Solution Manual for A Survey of Mathematics with Applications, 11th Edition

INSTRUCTOR’S SOLUTIONS MANUAL JAMES LAPP A SURVEY OF MATHEMATICS WITH A PPLICATIONS ELEVENTH EDITION Allen R. Angel Monroe Community College Christine D. Abbott Monroe Community College Dennis C. Runde State College of Florida 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 © 2021, 2017, 2013 by Pearson Education, Inc. 221 River Street, Hoboken, NJ 07030. All rights reserved. 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: 978-0-13-574072-9 ISBN-10: 0-13-574072-X Table of Contents Chapter 1: Critical Thinking Skills 1.1: Inductive and Deductive Reasoning ……………………………………………………………………………………..1 1.2: Estimation Techniques ……………………………………………………………………………………………………….3 1.3: Problem-Solving Procedures ……………………………………………………………………………………………….5 Review Exercises …………………………………………………………………………………………………………….11 Chapter Test ……………………………………………………………………………………………………………………14 Chapter 2: Sets 2.1: 2.2: 2.3: 2.4: 2.5: 2.6: Set Concepts ……………………………………………………………………………………………………………………17 Subsets……………………………………………………………………………………………………………………………19 Venn Diagrams and Set Operations ……………………………………………………………………………………21 Venn Diagrams with Three Sets and Verification of Equality of Sets ……………………………………..26 Applications of Sets …………………………………………………………………………………………………………32 Infinite Sets …………………………………………………………………………………………………………………….35 Review Exercises …………………………………………………………………………………………………………….37 Chapter Test ……………………………………………………………………………………………………………………40 Chapter 3: Logic 3.1: 3.2: 3.3: 3.4: 3.5: 3.6: 3.7: Statements and Logical Connectives …………………………………………………………………………………..43 Truth Tables for Negation, Conjunction, and Disjunction ……………………………………………………..45 Truth Tables for the Conditional and Biconditional ………………………………………………………………54 Equivalent Statements ………………………………………………………………………………………………………65 Symbolic Arguments ………………………………………………………………………………………………………..73 Euler Diagrams and Syllogistic Arguments …………………………………………………………………………80 Switching Circuits ……………………………………………………………………………………………………………83 Review Exercises …………………………………………………………………………………………………………….87 Chapter Test ……………………………………………………………………………………………………………………93 Chapter 4: Systems of Numeration 4.1: 4.2: 4.3: 4.4: 4.5: Additive, Multiplicative, and Ciphered Systems of Numeration……………………………………………..97 Place-Value or Positional-Value Numeration Systems ………………………………………………………….99 Other Bases …………………………………………………………………………………………………………………..101 Perform Computations in Other Bases ………………………………………………………………………………106 Early Computational Methods………………………………………………………………………………………….111 Review Exercises …………………………………………………………………………………………………………..115 Chapter Test ………………………………………………………………………………………………………………….119 Chapter 5: Number Theory and the Real Number System 5.1: 5.2: 5.3: 5.4: 5.5: 5.6: 5.7: 5.8: Number Theory ……………………………………………………………………………………………………………..121 The Integers …………………………………………………………………………………………………………………..125 The Rational Numbers…………………………………………………………………………………………………….128 The Irrational Numbers …………………………………………………………………………………………………..133 Real Numbers and Their Properties ………………………………………………………………………………….136 Rules of Exponents and Scientific Notation ……………………………………………………………………….138 Arithmetic and Geometric Sequences ……………………………………………………………………………….141 Fibonacci Sequence ………………………………………………………………………………………………………..146 Review Exercises …………………………………………………………………………………………………………..148 Chapter Test ………………………………………………………………………………………………………………….152 Chapter 6: Algebra, Graphs, and Functions 6.1: Order of Operations and Solving Equations ……………………………………………………………………….153 6.2: Formulas ……………………………………………………………………………………………………………………….157 6.3: Applications of Algebra ………………………………………………………………………………………………….162 6.4: Variation ……………………………………………………………………………………………………………………….167 6.5: Solving Linear Inequalities ……………………………………………………………………………………………..171 6.6: Graphing Linear Equations ……………………………………………………………………………………………..176 6.7: Solving Systems of Linear Equations………………………………………………………………………………..184 6.8: Linear Inequalities in Two Variables and Systems of Linear Inequalities ………………………………195 6.9: Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula …………..204 6.10: Functions and Their Graphs …………………………………………………………………………………………….210 Review Exercises …………………………………………………………………………………………………………..217 Chapter Test ………………………………………………………………………………………………………………….227 Chapter 7: The Metric System 7.1: 7.2: 7.3: 7.4: Basic Terms and Conversions Within the Metric System …………………………………………………….231 Length, Area, and Volume ………………………………………………………………………………………………232 Mass and Temperature ……………………………………………………………………………………………………235 Dimensional Analysis and Conversions to and from the Metric System ………………………………..237 Review Exercises …………………………………………………………………………………………………………..241 Chapter Test ………………………………………………………………………………………………………………….243 Chapter 8: Geometry 8.1: 8.2: 8.3: 8.4: 8.5: 8.6: 8.7: Points, Lines, Planes, and Angles……………………………………………………………………………………..245 Polygons ……………………………………………………………………………………………………………………….248 Perimeter and Area …………………………………………………………………………………………………………252 Volume and Surface Area ……………………………………………………………………………………………….256 Transformational Geometry, Symmetry, and Tessellations ………………………………………………….260 Topology ………………………………………………………………………………………………………………………265 Non-Euclidean Geometry and Fractal Geometry ………………………………………………………………..266 Review Exercises …………………………………………………………………………………………………………..267 Chapter Test ………………………………………………………………………………………………………………….270 Chapter 9: Mathematical Systems 9.1: 9.2: 9.3: 9.4: Groups ………………………………………………………………………………………………………………………….273 Finite Mathematical Systems …………………………………………………………………………………………..274 Modular Arithmetic ………………………………………………………………………………………………………..277 Matrices ………………………………………………………………………………………………………………………..282 Review Exercises …………………………………………………………………………………………………………..288 Chapter Test ………………………………………………………………………………………………………………….290 Chapter 10: Consumer Mathematics 10.1: 10.2: 10.3: 10.4: 10.5: 10.6: Percent ………………………………………………………………………………………………………………………….293 Personal Loans and Simple Interest ………………………………………………………………………………….295 Compound Interest …………………………………………………………………………………………………………300 Installment Buying …………………………………………………………………………………………………………304 Buying a House with a Mortgage ……………………………………………………………………………………..310 Ordinary Annuities, Sinking Funds, and Retirement Investments …………………………………………317 Review Exercises …………………………………………………………………………………………………………..318 Chapter Test ………………………………………………………………………………………………………………….322 Chapter 11: Probability 11.1: Empirical and Theoretical Probabilities …………………………………………………………………………….325 11.2: Odds …………………………………………………………………………………………………………………………….328 11.3: Expected Value (Expectation) ………………………………………………………………………………………….332 11.4: Tree Diagrams ……………………………………………………………………………………………………………….336 11.5: OR and AND Problems ……………………………………………………………………………………………………340 11.6: Conditional Probability …………………………………………………………………………………………………..344 11.7: The Fundamental Counting Principle and Permutations ………………………………………………………347 11.8: Combinations ………………………………………………………………………………………………………………..349 11.9: Solving Probability Problems by Using Combinations ………………………………………………………..351 11.10: Binomial Probability Formula ………………………………………………………………………………………….353 Review Exercises …………………………………………………………………………………………………………..355 Chapter Test ………………………………………………………………………………………………………………….359 Chapter 12: Statistics 12.1: 12.2: 12.3: 12.4: 12.5: 12.6: Sampling Techniques and Misuses of Statistics ………………………………………………………………….361 Frequency Distributions and Statistical Graphs ………………………………………………………………….363 Measures of Central Tendency and Position ………………………………………………………………………368 Measures of Dispersion …………………………………………………………………………………………………..374 The Normal Curve ………………………………………………………………………………………………………….379 Linear Correlation and Regression ……………………………………………………………………………………385 Review Exercises …………………………………………………………………………………………………………..395 Chapter Test ………………………………………………………………………………………………………………….400 Chapter 13: Graph Theory 13.1: 13.2: 13.3: 13.4: Graphs, Paths, and Circuits ……………………………………………………………………………………………..403 Euler Paths and Euler Circuits………………………………………………………………………………………….404 Hamilton Paths and Hamilton Circuits ………………………………………………………………………………408 Trees …………………………………………………………………………………………………………………………….411 Review Exercises …………………………………………………………………………………………………………..415 Chapter Test ………………………………………………………………………………………………………………….417 Chapter 14: Voting and Apportionment 14.1: 14.2: 14.3: 14.4: Voting Methods ……………………………………………………………………………………………………………..419 Flaws of the Voting Methods …………………………………………………………………………………………..423 Apportionment Methods………………………………………………………………………………………………….428 Flaws of the Apportionment Methods ……………………………………………………………………………….433 Review Exercises …………………………………………………………………………………………………………..438 Chapter Test ………………………………………………………………………………………………………………….445 Section 1.1: Inductive and Deductive Reasoning 1 Chapter One: Critical Thinking Skills Section 1.1: Inductive and Deductive Reasoning 1. Natural 2. Divisible 13. 14. 3. Counterexample 15. 4. Hypothesis 5. Inductive 16. 6. Deductive 17. 9, 11, 13 (Add 2 to previous number.) 7. Deductive 18. 16, 19, 22 (Add 3 to the previous number.) 8. Inductive 19. 5, –5, 5 (Alternate 5 and –5.) 9. 5  3  15 20. 16, –32, 64 (Multiply previous number by –2.) 10. 19  10  190 21. 1 1 1 , , (Increase denominator value by 1.) 5 6 7 22. 1 1 1 (Increase denominator value by 2.) , , 9 11 13 11. 1 5 (1 4) 10 (4  6) 10 (4  6) 5 (1 4) 1 12. 100, 000  105 23. 21, 28, 36 (15  6  21, 21  7  28, 28  8  36) 24. 16, 22, 29 (Add 1, then 2, then 3, ….) 25. 34, 55, 89 (Each number in the sequence is the sum of the previous two numbers.) 26. 76, 123, 199 (Each number in the sequence is the sum of the previous two numbers.) 27. There are three letters in the pattern. 39  3  117, so, the 117th entry is the second R in the pattern. Therefore, the 118th entry is Y. 28. a) Answers will vary. b) The sum of the digits is 9. c) The sum of the digits in the product when a one- or two-digit number is multiplied by 9 is divisible by 9. 29. a) 36, 49, 64 b) Square the numbers 6, 7, 8, 9 and 10. c) 8  8  64, 9  9  81; 72 is not a square number since it falls between the two square numbers 64 and 81. 30. a) 28 and 36 b) To find the 7th triangular number, add 7 to the 6th triangular number. To find the 8th triangular number, add 8 to the 7th triangular number. To find the 9th triangular number, add 9 to the 8th triangular number. To find the 10th triangular number, add 10 to the 9th triangular number. To find the 11th triangular number, add 11 to the 10th triangular number. c) 36  9  45, 45  10  55, 55  11  66; 66  12  78; 72 is not a triangular number since it falls between the consecutive triangular numbers 66 and 78. 31. Blue: 1, 5, 7, 10, 12; Purple: 2, 4, 6, 9, 11; Yellow: 3, 8 32. a) 19 (Each new row has two additional triangles.) b) 1  3  5  7  9  11  13  15  17  19  100 33. a) $3700 b) We are using observation of specific cases to make a prediction. Copyright © 2021 Pearson Education, Inc. 2 Chapter 1: Critical Thinking Skills 34. a) $410 b) We are using observation of specific cases to make a prediction. 36. 35. 37. a) You should obtain the original number. b) You should obtain the original number. c) Conjecture: The result is always the original number. d) n  3n  3n + 6  3n + 6 3n 6 = + = n + 2  n + 2-2 = n 3 3 3 38. a) You should obtain twice the original number. b) You should obtain twice the original number. c) Conjecture: The result is always twice the original number. d) n  4n  4n  6  4n  6 4n 6    2n  3  2n  3  3  2n 2 2 2 39. a) You should obtain the number 5. b) You should obtain the number 5. c) Conjecture: The result is always the number 5. d) n  n  1  n  (n  1)  2n  1  2n  1  9  2n  10  2n  10 2n 10    n5  n5n  5 2 2 2 40. a) You should obtain the number 0. b) You should obtain the number 0. c) Conjecture: The result is always the number 0. d) n  n  10  n  10  n  10   5  n  10  n  10  10  n  n  n  0  5  5 41. 3  5  15 is one counterexample. 42. 10  11  12 = 33, which is not a three digit number. 43. Two is a counting number. The sum of 2 and 3 is 5. Five divided by two is 5 2, which is not an even number. 44. 900 is a three-digit number. The product of 900 and 900 is 810,000, which is not a five-digit number. 45. One and two are counting numbers. The difference of 1 and 2 is 1  2  1 , which is not a counting number. 46. The sum of the odd numbers 1 and 5 is 6, which is not divisible by 4. 47. a) The sum of the measures of the interior angles should be 180. b) Yes, the sum of the measures of the interior angles should be 180. c) Conjecture: The sum of the measures of the interior angles of a triangle is 180. 48. a) The sum of the measures of the interior angles should be 360. b) Yes, the sum of the measures of the interior angles should be 360. c) Conjecture: The sum of the measures of the interior angles of a quadrilateral is 360. 49. Inductive reasoning: a general conclusion is obtained from observation of specific cases. Copyright © 2021 Pearson Education, Inc. Section 1.2: Estimation Techniques 3 50. Inductive reasoning: a general conclusion is obtained from observation of specific cases. 51. 129; The numbers in the positions of each inner 4  4 square are determined as follows. a b c abc 52. 1881, 8008, 8118 (They look the same when looked at in a mirror.) 53. c Section 1.2: Estimation Techniques Answers in this section will vary depending on how you round your numbers. The answers may differ from the answers in the back of the textbook. However, your answers should be something near the answers given. All answers are approximate.) 1. Estimation 10. 