Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Mathematical Excursions (MindTap Course List)
Recreational Logic Consider a checkerboard with two red squares on opposite corners removed, as shown below. Determine whether it is possible to completely cover the checkerboard with 31 dominoes if each domino is placed horizontally or vertically and each domino covers exactly two squares. If it is possible, show how to do it. If it is not possible, explain why it cannot be done.Write an argument that is an example of circulus in probando.2EE3EE4EEAlgebraic arguments often consist of a list of statements. In a valid algebraic argument, each statement (after the premises) can be deduced from the previous statements. The following argument that 1=2 contains exactly one step that is not valid. Identify the step and explain why it is not valid. Let a=bPremise.a2=abMultiplyeachsidebya.a2b2=abb2Subtractb2fromeachside.(a+b)(ab)=b(ab)Factoreachside.a+b=bDivideeachsideby(a-b).b+b=bSubstitutebfora.2b=bCollectliketerms.2=1Divideeachsidebyb.Use the indicated letters to write each argument in symbolic form. If you can read this bumper sticker (r), youre too close (c). You can read the bumper sticker. Therefore, youre too close.Use the indicated letters to write each argument in symbolic form. If Lois Lane marries Clark Kent (m, then Superman will get a new uniform (u). Superman does not get a new uniform. Therefore, Lois Lane did not marry Clark Kent.3ES4ES5ESUse the indicated letters to write each argument in symbolic form. If we check the sports results on ESPN (~c). we will know who won the match (w). We know who won the match. Therefore, we checked the sports results on ESPN.Use the indicated letters to write each argument in symbolic form. If the power goes off (~p), then the air conditioner will not work (~a). The air conditioner is working. Therefore, the power is not off.Use the indicated letters to write each argument in symbolic form. If it snowed (s), then I did not go to my chemistry class (~c). I went to my chemistry class. Therefore, it did not snow.Use a truth table to determine whether the argument is valid or invalid. p~q~qpUse a truth table to determine whether the argument is valid or invalid. ~pq~pqUse a truth table to determine whether the argument is valid or invalid. p~q~qp12ES13ES14ES15ESUse a truth table to determine whether the argument is valid or invalid. (pq)(pq)pq17ES18ES19ESUse a truth table to determine whether the argument is valid or invalid. (pq)(r~q)p~r21ES22ES23ES24ESUse the indicated letters to write the argument in symbolic form. Then use a truth table to determine whether the argument is valid or invalid. If you finish your homework (h), you may attend the reception (r). You did not finish your homework. Therefore, you cannot go to the reception.Use the indicated letters to write the argument in symbolic form. Then use a truth table to determine whether the argument is valid or invalid. The X Games will be held in Oceanside (a) if and only if the city of Oceanside agrees to pay $200,000 in prize money (a). If San Diego agrees to pay $300,000 in prize money (s), then the city of Oceanside will not agree to pay $200,000 in prize money. Therefore, if the X Games were held in Oceanside, then San Diego did not agree to pay $300,000 in prize money.27ESUse the indicated letters to write the argument in symbolic form. Then use a truth table to determine whether the argument is valid or invalid. If the winds are from the east (e), then we will not have a big surf (-s). We do not have a big surf. Therefore, the winds are from the east.Use the indicated letters to write the argument in symbolic form. Then use a truth table to determine whether the argument is valid or invalid. If I master college algebra (c), then I will be prepared for trigonometry (r). I am prepared for trigonometry. Therefore, I mastered college algebra.Use the indicated letters to write the argument in symbolic form. Then use a truth table to determine whether the argument is valid or invalid. If it is a blot (b), then it is not a clot (~c). If it is a zlot (z), then it is a clot. It is a blot. Therefore, it is not a zlot.Determine whether the argument is valid or invalid by comparing its symbolic form with the standard forms given in Tables 3.15 and 3.16. For each valid argument, state the name of its standard form. If you take Art 151 in the fall, you will be eligible to take Art 152 in the spring. You were not eligible to take Art 152 in the spring. Therefore, you did not take Art 151 in the fall.Determine whether the argument is valid or invalid by comparing its symbolic form with the standard forms given in Tables 3.15 and 3.16. For each valid argument, state the name of its standard form. He will attend Stanford or Yak. He did not attend Yale. Therefore, he attended Stanford.Determine whether the argument is valid or invalid by comparing its symbolic form with the standard forms given in Tables 3.15 and 3.16. For each valid argument, state the name of its standard form. If I had a nickel for every logic problem I have solved, then I would be rich. I have not received a nickel for every logic problem I have solved. Therefore, I am not rich.Determine whether the argument is valid or invalid by comparing its symbolic form with the standard forms given in Tables 3.15 and 3.16. For each valid argument, state the name of its standard form. If it is a dog, then it has fleas. Ii has fleas. Therefore, it is a dog.Determine whether the argument is valid or invalid by comparing its symbolic form with the standard forms given in Tables 3.15 and 3.16. For each valid argument, state the name of its standard form. If we serve salmon, then Vicky will