Analysis with an Introduction to Proof – 6th
Edition
ST
SOLUTIONS
UV
IA
MANUAL
_A
PP
Steven R. Lay
RO
Richard G. Ligo
VE
Comprehensive Solutions Manual for Instructors
D?
and Students
??
© Steven R. Lay & Richard G. Ligo
All rights reserved. Reproduction or distribution without permission is prohibited.
©STUDYSTREAM
, 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.
ST
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
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.
UV
IA
Analysis
_A
with an Introduction to Proof
6th Edition
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.
VE
(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).
D?
(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
, Section 1.1 • Logical Connectives 5
(f ) (an) is monotone and bounded, but (an) is not convergent.
(g) f is injective, and S is not finite and not denumerable.
4. (a) The function f (x) = x2 – 9 is not continuous at x = 3.
ST
(b) The relation R is not reflexive and not symmetric.
(c) Four and nine are not relatively prime.
(d) x is not in A and x is in B.
(e) x < 7 and f (x) is in C.
(f ) (an) is convergent, but (an) is not monotone or not bounded.
UV
(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.
(b) Antecedent: linearity; consequent: continuity.
(c) Antecedent: a sequence is Cauchy; consequent: it is bounded.
(d) Antecedent: y > 5; consequent: x < 3.
IA
6. (a) Antecedent: it is Cauchy; consequent: a sequence is convergent.
(b) Antecedent: boundedness; consequent: convergence.
(c) Antecedent: orthogonality; consequent: invertability.
(d) Antecedent: K is closed and bounded; consequent: K is compact.
_A
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 ⇒ T is T. (e) F ⇒ F is T.
(f) T ⇒ F is F. (g) (T ∧ F) ⇒ T is T. (h) (T ∨ F) ⇒ F is F. (i) (T ∧ F) ⇒ 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 ⇒ F is F. (e) F ⇒ F is T.
PP
(f) F ⇒ T is T.
(g) (F ∨ T) ⇒ F is F. (h) (T ⇒ F) ⇒ T is T. (i) (T ∧ T) ⇒ F is F. (j) ~ (F ∧ T) is T.
11. Answers in book: (a) p ∧ ~ q; (b) ( p ∨ q) ∧ ~ ( p ∧ q); (c) ~ q ⇒ p; (d) ~ p ⇒ q; (e) p ⇔ ~ q.
12. (a) n ∧ ~ m; (b) ~ m ∧ ~ n or ~ (m ∨ n); (c) n ⇒ m; (d) m ⇒ ~ n; (e) ~ (m ∧ n).
RO
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.
VE
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.
D?
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) ≠ 7.
Copyright © 2024 by Pearson Education, Inc. or its affiliates
, Section 1.2 • Quantifiers 6
(e) ∃ x in A ∋ ∀ y > 2, f ( y) ≤ 0 or f ( y) ≥ f (x).
(f ) ∃ x ∋ x > 3 and ∀ ε > 0, x2 ≤ 9 + ε.
4. (a) Someone does not like Robert.
ST
(b) No students work part-time.
(c) Some square matrices are triangular.
(d) ∀ x in B, f (x) ≤ k.
(e) ∃ x ∋ x > 5 and 3 ≤ f (x) ≤ 7.
(f ) ∃ x in A ∋∀ y in B, f (y) ≤ f (x).
UV
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.
(c) True. Let x = 5. (d) False. Choose x ≠ ± 5 such as x = 2.
(e) True. Let x = 1, or any other real number.
(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.
IA
6. (a) True. Let x = 5. (b) False. Let x = 3. (c) True. Choose x ≠ ± 3 such as x = 2.
(d) False. Let x = 3. (e) False. The square of a real number cannot be negative.
(f ) False. Let x = 1, or any other real number. (g) True. Let x = 1, or any other real number.
_A
(h) True. x – x = x + (– x) and a number plus its additive inverse is zero.
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.
8. The best answer is (c).
PP
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.
RO
10. (a) True. Given any x, let y = 0.
(b) False. Let x = 0. Then for all y we have xy = 0 ≠ 1.
(c) False. Let y = 0. Then for all x we have xy = 0 ≠ 1.
(d) True. Given any x, let y = 1. Then xy = x.
VE
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 ≠ 0.)
(e) True. Let x ≤ 0.
D?
(f ) True. Take z ≤ 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 ≠ 0 will work.)
(d) False. Let x = 1 and y = 0. (Any x ≠ 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 ≤ y.
