MTH101_Chap7 - 372 C H A P T E R 7 Logic, Sets, and...

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Unformatted text preview: 372 C H A P T E R 7 Logic, Sets, and Counting 3' Exercise 7-3 l A Problem 1—12 refer to the following Venn diagram. Find the number of elements in each of the indicated sets. 20. The 14 colleges of interest to a high school senior include 6 that are expensive (tuition more than $205!”, per year), 7 that are far from home (more than 200 miles away), and 2 that are both expensive and far from home. i l i i (A) If the student decides to select a college that is 11 expensive and within 200 miles of home, how man selections are possible? = (B) If the student decides to attend a college that is n } expensive and within 200 miles from home during his first two years of college, and then will transfe to a college that is not expensive but is far from ‘ 1‘ A 2' B 3‘ U 7 home, how many selections of two colleges are , 4. A’ S. B ’ 6. A O B possible? 7. AUB 8.AflB’ 9. A’DB 10. A. n B. 11. (A n B): 12. (A U B)! In Problems 21-26, use the given information to determiner - number of elements in each of the four disjoint subsets in th Solve Problems 13—16 two ways: (A) using a tree diagram and fozlowmg verm diagram (B) using the multiplication principle. 13. How many ways can 2 coins turn up—heads, H, or tails, T—if the combined outcome (H, T) is to be distin- guished from the outcome (T, H)? 14. How many 2-letter code words can be formed from the first 3 letters of the alphabet if no letter can be used more than once? 15. A coin is tossed with possible outcomes heads, H, or tails, T. Then a single die is tossed with possible outcomes 1, 2, 21. n(A) = 80, n(B) = 50, 3, 4, 5, or 6. How many combined outcomes are there? n(A 0 B) = 20, n( U) = 200 ‘ 16. In how many ways can 3 coins turn up—heads, H, or 22. n(A) = 45, n(B) = 35, ‘ tails, T—if combined outcomes such as (H,T,H), (H, H,T), n(A n B) = 15, n( U) = 100 and (T, H, H) are to be considered different? 23_ n(A) __. 25 "(3) = 55 17. An entertainment guide recommends 6 restaurants and n(A U B) = 60, n(U) = 100 , 3 plays that appeal to a couple. M n(A) = 70 "(3) ___ 90 , (A) If the couple goes to dinner or to a play, how many n(A U B) = 120, n(U) = 200 selections are possible? 25 n(A.) = 65 "(3’) = 40 (B) If the couple goes to dinner and then to a play, how n(A! n B! ) ’= 25’ n( U) ; 150 many combined selections are possible? 26 n(A,) _ 35 "(In _ 75 i 18. A college offers 2 introductory courses in history,3 in . "(A U B!) ,2 95, "((1) =1 120 science, 2 in mathematics, 2 in philosophy, and 1 in English. aim-uM-‘~:fi~§hfixgmaflehmyuawu' hm 1.4).: v ~> ’ i: “< A; W11” 3 (A) If a freshman takes one course in each area during 1" Pmblems 27—32’ use the gwe" infommio" to complete her first semester, how many course selections are fallowmg table' I possible? 1 (B) If a part-time student can afford to take only one in- A A! Tow" troductory course, how many selections are possible? 3 ‘?’ ? v Q B' 'é a '2 B ‘ Totals '2 o 7 19. A county park system rates its 20 golf courses in increas- ’ ing order of difficulty as bronze, silver, or gold. There are only two gold courses, and twice as many bronze as silver. . 27. n(A) = 70, n(B) = 90, n(AflB) = 30, n(U) = 200 (A) If a golfer decides to play’ a round at a silver or gold 28. n(A) = 55, "(3) = 65’ course, how many selections are possrble? "(A n B) = 35 n( U) = 100 (B) If a golfer decides to play one round per week for 3 ’ weeks, first on a bronze course, then silver, then 29. n(A) = 45. "(3) = 55. gold, how many combined selections are possible? n(A U B) = 80, n(U) = 100 n(A) = 80, n(B) = 70. maum=1w,mw=2m _.',,(A') = 15, n(B’) : 24, ty'nm' UB') = 32. n(U) = 90 '.n(A') = 81, n(B') = 90, , MATH?) = 63, n(U) = 180 Women’s 33 and 34, disortss the validity of each statement. ,1)"; statement is always true, explain why. If no; give a , (A) If A or B is the empty set, then A and B are disjoint. (B) If A and B are disjoint, then A or B is the empty set. - (A) HA and B are disjoint, then n(A 08) = n(A) + n(B). (B) If n(A U B) = n(A) + n(B), thenA and B are disjoint. color, 3 choices of transmission, 4 types of interior, and 2 types of engine. How many different