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All Textbook Solutions for Mathematical Excursions (MindTap Course List)
81ES82ESIn Exercises 83 to 86, use the extension of set-builder notation concepts and the roster method to write each of the given sets. D=3x21xNandx584ES85ES86ES87ES88ES89ES90ESLet K = {(1, 0.4), (2, 0.6), (3,0.8), (4, 1)) and J = {( 1, 0.3), (2, 0.6), (3, 0.5), (4, 0.1 )} be fuzzy sets defined on X = { 1, 2, 3, 4). Is J K? Explain.2EE3EE4EE5EE1ES2ES3ES4ES5ES6ES7ES8ES9ES10ES11ES12ES13ES14ES15ES16ES17ES18ES19ES20ES21ES22ES23ES24ES25ES26ES27ES28ES29ES30ES31ES32ES33ES34ES35ES36ES37ES38ES39ES40ES41ES42ES43ES44ES45ES46ES47ES48ES49ES50ES51ESIn Exercises 47 to 54, find the number of subsets of the given set. The set of all letters of our alphabet53ES54ESSuppose you have a nickel, two dimes, and a quarter. One of the dimes was minted in 1976, and the other one was minted in 1992. a. Assuming you choose at least one coin, how many different sets of coins can you form? b. Assuming you choose at least one coin, how many different sums of money can you produce? c. Explain why the answers in part a and part b are not the same.56ESSandwich Choices A delicatessen makes a roast-beef-on-sourdough sandwich for which you can choose from eight condiments. a. How many different types of roast-beef-on-sourdough sandwiches can the delicatessen prepare? b. What is the minimum number of condiments the delicatessen must have availableif it wishes to offer at least 2000 different types of roast-beef-on-sourdough sandwiches?Upgrade Options A company that builds homes advertises that, by choosing from us upgrade options, each of its new homes is available in 256 different variations. How many upgrade options does the company offer?Omelet Choices A restaurant provides a brunch where the omelets are individually prepared. Each guest is allowed to choose from 10 different ingredients. a. How many different types of omelets can the restaurant prepare? b. What is the minimum number of ingredients that must be available if the restaurant wants to advertise that it offers over 4000 different omelets?Truck Options A truck company makes a pickup truck with 12 upgrade options. Some of the options are air conditioning, chrome wheels, and a satellite radio. a. How many different versions of this truck can the company produce? b. What is the minimum number of upgrade options the company must be able to provide if it wishes to offer at least 14,000 different versions of this truck?61ES62ESVoting Coalitions Five people, designated A, B, C, D, and E, serve on a committee. To pass a motion, at least three of the committee members must vote for the motion. In such a situation, any set of three or more voters is called a winning coalition because if this set of people votes for a motion, the motion will pass. Any nonempty set of two or fewer voters is called a losing coalition. a. List all the winning coalitions. b. List all the losing coalitions.64ESEn Excursion Exercise I of Section 2.1, we defined the following fuzzy sets. Mark: M = {(A, 1), (B, 0.75), (C, 0.5), (D, 0.5), (F, 0)) Erica: E = {(A, 1), (B, 0), (C, 0), (D, 0), (F, 0)} Larry: L = {(A, 1), (B, 1), (C, 1), (D, 1), (F, 0)} Jennifer: J = {(A, 1), (B, 0.8), (C, 0.6), (D, 0.1), (F, 0)] Use these fuzzy sets to find each of the following. M JEn Excursion Exercise I of Section 2.1, we defined the following fuzzy sets. Mark: M = {(A, 1), (B, 0.75), (C, 0.5), (D, 0.5), (F, 0)) Erica: E = {(A, 1), (B, 0), (C, 0), (D, 0), (F, 0)} Larry: L = {(A, 1), (B, 1), (C, 1), (D, 1), (F, 0)} Jennifer: J = {(A, 1), (B, 0.8), (C, 0.6), (D, 0.1), (F, 0)] Use these fuzzy sets to find each of the following. M J3EEEn Excursion Exercise I of Section 2.1, we defined the following fuzzy sets. Mark: M = {(A, 1), (B, 0.75), (C, 0.5), (D, 0.5), (F, 0)) Erica: E = {(A, 1), (B, 0), (C, 0), (D, 0), (F, 0)} Larry: L = {(A, 1), (B, 1), (C, 1), (D, 1), (F, 0)} Jennifer: J = {(A, 1), (B, 0.8), (C, 0.6), (D, 0.1), (F, 0)] Use these fuzzy sets to find each of the following. J L5EE6EE7EE1ES2ES3ES4ES5ES6ES7ES8ES9ES10ES11ES12ES13ES14ES15ES16ES17ES18ES19ES20ES21ES22ES23ES24ESIn Exercises 21 to 28, write a sentence that describes the given mathematical expression. T(JK)26ES27ES28ES29ES30ES31ES32ES33ES34ES35ES36ES37ES38ES39ES40ES41ES42ES43ES44ES45ES46ES47ES48ES49ESSubtractive Color Mixing Artists who paint with pigments use subtractive color mixing to produce different colors. In