join us for lunch. If Vicky joins us for lunch, then Marilyn will not join us for lunch. Therefore, if we serve salmon, Marilyn will not join us for lunch.Determine whether the argument is valid or invalid by comparing its symbolic form with the standard forms given in Tables 3.15 and 3.16. For each valid argument, state the name of its standard form. If I go to college, then I will not be able to work for my Dad. I did not go to college. Therefore, I went to work for my Dad.Determine whether the argument is valid or invalid by comparing its symbolic form with the standard forms given in Tables 3.15 and 3.16. For each valid argument, state the name of its standard form. If my cat is left alone in the apartment, then she claws the sofa. Yesterday I left my cat alone in the apartment. Therefore, my cat clawed the sofa.38ESDetermine whether the argument is valid or invalid by comparing its symbolic form with the standard forms given in Tables 3.15 and 3.16. For each valid argument, state the name of its standard form. If Rita buys a new car, then she will not go on the cruise. Rita went on the cruise. Therefore, Rita did not buy a new car.40ESUse a sequence of valid arguments to show that each argument is valid. ~prrt~tpUse a sequence of valid arguments to show that each argument is valid. r~ss~tr~tUse a sequence of valid arguments to show that each argument is valid. If we sell the boat (s), then we will not go to the river (~r). If we dont go to the river, then we will go camping (c). If we do not buy a tent (~t), then we will not go camping. Therefore, if we sell the boat, then we will buy a tent.Use a sequence of valid arguments to show that each argument is valid. If it is an ammonite (a), then it is from the Cretaceous period (c). If it is not from the Mesozoic era (~m), then it is not from the Cretaceous period. If it is from the Mesozoic era, then it is at least 65 million years old (s). Therefore, lilt is an ammonite, then it is at least 65 million years old.Use a sequence of valid arguments to show that each argument is valid. If the computer is not operating (~o), then I will not be able to finish my report (~f). If the office is closed (c), then the computer is not operating. Therefore, if I am able to finish my report, then the office is open.Use a sequence of valid arguments to show that each argument is valid. If he reads the manuscript (r), he will like it (l). If he likes it, he will publish it (p). If he publishes it, then you will get royalties (m). You did not get royalties. Therefore, he did not read the manuscript.47ES48ESUse all of the premises to determine a valid conclusion for the given argument. If it is a theropod, then it is not herbivorous. If it is not herbivorous, then it is not a sauropod, It is a sauropod. Therefore, _____________Use all of the premises to determine a valid conclusion for the given argument. If you buy the car, you will need a loan. You do not need a loan or you will make monthly payments. You buy the car. Therefore, _________________________Recreational Logic Arc You Smarter Than a 5th Grader? is a popular television program that requires adult contestants to answer grade-school level questions. The show is hosted by Jeff Foxworthy. Here is an arithmetic problem that many 5th grade students can solve in less than 10 seconds after the problem has been stated. Start with the number 32. Add 46 to the starting number and divide the result by 2. Multiply the previous result by 3 and add 27 to that product. Take the square root of the previous sum and then quadruple that result. Multiply the latest result by 5 and divide that product by 32. Multiply the latest result by 0 and then add 24 to the result. Take half of the previous result. What is the final answer? Explain how to solve this arithmetic problem in less than 10 seconds.An Argument by Lewis Carroll The following argument is from Symbolic Logic by Lewis Carroll, written in 1896. Use symbolic logic to determine whether the argument is valid or invalid. Babies are illogical. Nobody is despised who can manage a crocodile. Illogical persons are despised. Hence, babies cannot manage crocodiles.Solve the following cryptarithms. Assume that no leading digit is a 0. (Source: http://cryptarithms.awardspace.us/)3 1. 2. 3. 4.Solve the following cryptarithms. Assume that no leading digit is a 0. (Source: http://cryptarithms.awardspace.us/)3 2.Solve the following cryptarithms. Assume that no leading digit is a 0. (Source: http://cryptarithms.awardspace.us/)3 3.Solve the following cryptarithms. Assume that no leading digit is a 0. (Source: http://cryptarithms.awardspace.us/)3 4.Use an Euler diagram to determine whether the argument is valid or invalid. AllfrogsarepoeticalKermitisafrogKermitispoeticalUse an Euler diagram to determine whether the argument is valid or invalid. AllOrcocookieshaveafillingAllFigNewtonshaveafillingAllFigNewtonsareOreocookiesUse an Euler diagram to determine whether the argument is valid or invalid. SomeplantshaveflowersAllthingsthathaveflowersarebeautifulSomeplantsarebeautifulUse an Euler diagram to determine whether the argument is valid or invalid. NosquaresaretrianglesSometrianglesareequilateralNosquaresareequilateralUse an Euler diagram to determine whether the argument is valid or invalid. NorockerwoulddothemariachiAllbaseballfansdothemariachiNorockerisabaseballfanUse an Euler diagram to determine whether the argument is valid or invalid. NuclearenergyisnotsafeSomeelectricenergyissafeNoelectricenergyisnuclearenergy `Use an Euler diagram to determine whether the argument is valid or invalid. SomebirdsbiteAllthingsthatbitearedangerousSomebirdsaredangerousUse an Euler diagram to determine whether the argument