GMAT考试Testprep数学精解2

　　If I eat nuts， then I break out in hives. This in turn can be symbolized as N――H.

　　Next， we interpret the clause there is a blemish on my hand to mean hives， which we symbolize as H. Substituting these symbolssintosthe argument yields the following diagram：

　　N――H

　　H

　　Therefore， N

　　The diagram clearly shows that this argument has the same structure as the g

　　iven argument. The answer， therefore， is 。

　　Denying the Premise Fallacy

　　A――B

　　~A

　　Therefore， ~B

　　The fallacy of denying the premise occurs when an if-then statement is prese

　　nted， its premise denied， and then its conclusion wrongly negated.

　　Example：

　　The senator will be reelected only if he opposes the new tax bill. But he wa

　　s defeated. So he must have supported the new tax bill.

　　The sentence The senator will be reelected only if he opposes the new tax b

　　ill contains an embedded if-then statement： If the senator is reelected， then he opposes the new tax bill. This in turn can be symbolized as R――~T. The sentence But　the senator was defeated can be reworded as He was not reelected， which　in turn can be symbolized as ~R. Finally， the sentence He must have supported the new tax bill can be symbolized as T. Using these symbols the argument can be diagrammed as follows：

　　R――~T

　　~R

　　Therefore， T

　　 This diagram clearly shows that the argument is committing the fallacy　of denying the premise. An if-then statement is made; its premise is negated; then its conclusion is negated.

　　Transitive Property

　　A――B

　　B――C

　　Therefore， A――C

　　These arguments are rarely difficult， provided you step back and take a birds-eye view. It may be helpful to view this structure as an inequality in mathematics. For example， 5 4 and 4 3， so 5 3.

　　Notice that the conclusion in the transitive property is also an if-then statement. So we dont know that C is true unless we know that A is true. However， if we add the premise A is true to the diagram， then we can conclude that C is true：

　　A――B

　　B――C

　　A

　　Therefore， C

　　As you may have anticipated， the contrapositive can be generalized to the transitive property：

　　A――B

　　B――C

　　~C

　　Therefore， ~A

　　Example：

　　If you work hard， you will be successful in America. If you are successful in America， you can lead a life of leisure. So if you work hard in America， you can live a life of leisure.

　　Let W stand for you work hard， S stand for you will be successful in America， and L stand for you can lead a life of leisure. Now the first sentence translates as W――S， the second sentence as S――L， and the conclusion as W――L. Combining these symbol statements yields the following diagram：

　　W――S

　　S――L

　　Therefore， W――L

　　The diagram clearly displays the transitive property.

　　DeMorgans Laws

　　~ = ~A or ~B

　　~ = ~A ~B

　　If you have taken a course in logic， you are probably familiar with these formulas. Their validity is intuitively clear： The conjunction AB is false when either， or both， of its parts are false. This is precisely what ~A or ~B　says. And the disjunction A or B is false only when both A and B are false，which is precisely what ~A and ~B says.

　　You will rarely get an argument whose main structure is based on these rules――they are too mechanical. Nevertheless， DeMorgans laws often help simplify，clarify，or transform parts of an argument. They are also useful with games.

　　Example：

　　It is not the case that either Bill or Jane is going to the party.

　　This argument can be diagrammed as ~， which by the second of DeMorgans laws simplifies to 。 This diagram tells us that neither of them is going to the party.

　　A unless B

　　~B――A

　　A unless B is a rather complex structure. Though surprisingly we use it with little thought or confusion in our day-to-day speech.

　　To see that A unless B is equivalent to ~B――A， consider the following situation：

　　Biff is at the beach unless it is raining.

　　Given this statement， we know that if it is not raining， then Biff is at the　beach. Now if we symbolize Biff is at the beach as B， and it is rainingas R， then the statement can be diagrammed as ~R――B.

　　CLASSIFICATION

　　In Logic II， we studied deductive arguments. However， the bulk of arguments　on the GMAT are inductive. In this section we will classify and study the major types of inductive arguments.