a subtractive color mixing system, the primary colors are cyan C, magenta M, and yellow Y. The following figure shows that when the three primary colors are mixed in equal amounts, using subtractive color mixing, they form black, K. Using set notation, we state this as C M Y = K. In subtractive color mixing, the colors red R, blue B. and green G are the secondary colors. As mentioned previously, a secondary color is produced by mixing equal amounts of exactly two of the primary colors. In Exercises 50 to 52, determine which color is represented by each of the following. Assume the colors are being mixed using subtractive color mixing. (Use C for cyan, M for magenta, and Y for yellow.) C M Y51ESSubtractive Color Mixing Artists who paint with pigments use subtractive color mixing to produce different colors. In a subtractive color mixing system, the primary colors are cyan C, magenta M, and yellow Y. The following figure shows that when the three primary colors are mixed in equal amounts, using subtractive color mixing, they form black, K. Using set notation, we state this as C M Y = K. In subtractive color mixing, the colors red R, blue B. and green G are the secondary colors. As mentioned previously, a secondary color is produced by mixing equal amounts of exactly two of the primary colors. In Exercises 50 to 52, determine which color is represented by each of the following. Assume the colors are being mixed using subtractive color mixing. (Use C for cyan, M for magenta, and Y for yellow.) C M Y53ES54ES55ES56ES57ES58ES59ES60ES61ES62ESA Survey Aspecial interest group plans to conduct a survey of households concerning a ban on handguns. The special interest group has decided to use the following Venn diagram to help illustrate the results of the survey. Note: A rifle is a gun, but it is not a handgun. a. Shade in the regions that represent households that oi a handgun and do not support the ban on handguns. b. Shade in the region that represents households that own only a rifle and support the ban on handguns. c. Shade in the region that represents households that do not own a gun and do not support the ban on handguns.A Music Survey Theadministrators of an Internet music site plan to conduct a survey of college students to determine how the students acquire music. The administrators have decided to use the following Venn diagram to help tabulate the results of the survey. a. Shade in the region that represents students who acquire music from CDs and the Internet, hut not from DVDs. b. Shade in the regions that represent students who acquire music from CDs or the Internet. c Shade in the regions that represent students who acquire music from both CDs and DVDs.65ES66ES67ES68ES69ES70ES71ES72ES73ES74ES75ES76ES77ES78ES79ESIn an article in New Scientist magazine, Anthony W. F. Edwards illustrated how to construct Venn diagrams that involve many sets. Search the Internet to find Edwardss method of constructing a Venn diagram for five sets and a Venn diagram for six sets. Use drawings to illustrate Edwardss method of constructing a Venn diagram for five sets and a Venn diagram for six sets. (Source: http://www.combinatorics.org/Surveys/ds5/VennWhatEJC.html)A selection committee consists of Ryan, Susan. and Trevor. To decide on issues, they use a one-person, one-vote majority voting system. a. Find all winning coalitions. b. Find all losing coalitions.A selection committee consists of three people designated by M. N. and P. Ms vote has a weight of 3, Ns vote has a weight of 2, and Ps vote has a weight of I. The quota for this weighted voting system is 4. Find all winning coalitions.Determine the minimal winning coalitions for the voting system in Excursion Exercise 2.1ES2ESIn Exercises I to 10, let U = English, French, History, Math, Physics, Chemistry, Psychology, Drama), A = {English, History, Psychology, Drama}, B = {Math, Physics, Chemistry, Psychology, Drama}, and C = {French, History, Chemistry}. Find each of the following. n(B)+n(C)4ES5ES6ESIn Exercises I to 10, let U = English, French, History, Math, Physics, Chemistry, Psychology, Drama), A = {English, History, Psychology, Drama}, B = {Math, Physics, Chemistry, Psychology, Drama}, and C = {French, History, Chemistry}. Find each of the following. n(A)+n(B)+n(C)8ES9ES10ES11ES12ES13ES14ES15ESGiven n(A) = 640, n(B) = 280, and n(A U B) = 765, find n(A ∩ B).17ESIn Exercises 17 and 18, use the given information to find the