is valid or invalid. AllfishcanswimThatbarracudacanswimThatbarracudaisafishUse an Euler diagram to determine whether the argument is valid or invalid. AllmenbehavebadlySomehockeyplayersbehavebadlySomehockeyplayersaremenUse an Euler diagram to determine whether the argument is valid or invalid. AllgrassisgreenThatgroundcoverisnotgreenThatgroundcoverisnotgrassUse an Euler diagram to determine whether the argument is valid or invalid. MostteenagersdrinksodaNoCEOsdrinksodaNoCEOisateenagerUse an Euler diagram to determine whether the argument is valid or invalid. SomestudentslikehistoryVeinisastudentVeinlikeshistoryUse an Euler diagram to determine whether the argument is valid or invalid. NomathematicstestisfunAllfunthingsareworthyourtimeNomathematicstestisworthyourtimeUse an Euler diagram to determine whether the argument is valid or invalid. AllprudentpeopleshunsharksNoaccountantisimprudentNoaccountantfailstoshunsharks15ES16ESUse an Euler diagram to determine whether the argument is valid or invalid. Allprimenumbersareodd2isaprimenumber2isanoddnumberUse an Euler diagram to determine whether the argument is valid or invalid. AllLewisCarrollargumentsarevalidSomevalidargumentsaresyllogismsSomeLewisCarrollargumentsaresyllogismsUse an Euler diagram to determine whether the argument is valid or invalid. AllaerobicsclassesarefunJansclassisfunJansclassisanaerobicsclass20ESUse all of the premises in each argument to determine a valid conclusion for the argument. Allreubensandwichesaregood.Allgoodsandwicheshavepastrami.Allsandwicheswithpastramineedmustard.?Use all of the premises in each argument to determine a valid conclusion for the argument. Allcatsarestrange.Boomerisnotstrange.?23ES24ESUse all of the premises in each argument to determine a valid conclusion for the argument. Somehorsesarefrisky.Allfriskyhorsesaregrey.?26ESExamine the following three premises: 1. All people who have an Xbox play video games. 2. All people who play video games enjoy life. 3. Some mathematics professors enjoy life. Now consider each of the following six conclusions. For each conclusion, determine whether the argument formed by the three premises and the conclusion is valid or invalid. a. Some mathematics professors have an Xbox. b.Some mathematics professors play video games. c. Some people who play video games are mathematics professors. d. Mathematics professors never play video games. e. All people who have an Xbox enjoy life. f. Some people who enjoy life arc mathematics professors.Examine the following three premises: 1. All people who drive pickup trucks like Garth Brooks. 2. All people who like Garth Brooks like country western music. 3. Some people who like heavy metal music like Garth Brooks. Now consider each of the following five conclusions. For each conclusion, determine whether the argument formed by the three premises and the conclusion is valid or invalid. a. Some people who like heavy metal music drive a pickup truck. b. Some people who like heavy metal music like country western music. c. Some people who like Garth Brooks like heavy metal music. d. All people who drive a pickup truck like country western music. e. People who like heavy metal music never drive a pickup truck.29ESBilateral Diagrams Lewis Carroll s method of analyzing syllogisms. Write a few paragraphs that explain his method and its advantages.Determine whether each sentence is a statement. Assume that a and b are real numbers. How much is a ticket to London?Determine whether each sentence is a statement. Assume that a and b are real numbers. 91 is a prime number.Determine whether each sentence is a statement. Assume that a and b are real numbers. abDetermine whether each sentence is a statement. Assume that a and b are real numbers. a20Determine whether each sentence is a statement. Assume that a and b are real numbers. Lock the car.6REWrite each sentence in symbolic form. Represent each simple statement of the sentence with the letter indicated in parentheses. Also state whether the sentence is a conjunction, a disjunction, a negation, a conditional, or a biconditional. Today is Monday (m) and it is my birthday (b).Write each sentence in symbolic form. Represent each simple statement of the sentence with the letter indicated in parentheses. Also state whether the sentence is a conjunction, a disjunction, a negation, a conditional, or a biconditional. If x is divisible by 2 (d), then x is an even number (e).Write each sentence in symbolic form. Represent each simple statement of the sentence with the letter indicated in parentheses. Also state whether the sentence is a conjunction, a disjunction, a negation, a conditional, or a biconditional. I am going to the dance (g) if and only if I have a date (d).Write each sentence in symbolic form. Represent each simple statement of the sentence with the letter indicated in parentheses. Also state whether the sentence is a conjunction, a disjunction, a negation, a conditional, or a biconditional. All triangles (t) have exactly three sides (s).Write the negation of each quantified statement. Start each negation with Some. No. or All. Some dogs bite.Write the negation of each quantified statement. Start each negation with Some. No. or All. Every dessert at the Cove restaurant is good.Write the negation of each quantified statement. Start each negation with Some. No. or All. All winners receive a prize.Write the negation of each quantified statement. Start each negation with Some. No. or All. Some cameras do not use film.Write the negation of each quantified statement. Start each negation with Some. No. or All. No student finished the assignment.16RE17REDetermine whether each statement is true or false. 