variations of this model car are possible? A delicatessen serves meat sandwiches with the follow- ing options: 3 kinds of bread, 5 kinds of meat, and lettuce or sprouts How many different sandwiches are possible, assuming that one item is used out of each category? How many 4-letter code words are possible from the first 6 letters of the alphabet if no letter is repeated? If letters can be repeated? If adjacent letters must be different? How many 5-letter code words are possible from the first 7 letters of the alphabet if no letter is repeated? If letters can be repeated? If adjacent letters must be different? A combination lock has 5 wheels, each labeled with the 10 digits from O to 9. How many S-digit opening combi- nations are possible if no digit is repeated? If digits can be repeated? If successive digits must be different? A small combination lock on a suitcase has 3 wheels, each labeled with the 10 digits from 0 to 9, How many 3-digit combinations are possible if no digit is repeated? If digits can be repeated? If successive digits must be 723,? different? ,e...~e....m_.—RW v ., , v, ,< 41. How many different license plates are possible if each contains 3 letters (out of the 26 letters of the alphabet) A particular new car model is available with 5 choices of ‘ S e ct i o n 7 ~ 3 Basic Counting Principles 373 followed by 3 digits (from 0 to 9)? How many of these license plates contain no repeated letters and no repeated digits? 42.‘ How many S—digit ZIP code numbers are possible? How many of these numbers contain no repeated digits? In Example 3, does it make any difference in which order the selection operations are performed? That is, if we select a jacket color first and then select a size, are there as many combined choices available as selecting a size first and then a color? Justify your answer using tree diagrams and the multiplication principle. Explain how three sets, A, B, and C, can be related to each other in order for the following equation to hold (Venn diagrams may be helpful): n(AUBUC) = n(A) + n(B) + n(C) — n(Ar‘IC) —— n(BnC) C Problems 45—48 refer to the following Venn diagram. U Which of the numbers x, y, z, or w must equal 0 if B C A? Which of the numbers my, z. or w must equal 0 if A and B are disjoint? Which of the numbers x, y, z, or w must equal 0 if A U B = An 3? Which of the numbers ray. 2, or w must equal 0 if A U B = U? A group of 75 people includes 32 who play tennis, 37 who play golf, and 8 who play both tennis and golf. How many people in the group play neither sport? 49. 50. A class of 30 music students includes 13 who play the piano, 16 who play the guitar, and 5 who play both the piano and the guitar. How many students in the class play neither instrument? 5]. A group of 100 people touring Europe includes 42 people who speak French, 55 who speak German, and 17 who speak neither language. How many people in the group speak both French and German? A high school football team with 40 players includes 16 players who played offense last year, 17 who played defense. and 12 who were not on last year‘s team. How many players from last year played both offense and defense? 52. 374 C H A P T E R 7 Logic, Sets, and Counting jv Applications 53. Management selection. A management selection service two premium channels, HBO and Showtime. If 2,450 su classifies its applicants (using tests and interviews) as high- sci-fibers receive HBO, 1,940 receive Showtime, and 5,1 IQ,middle-IQ, or low-IQ and as aggressive or passive. How do not receive any premium channel, how many subset] . ’ many combined classifications are possible? receive both HBO and Showtime? (A) Solve by using a tree diagram. i 60. Communications. A local telephone company offers i“ (B) Solve by using the multiplication principle. 