400.15  297.87  400  300  100 2. Equal 11. 22%  9116  20%  9000  0.20  9000  1800 3. 26.9  67.3  219  143.3  30  67  220  143  460 12. 11% of 8221  10% of 8000  0.10  8000  800 4. 86  47.2  289.8  532.4  12.8  90  50  290  530  10  970 5. 197,500  4.063  200, 000  4.000  50, 000 6. 405 400   8000 0.049 0.05 13. $91.35 $90   $30 3 3 14. $210 $200   $25 8 8 7. 1776  0.0098  1800  0.01  18 15. 12 months  $47  12  $50  $600 8. 0.63  1523  0.6  1500  900 16. 19 gallons  $3.11  20  $3  $60 9. 23.97  7.05  24  7  17 17. $7.99  $4.23  $16.82  $3.51  $20.12  $8  $4  $17  $4  $20  $53 18. $1.29  $6.86  $12.43  $25.62  $8.99  $1  $7  $12  $26  $9  $55 19. 95lb  127 lb  210 lb  100  100  200  400 lb 20. 16.1 lb  7.9 lb  21.4 lb  11.6 lb  3.7 lb  16  8  21  12  4  61 lb 21. 15% of $26.32  15% of $26  0.15  $26  $3.90 22. $400 $400   16 $23 $25 23. $595  $289  $120  $110  230  $600  $300  $100  $100  $200  $1300 24. Team A: 189  172  191  190  170  190  550 Team B: 183  229  167  180  230  170  580 580  550  30 lb 25. 11  8  $1.50  10  8  $1.50  10  $12  $120 26. 9 min, 55 sec/mi  26.2 mi  10 min  26 mi  260 min/mi; 27. 260 min 1  4 hours 60 min 3 599 Mexican pesos 600   $30 20.14 Mexican pesos / U.S. dollars 20 Copyright © 2021 Pearson Education, Inc. 4 Chapter 1: Critical Thinking Skills 28. $973  6 $61  6 $97   6 $200  $970  6 $60  6 $100  6 $200  $970  $360  $600  $1200  $3130 29. 90 miles 30. 75 miles 31. a) 23% of 700  25% of 700  0.25  700  175 b) 12% of 700  10% of 700  0.10  700  70 c) 21% of 700  20% of 700  0.20  700  140 32. a) 18% of 2987  18% of 3000  0.18  3000  540 b) 42% of 2987  42% of 3000  0.42  3000  1260 c) 7% of 2987  7% of 3000  0.07  3000  210 33. a) 5 million b) 98 million c) 98 million  33 million  65 million d) 19 million  79 million  84 million  65 million  33 million  280 million 34. a) 19% b) 25% c) 20% of 179 lb  20% of 180  0.2  180  36 lb 35. a) 85% b) 68%  53%  15% c) 85% of 70 million acres = 59,500,000 acres d) No, since we are not given the area of each state. 36. a) 2  410  4 545  2 400  4 500  800  2000  2800 calories b) Running: 4 920  4 900  3600 calories Walking: 4 330  4 300  1200 calories Difference: 3600  1200  2400 calories c) 3 545  3 545  3 550  3 550  1650  1650  3300 calories per week 3300 calories per week 52 weeks   3000  50  150, 000 calories 37. 20 44. 25% 38. 25 45. 9 square units 39. 120 bananas 46. 12 square units 40. 100 grapes 47. 160 feet 41. 150 42. 315 43. 10% 48. 4(60)  240 in. or 240  20 ft 12 49. – 57. Answers will vary. Copyright © 2021 Pearson Education, Inc. Section 1.3: Problem-Solving Procedures 5 58. Cost of one vitamin C tablet: Cost of one fish oil tablet: $24.99 $25   $0.05 500 500 $47.99 $48   $0.40 120 120 Cost of one vitamin D tablet: $17.99 $18   $0.10 180 180 Cost of one multivitamin tablet: $19.99 $20   $0.25 80 80 Roberto’s daily cost: 5  $0.05  4  $0.40  1  $0.10  1  $0.25  $1.40 59. There are 118 ridges around the edge. 60. a) Answers will vary. b) 11.6 days. There are 24  60  60  86, 400 seconds in a day, and 1, 000, 000 / 86, 400  11.5740. 62. Answers will vary. 63. Answers will vary. Section 1.3: Problem-Solving Procedures 1. 1 in. 4.25 in.  x mi 12 mi x  12  4.25 4. x  56 yd 3. 3 ft x ft  1.2 ft 15.36 ft 3 15.36  1.2 x 46.03 1.2 x  1.2 1.2 46.03 x  38.4 ft 1.2  x bags 35, 000 ft 2 4000 x 35, 000  4000 4000 35, 000 x  8.75 bags 4000 1 in. 22.4 in.  x yd 2.5 yd 1x  2.5 22.4 4000 ft 2 4000 x  135, 000 x  51 mi 2. 1 bag 5. 4  $113  $452 9.25% of $452  0.0925  $452  $41.81 $452  $41.81  $493.81 6. 2  $187  $374 9.5% of $374  0.095  $374  $35.53 $374  $35.53  $409.53 7. a) Entertainment/Miscellaneous: 19.1% of $1950  0.191  $1950  $372.45 Food: 12.7% of $1950  0.127  $1950  $247.65 $372.45  $247.65  $124.80 b) Housing: 34.4% of $1950  0.344  $1950  $670.80 Transportation: 16% of $1950  0.16  $1750  $312.00 $670.80  $312.00  $358.80 8. a) Business: 19% of 2,895, 000  0.19  2,895, 000  550, 050 Engineering: 5% of 2,895, 000  0.05  2,895, 000  144, 750 550, 050  144, 750  405,300 degrees Copyright © 2021 Pearson Education, Inc. 6 Chapter 1: Critical Thinking Skills 8. (continued) b) Health Professions: 11% of 2,895, 000  0.11  2,895, 000  318, 450 Psychology: 6% of 2,895, 000  0.06  2,895, 000  173, 700 318, 450  173, 700  144, 750 degrees 9. 43 rides  $2.00 per ride  $86.00; In order for the cost of rides with the $84.50 MetroCard to be less than the cost of the rides without the CharlieCard, Marcelo would have to take 43 bus rides per month. $84.50 Note that  $1.97 per ride. 43rides 10. 45 rides  $2.75 per ride  $123.75; In order for the cost of rides with the $121.50 MetroCard to be less than the cost of the rides without the MetroCard, Karissa would have to take 45 subway rides per month. $121.50  $2.70 per ride Note that 45 rides 11. $250  $130 18  $250  $2340  $2590 Savings: $2590  $2500  $90 12. $349.72  $32.39(4)  $349.72  129.56  $220.16 13. 15-year mortgage: $887.63(12)(15)  $159, 773.40 30-year mortgage: $572.90(12)(30)  $206, 244.00 Savings: $206, 244.00  $159, 773.40  $46, 470.60 14. Points needed for 80 average: 80(5)  400 points Wallace’s points so far: 79  93  91  68  331 points Grade needed on fifth exam: 400  331  69 15. a) 10  10  10  10  10, 000 b) 1 in 10,000 16. a) 460  18.4 min 25 c) 1400  40 min 35 b) 1550  31 min 50 d) 1550 2200 3750    47 min 80 80 80 17. 38, 687.0 mi  38, 451.4 mi  235.6 mi; 235.6 mi  18.7 mpg 12.6 gal 18. a) 40  $8.50  52  $17, 680 b) Each week he makes 40  $8.50  $340, it will take him $1275  3.75 weeks. $340 19. By mail: $52.80  $5.60  $8.56  4  $66.96  4  $267.84 Tire store: $324  0.08  $324  $324  $25.92  $349.92 Savings: $349.92  $267.84  $82.08 20. $1750  $50 15  $1750  $750  $1000 and $1000  25 hours $40 Copyright © 2021 Pearson Education, Inc. Section 1.3: Problem-Solving Procedures 7 21. 15,000 ft  3000 ft = 12,000 ft decrease in elevation. Temperature increases 2.4 F for every 1000 ft decrease in elevation. 6 F  28.8 F  22.8 F and 6 F  28.8 F  22.8 F; The precipitation at the airport will be snow. 22. a) $620 0.12  $74.40 b) $1200 0.22  $264 c) The store lost $1200  $1000  $200 on the purchase, so the store’s profit is $264  $200  $64. 