35 and 7 is a prime number.19RE20RE21REDetermine the truth value of the statement given that p is true, q is false. and r is false. (pq)(pq)Determine the truth value of the statement given that p is true, q is false. and r is false. (pq)(rq)24RE25RE26RE27RE28RE29RE30RE31RE32RE33REConstruct a truth table for the given statement. (pq)(qr)Make use of Dc Morgans laws to write the given statement in an equivalent form. It is not true that. Bob failed the English proficiency test and he registered for a speech course.Make use of Dc Morgans laws to write the given statement in an equivalent form. Ellen did not go to work this morning and she did not take her medication.37REMake use of Dc Morgans laws to write the given statement in an equivalent form. Gina enjoyed the movie, but she did not enjoy the party.39RE40RE41RE42RE43RE44RE45RE46REIdentify the antecedent and the consequent of each conditional statement. If he has talent, he will succeed.Identify the antecedent and the consequent of each conditional statement. If I had a credential, I could get the job.Identify the antecedent and the consequent of each conditional statement. I will follow the exercise program provided I join the fitness club.Identify the antecedent and the consequent of each conditional statement. I will attend only if it is free.Write each conditional statement in its equivalent disjunctive form. If she were tall, she would be on the volleyball team.Write each conditional statement in its equivalent disjunctive form. If he can stay awake, he can finish the report.Write each conditional statement in its equivalent disjunctive form. Rob will start, provided he is not ill.54RE55REWrite the negation of each conditional statement in its equivalent conjunctive form. The tomatoes will get big only if you provide them with plenty of water.57REWrite the negation of each conditional statement in its equivalent conjunctive form. If Ryan enrolls at a university, then he will enroll at Yale.59RE60RE61REDetermine whether the given statement is true or false. Assume that x and i are real numbers. If x2=y2, then x=y.Write each statement in If p, then q form. Every nonrepeating, nonterminating decimal is an irrational number.Write each statement in If p, then q form. Being well known is a necessary condition for a politician.65RE66REWrite the a. converse, b. inverse, and c. contrapositive of the given statement. If x+47, then x3.Write the a. converse, b. inverse, and c. contrapositive of the given statement. All recipes in this book can be prepared in less than 20 ml flutes.Write the a. converse, b. inverse, and c. contrapositive of the given statement. If a and b are both divisible by 3, then (a+b) is divisible by 3.Write the a. converse, b. inverse, and c. contrapositive of the given statement. If you build it, they will come.Write the a. converse, b. inverse, and c. contrapositive of the given statement. Every trapezoid has exactly two parallel sides.Write the a. converse, b. inverse, and c. contrapositive of the given statement. If they like it, they will return.73RE74REDetermine the original statement if the given statement is related to the original statement in the manner indicated. 75. comerse: If x2, then x is an odd prime number.Determine the original statement if the given statement is related to the original statement in the manner indicated. Negalion: The senator will attend the meeting and she will not vote on the motion.77REDetermine the original statement if the given statement is related to the original statement in the manner indicated. 78. contrapositive: If Ginny cant rollerblade, then I cant rollerblade.79REUse a truth table to determine whether the argument is valid or invalid. pqpp81RE82RE83REDetermine whether the argument is valid or invalid by comparing its symbolic form with the symbolic forms in Tables 3.15 and 3.16, pages 145 and 146. If Mike is a CEO, then he will be able to afford to make a donation. If Mike can afford to make a donation, then he loves to ski. Therefore, if Mike does not love to ski, he is not a CEO.Determine whether the argument is valid or invalid by comparing its symbolic form with the symbolic forms in Tables 3.15 and 3.16, pages 145 and 146. If we wish to win the lottery, we must buy a lottery ticket. We did not win the lottery. Therefore, we did not buy a lottery ticket.86REDetermine whether the argument is valid or invalid by comparing its symbolic form with the symbolic forms in Tables 3.15 and 3.16, pages 145 and 146. If we are going to have a caesar salad, then we need to buy some eggs. We did not buy eggs. Therefore, we are not going to have a caesar salad.88RE89REUse an Euler diagram to determine whether the argument is valid or invalid. Somedogshavetails.Somedogsarebig.Somebigdogshavetails.Use an Euler diagram to determine whether the argument is valid or invalid. All Italian villas are wonderful. Some wonderful villas are expensive. Therefore, some Italian villas are expensive.Use an Euler diagram to determine whether the argument is valid or invalid. All logicians like to sing Its a small world after all. Some logicians have been presidential candidates. Therefore, some presidential candidates like to sing Its a small world after all.Determine whether each sentence is a statement. a. Look for the cat. b. Clark Kent is afraid of the dark.Write the negation of each statement. Start each negation with Some, No. or All. a. Some trees are not green. b. No apartments are available.Determine whether each statement is true or false. a. 54 b. 224T5T6TUse one of De Morgans laws to write the following in an equivalent form. Elle did not eat breakfast and she did not take a lunch break.8TWrite pq in its equivalent disjunctive form.10TWrite the