Solution Manual for Using and Understanding Mathematics: A Quantitative Reasoning Approach, 7th edition

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SOLUTIONSMANUALJAMESLAPPUSING&UNDERSTANDINGMATHEMATICS:AQUANTITATIVEREASONINGAPPROACHSIXTHEDITIONJeffrey BennettUniversity of Colorado at BoulderWilliam BriggsUniversity of Colorado at Denver

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Table of ContentsChapter 1: Thinking CriticallyUnit 1A: Living in the Media Age ..................................................................................... 1Unit 1B: Propositions and Truth Values ............................................................................ 3Unit 1C: Sets and Venn Diagrams ..................................................................................... 8Unit 1D: Analyzing Arguments ....................................................................................... 16Unit 1E: Critical Thinking in Everyday Life ................................................................... 20Chapter 2: Approaches to Problem SolvingUnit 2A: Working with Units........................................................................................... 23Unit 2B: Problem Solving with Units .............................................................................. 29Unit 2C: Problem-Solving Guidelines and Hints............................................................. 38Chapter 3: Numbers in the Real WorldUnit 3A: Uses and Abuses of Percentages ....................................................................... 45Unit 3B: Putting Numbers in Perspective ........................................................................ 49Unit 3C: Dealing with Uncertainty .................................................................................. 55Unit 3D: Index Numbers: The CPI and Beyond .............................................................. 59Unit 3E: How Numbers Deceive: Polygraphs, Mammograms, and More....................... 62Chapter 4: Managing MoneyUnit 4A: Taking Control of Your Finances ..................................................................... 67Unit 4B: The Power of Compounding ............................................................................. 70Unit 4C: Savings Plans and Investments ......................................................................... 76Unit 4D: Loan Payments, Credit Cards, and Mortgages.................................................. 82Unit 4E: Income Taxes..................................................................................................... 91Unit 4F: Understanding the Federal Budget .................................................................... 96Chapter 5: Statistical ReasoningUnit 5A: Fundamentals of Statistics............................................................................... 101Unit 5B: Should You Believe a Statistical Study?......................................................... 104Unit 5C: Statistical Tables and Graphs .......................................................................... 107Unit 5D: Graphics in the Media ..................................................................................... 112Unit 5E: Correlation and Causality ................................................................................ 116Chapter 6: Putting Statistics to WorkUnit 6A: Characterizing Data......................................................................................... 121Unit 6B: Measures of Variation ..................................................................................... 125Unit 6C: The Normal Distribution ................................................................................. 129Unit 6D: Statistical Inference......................................................................................... 132

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Chapter 7: Probability: Living with the OddsUnit 7A: Fundamentals of Probability ........................................................................... 137Unit 7B: Combining Probabilities.................................................................................. 140Unit 7C: The Law of Large Numbers ............................................................................ 143Unit 7D: Assessing Risk ................................................................................................ 146Unit 7E: Counting and Probability................................................................................. 150Chapter 8: Exponential AstonishmentUnit 8A: Growth: Linear versus Exponential ................................................................ 155Unit 8B: Doubling Time and Half-Life.......................................................................... 157Unit 8C: Real Population Growth .................................................................................. 162Unit 8D: Logarithmic Scales: Earthquakes, Sounds, and Acids .................................... 165Chapter 9: Modeling Our WorldUnit 9A: Functions: The Building Blocks of Mathematical Models ............................. 169Unit 9B: Linear Modeling .............................................................................................. 174Unit 9C: Exponential Modeling ..................................................................................... 179Chapter 10: Modeling with GeometryUnit 10A: Fundamentals of Geometry........................................................................... 187Unit 10B: Problem Solving with Geometry................................................................... 190Unit 10C: Fractal Geometry........................................................................................... 196Chapter 11: Mathematics and the ArtsUnit 11A: Mathematics and Music ................................................................................ 201Unit 11B: Perspective and Symmetry ............................................................................ 203Unit 11C: Proportion and the Golden Ratio................................................................... 206Chapter 12: Mathematics and PoliticsUnit 12A: Voting: Does the Majority Always Rule?..................................................... 209Unit 12B: Theory of Voting ........................................................................................... 214Unit 12C: Apportionment: The House of Representatives and Beyond ........................ 218Unit 12D: Dividing the Political Pie .............................................................................. 228