23. a) Since $876 is less than 10%  $9525  $952.50, Shenile’s tax was 10% of her taxable income. Her taxable $876  $8760. income was 0.10 b) Since $2017.50 is greater than $952.50 and less than $4453.50, Logan’s taxable income was in the 12% bracket. Since $2017.50  $962.50  $1065 was 12% of his taxable income above $9525, he made an $1065  $8875. His taxable income was $9625  $8875  $18, 400. additional 0.12 24. a) Since $4541.50 is greater than $4453.50 and less than $14,089.50, Lewis’ taxable income was in the 22% bracket. Since $4541.50  $4453.50  $88 was 22% of his taxable income above $38,700, he made an $88  $400. His taxable income was $38, 700  $400  $39,100. additional 0.22 b) Since $14,593.50.50 is greater than $14,089.50 and less than $32,089, Leticia’s taxable income was in the 24% bracket. Since $14,593.50  $14, 089.50  $504 was 24% of his taxable income above $82,500, she $504  $2100. Her taxable income was $82,500  $2100  $84, 600. made an additional 0.24 25. a) 1  60  24  365  525, 600 oz and b) 525, 600 oz  4106.25 gal 128 4106.25  $11.20  4.10625  $11.20  $45.99 1000 26. a) 0.1  60  60  24  365  3,153, 600 cm3 b) 27. a) Volume of basin (30 cm)(20 cm)(20 cm) 12, 000 cm3    1.4; It would take about 1.4 days. Daily water loss 0.1  60  60  24 8640 cm3 20, 000 20, 000   961.538  925.926  35.612  35.61 gal 20.8 21.6 b) 35.61  $3.00  $106.83 c) 140, 000, 000  35.61  4,985, 400, 000 gal 28. a) Parking: $30  6  $180 Tax: 9.5% of $180  0.095  $180  $17.10 Total: $180  $17.10  $197.10 b) Parking: $17.95  6  $107.70 Tax: 8.75% of $107.70  0.0875  $107.70  $9.42 Total: $107.70  $9.42  $117.12 c) She can save $197.10  $117.12  $79.98. Copyright © 2021 Pearson Education, Inc. 8 Chapter 1: Critical Thinking Skills 29. Cost after 1 year: $999  0.02($999)  $999  $19.98  $1018.98 Cost after second year: $1018.98  0.02($1018.98)  $1018.98  $20.38  $1039.36 30. Value after first year: $1000  0.10($1000)  $1000  $100  $1100 Value after second year: $1100  0.10 $1100  $1100  $110  $990 $990 is less than the initial investment of $1000. 31. After paying the $100 deductible, Yungchen must pay 20% of the cost of x-rays. First x-ray: $100  0.20 $620  $100  $124  $224 Second x-ray: 0.20 $980  $196 Total: $224  $196  $420 32. $3000 is the difference between one-fourth of the cost and one-fifth of the cost. 1 1 1 and 20  $3000  $60, 000   4 5 20 33. a) water/milk: 3(1)  3 cups 1 3 salt: 3    tsp 8 8 Cream of wheat: 3 3  9 tbsp  b) water/milk: 9 cup (because 16 tbsp = 1 cup) 16 3 23 4  4  23  2 7 cups 2 2 8 8 23 1 1 3  3 salt: 4 2  4  tsp 2 2 8 1 3 5  5 cream of wheat: 2 4  4  cups =5 tbsp 2 2 8 3 15 4 11 3 c) water/milk: 3  1     2 cups 4 4 4 4 4 salt: 1 1 4 1 3     tsp 2 8 8 8 8 3 3 12 3 9 cup = 9 tbsp     4 16 16 16 16 d) Differences exist in water/milk because the amount for 4 servings is not twice that for 2 servings. cream of wheat: Differences also exist in Cream of Wheat because 1 cup is not twice 3 tbsp. 2 Copyright © 2021 Pearson Education, Inc. Section 1.3: Problem-Solving Procedures 9 34. a) rice: 1 4  2 cups 2 water: 1 salt: 1 4 16 1 4  4   5 cups 3 3 3 3 1 4  1 tsp 4 butter/margarine: 1 4  4 tsp b) rice: 1 2  2 cups water: 2 salt: 1 9 18 2 1 2  2   4  4 cups 4 4 4 4 2 1 2  1 tsp 2 butter/margarine: 2  2  4 tsp c) rice: 1 1 1 3 4  1     2 cups 2 2 2 2 2 1 1 4 10 14 2  4 cups water: 1  3    3 3 3 3 3 3 salt: 1 3 4    1 tsp 4 4 4 butter/margarine: 1 tsp  1 tbsp  1 tsp  3 tsp  4 tsp d) rice: 3  1  2 cups water: 6  2 1 24 9 15 3     3 cups 4 4 4 4 4 1 1 salt: 1   1 tsp 2 2 butter/margarine: 2 tbsp  2 tsp  2 3tsp   2 tsp  6 tsp  2 tsp  4 tsp e) Differences exist in water because the amount for 4 servings is not twice that for 2 servings. 35. a) 20 DVDs 1 0 12 DVDs 1 2 Total Number of DVDs 20  12  32 12  12  24 b) $240  $180  $420 36. Mark will win. 37. 1 ft 2  12 in.  12 in.  144 square inches 38. 1 ft 3  12 in.  12 in.  12 in.  1728 cubic inches 39. Area of original rectangle: lw Area of new rectangle: (2l )(2w)  4(lw) Thus, if the length and width of a rectangle are doubled, the area is 4 times as large. Copyright © 2021 Pearson Education, Inc. 10 Chapter 1: Critical Thinking Skills 40. 20 ft  20 ft = 400 ft 2 ; 5ft  5ft = 25 ft 2 ; 400 ft 2 25ft 2  16squares 41. Volume of original cube: lwh Volume of new cube: (2l )(2 w)(2h)  8(lwh) Thus, if the length, width, and height of a cube are doubled, the volume is 8 times as large. 42. 11 ft is one-sixth of the pole, so the length is 6  11 ft  66 ft. 43. 10 pieces 1000 pieces  $10 x 1000 x  10 10 1000 x 100  1000 1000 100  $0.10  10¢ x 1000 44. Left side Right side 1(6)  6 1(2)  2 2(2)  4 1(3)  3 (6)  6 6  (4)  10 2  3  6  11 Place it at –1, so the left side would total 10  (1)  1. 45. You have three ties, each a different color. 46. There are 10 palindromes: 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992. 47. a) refresh b) workout 51. 48. Answers will vary. 52. 49. 50. Eight pieces 53. 8  6  2  4  20; 3  7  5  1  16; 10  14  12  8  44 The sum of the four corner entries is 4 times the number in the center of the middle row. 54. 15,12,33; Multiply the number in the center of the middle row by 3. 55. 45,36,99; Multiply the number in the center of the middle row by 9. 56. 35  15  20 cubes 57. 3  2  1  6 ways 58. Each salesperson shakes hands with four people. Copyright © 2021 Pearson Education, Inc. Review Exercises 11 59. Other answers are possible, but 1 and 8 must appear in the center. 61. Other answers are possible. 60. The diagram shows the number of times each part is used. 62. With umbrella policy: Mustang reduced premium: $1648  $90  $1558 Focus reduced premium: $1530  0.12($1530)  $1530  $183.60  $1346.40 Total for umbrella policy: $1558  $1346.40  $450  $3354.40 Without umbrella policy: $1648  $1530  $3178 Net amount for umbrella policy: $3354.40  $3178  $176.40 63. Mark plays the drums. 64. 16  16  4  4  4  44 65. The areas of the colored regions are 1  1, 1  1, 2  2, 3  3, 5  5, 8  8, 13  13, 21  21. 1  1  4  9  25  64  169  441  714 square units 66. Let x be the amount Samantha had to start. 1 1 After first store: x  x  20  x  20 2 2 After second store: 1 1 1   x  20   20  x  30 2 2 4 1 x  30  0 4 x  120 The original amount was $120. 67. Thomas would have opened the box labeled grapes and cherries. Because all the boxes are labeled incorrectly, whichever fruit he pulls from the box of grapes and cherries, will be the only fruit in that box. If he pulled a grape, he labeled the box grape. If he pulled a cherry, he labeled the box cherries. That left two boxes whose original labels were incorrect. Because all labels must be changed, there was only one way for Thomas to assign the two remaining labels. Review Exercises 1. 23, 28, 33 (Add 5 to previous number.) 2. 16, 13, 10 (Subtract 3 from previous number) 3. 64, –128, 256 (Multiply previous number by –2.) 4. 25, 32, 40 (19  6  25, 25  7  32, 32  8  40) 5. 10, 4, –3 (Subtract 1, then 2, then 3, … ) 6. 3 3 3 (Multiply previous number by 12 . ) , , 8 16 32 Copyright © 2021 Pearson Education, Inc. 12 Chapter 1: Critical Thinking Skills 7. 8. 9. c 10. a) The final number is twice the original number. b) The final number is twice the original number. c) Conjecture: The final number is twice the original number. d) n  10n  10n  5  10n  5 10n 5    2n  1  2n  1  1  2n 5 5 5 11. This process will always result in an answer of 3. n  n  5  6 (n  5)  6n  30  6n  30  12  6n  18  6n  18 6n 18 3n  9 3n 9    3n  9     n  3 , n  3  n  3 2 2 2 3 3 3 12. 