10-000 yystomcrs two Special services: call forwarding : _ call wattmg. If 3,770 customers use call forwarding, 3,250 54. Management selection. A corporation plans to fill 2 differ- can waiting, and 4 530 do not use either of these services, h 3‘" Positions for Vice‘PreSldcmi V1 and V2, from adminis‘ many customers use both call forwarding and call waitin trative officers in 2 of its manufacturing plants Plant A has 6 officers and plant 3 has 8. How many ways can thesc 2 po_ 61. Minimum wage. The table below gives the number of . sitions be filled if the V} position is to be filled from plant and female workers “fining at 01' below the minimum “’3 A and the Vz position from plant B? How many ways can for several age Categories- the 2 positions be filled if the selection is made without _ regard to Plant? Workers per AgeGroup 55. Transportation. A sales representative who lives in city A w wishes to start from home and fly to 3 different cities, 8, C, 16—19 20—24 25+ To and D. If there are 2 choices of local transportation (drive " -- e I » : ~ _ her own car to and from the airport or use a taxi for both trips), and if all cities are interconnected by airlines, how many travel plans can be constructed to visit each city exactly once and return home? (A) How many males are age 20—24 and below minim wage? . (B) How many females are age 20 or older and at I l 56. Transportation. A manufacturing company in city A wishes mum wage? to truck its product to 4 different cities, B, C, D, and E. If the ((3)5 How many workers are age 16—19 or males at l.' 1 cities are all interconnected by roads, how many different mum wage? l route plans can be constructed so that a single truck. starting H . . from A, will visit each city exactly once, then return home? (D) ow many workers are below mlmmum wage? 62. M ' ‘ . R f t t l ' . 57. Market research. A survey of 1,200 people in a certain city lmmum wage 6 er 0 he tab 6 m Promem 61 i,..- .gvm? r HIM” ‘u‘nsfi indicates that 850 own microwave ovens, 740 own DVD (A) How many females are age 16—19 and at minim players, and 580 own microwave ovens and DVD players. wage? (A) How many people in the survey own either a (B) How many males are age 16—24 and below mi ii microwave oven or a DVD player? wage? ‘ (B) How many own neither a microwave oven nor a DVD (C) Hva many workers are age 20‘24 01' females bgl I mmtmum wage? i player? (C) How many own a microwave oven and do not own a (D) How many workers are at mlmmum Wage? DVD Player? 63. Medicine. A medical researcher classifies subjects accord 58. Market research. A survey of 800 small businesses indicates ing to male or female; smoker or nonsmOker; and und that 250 own photocopiers, 420 own fax machines, and 180 weight, average weight, or overweight. How many combin own photocopiers and fax machines classifications are possible? (A) How many businesses in the survey own either a pho- (A) some “Sing a tree diagram tocopier or a fax machine? (B) Solve using the multiplication principle. (B) How many own neither a photocopier nor a fax ma- 64. Family planning. A couple is planning to have 3 children chine? How many boy—girl combinations are possible? l (C) How many own a fax machine and do not own a pho— bemfingogbmed ontcomes§11°h as (B, Bi G), (Bi Gr 3,) an , , . » . tocopier? 59. Communications. A cable television company has 8,000 (A) 301% by “Sing a “‘36 diagram- subscribers in a suburban community. The company offers (B) - Solve by using the multiplication principle. 65. [2:i;’13c5. A politician running for a third term in office is binning to contact all contributors to her first two cam- ;:.. If [.475 individuals contributed to the first campaign. contributed to the second campaign, and 920 con- " pied to the first and second campaigns. how many indi. visuals have contributed to the first or second campaign? 7 “ <~ Permutations and Combinations 375 66. Politics. If 11,457 people voted for a politician in his first election, 15.322 voted for him in his second election. and 9,345 voted for him in the first and second elections, how many people voted for this politician in the first or second election? Section 7-4 DEFlNlTlON PERMUTATIONS AND COMBINATIONS Factorials Permutations , Combinations 1 Applications The multiplication principle discussed in the preceding section can be used to develop two additional devices for counting that are extremely useful in more complicated counting problems. Both of these devices use a function called a factorial function, which we introduce first. Factorials When using the multiplication principle, we encountered expressions such as 26-25-24 or 8-7-6 where each natural number factor is decreased by 1 as we move from left to right. The factors in the following product continue to decrease by 1 until a factor of 1 is reached: 5-4-3-2-1 Products of this type are encountered so frequently in counting problems that it is useful to be able to express them in a concise notation. The product of the first n natural numbers is called I: factorial and is denoted by n!. Also, we define zero factorial, 0!, to be 1. Symbolically, Factorial For n a natural number, ni:n(n-l)(n—2) -----2-l 1?? 