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UNIT 1A: LIVING IN THE MEDIA AGE1UNIT 1ATIME OUT TO THINKPg. 7. Not guilty does not mean innocent; it means notenough evidence to prove guilt. If defendants wererequired to prove innocence, there would be manycases where they would be unable to provide suchproof even though they were, in fact, innocent.This relates to the fallacy of appeal to ignorance inthe sense that lack of proof of guilt does not meaninnocence, and lack of proof of innocence does notmean guilt.Pg. 9.Opinionswillvary.Oneargumentisthatcharacter questions should be allowed in court ifanswerstothosequestionsmay showbiasorulterior motives for testimony given by a witness.This would be a good topic for a discussion eitherduring or outside of class.QUICK QUIZ1.a. By the definition used in this book, an argumentalwayscontainsatleastonepremiseandaconclusion.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 thepremise and the conclusion say essentially thesame thing.5.b. Using the fact that a statement is unproved toimply that it is false is appeal to ignorance.6.b. “I don’t support the President’s tax plan” is theconclusion because the premise “I don’t trust hismotives” supports that conclusion.7.b. This is a personal attack because the premise (Idon’t trust his motives) attacks the character of thePresident, and says nothing about the substance ofhis tax plan.8.c. This is limited choice because the argumentdoes not allow for the possibility that you are a fanof, say, boxing.9.b. Just becauseAprecededBdoes not necessarilyimply thatAcausedB.10.a. By definition, a straw man is an argument thatdistorts (or misrepresents) the real issue.DOES IT MAKE SENSE?5.Does not make sense. Raising one’s voice hasnothing to do with logical arguments.6.Does not make sense. Logical arguments alwayscontain at least one premise and a conclusion.7.Makes sense.A logicalpersonwouldnotputmuch faith in an argument that uses premises hebelieves to be false to support a conclusion.8.Makes sense. There’s nothing wrong with statingthe conclusion of an argument before laying outthe premises.9.Does not make sense. One can disagree with theconclusion of a well-stated argument regardless ofwhether it is fallacious.10.Makes sense. Despite the fact that an argumentmay be poorly constructed and fallacious, it stillmay have a believable conclusion.BASIC SKILLS AND CONCEPTS11.a.Premise:Apple’s iPhone outsells all other smartphones.Conclusion:It must be the best smartphone on the market.b. The fact that many people buy the iPhone doesnot necessarily mean it is the best smart phone.12.a.Premise: I became sick soon after eating atBurger Hut.Conclusion:Burger Hut food mademe sick.b. The argument doesn’t prove that Burger Hutfood was the cause of the sickness.13.a.Premise:Decadesofsearchinghavenotrevealed life on other planets.Conclusion:Life inthe universe must be confined to Earth.b. Failure to find life does not imply that life doesnot exist.14.a.Premise: I saw three people use food stamps tobuy expensive steaks.Conclusion:Abuse of foodstamps is widespread.b.Theconclusionisbasedon relativelylittleevidence.15.a.Premise: He refused to testify.Conclusion: Hemust be guilty.b. There are many reasons that someone mighthave for refusing to testify (being guilty is onlyone of them), and thus this is the fallacy of limitedchoice.16.a.Premise:Thousands of unarmed people, manyof them children, are killed by firearms every year.Conclusion:Thesaleofallgunsshouldbebanned.b. The conclusion is reached on the basis of anemotional statement.