　　An argument is deductive if its conclusion necessarily follows from its premises――otherwise it is inductive. In an inductive argument， the author presents the premises as evidence or reasons for the conclusion. The validity of the conclusion depends on how compelling the premises are. Unlike deductive arguments， the conclusion of an inductive argument is never certain. The truth of the conclusion can range from highly likely to highly unlikely. In reasonable arguments， the conclusion is likely. In fallacious arguments， it is improbable. We will study both reasonable and fallacious arguments.

　　We will classify the three major types of inductive reasoning――generalization， analogy， and causal――and their associated fallacies.

　　Generalization

　　Generalization and analogy， which we consider in the next section， are the main tools by which we accumulate knowledge and analyze our world. Many people define generalization as inductive reasoning. In colloquial speech， the　phrase to generalize carries a negative connotation. To argue by generalization， however， is neither inherently good nor bad. The relative validity of　a generalization depends on both the context of the argument and the likelihood that its conclusion is true. Polling organizations make predictions by　generalizing information from a small sample of the population， which hopefully represents the general population. The soundness of their predictions depends on how representative the sample is and on its size. Clearly， the less comprehensive a conclusion is the more likely it is to be true.

　　Example：

　　During the late seventies when Japan was rapidly expanding its share of the　American auto market， GM surveyed owners of GM cars and asked them whether they would be more willing to buy a large， powerful car or a small， economical car. Seventy percent of those who responded said that they would prefer a　large car. On the basis of this survey， GM decided to continue building large cars. Yet during the80s， GM lost even more of the market to the Japanese

　　

　　Which one of the following， if it were determined to be true， would best explain this discrepancy.

　　 Only 10 percent of those who were polled replied.

　　 Ford which conducted a similar survey with similar results continued to　build large cars and also lost more of their market to the Japanese.

　　 The surveyed owners who preferred big cars also preferred big homes.

　　 GM determined that it would be more profitable to make big cars.

　　 Eighty percent of the owners who wanted big cars and only 40 percent of　the owners who wanted small cars replied to the survey.

　　The argument generalizes from the survey to the general car-buying population， so the reliability of the projection depends on how representative the sample is. At first glance， choice seems rather good， because 10 percent does not seem large enough. However， political opinion polls are typically based on only .001 percent of the population. More importantly， we dont know　what percentage of GM car owners received the survey. Choice simply states that Ford made the same mistake that GM did. Choice is irrelevant. Choice ， rather than explaining the discrepancy， gives even more reason for　GM to continue making large cars. Finally， choice points out that part　of the survey did not represent the entire public， so is the answer.

　　Analogy

　　To argue by analogy is to claim that because two things are similar in some　respects， they will be similar in others. Medical experimentation on animals　is predicated on such reasoning. The argument goes like this： the metabolism of pigs， for example， is similar to that of humans， and high doses of saccharine cause cancer in pigs. Therefore， high doses of saccharine probably cause cancer in humans.

　　Clearly， the greater the similarity between the two things being compared the stronger the argument will be. Also the less ambitious the conclusion the　stronger the argument will be. The argument above would be strengthened by changing probably to may. It can be weakened by pointing out the dissimil　arities between pigs and people.

　　Example：

　　Just as the fishing line becomes too taut， so too the trials and tribulations of life in the city can become so stressful that ones mind can snap.

　　Which one of the following most closely parallels the reasoning used in the　argument above?

　　 Just as the bow may be drawn too taut， so too may ones life be wasted pursuing self-gratification.

　　 Just as a gamblers fortunes change unpredictably， so too do ones career opportunities come unexpectedly.

　　 Just as a plant can be killed by over watering it， so too can drinking too much water lead to lethargy.

　　 Just as the engine may race too quickly， so too may life in the fast lane lead to an early death.

　　 Just as an actor may become stressed before a performance， so too may dwelling on the negative cause depression.

　　The argument compares the tautness in a fishing line to the stress of city life; it then concludes that the mind can snap just as the fishing line can.