a. converse, b. inverse, and c. contrapositive of the following statement. If x+711, then x4.12. Write the symbolic form of direct reasoning.Write the symbolic form of transitive reasoning.Write the symbolic form of contrapositive reasoning.15. Write the symbolic form of the fallacy of the inverse.16T17TDetermine whether the argument is valid or invalid. Explain how you made your decision. If we wish to win the talent contest, we must practice. We did not win the contest. Therefore, we did not practice.19TDetermine whether the argument is valid or invalid. Explain how you made your decision. Nowizardcanglowinthedark.Somelizardscanglowinthedark.Nowizardisalizard.21TDetermine whether the argument is valid or invalid. Explain how you made your decision. If I cut my night class, then I will go to the party. I went to the party. Therefore, I cut my night class.Verify this apportionment using the Jefferson method. You will have to experiment with various modified standard divisors until you reach the given representation. See the Calculator Note on page 173.Find the apportionment that would have resulted if the Hamilton method had been used.3EEFind the apportionment that would have resulted if the Huntington-Hill method (the one used for the 2010 census) had been used in 1790. See our companion site at CengageBrain.com for a spreadsheet that will help with the calculations.Explain how to calculate the standard divisor of an apportionment for a total population p with n items to apportion.Teacher Aides A total of 25 teacher aides are to be apportioned among seven classes at a new elementary school. The enrollments in the seven classes are shown in the following table. a. Determine the standard divisor. What is the meaning of the standard divisor in the context of this exercise? b. Use the Hamilton method to determine the number of teacher aides to be apportioned to each class. c. Use the Jefferson method to determine the number of teacher aides to be apportioned to each class. Is this apportionment in violation of the quota rule? d. How do the apportionment results produced using the Jefferson method compare with the results produced using the Hamilton method?In the Hamilton apportionment method, explain how to calculate the standard quota for a particular state (group).4ESGoverning Boards The following table shows how the average constituency changes for two regional governing boards. Joshua and Salinas. when a new representative is added to each board. a. Determine the relative unfairness of an apportionment that gives a new board member to Joshua rather than to Salinas. Round to the nearest thousandth. b. Determine the relative unfairness of an apportionment that gives a new board member to Salinas rather than to Joshua. Round to the nearest thousandth. c. Using the apportionment principle, determine which regional governing board should receive the new board member.Forest Rangers The table below shows how the average constituency changes when two different national parks. Evergreen State Park and Rust Canyon Preserve, add a new forest ranger. a. Determine the relative unfairness of an apportionment that gives a new forest ranger to Evergreen rather than to Rust Canyon. Round to the nearest thousandth. b. Determine the relative unfairness of an apportionment that gives a new forest ranger to Rust Canyon rather than to Evergreen. Round to the nearest thousandth. c. Using the apportionment principle, determine which national park should receive the new forest ranger.Sales Associates The table below shows the number of sales associates and the average number of customers per day at a companys t department stores. The company is planning to add a new sales associate to one of the stores. Use the apportionment principle to determine which store should receive the new employee.Hospital Interns The table below shows the number of interns and the average number of patients admitted each day at two different hospitals. The hospital administrator is planning to add a new intern to one of the hospitals. Use the apportionment principle to determine which hospital should receive the new intern.House of Representatives The U.S. House of Representatives currently has 435 members to represent the 308,745.538 citizens of the U.S. as determined by the 2010 census. a. Calculate the standard divisor for the apportionment of these representatives and explain the meaning of this standard divisor in the context of this exercise. b. According to the 2010 census. the population of Delaware was 897,934. Delaware currently has only one representative in the House of Representatives. Is Delaware currently overrepresented or underrepresented in the House of Representatives? Explain. c. According to the 2010 census, the population of Vermont was 625.741. Vermont currently has only one representative in the House of Representatives. Is Vermont currently overrepresented or underrepresenled in the House of Representatives? Explain.College Enrollment The following table shows the enrollment for each of the four divisions of a college. The four divisions are liberal arts, business, humanities, and science. There are 180 new computers that are to be apportioned among the divisions based on the enrollments. Total 15.809 a. What is the standard divisor for an apportionm ent of the computers? What is the meaning of the standard divisor in the context of this exercise? b. Use the Hamilton method to determine the number of computers to be apportioned to each division. c. If the computers are to be apportioned using the Jefferson method, explain why neither 86 nor 87 can be used as a modified standard divisor. Explain why 86.5 can be used as a modified standard divisor. d. Explain