12  22  5 and 5 is an odd number. Other answers are possible. Answers for Exercises 13–25 will vary depending on how you round your numbers. The answers may differ from the answers in the back of the textbook. However, your answers should be something near the answers given. All answers are approximate. 13. 205,123  4002  200, 000  4000  800, 000, 000 26. $50  $40(12)  $530 Savings: $530  $500  $30 14. 215.9  128.752  3.6  861  792  200  100  0  900  800  2000 27. 4($1.99)  $7.96 for four six-packs 15. 21% of 2095  20% of 2000  0.20  2000  400 28. Freemac: $15  4  2  $120 Savings: $7.96  $4.99  $2.97 Sylvan: $25  4 hours  $100 16. Answers will vary. $120  $100  $20 17. 48 bricks  $3.97  50  4 = $200 Sylvan Rental is less expensive by $20. 18. 8% of $21, 000  8% of 20,000  0.08  20, 000  $1600 19. 1.1 mi 1 mi 3 mi    3 mph 22 min 20 min 60 min 20. $2.49  $0.79  $1.89  $0.10  $2.19  $6.75  $2  $1  $2  $0  $2  $7  $14.00 20 1 in.  20   in.  20 0.1 mi 4 4  2 mi 29. Cost per person with 5 people: $445  $89 5 Cost per person with 5 people: $510  $85 6 Savings: $89  $85  $4 30. a) 21. 5 in.  30 lb 2500 ft 22. Approximately 70F  30F  40F 23. Approximately 90F  80F  10F b) 24. 13 square units 2  x 24, 000 ft 2 30  24, 000 x 2500 x  288 lb 150 lb  5 bags; 5  2500  12,500 ft 2 30 lb/bag 25. Length: 1.75 in., 1.75(1.25)  21.875  22 ft Height: 0.625 in., 0.625(12.5)  7.8125  8 ft Copyright © 2021 Pearson Education, Inc. Review Exercises 13 31. 10% of $1030  0.10  $1030  $103 7  $103  $721 32. Savings: $721  $60  $661 1.5 mg x mg  10 lb 52 lb 10 x  52 1.5 10 x 78  10 10 x  7.8 mg 33. $5500  0.30 $5500  $5500  $1650  $3850 take-home; 28% of $3850  0.28  $3850  $1078 34. 9 A.M. Eastern is 6 A.M. Pacific, from 6 A.M. Pacific to 1:35 P.M. Pacific is 7 hr 35 min, 7 hr 35 min  50 min stop = 6 hr 45 min. 35. 3 P.M.  4 hr  11 A.M.; July 26, 11:00 A.M. 36. a) 65 mi 1.6 km 104 km    104 km/hr 1hr mi 1hr b) 90 km 1mi 90 mi    56.25 mi/hr 1hr 1.6 km 1.6 km 37. Each figure has an additional two dots. To get the hundredth figure, 97 more figures must be drawn, 97(2)  194 dots added to the third figure. Thus, 194  7  201. 39. 38. 40. 59 min 59 sec; Since it doubles every second, the jar was half full 1 second earlier than 1 hour. 41. 6 42. Nothing. Each friend paid $9 for a total of $27; $25 to the hotel, $2 to the clerk. $25 for the room + $3 for each friend + $2 for the clerk = $30 43. Let x  the score on the fifth exam 93  88  81  86  x  80 5 348  x  80, 5 348  x  400, x  52 44. Yes; 3 quarters and 4 dimes, or 1 half dollar, 1 quarter and 4 dimes, or 1 quarter and 9 dimes. 45. 6 cm  6 cm  6 cm  216 cm3 46. Place six coins in each pan with one coin off to the side. If it balances, the heavier coin is the one on the side. If the pan does not balance, take the six coins on the heavier side and split them into two groups of three. Select the three heavier coins and weigh two coins. If the pan balances, it is the third coin. If the pan does not balance, you can identify the heavier coin. 47. 1  500  501, 2  499  501,  ; There are 250 such pairs and 250(501)  125, 250. 48. 4 green weigh the same as 8 blue, 2 yellow weigh the same as 5 blue, 2 white weigh the same as 3 blue, 8  5  3  16 blue Copyright © 2021 Pearson Education, Inc. 14 Chapter 1: Critical Thinking Skills 49. There are 90 such numbers: 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202,h …, 292, 303, …, 393, 404, …, 494, 505, …, 595, 606, …, 696, 707, …, 797, 808, …, 898, 919, …, 999. 50. The fifth figure will be an octagon with sides of equal length. Inside the octagon will be a seven-sided figure with sides of equal length. The figure will have one antenna. 51. 61 orange tiles will be required. The sixth figure will have 6 rows of 6 tiles and 5 rows of 5 tiles (6  6  5  5  36  25  61). 52. Some possible answers are given below. There are other possibilities. 53. a) 2 b) There are 3 choices for the first spot. Once that person is standing, there are 2 choices for the second spot and 1 for the third. Thus, 3  2  1  6. c) 4  3  2  1  24 d) 5  4  3  2  1  120 e) n  n  1 n  21, (or n !), where n  the number of people in line. Chapter Test 1. 26, 32, 38 (Add 6 to previous number.) 2. 1 1 1 , , (Add 1 to the denominator of the previous number.) 5 6 7 3. a) The result is the original number plus 1. b) The result is the original number plus 1. c) Conjecture: The result will always be the original number plus 1. d) n  5n  5n  10  5n  10 5n 10    n  2  n  2 1  n 1 5 5 5 4. 0.51  96, 000  0.5  100, 000  50, 000 5. 188, 000 200, 000   2, 000, 000 0.11 0.1 6. 9 square units 8. a) 325,000 visitors b) 100,000 visitors 130 lb 2.0635  2.0635;  0.032754; 63 in. 63 in. 0.032754  703  23.03 b) He is in the at-risk range. 7. a) 9. $50.40  $35.00  $15.40; Copyright © 2021 Pearson Education, Inc. $15.40  77 miles $0.20 Chapter Test 15 $15  5.79, a maximum of 5 six-packs can be purchased. There will be $15.00  (5  $2.59) $2.59 $2.05  $15.00  $12.95  $2.05 remaining, so  2.5625, or two additional individual cans can be purchased. $0.80 The following table was created using similar calculations for the other possible number of six-packs. 10. Since Six-packs 5 4 3 2 1 0 Individual cans 2 5 9 12 15 18 Number of cans 6  5  2  32 6  4  5  29 6  3  9  27 6  2  12  24 6  1  15  21 6  0  18  18 The maximum number of cans is 32. 11. 3 cuts  2.5 min 1  7  7.5 min 1 cut 2 12. 10  13. $12.75  40  $510 3  10  0.75  7.5 miles 4 14. $12.75  1.5  10  $191.25 $510  $191.25  $701.25 $652.25  $701.25  $49.00 She was underpaid by $49.00. 15. Mary drove the first 15 miles at 60 mph which took 15 1  hour and drove the second 15 miles at 30 mph 60 4 15 1 3  hour for a total time of hour, or 45 minutes. If she drove the entire 30 miles at 45 mph, the trip 30 2 4 30 2 3 would take  hour, or 40 minutes, which is less than hour. 45 3 4 took 16. 6 lb 1 3 1 1  3; 3  tsp = tsp or 1 tsp  tbsp 2 lb 2 2 2 2 17. Area of lawn including walkway: (10  2)  (12  2)  12  14  168 m 2 Area of lawn only: 10  12  120 m 2 Area of walkway: 168  120  48 m 2 18. 243 jelly beans; 260  17  243, 234  9  243, and 274  31  243 19. a) 3  $3.99  $11.97 b) 9($1.75  0.75)  11.8125  $11.81 c) $11.97  $11.81  $0.16; Using the coupon is least expensive by $0.16. 20. 4  3  2  1  24; The first position can hold any of four letters, the second any of the three remaining letters, and so on. Copyright © 2021 Pearson Education, Inc. Section 2.1: Set Concepts 17 Chapter Two: Sets Section 2.1: Set Concepts 1. Set 6. Equivalent 2. Ellipsis 7. Equal 3. Description, Roster form, Set-builder notation 8. Cardinal 4. Finite 9. Empty or null 5. Infinite 10. { }, 11. Not well defined, “best” is interpreted differently by different people. 12. Not well defined, “most interesting” is interpreted differently by different people. 13. Well defined, the contents can be clearly determined. 