S OLUTIONS M ANUAL J AMES L APP U SING & U NDERSTANDING M ATHEMATICS : A Q UANTITATIVE R EASONING A PPROACH S IXTH E DITION Jeffrey Bennett University of Colorado at Boulder William Briggs University of Colorado at Denver Table of Contents Chapter 1: Thinking Critically Unit 1A: Living in the Media Age ..................................................................................... 1 Unit 1B: Propositions and Truth Values ............................................................................ 3 Unit 1C: Sets and Venn Diagrams ..................................................................................... 8 Unit 1D: Analyzing Arguments ....................................................................................... 16 Unit 1E: Critical Thinking in Everyday Life ................................................................... 20 Chapter 2: Approaches to Problem Solving Unit 2A: Working with Units........................................................................................... 23 Unit 2B: Problem Solving with Units .............................................................................. 29 Unit 2C: Problem-Solving Guidelines and Hints............................................................. 38 Chapter 3: Numbers in the Real World Unit 3A: Uses and Abuses of Percentages ....................................................................... 45 Unit 3B: Putting Numbers in Perspective ........................................................................ 49 Unit 3C: Dealing with Uncertainty .................................................................................. 55 Unit 3D: Index Numbers: The CPI and Beyond .............................................................. 59 Unit 3E: How Numbers Deceive: Polygraphs, Mammograms, and More....................... 62 Chapter 4: Managing Money Unit 4A: Taking Control of Your Finances ..................................................................... 67 Unit 4B: The Power of Compounding ............................................................................. 70 Unit 4C: Savings Plans and Investments ......................................................................... 76 Unit 4D: Loan Payments, Credit Cards, and Mortgages.................................................. 82 Unit 4E: Income Taxes..................................................................................................... 91 Unit 4F: Understanding the Federal Budget .................................................................... 96 Chapter 5: Statistical Reasoning Unit 5A: Fundamentals of Statistics............................................................................... 101 Unit 5B: Should You Believe a Statistical Study?......................................................... 104 Unit 5C: Statistical Tables and Graphs .......................................................................... 107 Unit 5D: Graphics in the Media ..................................................................................... 112 Unit 5E: Correlation and Causality ................................................................................ 116 Chapter 6: Putting Statistics to Work Unit 6A: Characterizing Data......................................................................................... 121 Unit 6B: Measures of Variation ..................................................................................... 125 Unit 6C: The Normal Distribution ................................................................................. 129 Unit 6D: Statistical Inference......................................................................................... 132 Chapter 7: Probability: Living with the Odds Unit 7A: Fundamentals of Probability ........................................................................... 137 Unit 7B: Combining Probabilities.................................................................................. 140 Unit 7C: The Law of Large Numbers ............................................................................ 143 Unit 7D: Assessing Risk ................................................................................................ 146 Unit 7E: Counting and Probability................................................................................. 150 Chapter 8: Exponential Astonishment Unit 8A: Growth: Linear versus Exponential ................................................................ 155 Unit 8B: Doubling Time and Half-Life.......................................................................... 157 Unit 8C: Real Population Growth .................................................................................. 162 Unit 8D: Logarithmic Scales: Earthquakes, Sounds, and Acids .................................... 165 Chapter 9: Modeling Our World Unit 9A: Functions: The Building Blocks of Mathematical Models ............................. 169 Unit 9B: Linear Modeling .............................................................................................. 174 Unit 9C: Exponential Modeling ..................................................................................... 179 Chapter 10: Modeling with Geometry Unit 10A: Fundamentals of Geometry........................................................................... 