why the modified standard divisor - used in the Jefferson method cannot be larger than the standard divisor. e. Use the Jefferson method to determine the number of computers to be apportioned to each division. Is this apportionment in violation of the quota rule? f. How do the apportionment results produced using the Jefferson method compare with the results prod uced using the Hamilton method?Medical Care A hospital district consists of six hospitals. The district administrators have decided that 48 new nurses should be apportioned based on the number of beds in each of the hospitals. The following table shows the number of beds in each hospital. a. Determine the standard divisor. What is the meaning of the standard divisor in the context of this exercise? b. Use the Hamilton method to determine the number of nurses to be apportioned to each hospital. c. Use the Jefferson method to determine the number of nurses to be apportioned to each hospital. d. How do the apportionment results produced using the Jefferson method compare with the results produced using the Hamilton method?What is the Alabama paradox?What is the population paradox?What is the new states paradox?15. What is the Balinski-Young Impossibility Theorem?Apportionment of Projectors Consider the apportionment of 27 digital projectors for a school district with four campus locations labeled A. B. C and D. The following table shows the apportionment of the projectors using the Hamilton method. a. If the number of projectors to be apportioned increases from 27 to 28, what will be the apportionment if the Hamilton method is used? Will the Alabama paradox occur? Explain. b. If the number of projectors to be apportioned using the Hamilton method increases from 28 to 29, will the Alabama paradox occur? Explain.Hotel Management A company operates four resorts. The CEO of the company decides to use the Hamilton method to appartion 115 new LCD television sets the resorts based on the number of guest rooms at each resort. a. If the number of television sets to be apportioned by the Hamilton method increases from 115 to 116, will the Alabama paradox occur? b. If the number of television sets to be apportioned by the Hamilton method increases from 116 to 117, will the Alabama paradox occur? c. If the number of television sets to be apportioned by the Hamilton method increases from 117 to 118, will the Alabama paradox occur?18ESManagement Scientific Research Corporation has offices in Boston and Chicago. The number of employees at each office is shown in the following table. There are 22 vice presidents to be apportioned between the offices. a. Use the Hamilton method to find each offices apportionment of vice presidents. b. The corporation opens an additional office in San Francisco with 135 employees and decides to have a total of 24 vice presidents. If the vice presidents are reapportioned using the Hamilton method, will the new states paradox occur? Explain.20ESElementary School Teachers The following table shows the number of fifth and sixth grade teachers in a school district and the number of students in the two grades. The number of teachers for each of the grade levels was determined by using the Huntington-Hill apportionment method. The district has decided to hire a new teacher for either the fifth or the sixth grade. a. Use the apportionment principle to determine to which grade the new teacher should be assigned. b. Use the Huntington-Hill apportionment principle to determine to which grade the new teacher should be assigned. How does this result compare with the result in part a?Social Workers The following table shows the number of social Workers and the number of cases (the case load) handled by the social workers for two offices. The number of social workers for each office was determined by using the Huntington-Hill apportionment method. A new social worker is to be hired for one of the offices. a. Use the apportionment principle to determine to which office the social worker should be assigned. b. Use the Huntington-Hill apportionment principle to determine to which office the new social worker should be assigned. How does this result compare with the result in part a? c. The results of part b indicate that the new social worker should be assigned to the Valley office. At this moment the Hill Street office has 20 social workers and the Valley office has 25 social workers. Use the Huntington-Hill apportionment principle to determine to which office the next new social worker should be assigned. Assume the case loads remain the same.Computer Usage The table below shows the number of computers that are assigned to four different schools and the number of students in those schools. Use the Huntington-Hill apportionment principle to determine to which school a new computer should be assigned.The population of Illinois increased by over 400.000 from 2(XK) to 2010, yet the slate lost a seat in the I-louse of Representatives. How must lllinoiss population increase have compared to the changes of other states populations?25ES26ESComputer Usage Use the Webster method to apportion the computers in Exercise 10, page l4. How do the apportionments results produced using the Webster method compare with the results produced using the a. Hamilton method? b. Jefferson method?Demographics The table below shows the populations of five European countries. A committee of 20 people from these countries is to be formed using the Webster method of apportionment. a. Explain why 13.500,000 cannot be used as a modified standard divisor. b. Explain why 13.750.000 can be used as a modified standard divisor. c. Use the Webster apportionment method to determine the apportionment