14. Well defined, the contents can be clearly determined. 15. Well defined, the contents can be clearly determined. 16. Not well defined, “most interesting” is interpreted differently by different people. 17. Infinite, the number of elements in the set is not a natural number. 18. Finite, the number of elements in the set is a natural number. 19. Infinite, the number of elements in the set is not a natural number. 20. Infinite, the number of elements in the set is not a natural number. 21. Infinite, the number of elements in the set is not a natural number. 22. Finite, the number of elements in the set is a natural number. 23.  Hawaii  32.  Alaska, Hawaii  24.  January, June, July  33. {Google Play Games, Pokémon Go} 25.  11,12,13,14,  ,177  34. {ApplsLoading} 35. {Solitaire by MobilityWare, Candy Crush Soda Saga, ApplsLoading} 26. C  4 36. { } or  27. B   2, 4, 6, 8,  28. { } or  37. 2015, 2016, 2017, 2018 29. { } or  38. 2011, 2012, 2013, 2014 30. Idaho, Oregon 39. { } or  31. E   14, 15, 16, 17,  , 84 40. 2013, 2014 41. B   x | x  N and 6  x  15 or B   x | x  N and 7  x  14 42. A   x | x  N and x  10 or A   x | x  N and x  9 43. C   x | x  N and x is a multiple of 3 46. A   x | x is Independence Day 44. D   x | x  N and x is a multiple of 5 47. C   x | x is February 45. E   x | x  N and x is odd Copyright © 2021 Pearson Education, Inc. 18 Chapter 2: Sets 48. F   x | x  N and 14  x  101 or F   x | x  N and 15  x  100 49. Set A is the set of natural numbers less than or equal to 7. 50. Set D is the set of natural numbers that are multiples of 3. 51. Set V is the set of vowels in the English alphabet 52. Set D is the set of the months of a year. 53. Set T is the set of species of trees. 54. Set E is the set of natural numbers greater than or equal to 4 and less than 11. 55. Set S is the set of seasons. 56. Set R is the set of presidents whose face is on Mount Rushmore. 57. Facebook, Instagram, Facebook Messenger 63. 2013, 2014, 2015 58. Tumblr, WhatsApp, Google Hangouts 64. 2014, 2015 59. Twitter, Pinterest, Snapchat 65. False; e is a set, and not an element of the set. 60. Instagram 66. True; b is an element of the set. 61. 2016, 2017, 2018, 2019 67. False; h is not an element of the set. 68. True; February is an element of the set. 62. 2010, 2011 69. False; 3 is an element of the set. 70. False; Tokyo is a city in Japan, not the United States. 71. True; 9 is an odd natural number. 74. n  B   6 72. False; 2 is an even natural number. 75. n C   0 73. n  A  4 76. n  D   5 77. Both; A and B contain exactly the same elements. 78. Equivalent; both sets contain the same number of elements, 3. 79. Neither; the sets have a different number of elements. 80. Neither; not all cats are Siamese. 81. Equivalent; both sets contain the same number of elements, 4. 82. Equivalent; both sets contain the same number of elements, 50. 83. a) Set A is the set of natural numbers greater than 2. Set B is the set of all numbers greater than 2. b) Set A contains only natural numbers. Set B contains other types of numbers, including fractions and decimal numbers. c) A   3, 4, 5, 6,  d) No; because there are an infinite number of elements between any two elements in set B, we cannot write set B in roster form. Copyright © 2021 Pearson Education, Inc. Section 2.2: Subsets 19 84. a) Set A is the set of natural numbers greater than 2 and less than or equal to 5. Set B is the set of all numbers greater than 2 and less than or equal to 5. b) Set A contains only natural numbers. Set B contains other types of numbers, including fractions and decimal numbers. c) A   3, 4, 5  d) No; because there are an infinite number of elements between any two elements in set B, we cannot write set B in roster form. 85. Cardinal; 7 tells how many. 86. Ordinal; 25 tells the relative position of the chart. 87. Ordinal; sixteenth tells Lincoln’s relative position. 88. Cardinal; 35 tells how many dollars she spent. 89. Answers will vary. 90. Answers will vary. Examples: The set of elephants that can fly. The set of dogs that can talk. The set of books that have no pages. 91. Answers will vary. 92. Answers will vary. Here are some examples. a) The set of men. The set of actors. The set of people over 12 years old. The set of people with two legs. The set of people who have been in a movie. b) The set of all the people in the world. Section 2.2: Subsets 1. Subset 2. Proper 3. 2n , where n is the number of elements in the set. 4. 2n  1, where n is the number of elements in the set. 5. True; table is a subset of sofa, chair, table . 6. True; Pacific is a subset of Atlantic, Pacific, Indian, Artic . 7. False; apple is not in the second set. 8. False; cat is not in the second set. 9. True; AT&T, Verizon is a proper subset of Verizon, Sprint, T-Mobile, Verizon . 10. True; American, Southwest is a proper subset of Delta, American, Southwest . 11. False; no subset is a proper subset of itself. 12. False; no set is a proper subset of itself. 13. True; book is an element of book, magazine, newspaper . 14. True; necklace is an element of necklance, ring, bracelet, earring . 15. False; cookie is a set, not an element. 16. False; { } is a set, not an element. 17. True; tiger is not an element of zebra, giraffe, polar bear . 18. True; the empty set is a subset of every set, including itself. Copyright © 2021 Pearson Education, Inc. 20 Chapter 2: Sets 19. True; chair is a proper subset of sofa, table, chair . 20. True; a, b, c  c, b, a . 22. True; { } and  each represent the empty set. 21. False; the set  contains the element . 23. False; the set 0 contains the element 0. 24. True; the empty set is a subset of every set, including itself. 25. False; 0 is a number and { } is a set. 26. True; the elements of the set are themselves sets. 27. B  A, B  A 33. A  B, A  B, B  A 28. A  B, A  B 34. B  A, B  A 29. B  A, B  A 35. { } is the only subset. 30. A  B, A  B 36. { },{} 31. None 37.   , cow , horse , cow, horse 32. B  A, B  A 38.   ,  steak  ,  pork  ,  chicken ,  steak, pork  ,  steak, chicken  ,  pork, chicken  ,  steak, pork, chicken    , a , b , c , d  , a, b , a, c , a, d  , b, c , b, d  , c, d  , a, b, c , a, b, d  , a, c, d  , b, c, d  , a, b, c, d  b) All the sets in part (a) are proper subsets of A except a, b, c, d  . 39. a) 40. a) 29  2  2  2  2  2  2  2  2  2  512 subsets b) 29  1  512  1  511 proper subsets 41. False; A could be equal to B. 43. True; every set is a subset of itself. 42. True; every proper subset is a subset. 44. False; no set is a proper subset of itself. 45. True;  is a proper subset of every set except itself. 46. True;  is a subset of every set. 48. False; a set cannot be a proper subset of itself. 47. True; every set is a subset of the universal set. 49. True;  is a proper subset of every set except itself and U  . 50. False; the only subset of  is itself and U  . 