　　So we are looking for an answer-choice that compares two things and draws a　conclusion based on their similarity. Notice that we are looking for an argument that uses similar reasoning， but not necessarily similar concepts. In fact， an answer-choice that mentions either tautness or stress will probably　be a same-language trap.

　　Choice uses the same-language trap――notice too taut. The analogy between a taut bow and self-gratification is weak， if existent. Choice offers　a good analogy but no conclusion. Choice offers both a good analogy and a conclusion; however， the conclusion， leads to lethargy， understates the　scope of what the analogy implies. Choice offers a strong analogy and a　conclusion with the same scope found in the original： the engine blows， the person dies the line snaps， the mind snaps. This is probably the best　answer， but still we should check every choice. The last choice， ， uses language from the original， stressful， to make its weak analogy more tempting. The best answer， therefore， is 。

　　Causal Reasoning

　　Of the three types of inductive reasoning we will discuss， causal reasoning　is both the weakest and the most prone to fallacy. Nevertheless， it is a useful and common method of thought.

　　To argue by causation is to claim that one thing causes another. A causal argument can be either weak or strong depending on the context. For example， to claim that you won the lottery because you saw a shooting star the night before is clearly fallacious. However， most people believe that smoking causes cancer because cancer often strikes those with a history of cigarette use.

　　Although the connection between smoking and cancer is virtually certain， as　with all inductive arguments it can never be 100 percent certain. Cigarette　companies have claimed that there may be a genetic predisposition in some people to both develop cancer and crave nicotine. Although this claim is highly improbable， it is conceivable.

　　There are two common fallacies associated with causal reasoning：

　　Confusing Correlation with Causation.

　　To claim that A caused B merely because A occurred immediately before B is clearly questionable. It may be only coincidental that they occurred together， or something else may have caused them to occur together. For example， the　fact that insomnia and lack of appetite often occur together does not mean　that one necessarily causes the other. They may both be symptoms of an underlying condition.

　　2. Confusing Necessary Conditions with Sufficient Conditions.

　　A is necessary for B means B cannot occur without A. A is sufficient for B means A causes B to occur， but B can still occur without A. For example，　a small tax base is sufficient to cause a budget deficit， but excessive spending can cause a deficit even with a large tax base. A common fallacy is to　assume that a necessary condition is sufficient to cause a situation. For example， to win a modern war it is necessary to have modern， high-tech equipment， but it is not sufficient， as Iraq discovered in the Persian Gulf War.

　　SEVEN COMMON FALLACIES

　　Contradiction

　　A Contradiction is committed when two opposing statements are simultaneously　asserted. For example， saying it is raining and it is not raining is a contradiction. Typically， however， the arguer obscures the contradiction to the point that the argument can be quite compelling. Take， for instance， the following argument：

　　We cannot know anything， because we intuitively realize that our thoughts are unreliable.

　　This argument has an air of reasonableness to it. But intuitively realize　means to know. Thus the arguer is in essence saying that we know that we dont know anything. This is self-contradictory.

　　Equivocation

　　Equivocation is the use of a word in more than one sense during an argument.

　　This technique is often used by politicians to leave themselves an out. If someone objects to a particular statement， the politician can simply claim　the other meaning.

　　Example：

　　Individual rights must be championed by the government. It is right for one　to believe in God. So government should promote the belief in God.

　　In this argument， right is used ambiguously. In the phrase individual rights it is used in the sense of a privilege， whereas in the second sentence right is used to mean proper or moral. The questionable conclusion is possible　only if the arguer is allowed to play with the meaning of the critical word　right.

　　Circular Reasoning

　　Circular reasoning involves assuming as a premise that which you are trying to prove. Intuitively， it may seem that no one would fall for such an argument. However， the conclusion may appear to state something additional， or the　argument may be so long that the reader may forget that the conclusion was　stated as a premise.

　　Example：

　　The death penalty is appropriate for traitors because it is right to execute　those who betray their own country and thereby risk the lives of millions.

　　This argument is circular because right means essentially the same thing as appropriate. In effect， the writer is saying that the death penalty is a　ppropriate because it is appropriate.

