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INSTRUCTOR’S
IF
SOLUTIONS MANUAL
IE
D
BR
A NALYSIS WITH AN
AI
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I NTRODUCTION TO PROOF
BO
FIFTH EDITION
O
ST
ER
Steven Lay
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
VERIFIEDBRAINBOOSTER
, VE
R
IF
IE
D
BR
AI
N
BO
O
ST
ER
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages
in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2014, 2005, 2001 Pearson Education, Inc.
Publishing as Pearson, 75 Arlington Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-74748-8
ISBN-10: 0-321-74748-8
www.pearsonhighered.com
VERIFIEDBRAINBOOSTER
, VE
Instructor’s Solutions Manual
R
to accompany
IF
IE
D
BR
ANALYSIS
AI
with an Introduction
N
to Proof
BO
5th Edition
O
ST
ER
Steven R. Lay
Lee University
&RS\ULJKW3HDUVRQ(GXFDWLRQ,QF
Upper Saddle River, New Jersey 07458
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
, VE
2
R
IF
IE
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
D
exercises that have hints (or answers) in the back of the book are numbered in bold print, and the hints
BR
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
AI
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
N
are the exercises that are used in some way in subsequent sections. There is a table on page 3 that
BO
indicates where starred exercises are used later. The following notations are used throughout this
manual:
= the set of natural numbers {1, 2, 3, 4, …}
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= the set of rational numbers
ST
= the set of real numbers
= “for every”
= “there exists”
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= “such that”
I have tried to be accurate in the preparation of this manual. Undoubtedly, however, some
mistakes will inadvertently slip by. I would appreciate having any errors in this manual or the text
brought to my attention.
Steven R. Lay
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
, VE
3
R
Table of Starred Exercises
IF
IE
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.
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Starred Starred
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Exercise Later Use Exercise Later use
2.1.26 T3.4.11 4.3.14 4.4.5
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2.2.10 2.4.26 4.4.10 8.2.14
2.3.32 2.5.3 4.4.16 8.3.9
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3.1.3 E7.1.7 4.4.17 T8.3.3
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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
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3.1.8 P8.1.3 5.1.19 5.2.17
3.1.24 4.1.7f, E5.3.7 5.2.10 T7.2.8
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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
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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
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
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
, VE
Section 1.1 x Logical Connectives 4
R
This work is protected by United States copyright laws and is provided solely for
IF
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)
IE
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
the accompanying text in their classes. All recipients of this work are expected to
D
abide by these restrictions and to honor the intended pedagogical purposes and
the needs of other instructors who rely on these materials.
BR
AI
N
BO
Analysis
O
ST
with an Introduction to Proof
5th Edition
ER
by Steven R. Lay
Chapter 1 – Logic and Proof
Solutions to Exercises
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.
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 º 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 .
(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) t 5.
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
, VE
Section 1.1 x Logical Connectives 5
(f ) (an) is monotone and bounded, but (an) is not convergent.
R
(g) f is injective, and S is not finite and not denumerable.
IF
4. (a) The function f (x) = x2 – 9 is not continuous at x = 3.
(b) The relation R is not reflexive and not symmetric.
IE
(c) Four and nine are not relatively prime.
(d) x is not in A and x is in B.
D
(e) x < 7 and f (x) is in C.
(f ) (an) is convergent, but (an) is not monotone or not bounded.
BR
(g) f is continuous and A is open, but f – 1(A) is not open.
5. Answers in book: (a) Antecedent: M is singular; consequent: M has a zero eigenvalue.
AI
(b) Antecedent: linearity; consequent: continuity.
(c) Antecedent: a sequence is Cauchy; consequent: it is bounded.
N
(d) Antecedent: y > 5; consequent: x < 3.
BO
6. (a) Antecedent: it is Cauchy; consequent: a sequence is convergent.
(b) Antecedent: boundedness; consequent: convergence.
(c) Antecedent: orthogonality; consequent: invertability.
O
(d) Antecedent: K is closed and bounded; consequent: K is compact.
ST
7 and 8 are routine.
9. Answers in book: (a) T T is T. (b) F T is T. (c) F F is F. (d) T º T is T. (e) F º F is T.
ER
(f) T º F is F. (g) (T F) º T is T. (h) (T F) º F is F. (i) (T F) º F is T. (j) ~ (F T) is F.