187 Unit 10B: Problem Solving with Geometry................................................................... 190 Unit 10C: Fractal Geometry........................................................................................... 196 Chapter 11: Mathematics and the Arts Unit 11A: Mathematics and Music ................................................................................ 201 Unit 11B: Perspective and Symmetry ............................................................................ 203 Unit 11C: Proportion and the Golden Ratio................................................................... 206 Chapter 12: Mathematics and Politics Unit 12A: Voting: Does the Majority Always Rule?..................................................... 209 Unit 12B: Theory of Voting ........................................................................................... 214 Unit 12C: Apportionment: The House of Representatives and Beyond ........................ 218 Unit 12D: Dividing the Political Pie .............................................................................. 228 UNIT 1A: LIVING IN THE MEDIA AGE 1 UNIT 1A TIME OUT TO THINK Pg. 7. Not guilty does not mean innocent; it means not enough evidence to prove guilt. If defendants were required to prove innocence, there would be many cases where they would be unable to provide such proof even though they were, in fact, innocent. This relates to the fallacy of appeal to ignorance in the sense that lack of proof of guilt does not mean innocence, and lack of proof of innocence does not mean guilt. Pg. 9. Opinions will vary. One argument is that character questions should be allowed in court if answers to those questions may show bias or ulterior motives for testimony given by a witness. This would be a good topic for a discussion either during or outside of class. QUICK QUIZ 1. a . By the definition used in this book, an argument always contains at least one premise and a conclusion. 2. c . By definition, a fallacy is a deceptive argument. 3. b . An argument must contain a conclusion. 4. a . Circular reasoning is an argument where the premise and the conclusion say essentially the same thing. 5. b . Using the fact that a statement is unproved to imply that it is false is appeal to ignorance. 6. b . “I don’t support the President’s tax plan” is the conclusion because the premise “I don’t trust his motives” supports that conclusion. 7. b . This is a personal attack because the premise (I don’t trust his motives) attacks the character of the President, and says nothing about the substance of his tax plan. 8. c . This is limited choice because the argument does not allow for the possibility that you are a fan of, say, boxing. 9. b . Just because A preceded B does not necessarily imply that A caused B . 10. a . By definition, a straw man is an argument that distorts (or misrepresents) the real issue. DOES IT MAKE SENSE? 5. Does not make sense. Raising one’s voice has nothing to do with logical arguments. 6. Does not make sense. Logical arguments always contain at least one premise and a conclusion. 7. Makes sense. A logical person would not put much faith in an argument that uses premises he believes to be false to support a conclusion. 8. Makes sense. There’s nothing wrong with stating the conclusion of an argument before laying out the premises. 9. Does not make sense. One can disagree with the conclusion of a well-stated argument regardless of whether it is fallacious. 10. Makes sense. Despite the fact that an argument may be poorly constructed and fallacious, it still may have a believable conclusion. BASIC SKILLS AND CONCEPTS 11. a. Premise: Apple’s iPhone outsells all other smart phones. Conclusion: It must be the best smart phone on the market. b. The fact that many people buy the iPhone does not necessarily mean it is the best smart phone. 12. a. Premise : I became sick soon after eating at Burger Hut. Conclusion: Burger Hut food made me sick. b. The argument doesn’t prove that Burger Hut food was the cause of the sickness. 13. a. Premise: Decades of searching have not revealed life on other planets. Conclusion: Life in the universe must be confined to Earth. b. Failure to find life does not imply that life does not exist. 14. a. Premise : I saw three people use food stamps to buy expensive steaks. Conclusion: Abuse of food stamps is widespread. b. The conclusion is based on relatively little evidence. 15. a. Premise : He refused to testify. Conclusion : He must be guilty. b. There are many reasons that someone might have for refusing to testify (being guilty is only one of them), and thus this is the fallacy of limited choice. 16. a. Premise: Thousands of unarmed people, many of them children, are killed by firearms every year. Conclusion: The sale of all guns should be banned. b. The conclusion is reached on the basis of an emotional statement. 2 CHAPTER 1: THINKING CRITICALLY 17. a. Premise: Senator Smith is supported by companies that sell genetically modified crop seeds. Conclusion: Senator Smith’s bill is a sham. b. A claim about Senator Smith’s personal behavior is used to criticize his bill. 18. a. Premise: Illegal immigration is against the law. Conclusion: Illegal immigrants are criminals. b. The conclusion is a restatement of the premise. 