of the 20 committee members.Which of she following apportionment methods can violate the quota rule? Hamilton method Jefferson method Webster method Huntington-Hill methodAccording to Michael Balinski and H. Peyton Young, which of the apportionment methods most closely satisfies the goal of one person, one vote?31. What method is presently used to apportion the members of the U.S. House of Representatives?32ESJohn Quincy Adams, the sixth president of the United States, proposed an apportionment method. Research this method, which is known as the Adams method of apportionment. Describe how this method works. Also indicate whether ii satisties the quota rule and whether it is susceptible to any paradoxes.In the Huntington-Hill method of apportionment, each state is first given one representative and then additional representatives are assigned, one at a time, to the state currently having the highest Huntington-Hill number. This way of implementing the Huntington-Hill apportionment method is time consuming. Another process for implementing the Huntington-Hill apportionment method consists of using a modified divisor and a special rounding procedure that involves the geometric means of two consecutive integers. Research this process for implementing the Huntington-Hill apportionment method. Apply this process to apportion 22 new security vehicles to each of the following schools, based on their student populations.35ESUsing the Borda method of voting, which flavor of iced tea is preferred by this group? Which is second? Which is third?Instead of using the normal Borda method, suppose the Borda method used in Exercise 1 of this Excursion assigned I point for first, 0 points for second, and — 1 point for third place. Does this alter the preferences you found in Exercise 1?3EESuppose the Borda method used in Exercise I of this Excursion assigned 20 points for first, 5 points for second, and 0 points for third place. Does this alter the preferences you found in Exercise 1?5EECan the assignment of points for first. Second, and third place change the preference order when the Borda method of voting is used?7EE1ESExplain why the plurality voting system may not be the best system to use in some situations.3ESExplain how the plurality with elimination voting method works.5ES6ESIs there a best voting method? Is one method more fair than the others?Explain why. if only two candidates are running, the plurality and Borda count methods will determine the same winner.Presidential Election The table below shows the popular vote and the Electoral College vote for the major candidates in the 2000 presidential election. a. Which candidate received the plurality of the popular vote? b. Did any candidate receive a majority of the popular vote? c. Who won the election?Breakfast Cereal Sixteen people were asked to rank three breakfast cereals in order of preference. Their responses are given below. If the plurality method of voting is used, which cereal is the groups first preference?Cartoon Characters A kindergarten class was surveyed to determine the childrens favourite cartoon characters among Dora the Explorer. SpongeBob Square Pants, and Buzz Lightyear. The students ranked the characters in order of preference; the results are shown in the preference schedule below. Rankings a. How many students are in the class? b. How many votes are required for a majority? c. Using plurality voting, which character is the childrens favorite?Catering A 15-person committee is having lunch catered for a meeting. Three caterers, each specializing in a different cuisine, are available. In order to choose a caterer for the group, each member is asked to rank the cuisine options in order of preference. The results are given in the preference schedule below. Rankings Using plurality voting, which caterer should be chosen?Movies Fifty consumers were surveyed about their movie-watching habits. They were asked to rank the likelihood that they would participate in each listed activity. The results are summarized in the table below. Using the Borda count method of Voting, which activity is the most popular choice among this group of consumers?Breakfast Cereal Use the Borda count method of voting to determine the preferred breakfast cereal in Exercise 10.15ESCatering Use the Borda count method of voting to determine which caterer the committee should hire in Exercise 12.Class Election A senior high school class held an election for class president. Instead of just voting for one candidate, he students were asked to rank all four candidates in order of preference. The results are shown below. Using the Borda count method, which student should be class president?Cell Phone Usage A journalist reviewing various cellular phone services surveyed 200 customers and asked each one to rank four service providers in order of preference. The groups results are shown below. Using the Borda count method, which provider is the favorite of these customers?Baseball Uniforms A Little League baseball team must choose the colors for its uniforms. The coach offered four different choices, and the players ranked them in order of preference, as shown in the table below. Using the plurality with elimination method, what colors should be used for the uniforms?Radio Stations A number of college students were asked to rank four radio stations in order of preference. The responses are given in the table below. Use plurality with elimination to determine the students favorite radio station among the four.Class Election Use plurality with elimination to choose the class president in Exercise 17.22ESCampus Club A campus club has money left over in its budget and must spend it before the school year