10. (a) T F is F. (b) F F is F. (c) F T is T. (d) T º F is F. (e) F º F is T. (f) F º T is T.
(g) (F T) º F is F. (h) (T º F) º T is T. (i) (T T) º F is F. (j) ~ (F T) is T.
11. Answers in book: (a) p ~ q; (b) ( p q) ~ ( p q); (c) ~ q º p; (d) ~ p º q; (e) p ~ q.
12. (a) n ~ m; (b) ~ m ~ n or ~ (m n); (c) n º m; (d) m º ~ n; (e) ~ (m n).
13. (a) and (b) are routine. (c) p q.
14. These truth tables are all straightforward. Note that the tables for (c) through (f ) have 8 rows because there are 3
letters and therefore 23 = 8 possible combinations of T and F.
Section 1.2 - Quantifiers
1. (a) True: See the comment before Example 1.2.1.
(b) False: The negation of a universal statement is an existential statement.
(c) True: See the comment before Example 1.2.1.
2. (a) False: It means there exists at least one.
(b) True: Example 1.2.1.
(c) True: See the comment after Practice 1.2.4.
3. (a) No pencils are red.
(b) Some chair does not have four legs.
(c) Someone on the basketball team is over 6 feet 4 inches tall.
(d) x > 2, f (x) z 7.
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
, VE
Section 1.2 x Quantifiers 6
(e) x in A y > 2, f ( y) d 0 or f ( y) t f (x).
R
(f ) x x > 3 and H > 0, x2 9 + H.
IF
4. (a) Someone does not like Robert.
(b) No students work part-time.
IE
(c) Some square matrices are triangular.
(d) x in B, f (x) k.
D
(e) x x > 5 and 3 d f (x) d 7.
(f ) x in A y in B, f (y) f (x).
BR
5. Hints in book: The True/False part of the answers.
(a) True. Let x = 3. (b) True. 4 is less than 7 and anything smaller than 4 will also be less than 7.
AI
(c) True. Let x = 5. (d) False. Choose x z r 5 such as x = 2.
(e) True. Let x = 1, or any other real number.
N
(f ) True. The square of a real number cannot be negative.
(g) True. Let x = 1, or any real number other than 0. (h) False. Let x = 0.
BO
6. (a) True. Let x = 5. (b) False. Let x = 3. (c) True. Choose x z r 3 such as x = 2.
(d) False. Let x = 3. (e) False. The square of a real number cannot be negative.
O
(f ) False. Let x = 1, or any other real number. (g) True. Let x = 1, or any other real number.
(h) True. x – x = x + (– x) and a number plus its additive inverse is zero.
ST
7. Answers in book: (a) You can use (ii) to prove (a) is true. (b) You can use (i) to prove (b) is true.
Additional answers: (c) You can use (ii) to prove (c) is false. (d) You can use (i) to prove (d) is false.
ER
8. The best answer is (c).
9. Hints in book: The True/False part of the answers.
(a) False. For example, let x = 2 and y = 1. Then x > y.
(b) True. For example, let x = 2 and y = 3. Then x y.
(c) True. Given any x, let y = x + 1. Then x y.
(d) False. Given any y, let x = y + 1. Then x > y.
10. (a) True. Given any x, let y = 0.
(b) False. Let x = 0. Then for all y we have xy = 0 z 1.
(c) False. Let y = 0. Then for all x we have xy = 0 z 1.
(d) True. Given any x, let y = 1. Then xy = x.
11. Hints in book: The True/False part of the answers.
(a) True. Let x = 0. Then given any y, let z = y. (A similar argument works for any x.)
(b) False. Given any x and any y, let z = x + y + 1.
(c) True. Let z = y – x.
(d) False. Let x = 0 and y = 1. (It is a true statement for x z 0.)
(e) True. Let x d 0.
(f ) True. Take z d y. This makes “z ! y ” false so that the implication is true. Or, choose z ! x + y.
12. (a) True. Given x and y, let z = x + y.
(b) False. Let x = 0. Then given any y, let z = y + 1.
(c) True. Let x = 1. Then given any y, let z = y. (Any x z 0 will work.)
(d) False. Let x = 1 and y = 0. (Any x z 0 will work.)
(e) False. Let x = 2. Given any y, let z = y + 1. Then “z ! y ” is true, but “z ! x + y ” is false.
(f ) True. Given any x and y, either choose z ! x + y or z d y.