51. True;  is a subset of every set. 52. False; U is not a subset of . 53. The number of different variations is equal to the number of subsets of a set with 4 elements, which is 24  2  2  2  2  16. 54. The number of different variations of the house is equal to the number of subsets of a set with 7 elements, which is 27  2  2  2  2  2  2  2  128. 55. The number of options is equal to the number of subsets of a set with 7 elements, which is 27  2  2  2  2  2  2  2  128. Copyright © 2021 Pearson Education, Inc. Section 2.3: Venn Diagrams and Set Operations 21 56. The number of different variations is equal to the number of subsets of a set with 6 elements, which is 26  2  2  2  2  2  2  64. 57. E  F since they are both subsets of each other. 58. If there is a one-to-one correspondence between boys and girls, then the sets are equivalent. 59. a) Yes. b) No, c is an element of set D. c) Yes, each element of a, b is an element of set D. 60. a) Each person has 2 choices, namely yes or no, so there are 24  2  2  2  2  16 outcomes. b) YYYY, YYYN, YYNY, YNYY, NYYY, YYNN, YNYN, YNNY, NYNY, NNYY, NYYN, YNNN, NYNN, NNYN, NNNY, NNNN c) 5 out of 16 61. A one element set has one proper subset, namely the empty set. A one element set has two subsets, namely itself and the empty set. One is one-half of two. Thus, the set must have one element. 62. Yes 64. No 63. Yes Section 2.3: Venn Diagrams and Set Operations 1. Complement 5. Cartesian 2. Union 6. Disjoint 3. Intersection 7. m  n 4. Difference 8. Four 9. 13. 10. 14. 11. 15. 12. 16. Copyright © 2021 Pearson Education, Inc. 22 Chapter 2: Sets 17. 19. 18. 20. 21. The set of retail stores in the United States that do not sell children’s clothing 22. The set of animals in the United States’ zoos that are not in the San Diego Zoo 23. The set of cities in the United States that do not have a professional sports team 24. The set of cities in the United States that do not have a symphony 25. The set of cities in the United States that have a professional sports team or a symphony 26. The set of cities in the United States that have a professional sports team and a symphony 27. The set of cities in the United States that have a professional sports team and do not have a symphony 28. The set of cites in the United States that have a professional sports team or do not have a symphony 29. The set of furniture stores in the U.S. that sell mattresses or leather furniture 30. The set of furniture stores in the U.S. that sell mattresses and outdoor furniture 31. The set of furniture stores in the U.S. that do not sell outdoor furniture and sell leather furniture 32. The set of furniture stores in the U.S. that sell mattresses, outdoor furniture, and leather furniture 33. The set of furniture stores in the U.S. that sell mattresses or outdoor furniture or leather furniture 34. The set of furniture stores in the U.S. that do not sell mattresses or do not sell leather furniture 35. A  a, b, e, g , h 41. A  B   a, b, e, g , h  b, c, e, f   c, d , f , i, j  a, d , g , h, i, j 36. B  b, c, e, f  37. A  B  a, b, e, g , h  b, c, e, f   b, e 38. U  a, b, c, d , e, f , g , h, i, j 39. A  B  a, b, e, g , h  b, c, e, f   a, b, c, e, f , g , h 40. A  B  a, b, c, e, f , g , h (See Exercise 39.)  d , i, j 42. A  B  b, e (See Exercise 40.)  A  B   b, e  a, c, d , f , g , h, i, j 43. A  B  a, b, e, g , h  b, c, e, f   a, g , h 44. A  B   a, b, e, g , h  b, c, e, f   A  B   a, b, c, e, f , g , h  d , i, j Copyright © 2021 Pearson Education, Inc.  a, b, e, g , h  a, d , g , h, i, j  b, e Section 2.3: Venn Diagrams and Set Operations 23 45. A   1, 2, 7,8,9  47. U  1, 2, 3, 4,5, 6, 7,8,9,10,11 46. B  2,3,5, 6,9 48. A  B  1, 2, 7,8,9  2,3,5, 6,9  2,9 49. A  B  3, 4,5, 6,10,11  2, 3, 5, 6, 9  2, 3, 4, 5, 6, 9, 10, 11  50. A  B   {1, 2, 7,8,9}  {1, 4, 7,8,10,11}  {1, 2, 4, 7,8,9,10,11} 51. A  B   3, 4, 5, 6, 10,11    2, 3, 5, 6,9    3, 5, 6 57. A  B   1, 2, 3, 4, 5, 6, 7  (See Exercise 55) 52.  A  B    4, 10, 11   A  B   1, 2, 3, 4, 5, 6, 7  8 53. A   B   4, 10, 11  58. A  B    1, 2, 4, 5, 7    2, 3, 5, 6  54.  A  B     2, 3, 4, 5, 6,9, 10,11  55. A  B   1, 2, 4, 5, 7    2, 3, 5, 6    1, 2, 3, 4, 5, 6, 7    3, 6, 8    1, 4, 7, 8    8  59.  A  B   8 (See Exercise 57)  A  B   B   8    2, 3, 5, 6     56. A  B   1, 2, 4, 5, 7    2, 3, 5, 6    2, 5  60.  A  B    A  B  =   (The intersection of a set and its complement is always empty.) 61.  B  A   A  B   8 (See Exercise 57)   B  A   B   A    8    2, 3, 5, 6    1, 2, 4, 5, 7    8    1, 4, 7, 8    3, 6, 8     8    1, 3, 4, 6, 7, 8    8  62. A  B   2, 5  (See Exercise 56) A   A  B    1, 2, 4, 5, 7    2, 5    3, 6, 8    2, 5    2, 3, 5, 6, 8  64. A   B    3, 6,8    1, 4, 7,8    3, 6  63. ( A  B) =  1, 4, 7    2, 3, 5, 6, 8  65. B  C   b, c, d , f , g    a, b, f , i, j    a, b, c, d , f , g , i, j  66. A  C   a, c, d , f , g , i    a, b, f , i, j    a, f , i  67. A  B    b, e, h, j , k    a, e, h, i, j , k    a, b, e, h, i, j , k  68. A  C  a, f , i (See Exercise 66);  A  C    a, f , i    b, c, d , e, g , h, j , k  69.  A  B   C   a, c, d , f , g , i    b, c, d , f , g    a, b, f , i, j    c, d , f , g    a, b, f , i, j    a, b, c, d , f , g , i, j  70. A  C  B    a, c, d , f , g , i    a, b, f , i, j    b, c, d , f , g    a, c, d , f , g , i    b, f    a, c, d , f , g , i    a, c, d , e, g , h, i, j , k    a, c, d , e, f , g , h, i, j , k  Copyright © 2021 Pearson Education, Inc. 24 Chapter 2: Sets   71.  A  C    A  B    a, c, d , f , g , i    a, b, f , i, j    a, c, d , f , g , i    b, c, d , f , g    b, e, h, j , k    a, b, f , i, j    c, d , f , g    a, b, e, f , h, i, j , k    c, d , f , g    a, b, c, d , e, f , g , h, i, j , k  , or U  72. C  B    A  B   b, f    a, c, d , f , g , i    b, c, d , f , g    b, f    b, e, h, j , k    b, c, d , f , g     b, f    b    b  73.  A  B   C  a, i  a, b, f , i, j  b, c, d , e, f , g , h, j , k   a, b, f , i, j  c, d , e, g , h, k  74. C  A  B  b, d , j  b, c, d , f , g   a, c, e, f , g , h, i, k   b, c, d , f , g   {a, e, h, i, k} 75. (1, a ), (1, b), (2, a ), (2, b), (3, a), (3, b) 78. n (1, a), (1, b), (2, a ), (2, b), (3, a), (3, b)  6 76. (a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3) 79. n (a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)  6 77. No; The ordered pairs are not the same. For example: 1, a    a,1 . 80. Yes; n( A  B )  n( B  A)  6 81. A  B   1, 3, 5, 7, 9    2, 4, 6, 8     82. A  B   1, 3, 5, 7, 9    2, 4, 6, 8    1, 2, 3, 4, 5, 6, 7, 8, 9 , or U 83.  B  C    2, 4, 6, 8    1, 2, 3, 4, 5    1, 2, 3, 4, 5, 6, 8    7, 9  84. A  C    1, 3, 5, 7, 9    1, 2, 3, 4, 5    1, 3, 5, 7, 9    6, 7, 8, 9   7, 9  85. A  B    1, 3, 5, 7, 9    2, 4, 6, 8    1, 3, 5, 7, 9   1, 3, 5, 7, 9    1, 3, 5, 7, 9  , or A 86.  B  C    2, 4, 6, 8    1, 2, 3, 4, 5    2, 4    1, 3, 5, 6, 7, 8, 9  87.  