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2CHAPTER 1: THINKING CRITICALLY17.a.Premise:SenatorSmithissupportedbycompaniesthatsellgeneticallymodifiedcropseeds.Conclusion:Senator Smith’s bill is a sham.b.AclaimaboutSenatorSmith’spersonalbehavior 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 intocollege, and a college diploma is necessary for agood career.Conclusion:Attendance should countin high school grades.b.Thepremise(whichisoftentrue)directsattention away from the conclusion.20.a.Premise:The mayor wants to raise taxes to fundsocial programs.Conclusion:She must not believein the value of hard work.b. The mayor is characterized (perhaps wrongly)by one quality, on which the conclusion is based.21.False22.True23.False24.TrueFURTHER APPLICATIONS25.Premise:Obesity among Americans has increasedsteadily,ashasthesaleofvideogames.Conclusion:Video games are compromising thehealth of Americans. This argument suffers fromthefalse causefallacy. It’s true obesity and videogame sales have increased steadily for the lastdecade, but we cannot conclude that the lattercaused the former simply because they happenedtogether.26.Premise:The Republican candidate leads by a2-to-1 margin.Conclusion:You should vote fortheRepublican.Thisisablatantappealtopopularity. No argument concerning the platformof the candidate is given.27.Premise:All the mayors of my home town havebeen men.Conclusion:Men are better qualifiedfor high office than women. The conclusion hasbeenreachedwithahastygeneralization,because a small number of male mayors were usedas evidence to support a claim about all men andwomen.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 isapersonalattackonthefather’spasttransgressions, which should play little part in thechild’s logical decision about whether to exercise.29.Premise:Mybabywasvaccinatedandlaterdevelopedautism.Conclusion:Ibelievethatvaccines cause autism.False causeis at play here,as the vaccination may have nothing to do with thedevelopmentofautism,eventhoughbothareoccurring at the same time.30.Premise:The state has no right to take a life.Conclusion:Thedeathpenaltyshouldbeabolished.Both the premise and conclusion sayessentiallythesamething;thisiscircularreasoning.31.Premise:Shakespeare’s plays have been read formanycenturies.Conclusion:EveryonelovesShakespeare.Both the premise and conclusionsay essentially the same thing; this iscircularreasoning.32.Premise:I’ve never heard of anyone getting sickfrom GMO foods.Conclusion:Claims that GMOfoods are unsafe are ridiculous. This is anappealto ignorance: the lack of knowledge of caseswhere GMO foods have caused health issues doesnot mean they don’t.33.Premise:After I last gave to a charity, an auditshowed that most of the money was used to pay itsadministrators in the front office.Conclusion:Iwill not give money to the earthquake relief effort.This is apersonal attackon charities. It can alsobe seen as anappeal to ignorance: the lack ofexamples of charities passing donations on to theintended recipients does not mean that a charitywill not pass on donations.34.Premise:Democrats don’t care about taxpayers’money.Conclusion:It’s not surprising that thePresident’s budgetcontainsspending increases.This islimited choice: the premise does not allowfor thepossibility that theDemocrats do careabout taxpayers’ money.35.Premise:The Congressperson is a member of theNational Rifle Association.Conclusion:I’m sureshe will not support a ban on assault rifles. This isapersonal attackon members of the NationalRifle Association. The argument also distorts theposition of the National Rifle Association (not allmembers would oppose a ban on assault rifles);this is astraw man.36.Premise:My three friends who drink wine havenever had heart attacks.Premise:My two friendswhohavehadheartattacksarenon-drinkers.Conclusion:Drinkingwineisclearlyagoodtherapy. The conclusion has been reached with ahasty generalization, because a small number ofwine drinkers were used as evidence to support aclaim about drinking wine.