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
R
INSTRUCTOR’S
IF
SOLUTIONS MANUAL
IE
D
BR
A NALYSIS WITH AN
AI
N
I NTRODUCTION TO PROOF
BO
FIFTH EDITION
O
ST
ER
Steven Lay
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
VERIFIEDBRAINBOOSTER
, VE
R
IF
IE
D
BR
AI
N
BO
O
ST
ER
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages
in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2014, 2005, 2001 Pearson Education, Inc.
Publishing as Pearson, 75 Arlington Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-74748-8
ISBN-10: 0-321-74748-8
www.pearsonhighered.com
VERIFIEDBRAINBOOSTER
, VE
Instructor’s Solutions Manual
R
to accompany
IF
IE
D
BR
ANALYSIS
AI
with an Introduction
N
to Proof
BO
5th Edition
O
ST
ER
Steven R. Lay
Lee University
&RS\ULJKW3HDUVRQ(GXFDWLRQ,QF
Upper Saddle River, New Jersey 07458
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
, VE
2
R
IF
IE
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
D
exercises that have hints (or answers) in the back of the book are numbered in bold print, and the hints
BR
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
AI
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
N
are the exercises that are used in some way in subsequent sections. There is a table on page 3 that
BO
indicates where starred exercises are used later. The following notations are used throughout this
manual:
= the set of natural numbers {1, 2, 3, 4, …}
O
= the set of rational numbers
ST
= the set of real numbers
= “for every”
= “there exists”
ER
= “such that”
I have tried to be accurate in the preparation of this manual. Undoubtedly, however, some
mistakes will inadvertently slip by. I would appreciate having any errors in this manual or the text
brought to my attention.
Steven R. Lay
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
, VE
3
R
Table of Starred Exercises
IF
IE
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.
D
Starred Starred
BR
Exercise Later Use Exercise Later use
2.1.26 T3.4.11 4.3.14 4.4.5
AI
2.2.10 2.4.26 4.4.10 8.2.14
2.3.32 2.5.3 4.4.16 8.3.9
N
3.1.3 E7.1.7 4.4.17 T8.3.3
BO
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
O
3.1.8 P8.1.3 5.1.19 5.2.17
3.1.24 4.1.7f, E5.3.7 5.2.10 T7.2.8
ST
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
ER
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
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
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
, VE
Section 1.1 x Logical Connectives 4
R
This work is protected by United States copyright laws and is provided solely for
IF
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)
IE
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
the accompanying text in their classes. All recipients of this work are expected to
D
abide by these restrictions and to honor the intended pedagogical purposes and
the needs of other instructors who rely on these materials.
BR
AI
N
BO
Analysis
O
ST
with an Introduction to Proof
5th Edition
ER
by Steven R. Lay
Chapter 1 – Logic and Proof
Solutions to Exercises
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.
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 º 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 .
(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) t 5.
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
, VE
Section 1.1 x Logical Connectives 5
(f ) (an) is monotone and bounded, but (an) is not convergent.
R
(g) f is injective, and S is not finite and not denumerable.
IF
4. (a) The function f (x) = x2 – 9 is not continuous at x = 3.
(b) The relation R is not reflexive and not symmetric.
IE
(c) Four and nine are not relatively prime.
(d) x is not in A and x is in B.
D
(e) x < 7 and f (x) is in C.
(f ) (an) is convergent, but (an) is not monotone or not bounded.
BR
(g) f is continuous and A is open, but f – 1(A) is not open.
5. Answers in book: (a) Antecedent: M is singular; consequent: M has a zero eigenvalue.
AI
(b) Antecedent: linearity; consequent: continuity.
(c) Antecedent: a sequence is Cauchy; consequent: it is bounded.
N
(d) Antecedent: y > 5; consequent: x < 3.
BO
6. (a) Antecedent: it is Cauchy; consequent: a sequence is convergent.
(b) Antecedent: boundedness; consequent: convergence.
(c) Antecedent: orthogonality; consequent: invertability.
O
(d) Antecedent: K is closed and bounded; consequent: K is compact.
ST
7 and 8 are routine.
9. Answers in book: (a) T T is T. (b) F T is T. (c) F F is F. (d) T º T is T. (e) F º F is T.
ER
(f) T º F is F. (g) (T F) º T is T. (h) (T F) º F is F. (i) (T F) º F is T. (j) ~ (F T) is F.
