7.2.9b
3.1.30b 3.3.11, E4.1.11, 4.3.14 5.2.13 T5.3.5, T6.1.7, 7.1.13
AN
3.2.6a 4.1.9a, T4.2.1, 6.2.23, 7.2.16, T9.2.9 5.2.16 9.2.15
3.2.6b T6.3.8 5.3.13b T6.2.8, T6.2.10
3.2.6c T4.1.14 6.1.6 6. 2.14, 6.2.19
3.2.7 T8.2.5 6.1.8 7.3.13
3.3.7 T7.2.4, 7.2.3 6.1.17b 6.4.9
IE
3.3.12 7.1.14, T7.2.4 6.2.8 T7.2.1
3.4.15 3.5.12, T4.3.12 6.3.13d 9.3.16
3.4.21 3.5.7 7.1.12 P7.2.5
??
3.5.8 9.2.15 7.1.13 7.2.5
3.6.12 5.5.9 7.1.16 7.2.17
4.1.6b E4.2.2 7.2.9a P7.3.7
??
4.1.7f T4.2.7, 4.3.10, E8.1.7 7.2.11 T8.2.13
4.1.9a 5.2.10, 9.2.17 7.2.15 7.3.20
4.1.11 E4.3.4 7.2.20 E7.3.9
4.1.12 5.1.15 8.1.7 E8.2.6
4.1.13 5.1.13 8.1.8 8.2.13
4.1.15b 4.4.11, 4.4.18, 5.3.12 8.1.13a 9.3.8
4.1.16 5.1.15 8.2.12 9.2.7, 9.2.8
4.2.17 E6.4.3 8.2.14 T8.3.4
4.2.18 5.1.14, T9.1.10 9.1.15a 9.2.9
All Chapters Included
All Answers Included
wertyui
, Section 1.1 • Logical Connectives 4
Analysis
with an Introduction to Proof
5th Edition
by Steven R. Lay
Chapter 1 – Logic and Proof
Solutions to Exercises
M
Section 1.1 – Logical Connectives
1. (a) False: A statement may be false.
(b) False: A statement cannot be both true and false.
EL
(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.
AN
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.
IE
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.
??
SOLUTIONS MANUAL
ANALYSIS WITH AN
INTRODUCTION TO PROOF
M
FIFTH EDITION
EL
All Chapters Included
AN
All Answers Included
IE
Steven Lay
??
??
, qwertyui
This manual is intended to accompany the 5th edition of Analysis with an Introduction to Proof
by Steven R. Lay (Pearson, 2013). 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. I have not tried
to be exhaustive in discussing each exercise, but rather to be suggestive.
Let me 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
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, …}
M
= the set of rational numbers
R = the set of real numbers
= “for every”
EL
= “there exists”
e = “such that”
I have tried to be accurate in the preparation of this manual. Undoubtedly, however, some
AN
mistakes will inadvertently slip by. I would appreciate having any errors in this manual or the text
brought to my attention.
Steven R. Lay
IE
??
??
All Chapters Included
All Answers Included
wertyui
, qwertyui
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 8.2.14
2.3.32 2.5.3 4.4.16 8.3.9
3.1.3 E7.1.7 4.4.17 T8.3.3
M
3.1.4 7.1.7 5.1.14 6.2.8
3.1.6 E8.1.1 5.1.16 T6.2.9
3.1.7 4.3.10, 4.3.15, E8.1.7, T9.2.9 5.1.18 5.2.14, 5.3.15
3.1.8 P8.1.3 5.1.19 5.2.17
EL
3.1.24 4.1.7f, E5.3.7 5.2.10 T7.2.8
3.1.27 3.3.14 5.2.11 7.2.9b
3.1.30b 3.3.11, E4.1.11, 4.3.14 5.2.13 T5.3.5, T6.1.7, 7.1.13
AN
3.2.6a 4.1.9a, T4.2.1, 6.2.23, 7.2.16, T9.2.9 5.2.16 9.2.15
3.2.6b T6.3.8 5.3.13b T6.2.8, T6.2.10
3.2.6c T4.1.14 6.1.6 6. 2.14, 6.2.19
3.2.7 T8.2.5 6.1.8 7.3.13
3.3.7 T7.2.4, 7.2.3 6.1.17b 6.4.9
IE
3.3.12 7.1.14, T7.2.4 6.2.8 T7.2.1
3.4.15 3.5.12, T4.3.12 6.3.13d 9.3.16
3.4.21 3.5.7 7.1.12 P7.2.5
??
3.5.8 9.2.15 7.1.13 7.2.5
3.6.12 5.5.9 7.1.16 7.2.17
4.1.6b E4.2.2 7.2.9a P7.3.7
??
4.1.7f T4.2.7, 4.3.10, E8.1.7 7.2.11 T8.2.13
4.1.9a 5.2.10, 9.2.17 7.2.15 7.3.20
4.1.11 E4.3.4 7.2.20 E7.3.9
4.1.12 5.1.15 8.1.7 E8.2.6
4.1.13 5.1.13 8.1.8 8.2.13
4.1.15b 4.4.11, 4.4.18, 5.3.12 8.1.13a 9.3.8
4.1.16 5.1.15 8.2.12 9.2.7, 9.2.8
4.2.17 E6.4.3 8.2.14 T8.3.4
4.2.18 5.1.14, T9.1.10 9.1.15a 9.2.9
All Chapters Included
All Answers Included
wertyui
, Section 1.1 • Logical Connectives 4
Analysis
with an Introduction to Proof
5th Edition
by Steven R. Lay
Chapter 1 – Logic and Proof
Solutions to Exercises
M
Section 1.1 – Logical Connectives
1. (a) False: A statement may be false.
(b) False: A statement cannot be both true and false.
EL
(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.
AN
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.
IE
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.
??