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UNIT 1B: PROPOSITIONS AND TRUTH VALUES337.Premise:TheRepublicansfavorrepealingtheestatetax,whichfallsmostheavilyonthewealthy.Conclusion:Republicansthinkthewealthy aren’t rich enough. (Implied here is thatyou should vote for Democrats). The argumentdistorts the position of the Republicans; this is astraw man.38.Premise: The Wyoming toad has not been seenoutside of captivity since 2002.Conclusion: It isextinct in the wild.Appeal to ignoranceis usedhere – the lack of proof of the existence of thewoodpecker does not imply it is extinct.39.Premise: My boy loves dolls, and my girl lovestrucks.Conclusion: There’s no truth to the claimthat boys prefer mechanical toys while girls prefermaternal toys. Using one child of each gender tocome up with a conclusion about all children ishasty generalization. It can also be seen as anappeal to ignorance: the lack of examples of boysenjoying mechanical toys (and girls maternal toys)does not mean that they don’t enjoy these toys.40.Premise: The Democrats want to raise gas mileagerequirementsonnewvehicles.Conclusion:Democrats think the government is the solution toall of our problems. The argument distorts theposition of the Democrats; this is astraw man.UNIT 1BTIME OUT TO THINKPg. 18. We needed 8 rows for 3 propositions; adding afourth proposition means two possible truth valuesfor each of those 8 rows, or 16 rows total. Theconjunction is true only if all four propositions aretrue.Pg. 20. The precise definitions of logic sometimesdiffer from our “everyday” intuition. There is nopossible way that Jones could personally eliminateall poverty on Earth, regardless of whether she iselected. Thus, at the time you heard her make thispromise, you would certainly conclude that shewasbeinglessthantruthful.Nevertheless,according to the rules of logic, the only way herstatement can be false is if she is elected, in whichcase she would be unable to follow through on thepromise. If she is not elected, her claim is true (atleast according to the laws of logic).QUICK QUIZ1.c. This is a proposition because it is a completesentence making a claim, which could be true orfalse.2.a. The truth value of a proposition’s negation (notp) can always be determined by the truth value ofthe proposition.3.c. Conditional statements are, by definition, in theform ofif p, then q.4.c. The table will require eight rows because therearetwopossibletruthvaluesforeachofthepropositionsx,y, andz.5.c.Becauseitisnotstatedotherwise,wearedealing with the inclusiveor(and thus eitherpistrue, orqis true, or both are true).6.a. The conjunctionp and qis true only when bothare true, and sincepis false,p and qmust also befalse.7.b. This is the correct rephrasing of the originalconjunction.8.c.Thisisthecontrapositiveoftheoriginalconjunction.9.b. Statements are logically equivalent only whenthey have the same truth values.10.a. Rewriting the statement inif p, then qformgives, “if you want to win, then you’ve got toplay.”DOES IT MAKE SENSE?7.Doesnotmakesense.Propositionsareneverquestions.8.Makes sense. The Mayor’s stance on banning gunsindicates he supports gun control.9.Makes sense. If restated inif p, then qform, thisstatement would read, “If we catch him, then hewill be dead or alive.” Clearly this is true, as itcoversallthepossibilities.(Onecouldarguesemantics,andsaythatadeadpersonisnotcaught, but rather discovered. Splitting hairs likethis might lead one to claim the statement does notmake sense).10.Does not make sense. The first statement is in theif p, then qform, and the second is the converse(i.e.if q, then p). Since the converse of anif…thenstatement is not logically equivalent to the originalstatement, this doesn’t make sense.11.Does not make sense. Not all statements fall underthe purview of logical analysis.12.Does not make sense. The converse of a statementis not always false if the original statement is true.

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4CHAPTER 1: THINKING CRITICALLYBASIC SKILLS AND CONCEPTS13.Since it’s a complete sentence that makes a claim(whethertrueorfalseisimmaterial),it’saproposition.14.No claim is made with this statement, so it’s not aproposition.15.No claim is made with this statement, so it’s not aproposition.16.This is a complete sentence that makes a claim, soit’s a proposition.17.Questions are never propositions.18.This is a proposition as we can assign a truth valueto it, and it’s a complete sentence.19.Asiaisnotinthenorthernhemisphere.Thestatement is false; the negation is true.20.Spain is not in North America. The statement isfalse; the negation is true.21.TheBeatleswerenotaGermanband.Thestatement is false; the negation is true.22.Brad Pitt is an American actor. The statement isfalse; the negation is true.23.Sarah did go to dinner.24.TheSenatorappearstoapproveofthedemonstrations. Whether he approves of them isdebatable, given the limited information.25.TheCongressmanvotedinfavorofdiscrimination.26.The Senate failed to push the bill through to stoplogging (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 lawprohibiting cell phones in public meetings, themayor appears to support the use of cell phones inpublic meetings.29.This is the truth table for the conjunctionq and r.qrq and rTTTTFFFTFFFF30.This is the truth table for the conjunctionp and s.psp and sTTTTFFFTFFFF31.“Cucumbers are vegetables” is true. “Apples arefruit” is true. Since both propositions are true, theconjunction is true.32.“12 + 6 = 18” is true, but “3×5 = 8” is false. Theconjunction is false because both propositions in aconjunction must be true for the entire statementto be true.33.“The Mississippi River flows through Louisiana”istrue.“TheColoradoRiverflowsthroughArizona” is true. Since both propositions are true,the conjunction is true.34.“Bach was a composer” is true, but “Bono is aviolinist” is false. The conjunction is false becauseboth propositions in a conjunction must be true forthe entire statement to be true.35.“Some people are happy” is true (in general), as is“Some people are short,” so the conjunction istrue.36.“Not all dogs are black” is true. “Not all cats arewhite” is also true, so the conjunction is true.37.This is the truth table forq and r and s.qrsq and r and sTTTTTTFFTFTFTFFFFTTFFTFFFFTFFFFF38.This is the truth table forp and q and r and s.pqrsp and q and r and sTTTTTTTTFFTTFTFTTFFFTFTTFTFTFFTFFTFTFFFFFTTTFFTTFFFTFTFFTFFFFFTTFFFTFFFFFTFFFFFF