number of elements in each of the regions labeled with a question mark. 18. n(A)= 610, n(B)= 440, n(C)= 1000, n(U)= 2900In a surveyof 600 investors, it was reported that 380 hadinvested in stocks, 325 had invested in bonds, and 75 had notinvested in either stocks or bonds. a. How manyinvestors had invested in both stocks and bonds? b. How many investors had invested only in stocks?Commuting A survey of 1500 commuters in New York City showed that 1140 take the subway, 680 take the bus, and 120 do not take either the bus or the subway. a. How many commuters take both the bus and the subway? b. How many commuters take only the subway?A team physician has determined that of all the athletes who were treated for minor back pain, 72% responded favorably to an analgesic, 59% responded favorably to a muscle relaxant, and 44% responded favorably to both forms of treatment. a. What percent of the athletes who were treated responded favorably to the muscle relaxant but not to the analgesic? b. What percent of the athletes who were treated did not respond favorably to either form of treatment?The management of a hotel conducted a survey. ft found that of the 2560 guests who were surveyed. 1785 tip the wait staff. 1219 tip the luggage handlers. 831 tip the maids. 275 tip the maids and the luggage handlers. 700 tip the wait staff and the maids. 755 tip the wait staff and the luggage handlers. 245 tip all three services. 210 do not tip these services. How many of the surveyed guests tip a. exactly two of the three services? b. only the wait staff? c. only one of the three services?A computer company advertises its computers in PC World, in PC Magazine, and on television. A survey of 770 customers finds that the number: of customers who are familiar with the companys con putters because of the different forms of advertising an as follows. 305, PC World 290, PC Magazine 390, television 110, PC World and PC Magazine 135, PC Magazine and television 150, PC World and television 85, all three sources How many of the surveyed customers know about the computers because of a. exactly one of these forms of advertising? b. exactly two of these forms of advertising? c. PC World and neither of the other two forms of advertising?During one month, a blood donation center found that 45.8% of the donors had the A antigen. 14.2% of the donors had the B antigen. 4.1% of the donors had the A antigen and the B antigen. 84.7% of the donors were Rh+. 87.2% of the donors had the B antigen or were Rh+. Find the percent of the donors that a. have the A antigen or the B antigen. b. have the B antigen and are Rh+.A special interest group has conducted a survey concerning a ban on handguns. Note: A rifle is a gun, but ills not a handgun. The survey yielded the following results for the 1000 households that responded. 271 own a handgun. 437 own a rifle. 497 supported the ban on handguns. 140 own both a handgun and a rifle. 202 own a rifle but no handgun and do not support the ban on handguns. 74 own a handgun and support the ban on handguns. 52 own both a handgun and a rifle and also support the ban on handguns. How many of the surveyed households? a. only own a handgun and do not support the ban on handguns? b. do not own a gun and support the ban on handguns? c. do not own a gun and do not support the ban on handguns?A survey of college students was taken to determine how the students acquired music. The survey showed the following results. 365 students acquired music from CDs. 298 students acquired music from the Internet. 268 students acquired music from DVDs. 212 students acquired music from both CDs and DVDs. 155 students acquired music from both CDs and the Internet. 36 students acquired music from DVDs, but not from CDs or the Internet. 98 students acquired music from CDs, DVDs, and the Internet. Of those surveyed, a. how many acquired music from CDs, but not from the Internet or DVDs? b. how many acquired music from the Internet., but not from CDs or DVDs? c. how many acquired music from CDs or the Internet? d. how many acquired music from the Internet and DVDs?A survey was completed by individuals who were currently on the Zone diet (Z). the South Beach diet (S), or the Weight Watchers diet (W). All persons surveyed were also asked whether they were currently in an exercise program (F), taking diet pills (P), or under medical supervision (M). The following table shows the results of the survey. Find the number of surveyed people in each of the following sets. a. S∩E b.Z∪M S∩ (E∪P)d.