19. a. Premise: Good grades are needed to get into college, and a college diploma is necessary for a good career. Conclusion: Attendance should count in high school grades. b. The premise (which is often true) directs attention away from the conclusion. 20. a. Premise: The mayor wants to raise taxes to fund social programs. Conclusion: She must not believe in the value of hard work. b. The mayor is characterized (perhaps wrongly) by one quality, on which the conclusion is based. 21. False 22. True 23. False 24. True FURTHER APPLICATIONS 25. Premise: Obesity among Americans has increased steadily, as has the sale of video games. Conclusion: Video games are compromising the health of Americans. This argument suffers from the false cause fallacy. It’s true obesity and video game sales have increased steadily for the last decade, but we cannot conclude that the latter caused the former simply because they happened together. 26. Premise: The Republican candidate leads by a 2-to-1 margin. Conclusion: You should vote for the Republican. This is a blatant appeal to popularity . No argument concerning the platform of the candidate is given. 27. Premise: All the mayors of my home town have been men. Conclusion: Men are better qualified for high office than women. The conclusion has been reached with a hasty generalization , because a small number of male mayors were used as evidence to support a claim about all men and women. 28. Premise: My father says I should exercise daily. Premise: He never exercised when he was young. Conclusion: I don’t need to take his advice. This is a personal attack on the father’s past transgressions, which should play little part in the child’s logical decision about whether to exercise. 29. Premise: My baby was vaccinated and later developed autism. Conclusion: I believe that vaccines cause autism. False cause is at play here, as the vaccination may have nothing to do with the development of autism, even though both are occurring at the same time. 30. Premise: The state has no right to take a life. Conclusion: The death penalty should be abolished. Both the premise and conclusion say essentially the same thing; this is circular reasoning . 31. Premise: Shakespeare’s plays have been read for many centuries. Conclusion: Everyone loves Shakespeare. Both the premise and conclusion say essentially the same thing; this is circular reasoning . 32. Premise: I’ve never heard of anyone getting sick from GMO foods. Conclusion: Claims that GMO foods are unsafe are ridiculous. This is an appeal to ignorance : the lack of knowledge of cases where GMO foods have caused health issues does not mean they don’t. 33. Premise: After I last gave to a charity, an audit showed that most of the money was used to pay its administrators in the front office. Conclusion: I will not give money to the earthquake relief effort. This is a personal attack on charities. It can also be seen as an appeal to ignorance : the lack of examples of charities passing donations on to the intended recipients does not mean that a charity will not pass on donations. 34. Premise: Democrats don’t care about taxpayers’ money. Conclusion: It’s not surprising that the President’s budget contains spending increases. This is limited choice : the premise does not allow for the possibility that the Democrats do care about taxpayers’ money. 35. Premise: The Congressperson is a member of the National Rifle Association. Conclusion: I’m sure she will not support a ban on assault rifles. This is a personal attack on members of the National Rifle Association. The argument also distorts the position of the National Rifle Association (not all members would oppose a ban on assault rifles); this is a straw man . 36. Premise: My three friends who drink wine have never had heart attacks. Premise: My two friends who have had heart attacks are non-drinkers. Conclusion: Drinking wine is clearly a good therapy. The conclusion has been reached with a hasty generalization , because a small number of wine drinkers were used as evidence to support a claim about drinking wine. UNIT 1B: PROPOSITIONS AND TRUTH VALUES 3 37. Premise : The Republicans favor repealing the estate tax, which falls most heavily on the wealthy. Conclusion : Republicans think the wealthy aren’t rich enough. (Implied here is that you should vote for Democrats). The argument distorts the position of the Republicans; this is a straw man . 38. Premise : The Wyoming toad has not been seen outside of captivity since 2002. Conclusion : It is extinct in the wild. Appeal to ignorance is used here – the lack of proof of the existence of the woodpecker does not imply it is extinct. 