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UNIT 1B: PROPOSITIONS AND TRUTH VALUES539.Oris used in the exclusive sense because youprobably can’t wear both a skirt and a dress.40.Oris used in the exclusive sense because youprobably can’t have both the salad and soup.41.The exclusiveoris used here as it is unlikely thatthestatementmeansyou might travel tobothcountries during the same trip.42.Oil changes are good for either 3 months or 5,000miles,whichevercomesfirst,sothisistheexclusive use ofor.43.Oris used in the inclusive sense because youprobably would be thrilled to attend both concertsor 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, sothis is the inclusiveor.45.This is the truth table for the disjunctionr or s.rsr or sTTTTFTFTTFFF46.This is the truth table for the disjunctionp or r.prp or rTTTTFTFTTFFF47.This is the truth table forp and(not p).pnot pp and(not p)TFFFTF48.This is the truth table forq or(not q).qnot qq or(not q)TFTFTT49.This is the truth table forp or q or r.pqrp or q or rTTTTTTFTTFTTTFFTFTTTFTFTFFTTFFFF50.This is the truth table forp or(not p)or q.p(not p)qp or(not p)or qTFTTTFFTFTTTFTFT51.“Oranges are vegetables” is false. “Oranges arefruits” is true. The disjunction is true because adisjunctionistruewhenatleastoneofitspropositions is true.52.Both “3×5 = 15” and “3 + 5 = 8” are true, andthus the disjunction is true, as all you need is oneproposition or the other to be true for the statementto be true.53.“The Nile River is in Africa” is true. “China is inEurope” is false. The disjunction is true because adisjunctionistruewhenatleastoneofitspropositions is true.54.“Bachelors are married” is false. “Bachelors aresingle” is true. The disjunction is true because atleast 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 leastone of the propositions is true.57.This is the truth table forif p, then r.prif p, then rTTTTFFFTTFFT58.This is the truth table forif q, then s.qsif q, then sTTTTFFFTTFFT59.Hypothesis:Eagles can fly.Conclusion:Eaglesare birds. Since both are true, the implication istrue, because implications are always true exceptin the case where the hypothesis is true and theconclusion is false.60.Hypothesis:London is in England.Conclusion:Chicago is in America. Since both are true, theimplication is true.

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6CHAPTER 1: THINKING CRITICALLY61.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 simpleif p, then qstatement is false).62.Hypothesis:London is in Mongolia.Conclusion:Chicago is in America. Since the hypothesis isfalse, the implication is true, no matter the truthvalue of the conclusion (which, in this case, istrue).63.Hypothesis:Pigs can fly.Conclusion:Fish canbrush their teeth. Since the hypothesis is false, theimplication is true, no matter the truth value of theconclusion (which, in this case, is false).64.Hypothesis:2×3 = 6Conclusion:2 + 3 = 6.Since the hypothesis is true, and the conclusion isfalse, the implication is false.65.Hypothesis:Butterfliescanfly.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 simpleif p, then qstatement is false).66.Hypothesis:Butterfliesarebirds.Conclusion:Butterflies can fly. Since the hypothesis is false,the implication is true, no matter the truth value ofthe 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 thatperson 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 acomputer.Inverse:IfJosédoesnotownacomputer,thenhedoesnotownaMac.Contrapositive:If José does not own a Mac, thenhe does not own a computer. The converse andinverse are always logically equivalent, and thecontrapositive is always logically equivalent to theoriginal statement.74.Converse:If the patient is breathing, then thepatient is alive.Inverse:If the patient is not alive,then the patient is not breathing.Contrapositive:Ifthe patient is not breathing, then the patient is notalive.Theconverseandinversearealwayslogicallyequivalent,andthecontrapositiveisalwayslogicallyequivalenttotheoriginalstatement.75.Converse:If Teresa works in Massachusetts, thenshe works in Boston.Inverse:If Teresa does notworkinBoston,thenshedoesnotworkinMassachusetts.Contrapositive:If Teresa does notwork in Massachusetts, then she does not work inBoston.Theconverseandinversearealwayslogicallyequivalent,andthecontrapositiveisalwayslogicallyequivalenttotheoriginalstatement.76.Converse:If the lights are on, then I am usingelectricity.Inverse:If I am not using electricity,then the lights are not on.Contrapositive:If thelights are not on, then I am not using electricity.The converse and inverse are always logicallyequivalent,andthecontrapositiveisalwayslogically equivalent to the original statement.77.Converse:If it is warm outside, then the sun isshining.Inverse:If the sun is not shining, then it isnot warm outside.Contrapositive:If it is not warmoutside, then the sun is not shining. The converseand inverse are always logically equivalent, andthe contrapositive is always logically equivalent tothe original statement.78.Converse:If the oceans rise, then the polar icecaps will have melted.Inverse:If the polar icecaps do not melt, then the oceans will not rise.Contrapositive:If the oceans do not rise, then thepolar ice caps will not have melted. The converseand inverse are always logically equivalent, andthe contrapositive is always logically equivalent tothe original statement.FURTHER APPLICATIONS79.If you die young, then you are good.80.If a man hasn’t discovered something that he willdie for, then he isn’t fit to live.81.If a free society cannot help the many who arepoor, then it cannot save the few who are rich.82.Ifyoudon’tlikesomething,thenyoushouldchange it. If you can’t change it, then you shouldchange your attitude.83.“If Sue lives in Cleveland, then she lives in Ohio,”where it is assumed that Sue lives in Cincinnati.(Answerswillvary.)BecauseSuelivesinCincinnati,thehypothesisisfalse,whiletheconclusion is true, and this means the implicationis true. The converse, “If Sue lives in Ohio, thenshelivesinCleveland,”isfalse,becausethehypothesis is true, but the conclusion is false.