　　Shifting The Burden Of Proof

　　It is incumbent on the writer to provide evidence or support for her position. To imply that a position is true merely because no one has disproved it is to shift the burden of proof to others.

　　Example：

　　Since no one has been able to prove Gods existence， there must not be a God

　　

　　There are two major weaknesses in this argument. First， the fact that Gods　existence has yet to be proven does not preclude any future proof of existence. Second， if there is a God， one would expect that his existence is independent of any proof by man.

　　Unwarranted Assumptions

　　The fallacy of unwarranted assumption is committed when the conclusion of an　argument is based on a premise that is false or unwarranted. An assumption is unwarranted when it is false――these premises are usually suppressed or vaguely written. An assumption is also unwarranted when　it is true but does not apply in the given context――these premises are usually explicit.

　　Example：

　　Either restrictions must be placed on freedom of speech or certain subversive elements in society will use it to destroy this country. Since to allow the latter to occur is unconscionable， we must restrict freedom of speech.

　　The conclusion above is unsound because subversives do not in fact want to destroy the country the author places too much importance on the freedom of speech the author fails to consider an accommodation between the two alternatives the meaning of freedom of speech has not been defined subversives are a true threat to our way of life

　　The arguer offers two options： either restrict freedom of speech， or lose the country. He hopes the reader will assume that these are the only options available. This is unwarranted. He does not state how the so-called subversive elements would destroy the country， nor for that matter， why they would　want to destroy it. There may be a third option that the author did not mention; namely， that society may be able to tolerate the subversives and it may even be improved by the diversity of opinion they offer. The answer is 。

　　Appeal To Authority

　　To appeal to authority is to cite an experts opinion as support for ones own opinion. This method of thought is not necessarily fallacious. Clearly， the reasonableness of the argument depends on the expertise of the person being cited and whether she is an expert in a field relevant to the argument.

　　Appealing to a doctors authority on a medical issue， for example， would be　reasonable; but if the issue is about dermatology and the doctor is an orthopedist， then the argument would be questionable.

　　Personal Attack

　　In a personal attack ， a persons character is challenged instead of her opinions.

　　Example：

　　Politician： How can we trust my opponent to be true to the voters? He isnt　true to his wife!

　　This argument is weak because it attacks the opponents character， not his positions. Some people may consider fidelity a prerequisite for public office History， however， shows no correlation between fidelity and great political leadership.

　　I would fly you to the moon and back

　　If youll be if youll be my baby

　　Got a ticket for a worldswhereswe belong

　　So would you be my baby

　　

　　INTRODUCTION DATA SUFFICIENCY

　　Most people have much more difficulty with the Data Sufficiency problems than with the Standard Math problems. However， the mathematical knowledge and s　kill required to solve Data Sufficiency problems is no greater than that required to solve standard math problems. What makes Data Sufficiency problems　appear harder at first is the complicated directions. But once you become familiar with the directions， youll find these problems no harder than standard math problems. In fact， people usually become proficient more quickly on　Data Sufficiency problems.

　　THE DIRECTIONS

　　The directions for Data Sufficiency questions are rather complicated. Before　reading any further， take some time to learn the directions cold. Some of the wording in the directions below has been changed from the GMAT to make it　clearer. You should never have to look at the instructions during the test.

　　Directions： Each of the following Data Sufficiency problems contains a question followed by two statements， numbered and 。 You need not solve the　problem; rather you must decide whether the information given is sufficient to solve the problem.

　　The correct answer to a question is

　　A if statement ALONE is sufficient to answer the question but statement alone is not sufficient;

　　B if statement ALONE is sufficient to answer the question but statement alone is not sufficient;

　　C if the two statements TAKEN TOGETHER are sufficient to answer the question， but NEITHER statement ALONE is sufficient;

　　D if EACH statement ALONE is sufficient to answer the question;

　　E if the two statements TAKEN TOGETHER are still NOT sufficient to answer the question.

　　Numbers： Only real numbers are used. That is， there are no complex numbers.

　　Drawings： The drawings are drawn to scale according to the information given　in the question， but may conflict with the information given in statements