39. Premise : My boy loves dolls, and my girl loves trucks. Conclusion : There’s no truth to the claim that boys prefer mechanical toys while girls prefer maternal toys. Using one child of each gender to come up with a conclusion about all children is hasty generalization . It can also be seen as an appeal to ignorance : the lack of examples of boys enjoying mechanical toys (and girls maternal toys) does not mean that they don’t enjoy these toys. 40. Premise : The Democrats want to raise gas mileage requirements on new vehicles. Conclusion : Democrats think the government is the solution to all of our problems. The argument distorts the position of the Democrats; this is a straw man . UNIT 1B TIME OUT TO THINK Pg. 18. We needed 8 rows for 3 propositions; adding a fourth proposition means two possible truth values for each of those 8 rows, or 16 rows total. The conjunction is true only if all four propositions are true. Pg. 20. The precise definitions of logic sometimes differ from our “everyday” intuition. There is no possible way that Jones could personally eliminate all poverty on Earth, regardless of whether she is elected. Thus, at the time you heard her make this promise, you would certainly conclude that she was being less than truthful. Nevertheless, according to the rules of logic, the only way her statement can be false is if she is elected, in which case she would be unable to follow through on the promise. If she is not elected, her claim is true (at least according to the laws of logic). QUICK QUIZ 1. c . This is a proposition because it is a complete sentence making a claim, which could be true or false. 2. a . The truth value of a proposition’s negation ( not p ) can always be determined by the truth value of the proposition. 3. c . Conditional statements are, by definition, in the form of if p, then q . 4. c . The table will require eight rows because there are two possible truth values for each of the propositions x , y , and z . 5. c . Because it is not stated otherwise, we are dealing with the inclusive or (and thus either p is true, or q is true, or both are true). 6. a . The conjunction p and q is true only when both are true, and since p is false, p and q must also be false. 7. b . This is the correct rephrasing of the original conjunction. 8. c . This is the contrapositive of the original conjunction. 9. b . Statements are logically equivalent only when they have the same truth values. 10. a . Rewriting the statement in if p, then q form gives, “if you want to win, then you’ve got to play.” DOES IT MAKE SENSE? 7. Does not make sense. Propositions are never questions. 8. Makes sense. The Mayor’s stance on banning guns indicates he supports gun control. 9. Makes sense. If restated in if p, then q form, this statement would read, “If we catch him, then he will be dead or alive.” Clearly this is true, as it covers all the possibilities. (One could argue semantics, and say that a dead person is not caught, but rather discovered. Splitting hairs like this might lead one to claim the statement does not make sense). 10. Does not make sense. The first statement is in the if p, then q form, and the second is the converse (i.e. if q, then p ). Since the converse of an if…then statement is not logically equivalent to the original statement, this doesn’t make sense. 11. Does not make sense. Not all statements fall under the purview of logical analysis. 12. Does not make sense. The converse of a statement is not always false if the original statement is true. 4 CHAPTER 1: THINKING CRITICALLY BASIC SKILLS AND CONCEPTS 13. Since it’s a complete sentence that makes a claim (whether true or false is immaterial), it’s a proposition. 14. No claim is made with this statement, so it’s not a proposition. 15. No claim is made with this statement, so it’s not a proposition. 16. This is a complete sentence that makes a claim, so it’s a proposition. 17. Questions are never propositions. 18. This is a proposition as we can assign a truth value to it, and it’s a complete sentence. 19. Asia is not in the northern hemisphere. The statement is false; the negation is true. 20. Spain is not in North America. The statement is false; the negation is true. 21. The Beatles were not a German band. The statement is false; the negation is true. 22. Brad Pitt is an American actor. The statement is false; the negation is true. 23. Sarah did go to dinner. 24. The Senator appears to approve of the demonstrations. Whether he approves of them is debatable, given the limited information. 25. The Congressman voted in favor of discrimination. 26. The Senate failed to push the bill through to stop logging (it did not overturn the President’s veto), so logging will continue. 27. Paul appears to support building the new dorm. 28. Since the mayor was trying to strike down a law prohibiting cell phones in public meetings, the mayor appears to support the use of cell phones in public meetings. 29. This is the truth table for the conjunction q and r . q r q and r T T T T F F F T F F F F 30. This is the truth table for the conjunction p and s . p s p and s T T T T F F F T F F F F 31. “Cucumbers are vegetables” is true. “Apples are fruit” is true. Since both propositions are true, the conjunction is true. 32. “12 + 6 = 18” is true, but “3 × 5 = 8” is false. The conjunction is false because both propositions in a conjunction must be true for the entire statement to be true. 33. “The Mississippi River flows through Louisiana” is true. “The Colorado River flows through Arizona” is true. Since both propositions are true, the conjunction is true. 34. “Bach was a composer” is true, but “Bono is a violinist” is false. The conjunction is false because both propositions in a conjunction must be true for the entire statement to be true. 35. “Some people are happy” is true (in general), as is “Some people are short,” so the conjunction is true. 36. “Not all dogs are black” is true. “Not all cats are white” is also true, so the conjunction is true. 37. This is the truth table for q and r and s . q r s q and r and s T T T T T T F F T F T F T F F F F T T F F T F F F F T F F F F F 38. This is the truth table for p and q and r and s . p q r s p and q and r and s T T T T T T T T F F T T F T F T T F F F T F T T F T F T F F T F F T F T F F F F F T T T F F T T F F F T F T F F T F F F F F T T F F F T F F F F F T F F F F F F UNIT 1B: PROPOSITIONS AND TRUTH VALUES 5 39. Or is used in the exclusive sense because you probably can’t wear both a skirt and a dress. 