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UNIT 1B: PROPOSITIONS AND TRUTH VALUES784.“If 2 + 2 = 4, then 3 + 3 = 6.” (Answers will vary.)The implication is true, because the hypothesis istrue and the conclusion is true. The converse, “If 3+ 3 = 6, then 2 + 2 = 4” is also true for the samereason.85.“If Ramon lives in Albuquerque, then he lives inNew Mexico” where it is assumed that Ramonlives in Albuquerque. (Answers will vary.) Theimplication is true, because the hypothesis is trueand the conclusion is true. The contrapositive, “IfRamon does not live in New Mexico, then he doesnot live in Albuquerque”, is logically equivalent tothe original conditional, so it is also true.86.“If Delaware is in America, then Maryland is inCanada.” (Answers will vary.) The hypothesis istrue, while the conclusion is false, and this meanstheimplicationisfalse.Intheinverse,“IfDelaware is not in America, then Maryland is notinCanada,”thehypothesisisfalse,whiletheconclusion is true, and this means the implicationis true.87.“If it is a fruit, then it is an apple.” (Answers willvary.) The implication is false because, when thehypothesis is true, the conclusion may be false (itcould be an orange). In the converse, “If it is anapple, then it is a fruit.”, when the hypothesis istrue, the conclusion is true, and this means theimplication 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, thenalimony you receive is not taxable.(3) If you pay alimony to another party, then it isnot deductible on your return.89.Believing is sufficient for achieving. Achieving isnecessary for believing.90.Ourspeciesbeingaloneintheuniverseissufficient for the universe having aimed ratherlow. The universe having aimed rather low isnecessaryforourspeciesbeingaloneintheuniverse.91.Forgetting that we are One Nation Under God issufficient for being a nation gone under. Being anationgoneunderisanecessaryresultofforgetting that we are One Nation Under God.92.Needing both of your hands for whatever it isyou’re doing is sufficient for your brain being inon it too. Your brain being in on it too is necessaryfor needing both of your hands for whatever it isyou’re doing.93.Following is a truth table for bothnot(p and q)and (not p)or(not q).pqp and qnot(p and q)(not p)or(not q)TTTFFTFFTTFTFTTFFFTTSince both statements have the same truth values(compare the last two columns of the table), theyare logically equivalent.94.Following is a truth table for bothnot(p or q) and(not p)and(not q).pqp or qnot(p or q)(not p)and(not q)TTTFFTFTFFFTTFFFFFTTSince both statements have the same truth values(compare the last two columns in the table), theyare logically equivalent.95.Following is a truth table for bothnot(p and q)and (not p)and(not q).pqp and qnot(p and q)(not p)and(not q)TTTFFTFFTFFTFTFFFFTTNote that the last two columns in the truth tabledon’tagree,andthusthestatementsarenotlogically equivalent.96.Following is a truth table fornot(p or q) and (notp)or(not q).pqp or qnot(p or q)(not p)or(not q)TTTFFTFTFTFTTFTFFFTTNote that the last two columns in the truth tabledon’tagree,andthusthestatementsarenotlogically equivalent.