10. (a) T F is F. (b) F F is F. (c) F T is T. (d) T º F is F. (e) F º F is T. (f) F º T is T.
(g) (F T) º F is F. (h) (T º F) º T is T. (i) (T T) º F is F. (j) ~ (F T) is T.
11. Answers in book: (a) p ~ q; (b) ( p q) ~ ( p q); (c) ~ q º p; (d) ~ p º q; (e) p ~ q.
12. (a) n ~ m; (b) ~ m ~ n or ~ (m n); (c) n º m; (d) m º ~ n; (e) ~ (m n).
13. (a) and (b) are routine. (c) p q.
14. These truth tables are all straightforward. Note that the tables for (c) through (f ) have 8 rows because there are 3
letters and therefore 23 = 8 possible combinations of T and F.
Section 1.2 - Quantifiers
1. (a) True: See the comment before Example 1.2.1.
(b) False: The negation of a universal statement is an existential statement.
(c) True: See the comment before Example 1.2.1.
2. (a) False: It means there exists at least one.
(b) True: Example 1.2.1.
(c) True: See the comment after Practice 1.2.4.
3. (a) No pencils are red.
(b) Some chair does not have four legs.
(c) Someone on the basketball team is over 6 feet 4 inches tall.
(d) x > 2, f (x) z 7.
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
, VE
Section 1.2 x Quantifiers 6
(e) x in A y > 2, f ( y) d 0 or f ( y) t f (x).
R
(f ) x x > 3 and H > 0, x2 9 + H.
IF
4. (a) Someone does not like Robert.
(b) No students work part-time.
IE
(c) Some square matrices are triangular.
(d) x in B, f (x) k.
D
(e) x x > 5 and 3 d f (x) d 7.
(f ) x in A y in B, f (y) f (x).
BR
5. Hints in book: The True/False part of the answers.
(a) True. Let x = 3. (b) True. 4 is less than 7 and anything smaller than 4 will also be less than 7.
AI
(c) True. Let x = 5. (d) False. Choose x z r 5 such as x = 2.
(e) True. Let x = 1, or any other real number.
N
(f ) True. The square of a real number cannot be negative.
(g) True. Let x = 1, or any real number other than 0. (h) False. Let x = 0.
BO
6. (a) True. Let x = 5. (b) False. Let x = 3. (c) True. Choose x z r 3 such as x = 2.
(d) False. Let x = 3. (e) False. The square of a real number cannot be negative.
O
(f ) False. Let x = 1, or any other real number. (g) True. Let x = 1, or any other real number.
(h) True. x – x = x + (– x) and a number plus its additive inverse is zero.
ST
7. Answers in book: (a) You can use (ii) to prove (a) is true. (b) You can use (i) to prove (b) is true.
Additional answers: (c) You can use (ii) to prove (c) is false. (d) You can use (i) to prove (d) is false.
ER
8. The best answer is (c).
9. Hints in book: The True/False part of the answers.
(a) False. For example, let x = 2 and y = 1. Then x > y.
(b) True. For example, let x = 2 and y = 3. Then x y.
(c) True. Given any x, let y = x + 1. Then x y.
(d) False. Given any y, let x = y + 1. Then x > y.
10. (a) True. Given any x, let y = 0.
(b) False. Let x = 0. Then for all y we have xy = 0 z 1.
(c) False. Let y = 0. Then for all x we have xy = 0 z 1.
(d) True. Given any x, let y = 1. Then xy = x.
11. Hints in book: The True/False part of the answers.
(a) True. Let x = 0. Then given any y, let z = y. (A similar argument works for any x.)
(b) False. Given any x and any y, let z = x + y + 1.
(c) True. Let z = y – x.
(d) False. Let x = 0 and y = 1. (It is a true statement for x z 0.)
(e) True. Let x d 0.
(f ) True. Take z d y. This makes “z ! y ” false so that the implication is true. Or, choose z ! x + y.
12. (a) True. Given x and y, let z = x + y.
(b) False. Let x = 0. Then given any y, let z = y + 1.
(c) True. Let x = 1. Then given any y, let z = y. (Any x z 0 will work.)
(d) False. Let x = 1 and y = 0. (Any x z 0 will work.)
(e) False. Let x = 2. Given any y, let z = y + 1. Then “z ! y ” is true, but “z ! x + y ” is false.
(f ) True. Given any x and y, either choose z ! x + y or z d y.
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