40. Or is used in the exclusive sense because you probably can’t have both the salad and soup. 41. The exclusive or is used here as it is unlikely that the statement means you might travel to both countries during the same trip. 42. Oil changes are good for either 3 months or 5,000 miles, whichever comes first, so this is the exclusive use of or . 43. Or is used in the inclusive sense because you probably would be thrilled to attend both concerts or the theater while in New York. 44. Most insurance policies that cover “fire or theft” allow for the coverage of both at the same time, so this is the inclusive or . 45. This is the truth table for the disjunction r or s . r s r or s T T T T F T F T T F F F 46. This is the truth table for the disjunction p or r . p r p or r T T T T F T F T T F F F 47. This is the truth table for p and ( not p ). p not p p and ( not p ) T F F F T F 48. This is the truth table for q or ( not q ). q not q q or ( not q ) T F T F T T 49. This is the truth table for p or q or r . p q r p or q or r T T T T T T F T T F T T T F F T F T T T F T F T F F T T F F F F 50. This is the truth table for p or ( not p ) or q . p ( not p ) q p or ( not p ) or q T F T T T F F T F T T T F T F T 51. “Oranges are vegetables” is false. “Oranges are fruits” is true. The disjunction is true because a disjunction is true when at least one of its propositions is true. 52. Both “3 × 5 = 15” and “3 + 5 = 8” are true, and thus the disjunction is true, as all you need is one proposition or the other to be true for the statement to be true. 53. “The Nile River is in Africa” is true. “China is in Europe” is false. The disjunction is true because a disjunction is true when at least one of its propositions is true. 54. “Bachelors are married” is false. “Bachelors are single” is true. The disjunction is true because at least one of the propositions is true. 55. “Trees walk” is false. “Rocks run” is also false. Since both are false, the disjunction is false. 56. “France is a country” is true. “Paris is a continent” is false. The disjunction is true because at least one of the propositions is true. 57. This is the truth table for if p, then r . p r if p, then r T T T T F F F T T F F T 58. This is the truth table for if q, then s . q s if q, then s T T T T F F F T T F F T 59. Hypothesis: Eagles can fly. Conclusion: Eagles are birds. Since both are true, the implication is true, because implications are always true except in the case where the hypothesis is true and the conclusion is false. 60. Hypothesis: London is in England. Conclusion: Chicago is in America. Since both are true, the implication is true. 6 CHAPTER 1: THINKING CRITICALLY 61. Hypothesis: London is in England. Conclusion: Chicago is in Bolivia. Since the hypothesis is true, and the conclusion is false, the implication is false (this is the only instance when a simple if p, then q statement is false). 62. Hypothesis: London is in Mongolia. Conclusion: Chicago is in America. Since the hypothesis is false, the implication is true, no matter the truth value of the conclusion (which, in this case, is true). 63. Hypothesis: Pigs can fly. Conclusion: Fish can brush their teeth. Since the hypothesis is false, the implication is true, no matter the truth value of the conclusion (which, in this case, is false). 64. Hypothesis: 2 × 3 = 6 Conclusion: 2 + 3 = 6. Since the hypothesis is true, and the conclusion is false, the implication is false. 65. Hypothesis: Butterflies can fly. Conclusion: Butterflies are birds. Since the hypothesis is true, and the conclusion is false, the implication is false (this is the only instance when a simple if p, then q statement is false). 66. Hypothesis: Butterflies are birds. Conclusion: Butterflies can fly. Since the hypothesis is false, the implication is true, no matter the truth value of the conclusion (which, in this case, is true). 67. If it rains ( p ), then I get wet ( q ). 68. If a person is a resident of Tel Aviv ( p ), then that person is a resident of Israel ( q ). 69. If you are eating ( p ), then you are alive ( q ). 70. If you are alive ( p ), then you eat ( q ). 71. If you are bald ( p ), then you are a male ( q ). 72. If she is an art historian ( p ), then she is educated ( q ). 73. Converse: If José owns a Mac, then he owns a computer. Inverse: If José does not own a computer, then he does not own a Mac. Contrapositive: If José does not own a Mac, then he does not own a computer. The converse and inverse are always logically equivalent, and the contrapositive is always logically equivalent to the original statement. 