ends. The members arrive at five different possibilities, and each member ranks them in order of preference. The results are shown in the table below. a. Using the plurality voting system, how should the club spend the money? b. Use the plurality with elimination method to determine how the money should be spent. c. Using the Borda count method of voting, how should the money be spent? d. In your opinion, which of the previous three methods seems most approp riate in this situation? Why?Recreation A company is planning its annual summer retreat and has asked its employees to rink live different choices of recreation in order of preference. The results are given in the table below. a. Using the plurality voting system, what activity should be planned for the retreat? b. Use the plurality with elimination method to determine which activity should be chosen. c. Using the Borda count method of voting, which activity should be planned?X-Men Movies Fans of the X-Men movies have been debating on an online forum regarding which of the films is the best. To see what the overall opinion is. visitors to the site can rank four films in order of preference. The results are shown in the preference schedule below. Using pairwise comparison, which film is the favorite of the visitors to the site who voted?26ESSchool Mascot A new college needs to pick a moscot for its football (earn. The students were asked to rank four choices in order of preference; the results are tallied below. Using the pairwise comparison method of voting, which mascot should be chosen?Election Five candidates are running for president of a charity organization. Interested persons were asked to rank the candidates in order of preference The results are given below. Use the pairwise comparison method to determine the president of the organization.29ESRadio Stations Use the pairwise comparison method to determine the favorite radio sation in Exercise 20.Does the winner in Exercise tic satisfy the Condorcet criterion?32. Does the winner in Exercise 12 satisfy the Condorcet criterion?33ES34ES35ES36ESElection Three candidates are running for mayor. A vote was taken in which the candidates were ranked in order of preference. The results are shown in the preference schedule below. a. Use the Borda count method to determine the winner of the election. b. Verify that the majority criterion has been violated. c. Identify the candidate who wins all head-to-head comparisons. d. Explain why the Condorcet criterion has been violated. e. If Marcia Beasley drops out of there for mayor (and voter preferences remain the same), determine the winner of the election again, using the Borda count method. f. Explain why the independence of irrelevant alternatives criterion has been violated.Film Competition Three films have been selected as finalists in a national student him competition. Seventeen judges have viewed each of the films and have ranked them in order of preference. The results are given in the preference schedule below. a. Using the plurality with elimination method, which film should win the competition? b. Suppose the first vote is declared invalid and a revote is taken. All of the judges preferences remain the same except for the votes represented by the last column of the table. The judges who cast these votes both decide to switch their first place vote to film C. so their preference now is film C first, then film A. and then film B. Which film now wins using the plurality with elimination method? c. Has the monotonicity criterion been violated?39. Election A campus club needs to elect four officers: a president. a vice president. a secretary, and a treasurer. The club has five volunteers. Rather than vote individually for each position, the club members will rank the candidates in order of preference. The votes will then be tallied using the Borda count method. The candidate receiving the highest number of points will be president, the candidate receiving the next highest number of points will be vice president, the candidate receiving the next highest number of points will be secretary, and the candidate receiving the next highest number of points will be treasurer. For the preference schedule shown below, determine who wins each position in the club.Scholarship Awards The members of a scholarship committee have ranked four finalists competing for a scholarship in order of preference. The results are shown in the preference schedule below. If you are one of the voting members and you want David Chang to win the scholarship, which voting method would you suggest that the committee use?Another method of voting is to assign a weight. or score, to each candidate rather than ranking the candidates in order. All candidates must receive a score, and two or more candidates can receive the same score from a voter. A score of 5 represents the strongest endorsement of a candidate. The scores range down to 1, which corresponds to complete disapproval of a candidate. A score of 3 represents indifference. The candidate with the most total points wins the election. The results of a sample election are given in the table. a. Find the winner of the election. b. If plurality were used (assuming that a persons vote would go to the candidate that he or she gave the highest score to), verify that a different winner would result.42ES43ESUsing the data in Example 1 on page 211, list all blocking coalitions.For the data in Example I on page 211, calculate the Banthaf power indices for A, B. C, and D using the alternative definition.3EE4EECreate a voting system with three members that is a dictatorship. Calculate the Banzhaf power index for each voter for this system using the alternative definition.