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8CHAPTER 1: THINKING CRITICALLY97.Following is a truth table for (p and q)or rand (p or r)and(p or q).pqrp and q(p and q)or rp or rp or q(p or r)and(p or q)TTTTTTTTTTFTTTTTTFTFTTTTTFFFFTTTFTTFTTTTFTFFFFTFFFTFTTFFFFFFFFFFSince 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 rand (p and r)or(q and r).pqrp or q(p or q)and rp and rq and r(p and r)or(q and r)TTTTTTTTTTFTFFFFTFTTTTFTTFFTFFFFFTTTTFTTFTFTFFFFFFTFFFFFFFFFFFFFSince the fifth and eighth columns agree, the statements are logically equivalent.99.Given the implicationif p, then q, the contrapositive is (not q)then(not p). The converse isif q, then pand theinverse of the converse isif(not q)then(not p), which is the contrapositive. Similarly, the contrapositive isalso the converse of the inverse.UNIT 1CTIME OUT TO THINKPg. 26. The set of students in the mathematics classcould be described by writing each student’s namewithin the braces, separated by commas. The setof countries you have visited would be writtenwith the names of the countries within the braces.Additional examples will vary.Pg. 32. The student should see that the statementsome teachers are not men leaves both questionsposed in the Time Out unanswered. Thus, from thestatement given, it is not possible to know whethersome teachers are men. From this, it also followsthat we cannot be sure that none of the teachersare men.Pg. 33. Changing the circle for boys to girls is fine,since a teenager is either one or the other. It wouldalso be fine to change the circle for employed tounemployed.Butthesetgirls,boys,andunemployed does not work because it offers noplace to record if the teenager is an honor student.Pg. 34. This question should convince the student thatthe variety of colors on TVs and monitors is madefrom just red, green, and blue. Higher-resolutionmonitorsusesmallerormoredenselypackedpixels (or both).Pg. 35. The two sets in this case are the opposites ofthe two sets chosen for Figure 1.24, so they workequally well.QUICK QUIZ1.b. The ellipsis is a convenient way to represent allthe other states in the U.S. without having to writethem all down.2.c.123isarationalnumber(aratiooftwointegers), but it is not an integer.3.a. When the circle labeled C is contained withinthe circle labeled D, it indicates that C is a subsetof D.

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Solution Manual for Using and Understanding Mathematics: A Quantitative Reasoning Approach, 7th edition - Page 13 preview image

UNIT 1C: SETS AND VENN DIAGRAMS94.b. Since the set of boys is disjoint from the set ofgirls, the two circles should be drawn as non-overlapping circles.5.a. Because all apples are fruit, the set A should bedrawn within the set B (the set of apples is asubset of the set of fruits).6.c.Somecrosscountryrunnersmayalsobeswimmers, so their sets should be overlapping.7.a. The X is placed in the region wherebusinessexecutivesandworkingmothersoverlaptoindicate that there is at least one member in thatregion.8.c. The region X is within bothmalesandathletes,but not withinRepublicans.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 labeledLow Birth Weight is 32.DOES IT MAKE SENSE?7.Does not make sense. More likely than not, thepayments go to two separate companies.8.Does not makes sense. The set of jabbers is asubset of the set of wocks, but this does not meanthere could be no wocks outside the set of jabbers.9.Does not make sense. The number of students in aclass is a whole number, and whole numbers arenot in the set of irrational numbers.10.Makes sense. The students that ate breakfast couldbe represented by the inside of the circle and thosethat did not eat breakfast would be represented bytheareaoutsideofthecircle,butinsidetherectangle, or vice versa.11.Does not make sense. A Venn diagram shows onlythe relationship between members of sets, but doesnot have much to say about the truth value of acategorical proposition.12.Does not make sense. A Venn diagram is used toshow the relationship between members of sets,but it is not used to determine the truth value foran opinion.BASIC SKILLS AND CONCEPTS13.23 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.πis a real number.20.8is a real number.21.–34.45 is a rational number.22.98is 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.Becausesomemenareattorneys,thecirclesshould overlap.38.Because some nurses are skydivers, the circlesshould overlap.39.Water is a liquid, and thus the set of water is asubset of the set of liquids. This means one circleshould be contained within the other.40.No reptile is a bacteria, so these sets are disjoint,and the circles should not overlap.