Copyright © 2024 by Pearson Education, Inc. or its affiliates
Edition
ST
SOLUTIONS
UV
IA
MANUAL
_A
PP
Steven R. Lay
RO
Richard G. Ligo
VE
Comprehensive Solutions Manual for Instructors
D?
and Students
??
© Steven R. Lay & Richard G. Ligo
All rights reserved. Reproduction or distribution without permission is prohibited.
©STUDYSTREAM
, 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.
ST
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
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.
UV
IA
Analysis
_A
with an Introduction to Proof
6th Edition
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.
VE
(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).
D?
(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
, Section 1.1 • Logical Connectives 5
(f ) (an) is monotone and bounded, but (an) is not convergent.
(g) f is injective, and S is not finite and not denumerable.
4. (a) The function f (x) = x2 – 9 is not continuous at x = 3.
ST
(b) The relation R is not reflexive and not symmetric.
(c) Four and nine are not relatively prime.
(d) x is not in A and x is in B.
(e) x < 7 and f (x) is in C.
(f ) (an) is convergent, but (an) is not monotone or not bounded.
UV
(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.
(b) Antecedent: linearity; consequent: continuity.
(c) Antecedent: a sequence is Cauchy; consequent: it is bounded.
(d) Antecedent: y > 5; consequent: x < 3.
IA
6. (a) Antecedent: it is Cauchy; consequent: a sequence is convergent.
(b) Antecedent: boundedness; consequent: convergence.
(c) Antecedent: orthogonality; consequent: invertability.
(d) Antecedent: K is closed and bounded; consequent: K is compact.
_A
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 ⇒ T is T. (e) F ⇒ F is T.
(f) T ⇒ F is F. (g) (T ∧ F) ⇒ T is T. (h) (T ∨ F) ⇒ F is F. (i) (T ∧ F) ⇒ 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 ⇒ F is F. (e) F ⇒ F is T.
PP
(f) F ⇒ T is T.
(g) (F ∨ T) ⇒ F is F. (h) (T ⇒ F) ⇒ T is T. (i) (T ∧ T) ⇒ F is F. (j) ~ (F ∧ T) is T.
11. Answers in book: (a) p ∧ ~ q; (b) ( p ∨ q) ∧ ~ ( p ∧ q); (c) ~ q ⇒ p; (d) ~ p ⇒ q; (e) p ⇔ ~ q.
12. (a) n ∧ ~ m; (b) ~ m ∧ ~ n or ~ (m ∨ n); (c) n ⇒ m; (d) m ⇒ ~ n; (e) ~ (m ∧ n).
RO
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.
VE
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.
D?
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) ≠ 7.
Copyright © 2024 by Pearson Education, Inc. or its affiliates
, Section 1.2 • Quantifiers 6
(e) ∃ x in A ∋ ∀ y > 2, f ( y) ≤ 0 or f ( y) ≥ f (x).
(f ) ∃ x ∋ x > 3 and ∀ ε > 0, x2 ≤ 9 + ε.
4. (a) Someone does not like Robert.
ST
(b) No students work part-time.
(c) Some square matrices are triangular.
(d) ∀ x in B, f (x) ≤ k.
(e) ∃ x ∋ x > 5 and 3 ≤ f (x) ≤ 7.
(f ) ∃ x in A ∋∀ y in B, f (y) ≤ f (x).
UV
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.
(c) True. Let x = 5. (d) False. Choose x ≠ ± 5 such as x = 2.
(e) True. Let x = 1, or any other real number.
(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.
IA
6. (a) True. Let x = 5. (b) False. Let x = 3. (c) True. Choose x ≠ ± 3 such as x = 2.
(d) False. Let x = 3. (e) False. The square of a real number cannot be negative.
(f ) False. Let x = 1, or any other real number. (g) True. Let x = 1, or any other real number.
_A
(h) True. x – x = x + (– x) and a number plus its additive inverse is zero.
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.
8. The best answer is (c).
PP
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.
RO
10. (a) True. Given any x, let y = 0.
(b) False. Let x = 0. Then for all y we have xy = 0 ≠ 1.
(c) False. Let y = 0. Then for all x we have xy = 0 ≠ 1.
(d) True. Given any x, let y = 1. Then xy = x.
VE
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 ≠ 0.)
(e) True. Let x ≤ 0.
D?
(f ) True. Take z ≤ 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 ≠ 0 will work.)
(d) False. Let x = 1 and y = 0. (Any x ≠ 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 ≤ y.
Copyright © 2024 by Pearson Education, Inc. or its affiliates