74. Converse: If the patient is breathing, then the patient is alive. Inverse: If the patient is not alive, then the patient is not breathing. Contrapositive: If the patient is not breathing, then the patient is not alive. The converse and inverse are always logically equivalent, and the contrapositive is always logically equivalent to the original statement. 75. Converse: If Teresa works in Massachusetts, then she works in Boston. Inverse: If Teresa does not work in Boston, then she does not work in Massachusetts. Contrapositive: If Teresa does not work in Massachusetts, then she does not work in Boston. The converse and inverse are always logically equivalent, and the contrapositive is always logically equivalent to the original statement. 76. Converse: If the lights are on, then I am using electricity. Inverse: If I am not using electricity, then the lights are not on. Contrapositive: If the lights are not on, then I am not using electricity. The converse and inverse are always logically equivalent, and the contrapositive is always logically equivalent to the original statement. 77. Converse: If it is warm outside, then the sun is shining. Inverse: If the sun is not shining, then it is not warm outside. Contrapositive: If it is not warm outside, then the sun is not shining. The converse and inverse are always logically equivalent, and the contrapositive is always logically equivalent to the original statement. 78. Converse: If the oceans rise, then the polar ice caps will have melted. Inverse: If the polar ice caps do not melt, then the oceans will not rise. Contrapositive: If the oceans do not rise, then the polar ice caps will not have melted. The converse and inverse are always logically equivalent, and the contrapositive is always logically equivalent to the original statement. FURTHER APPLICATIONS 79. If you die young, then you are good. 80. If a man hasn’t discovered something that he will die for, then he isn’t fit to live. 81. If a free society cannot help the many who are poor, then it cannot save the few who are rich. 82. If you don’t like something, then you should change it. If you can’t change it, then you should change your attitude. 83. “If Sue lives in Cleveland, then she lives in Ohio,” where it is assumed that Sue lives in Cincinnati. (Answers will vary.) Because Sue lives in Cincinnati, the hypothesis is false, while the conclusion is true, and this means the implication is true. The converse, “If Sue lives in Ohio, then she lives in Cleveland,” is false, because the hypothesis is true, but the conclusion is false. UNIT 1B: PROPOSITIONS AND TRUTH VALUES 7 84. “If 2 + 2 = 4, then 3 + 3 = 6.” (Answers will vary.) The implication is true, because the hypothesis is true and the conclusion is true. The converse, “If 3 + 3 = 6, then 2 + 2 = 4” is also true for the same reason. 85. “If Ramon lives in Albuquerque, then he lives in New Mexico” where it is assumed that Ramon lives in Albuquerque. (Answers will vary.) The implication is true, because the hypothesis is true and the conclusion is true. The contrapositive, “If Ramon does not live in New Mexico, then he does not live in Albuquerque”, is logically equivalent to the original conditional, so it is also true. 86. “If Delaware is in America, then Maryland is in Canada.” (Answers will vary.) The hypothesis is true, while the conclusion is false, and this means the implication is false. In the inverse, “If Delaware is not in America, then Maryland is not in Canada,” the hypothesis is false, while the conclusion is true, and this means the implication is true. 87. “If it is a fruit, then it is an apple.” (Answers will vary.) The implication is false because, when the hypothesis is true, the conclusion may be false (it could be an orange). In the converse, “If it is an apple, then it is a fruit.”, when the hypothesis is true, the conclusion is true, and this means the implication is true. 88. (1) If the payer does not know that you remarried, then alimony you receive is taxable. (2) If the payer knows that you remarried, then alimony you receive is not taxable. (3) If you pay alimony to another party, then it is not deductible on your return. 89. Believing is sufficient for achieving. Achieving is necessary for believing. 90. Our species being alone in the universe is sufficient for the universe having aimed rather low. The universe having aimed rather low is necessary for our species being alone in the universe. 91. Forgetting that we are One Nation Under God is sufficient for being a nation gone under. Being a nation gone under is a necessary result of forgetting that we are One Nation Under God. 92. Needing both of your hands for whatever it is you’re doing is sufficient for your brain being in on it too. Your brain being in on it too is necessary for needing both of your hands for whatever it is you’re doing. 93. Following is a truth table for both not ( p and q ) and ( not p ) or ( not q ). p q p and q not ( p and q ) ( not p ) or ( not q ) T T T F F T F F T T F T F T T F F F T T Since both statements have the same truth values (compare the last two columns of the table), they are logically equivalent. 94. Following is a truth table for both not ( p or q ) and ( not p ) and ( not q ). p q p or q not ( p or q ) ( not p ) and ( not q ) T T T F F T F T F F F T T F F F F F T T Since both statements have the same truth values (compare the last two columns in the table), they are logically equivalent. 