A  C   B   1, 3, 5, 7, 9    1, 2, 3, 4, 5    2, 4, 6, 8    1, 2, 3, 4, 5, 7, 9   2, 4, 6, 8    2, 4  88. A  B  { } (See Exercise 81)  A  B   C      1, 2, 3, 4, 5    1, 2, 3, 4, 5, 6, 7, 8, 9    1, 2, 3, 4, 5    1, 2, 3, 4, 5, 6, 7, 8, 9  , or U   89.  A  B    C   1, 3, 5, 7, 9    2, 4, 6, 8    1, 2, 3, 4, 5    2, 4, 6, 8    1, 3, 5, 7, 9    1, 2, 3, 4, 5    1, 2, 3, 4, 5, 6, 7, 8, 9    1, 2, 3, 4, 5    1, 2, 3, 4, 5  , or C 90. A  B  { } (See Exercise 81)    A  C    A  B    1, 3, 5, 7, 9    1, 2, 3, 4, 5       2, 4, 6, 8    1, 2, 3, 4, 5       2, 4       2, 4  Copyright © 2021 Pearson Education, Inc. Section 2.3: Venn Diagrams and Set Operations 25 91. A set and its complement will always be disjoint since the complement of a set is all of the elements in the universal set that are not in the set. Therefore, a set and its complement will have no elements in common. For example, if U  1, 2,3 , A  1, 2 , and A  3 , so A  A    . 92. n  A  B   0 when A and B are disjoint sets. For example, if U  1, 2,3, 4,5, 6 , A  1,3 , and B  2, 4 , then A  B  { } and n { }  0. 93. Let A   customers who owned dogs  and B   customers who owned cats n  A  B   n  A  n  B   n  A  B   27  38  16  49 94. Let A  students who are members of student government and B  students who are members of clubs n  A  B   n  A  n  B   n  A  B  46  30  n  B   4 46  n  B   26 20  n  B  So, n  B   n  A  20  4  16 students were only members of clubs. 95. a) A  B  a, b, c, d   b, d , e, f , g , h  a, b, c, d , e, f , g , h , n  A  B   8 A  B  a, b, c, d   b, d , e, f , g , h  b, d  , n  A  B   2. n  A  n  B   n  A  B   4  6  2  8 Therefore, n  A  B   n  A  n  B   n  A  B  . b) Answers will vary. c) Elements in the intersection of A and B are counted twice in n( A)  n( B ). 96. A  B  defines Region I, A  B defines Region II, A  B defines Region III, and A  B  or  A  B  defines Region IV. 97. A  B   1, 2, 3, 4,    4, 8, 12, 16,    1, 2, 3, 4,  , or A 98. A  B   1, 2, 3, 4,    4, 8, 12, 16,    4, 8, 12, 16,  , or B 99. B  C   4, 8, 12, 16,    2, 4, 6, 8,    2, 4, 6, 8,  , or C 100. B  C   4, 8, 12, 16,    2, 4, 6, 8,    4, 8, 12, 16,  , or B 101. A  C   1, 2, 3, 4,    2, 4, 6, 8,    2, 4, 6, 8,  , or C 102. A  C   1, 2, 3, 4,    2, 4, 6, 8,    0    2, 4, 6, 8,     103. B  C  C (See Exercise 99)  B  C   C  C   C   2, 4, 6, 8,    2, 4, 6, 8,    0, 1, 2, 3, 4,  , or U 104. A  C  C (See Exercise 101);  A  C   B   C  B    2, 4, 6, 8,    4, 8, 12, 16,    2, 4, 6, 8,    2, 6, 10, 14,    2, 6, 10, 14, 18,  Copyright © 2021 Pearson Education, Inc. 26 Chapter 2: Sets 105. A  A    106. A  A  U 110. A  U  A 111. If A  B  B , then B  A . 107. A    A 112. If A  B  B, then A  B. 108. A     113. If A  B  , then A and B are disjoint sets. 109. A  U  U 114. If A  B  , then A   and B  . Section 2.4: Venn Diagrams with Three Sets and Verification of Equality of Sets 1. 8 3. a) ( A  B )   A  B  2. a) V b) VI b) ( A  B )   A  B  4. Deductive 5. A  B  is represented by regions V and VI. If B  C contains 12 elements and region V contains 4 elements, then region VI contains 12  4  8 elements. 6. A  B is represented by regions II and V. If A  B contains 9 elements and region V contains 4 elements, then region II contains 9  4  5 elements. 7. 8. 9. 11. 10. 12. Copyright © 2021 Pearson Education, Inc. Section 2.4: Venn Diagrams with Three Sets and Verification of Equality of Sets 27 13. 14. 15. Canada, IV 35. VIII 16. Pakistan, VI 36. V 17. China, V 37. VI 18. Germany, VII 38. III 19. Spain, VIII 39. A   1, 3, 4,5, 9, 10  20. Brazil, II 40. U   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14  21. VI 41. C   4, 5, 6, 8, 9, 11  22. VIII 42. B    1, 2, 9, 10, 11, 13  23. III 24. IV 43. A  B   3, 4, 5  25. III 26. I 44. A  C   4, 5, 9  27. V 45.  B  C    1, 2, 3, 7, 9, 10, 11, 12, 13, 14  28. III 29. II 46. A  B  C   4, 5  30. VIII 47.  A  C    2, 7, 12, 13, 14  31. VII 48. A  ( B  C )   3, 4, 5, 9  32. VI 33. I 49.  A  B   2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14 34. VII 50. A   B  2, 11, 13  A  B  51.  A  B   Set A B A B Regions I, II II, III II  A  B  I, III, IV Set A A B B A  B  Regions I, II III, IV II, III I, IV I, III, IV Both statements are represented by the same regions, I, III, IV, of the Venn diagram. Therefore,  A  B   A  B  for all sets A and B. Copyright © 2021 Pearson Education, Inc. 28 Chapter 2: Sets  A  B  52.  A  B  Set A B A B Regions I, II II, III II  A  B  I, III, IV Set A A B A  B Regions I, II III, IV II, III II, III, IV Since the two statements are not represented by the same regions, it is not true that  A  B   A  B for all sets A and B. A B A  B  53. Set A A B B A  B  Set A B A B Regions I, II III, IV II, III I, IV I, III, IV Regions I, II II, III II Since the two statements are not represented by the same regions, it is not true that A  B   A  B for all sets A and B.  A  B   54. Set A A B B A  B   A  B  Regions I, II III, IV II, III I, IV I, III, IV Set A B Regions I, II II, III  A  B  I, II, IV Since the two statements are not represented by the same regions, it is not true that A  B   A  B  for all sets A and B. A  B  55. Set A A B B A  B  A  B Regions I, II III, IV II, III I, IV IV Set A B B A  B Regions I, II II, III I, IV I, II, IV Since the two statements are not represented by the same regions, it is not true that A  B   A  B  for all sets A and B. Copyright © 2021 Pearson Education, Inc. Section 2.4: Venn Diagrams with Three Sets and Verification of Equality of Sets 29  A  B  56.  A  B  Set A A B A  B Regions I, II III, IV II, III III  A  B  I, II, IV Set A B B A  B Regions I, II II, III I, IV I, II, IV Both statements are represented by the same regions, I, II, IV, of the Venn diagram. Therefore,  A  B   A  B  for all sets A and B.  A  B  C A  B  C  57. Set B C A Regions II, III, V, VI IV, V, VI, VII II, III, IV, V, VI, VII I, II, IV, V A  B  C  II, IV, V B C Set A B A B C Regions I, II, IV, V II, III, V, VI II, V IV, V, VI, VII  A  B  C II, IV, V, VI, VII Since the two statements are not represented by the same regions, it is not true that A   B  C    A  B   C for all sets A, B, and C. A  B  C  58. B  C   A Set B C B C A Regions II, III, V, VI IV, V, VI, VII V, VI I, II, IV, V Set B C B C A Regions II, III, V, VI IV, V, VI, VII V, VI I, II, IV, V A  B  C  I, II, IV, V, VI B  C   A I, II, IV, V, VI Both statements are represented by the same regions, I, II, IV, V, VI, of the Venn diagram. Therefore, A   B  C    B  C   A for all sets A, B, and C. A  B  C  59. B  C   A Set B C B C A Regions II, III, V, VI IV, V, VI, VII II, III, IV, V, VI, VII I, II, IV, V Set B C B C A Regions II, III, V, VI IV, V, VI, VII II, III, IV, V, VI, VII I, II, IV, V A  B  C  II, IV, V B  C   A II, IV, V Both statements are represented by the same regions, II, IV, V, of the Venn diagram. Therefore, A   B  C    B  C   A for all sets A, B, and C. Copyright © 2021 Pearson Education, Inc.