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10CHAPTER 1: THINKING CRITICALLY41.Some novelists are also athletes, so the circlesshould overlap.42.No atheist is a Catholic bishop, so these sets aredisjoint, and the circles should not overlap.43.No rational number is an irrational number, sothese sets are disjoint, and the circles should notoverlap.44.All limericks are poems, so one circle should beplaced within the other.45.b. The subject iswidows, and the predicate iswomen.c.d.No, the diagram does not show evidence thatthere is a woman that is not a widow.46.b.The subject isworms, andthe predicate isbirds.c.d. No, since the sets are disjoint, they would haveno common members.47.a. All U.S. presidents are people over 30 years old.b. The subject isU.S. presidents, and the predicateispeople over 30 years old.c.d. Yes, no U.S. presidents are outside the set ofpeople over 30.48.a. All children are people that sing.b. The subject ischildren, and the predicate ispeople who can sing.c.d. No, adults are not addressed.49.a. No monkey is a gambling animal.b. The subject ismonkeys, and the predicate isgambling animals.c.d. No, since the sets are disjoint, the would haveno common members.50.a. No plumbers are people who cheat.b. The subject isplumbers, and the predicate ispeople who cheat.c.d. No, since the sets are disjoint, the would haveno common members.51.a. All winners are people who smile.b. The subject iswinners, and the predicate ispeople who smile.

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UNIT 1C: SETS AND VENN DIAGRAMS1151.(continued)c.d. Yes, since all winners are inside the set ofpeople that smile, no frowner can be a winner.52.b. The subject ismovie stars, and the predicate isredheads.c.d. No, the diagram gives no evidence that thereare blonde movie stars.53.54.55.56.57.58.

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12CHAPTER 1: THINKING CRITICALLY59.a. There are 16 women at the party that are under30.b. There are 22 men at the party that are not under30.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 over30.c. There are 37 men at the party.d. There are 50 people at the party that are notunder 30.61.62.63.a. There are 20 people at the conference that areunemployed women with a college degree.b. There are 22 people at the conference that areemployed men.c. There are 8 people at the conference that areemployed women without a college degree.d. There are 34 people at the conference that aremen.64.a. There are 6 people at the conference that areemployed men without a college degree.b. There are 24 people at the conference that areunemployed women.c. There are 3 people at the conference that areunemployed men without a college degree.d. There are 77 people at the conference.65.a.b.Addthenumbersintheregionsthatarecontained in theAandBPcircles, to find that 95peopletookantibioticsorbloodpressuremedication.c. Add the number of people that are in theBPcircle, but outside thePcircle, to arrive at 23people.d. Add the number of people that are in thePcircle. There are 82 such people.e. Use the region that is common to theAandBPcircles, but not contained in thePcircle, to findthat 15 people took antibiotics and blood pressuremedicine, but not pain medication.f. Add the numbers in the regions that are in atleast one of the three circles, to find that 117people took antibiotics or blood pressure medicineor pain medicine.66.a.b.TheregioncommontobothTV/radioandnewspapersshows that 26people use at leastTV/radioandnewspapers(some of these also usetheInternet).c. Add the number of people that are in any of theregions contained within the two circlesTV/RadioandInternet. There are 109 such people.d.UsetheregionsthatarecontainedintheTV/radioorInternetcircles, but not contained inthenewspaperscircle. There are 61 such people.e.AddthenumberofpeoplethatareintheInternetcircle, but outside of theTV/radiocircle,to arrive at 51 people.f.AddthenumberofpeoplethatareintheTV/radiocircle,butoutsideofthenewspapercircle, to arrive at 32 people.FURTHER APPLICATIONS67.a.FavorableReviewNon-favorableReviewTotalComedy823 – 8= 1523Non-comedy22 – 12= 101245 – 23= 22Total8 + 10= 1815 + 12= 2745

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