0! = l Ill = n - (n W l)! Note: Many calculators have an 5 it! 3 key or its equivalent. “Enomputing Factoriai; (A)5!=5'4-3-2-1=120 7! 7-m_ (B)5i_ M ‘7 8! 8.7.6.4? (C 5: a =8-7-6=336 ! -51-50-4- (D) _5_Z_ = imifl = 2,598,960 _ 5-4-3-2-1-4?! S e c t i o n 7 A a Permutations and Combinations 385 33. How many ways can a 3-person subcommittee be selected 31. In a horse race, how many different finishes among the first 3 places are possible if 10 horses are running? (Exclude ties.) 46. If n is a positive integer greater than 3, then n! > 2 ". 2 9, i from a committee of 7 people? How many ways can a ' ' president, vice-president, and secretary be chosen from a 18'- » committee of 7 people? 4. — 15 l 34. Nine cards are numbered with the digits from 1 to 9. A 6 1000! 3-card hand is dealt, 1 card at a time. How many hands ' (1000 _ 2)! are possible where 13! (A) Order is taken into consideration? 8' (B) Order is not taken into consideration? lo 202 B ' l _ l 8' (20 8)' 35. Discuss the relative growth rates of xi, 3“, and x3. 12' C“ 36. Discuss the relative growth rates of x!, 2", and x2. 14' PM i 37. From a standard 52-card deck, how many S-card hands 16. C5233 i will have all hearts? 18 C133 38. From a standard SZ-card deck, how many 5-card hands ' C523 will have all face cards? All face cards, but no kings? [365.23 39. From a standard 52-card deck, how many 7-card hands 36523 have exactly 5 spades and 2 hearts? i 40. From a standard 52-eard deck, how many 5-card hands g will have 2 clubs and 3 hearts? In Problems 21—24, simplify each expression assuming that n is i an integer and n 2 3. i 41. A catering service offers 8 appetizers-10 main courses, ' ' g and 7 desserts A banquet committee is to select 3 appe- 21. __"_'__ 22. _";_ § tizers, 4 main courses, and 2 desserts. How many ways (" ‘ 3)! 2K" _ 2)! 3 can this be done? 23. (n + 2)! 24. (n + 2)! 42. Three departments have 12,15, and 18 members, respec- n!2! (n — 1)! tively. If each department is to select a delegate and an 3 alternate to represent the department at a conference, 3 how many ways can this be done? , In Problems 25—30, would you consider the selection to be a » permutation, a combination, or neither? Explain your In Problems 43 and 44, refer to the table in the graphing reasoning. calculator display below, which shows y1 = P,._, and yz = C", 25. The new university president named 3 new officers: a for n = 6. vice-president of finance, a vice-president of academic k affairs, and a vice-president of student affairs ? o 26. The university president selected 2 of her vice-presidents g to attend the dedication ceremony of a new branch ; 5 ‘ ' campus. g V 27. A student checked out 4 novels from the library to read v 2 over the holiday. 5 ’ 23_ A student did some holiday shopping by buying 4 books; 43. Discuss and explain the symmetry of the numbers in the 1 for his father, 1 for his mother, 1 for his younger sister, g yz 0011mm of the table. and 1 for his older brother. 44. Explain how the table illustrates the formula 29. A father ordered an ice cream cone (chocolate, vanilla, P” r = rm" r or strawberry) for each of his 4 children. i ’ ‘ 30. A book club meets monthly in a home of one of its 10 I In Problems 45—50, discuss the validity of each statement. If the members In December the club selects a host for each statement is true, explain why. I f not, give a counterexample. meenng 0f the “em year- I 45. Ifn is a positive integer, then n! < (n + 1)! l l 47. If n and r are positive integers and 1 < r < n, then ' . . P < . 