95. Following is a truth table for both not ( p and q ) and ( not p ) and ( not q ). p q p and q not ( p and q ) ( not p ) and ( not q ) T T T F F T F F T F F T F T F F F F T T Note that the last two columns in the truth table don’t agree, and thus the statements are not logically equivalent. 96. Following is a truth table for not ( p or q ) and ( not p ) or ( not q ). p q p or q not ( p or q ) ( not p ) or ( not q ) T T T F F T F T F T F T T F T F F F T T Note that the last two columns in the truth table don’t agree, and thus the statements are not logically equivalent. 8 CHAPTER 1: THINKING CRITICALLY 97. Following is a truth table for ( p and q ) or r and ( p or r ) and ( p or q ). p q r p and q ( p and q ) or r p or r p or q ( p or r ) and ( p or q ) T T T T T T T T T T F T T T T T T F T F T T T T T F F F F T T T F T T F T T T T F T F F F F T F F F T F T T F F F F F F F F F F Since the fifth and eighth column of the table don’t agree, these two statements are not logically equivalent. 98. Following is a truth table for ( p or q ) and r and ( p and r ) or ( q and r ). p q r p or q ( p or q ) and r p and r q and r ( p and r ) or ( q and r ) T T T T T T T T T T F T F F F F T F T T T T F T T F F T F F F F F T T T T F T T F T F T F F F F F F T F F F F F F F F F F F F F Since the fifth and eighth columns agree, the statements are logically equivalent. 99. Given the implication if p, then q , the contrapositive is ( not q ) then ( not p ). The converse is if q, then p and the inverse of the converse is if ( not q ) then ( not p ), which is the contrapositive. Similarly, the contrapositive is also the converse of the inverse. UNIT 1C TIME OUT TO THINK Pg. 26. The set of students in the mathematics class could be described by writing each student’s name within the braces, separated by commas. The set of countries you have visited would be written with the names of the countries within the braces. Additional examples will vary. Pg. 32. The student should see that the statement some teachers are not men leaves both questions posed in the Time Out unanswered. Thus, from the statement given, it is not possible to know whether some teachers are men. From this, it also follows that we cannot be sure that none of the teachers are men. Pg. 33. Changing the circle for boys to girls is fine, since a teenager is either one or the other. It would also be fine to change the circle for employed to unemployed. But the set girls, boys, and unemployed does not work because it offers no place to record if the teenager is an honor student. Pg. 34. This question should convince the student that the variety of colors on TVs and monitors is made from just red, green, and blue. Higher-resolution monitors use smaller or more densely packed pixels (or both). Pg. 35. The two sets in this case are the opposites of the two sets chosen for Figure 1.24, so they work equally well. QUICK QUIZ 1. b . The ellipsis is a convenient way to represent all the other states in the U.S. without having to write them all down. 2. c . 1 2 3 is a rational number (a ratio of two integers), but it is not an integer. 3. a . When the circle labeled C is contained within the circle labeled D, it indicates that C is a subset of D. UNIT 1C: SETS AND VENN DIAGRAMS 9 4. b . Since the set of boys is disjoint from the set of girls, the two circles should be drawn as non- overlapping circles. 5. a . Because all apples are fruit, the set A should be drawn within the set B (the set of apples is a subset of the set of fruits). 6. c . Some cross country runners may also be swimmers, so their sets should be overlapping. 7. a . The X is placed in the region where business executives and working mothers overlap to indicate that there is at least one member in that region. 8. c . The region X is within both males and athletes , but not within Republicans . 9. a . The central region is common to all three sets, and so represents those who are male, Republican, and an athlete. 10. c . The sum of the entries in the column labeled Low Birth Weight is 32. DOES IT MAKE SENSE? 7. Does not make sense. More likely than not, the payments go to two separate companies. 8. Does not makes sense. The set of jabbers is a subset of the set of wocks, but this does not mean there could be no wocks outside the set of jabbers. 9. Does not make sense. The number of students in a class is a whole number, and whole numbers are not in the set of irrational numbers. 10. Makes sense. The students that ate breakfast could be represented by the inside of the circle and those that did not eat breakfast would be represented by the area outside of the circle, but inside the rectangle, or vice versa. 11. Does not make sense. A Venn diagram shows only the relationship between members of sets, but does not have much to say about the truth value of a categorical proposition. 12. Does not make sense. A Venn diagram is used to show the relationship between members of sets, but it is not used to determine the truth value for an opinion. BASIC SKILLS AND CONCEPTS 13. 23 is a natural number. 14. –45 is an integer. 15. 2/3 is a rational number. 16. –5/2 is a rational number. 17. 1.2345 is a rational number. 18. 0 is a whole number. 