(Z∪S)∩ (M) w∩ (P∪M)f. w∪ P28ES29ES30ES31ES32ESUse two disjoint sets to show that 0+2=0.2EEUse sets toshow that 06=0.1ESEstablish a one-to-one correspondence between the set of natural numbers N={1,2,3,4,5, ...,n,...} and F = {5, 10, 15, 20,….. 5n,….} by stating a general rule that can be used to pair the elements of the sets.Establish a one-to-one correspondence between D = {l,3,5,...,2n -1, ...} and M = (3, 6, 9,…., 3n, ...) by stating a general rule that can be used to pair the elements of the sets.4ES5ESIn Exercises 4 to 10, state the cardinality of each set. The set Q of rational numbers7ES8ES9ES10ES11ESIn Exercises 11 to 14, determine whether the given sets are equivalent. The set of whole numbers and the set of real numbers13ES14ES15ESIn Exercises 15 to 18, show that the given set is an infinite set by placing it in a one-to-one correspondence with a proper subset of itself. B = {1l, 15, l9, 23, 27,31, .4n + 7, ,.}17ES18ESIn Exercises 19 to 26, show that the given set has a cardinality of K by establishing a one-to-one correspondence between the elements of the given set and the elements of N. 26. {0.1,0.01,0.001,0.0001,...10-n,..}20ES21ESIn Exercises 19 to 26, show that the given set has a cardinality of K by establishing a one-to-one correspondence between the elements of the given set and the elements of N. 23.{10,100,1000,...,10n,…}23ESIn Exercises 19 to 26, show that the given set has a cardinality of K by establishing a one-to-one correspondence between the elements of the given set and the elements of N. 21.{ 1,13,19,127,.....13n1,... }In Exercises 19 to 26, show that the given set has a cardinality of K by establishing a one-to-one correspondence between the elements of the given set and the elements of N. 20. {10,5,0,-5,-10,-15,...,-5n + 15,..}In Exercises 19 to 26, show that the given set has a cardinality of K by establishing a one-to-one correspondence between the elements of the given set and the elements of N. 19. {50,51,52,53,...,n + 49,...}27ES28ES29ESThe Hubert Hotel is an imaginary hotel created by the mathematician David Hubert (1862—1943). The hotel has an infinite number of rooms. Each room is numbered with a natural number—room 1, room 2, room 3, and so on. Search the Internet for information on Hilberts Hotel. Write a few paragraphs that explain some of the Interesting questions that arise when guests arrive to stay at the hotel. Mary Pat Campbell has written a song about a hotel with an infinite number of rooms. Her song is titled Hotel Aleph Null—yeah. Here are the lyrics for the chorus of her song, which i to be sung to the tune of Hotel California by the Eagles (Source: http//www.marypat.org/mathcamp/doc200l/hellrelays.html#hotel). 4 hotel Aleph Null—yeah Welcome to the Hotel Aleph Null—yeah What a lovely place (what a lovely place) Got a lot of space Packin em in at the Hotel Aleph Null—yeah Any time of year You can find space here1REIn Exercises 1 to 6, use the roster method to represent each set. The set of states in the United States that do not share a common border with another state3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36REIn Exercises 35 to 38, find the number of subsets of the given set. The set of the letters of uncopyrightable, which is the longest English word with no repeated letters38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52REIn a survey at a health club, 208 members indicated that they enjoy aerobic exercises, 145 indicated that they enjoy weight training, 97 indicated that they enjoy both aerobics and weight training, and 135 indicated that they do not enjoy either of these types of exercise. How many members were surveyed?A gourmet coffee bar conducted a survey to determine the preferences of its customers. Of the customers surveyed, 221 like espresso. 127 like cappuccino and chocolate-flavored coffee. 182 like cappuccino. 136 like espresso and chocolate-flavored coffee. 209 like chocolate-flavored coffee. 96 like all three types of coffee. 116 like espresso and cappuccino. 82 like none of these types of coffee. How many of the customers in the survey. a. like only chocolate-flavored coffee? b. like cappuccino and chocolate flavored coffee but not espresso? c. like espresso and cappuccino but not chocolate flavored coffee d. like exactly one of the three types of coffee?55RE56RE57RE58RE59RE60RE61RE62RE63RE64RE65RE66RE