SOLUTIONS MANUAL — Analysis with an Introduction to Proof, 5th Edition — Steven R. Lay — ISBN 9780137546138

Analysis with an Introduction to Proof
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– 5th Edition

SOLUTIONS
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MANUAL
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AP
Steven R. Lay
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Complete Solutions Manual for Instructors and
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Students

© Steven R. Lay
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All rights reserved. Reproduction or distribution without permission is prohibited.
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, Section 1.1 x Logical Connectives 4


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Analysis
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with an Introduction to Proof
5th Edition
AP
by Steven R. Lay



Chapter 1 – Logic and Proof
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Solutions to Exercises

Section 1.1 – Logical Connectives
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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.
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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 š ~ q.
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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.
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, Section 1.1 x 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.
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(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.
(g) f is continuous and A is open, but f – 1(A) is not open.
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5. Answers in book: (a) Antecedent: M is singular; consequent: M has a zero eigenvalue.
(b) Antecedent: linearity; consequent: continuity.
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(c) Antecedent: a sequence is Cauchy; consequent: it is bounded.
(d) Antecedent: y > 5; consequent: x < 3.

6. (a) Antecedent: it is Cauchy; consequent: a sequence is convergent.
(b) Antecedent: boundedness; consequent: convergence.
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(c) Antecedent: orthogonality; consequent: invertability.
(d) Antecedent: K is closed and bounded; consequent: K is compact.

7 and 8 are routine.
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9. Answers in book: (a) T š 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.
(f) T º F is F. (g) (T š F) º T is T. (h) (T › F) º F is F. (i) (T š F) º F is T. (j) ~ (F › T) is F.

10. (a) T š 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 š T) º F is F. (j) ~ (F š T) is T.
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11. Answers in book: (a) p š ~ q; (b) ( p › q) š ~ ( p š q); (c) ~ q º p; (d) ~ p º q; (e) p ” ~ q.

12. (a) n š ~ m; (b) ~ m š ~ n or ~ (m › n); (c) n º m; (d) m º ~ n; (e) ~ (m š n).

13. (a) and (b) are routine. (c) p š q.
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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.
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(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.
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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.
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, Section 1.2 x Quantifiers 6
(e)  x in A †  y > 2, f ( y) d 0 or f ( y) t f (x).
(f )  x † x > 3 and  H > 0, x2 ” 9 + H.

4. (a) Someone does not like Robert.
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(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 d f (x) d 7.
(f )  x in A † y in B, f (y) ” f (x).
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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 z r 5 such as x = 2.
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(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.

6. (a) True. Let x = 5. (b) False. Let x = 3. (c) True. Choose x z r 3 such as x = 2.
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(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.
(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.
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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).

9. Hints in book: The True/False part of the answers.
(a) False. For example, let x = 2 and y = 1. Then x > y.
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(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.
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(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.
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(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.
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(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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