19. π is a real number. 20. 8 is a real number. 21. –34.45 is a rational number. 22. 98 is a real number. 23. π /4 is a real number. 24. 123/456 is a rational number. 25. –13/3 is a rational number. 26. –145.01 is a rational number. 27. π /129 is a real number. 28. 13,579,023 is a natural number. 29. {January, February, March, …, November, December} 30. {14, 16, 18, . . . , 96, 98} 31. {New Mexico, Oklahoma, Arkansas, Louisiana} 32. {4, 7, 10, 13, 16, 19} 33. {9, 16, 25} 34. The set has no members. 35. {3, 9, 15, 21, 27} 36. {a, e, i, o, u} 37. Because some men are attorneys, the circles should overlap. 38. Because some nurses are skydivers, the circles should overlap. 39. Water is a liquid, and thus the set of water is a subset of the set of liquids. This means one circle should be contained within the other. 40. No reptile is a bacteria, so these sets are disjoint, and the circles should not overlap. 10 CHAPTER 1: THINKING CRITICALLY 41. Some novelists are also athletes, so the circles should overlap. 42. No atheist is a Catholic bishop, so these sets are disjoint, and the circles should not overlap. 43. No rational number is an irrational number, so these sets are disjoint, and the circles should not overlap. 44. All limericks are poems, so one circle should be placed within the other. 45. b. The subject is widows , and the predicate is women . c. d. No, the diagram does not show evidence that there is a woman that is not a widow. 46. b. The subject is worms , and the predicate is birds . c. d. No, since the sets are disjoint, they would have no common members. 47. a. All U.S. presidents are people over 30 years old. b. The subject is U.S. presidents , and the predicate is people over 30 years old . c. d. Yes, no U.S. presidents are outside the set of people over 30. 48. a. All children are people that sing. b. The subject is children , and the predicate is people who can sing . c. d. No, adults are not addressed. 49. a. No monkey is a gambling animal. b. The subject is monkeys , and the predicate is gambling animals . c. d. No, since the sets are disjoint, the would have no common members. 50. a. No plumbers are people who cheat. b. The subject is plumbers , and the predicate is people who cheat . c. d. No, since the sets are disjoint, the would have no common members. 51. a. All winners are people who smile. b. The subject is winners , and the predicate is people who smile . UNIT 1C: SETS AND VENN DIAGRAMS 11 51. (continued) c. d. Yes, since all winners are inside the set of people that smile, no frowner can be a winner. 52. b. The subject is movie stars , and the predicate is redheads . c. d. No, the diagram gives no evidence that there are blonde movie stars. 53. 54. 55. 56. 57. 58. 12 CHAPTER 1: THINKING CRITICALLY 59. a. There are 16 women at the party that are under 30. b. There are 22 men at the party that are not under 30. c. There are 44 women at the party. d. There are 81 people at the party. 60. a. There are 15 men at the party that are under 30. b. There are 28 women at the party who are over 30. c. There are 37 men at the party. d. There are 50 people at the party that are not under 30. 61. 62. 63. a. There are 20 people at the conference that are unemployed women with a college degree. b. There are 22 people at the conference that are employed men. c. There are 8 people at the conference that are employed women without a college degree. d. There are 34 people at the conference that are men. 64. a. There are 6 people at the conference that are employed men without a college degree. b. There are 24 people at the conference that are unemployed women. c. There are 3 people at the conference that are unemployed men without a college degree. d. There are 77 people at the conference. 65. a. b. Add the numbers in the regions that are contained in the A and BP circles, to find that 95 people took antibiotics or blood pressure medication. c. Add the number of people that are in the BP circle, but outside the P circle, to arrive at 23 people. d. Add the number of people that are in the P circle. There are 82 such people. e. Use the region that is common to the A and BP circles, but not contained in the P circle, to find that 15 people took antibiotics and blood pressure medicine, but not pain medication. f. Add the numbers in the regions that are in at least one of the three circles, to find that 117 people took antibiotics or blood pressure medicine or pain medicine. 66. a. b. The region common to both TV/radio and newspapers shows that 26 people use at least TV/radio and newspapers (some of these also use the Internet ). c. Add the number of people that are in any of the regions contained within the two circles TV/Radio and Internet . There are 109 such people. d. Use the regions that are contained in the TV/radio or Internet circles, but not contained in the newspapers circle. There are 61 such people. e. Add the number of people that are in the Internet circle, but outside of the TV/radio circle, to arrive at 51 people. f. Add the number of people that are in the TV/radio circle, but outside of the newspaper circle, to arrive at 32 people. FURTHER APPLICATIONS 67. a. Favorable Review Non-favorable Review Total Comedy 8 23 – 8 = 15 23 Non-comedy 22 – 12 = 10 12 45 – 23 = 22 Total 8 + 10 = 18 15 + 12 = 27 45