32. In a long-distance foot race, how many different finishes "" P "’ +1 I I . among the first 5 places are possible if 50 people are 48. If n and r are posmve integers and 1 < r < n, then running? (Exclude ties.) Cu, < Cn.r+l' i mun—— 385 C H A PT 5 R 7 Logic, Sets, and Counting 49. If n and r are positive integers and 1 < r < n, then Cm = n,n-rv 50. If n and rare positive integers and 1 < r < n, then Pm! = Prim-r 51. Eight distinct points are selected on the circumference of a circle. (A) How many chords can be drawn by joining the points in all possible ways? (B) How many triangles can be drawn using these 8 points as vertices? (C) How many quadrilaterals can be drawn using these 8 points as vertices? 52. Five distinct points are selected on the circumference of a circle. (A) How many chords can be drawn by joining the points in all possible ways? (B) How many triangles can be drawn using these 5 points as vertices? 53. How many ways can 2 people be seated in a row of 5 chairs? 3 people? 4 people? 5 people? 54. Each of 2 countries sends 5 delegates to a negotiating conference. A rectangular table is used with 5 chairs on each long side. If each country is assigned a long side of the table (operation 1), how many seating arrangements are possible? 55. A basketball team has 5 distinct positions. Out of 8 play- ers, how many starting teams are possible if (A) The distinct positions are taken into consideration? (B) The distinct positions are not taken into consideration? (C) The distinct positions are not taken into consideration, but either Mike or Ken (but not both) must start? Applications 61. Quality control. A computer store receives a shipment of 24 laser printers, including 5 that are defective. Three of these printers are selected to be displayed in the store. (A) How many selections can be made? (B) How many of these selections will contain no defective printers? 62. Quality control. An electronics store receives a shipment of 30 graphing calculators, including 6 that are defective. Four of these calculators are selected to be sent to a local high school. (A) How many selections can be made? (B) How many of these selections will contain no defective calculators? 63. Business closings. A jewelry store chain with 8 stores in Georgia, 12 in Florida, and 10 in Alabama is planning to close 10 of these stores (A) How many ways can this be done? 2 56. How many 4-person committees are possible from a E group of 9 people if t (A) There are no restrictions? 1% (B) Both I im and Mary must be on the committee? 3 (C) Either Jim or Mary (but not both) must be on the E committee? 3 57. Find the largest integer k such that your calculator can i compute k! without an overflow error. 1 l 58. Find the largest integer k such that your calculator can 4 compute C2,“,c without an overflow error. 59. Note from the table in the graphing calculator display below that the largest value of C,” when n = 20 is Cmo = 184,756. Use a similar table to find the largest value of C,” when n = 24. 60. Note from the table in the graphing calculator display that the largest value of C", when n = 21 is Cum = GIL“ = 352,716. Use a similar table to find the largest value of C", when n = 17. (B) The company decides to close 2 stores in Georgia, 5 ' Florida, and 3 in Alabama. In how many ways can thtS: be done? 64. Employee layoffs. A real estate company with 14 employ . ees in their central office, 8 in their north office, and 6 in their south office is planning to lay off 12 employees (A) How many ways can this be done? (B) The company decides to lay off 5 employees from the. south office. In how many ways can this be done? 65. Personnel selection. Suppose that 6 female and 5 male ap- plicants have been successfully screened for 5 positions. In how many ways can the following compositions be selected? (B) 4 females and 1 male (C) 5 females (D) 5 people regardless of sex i i (A) 3 females and 2 males (E) At least 4 females central office, 4 from the north office, and 3 from the. 65, memirmc selection. A 4-person grievance committee is to be selected out of 2 departments A and B, with 15 and 20 people. respectively. In how many ways can the following .3._~;»-.nnittees be selected? ( a.) 3 from A and 1 from B 1 2 from A and ‘2 from B Chapter 7 Review 387 samples. How many different examinations can be given if no 2 of the samples provided for the candidate have the same type? If 2 or more samples can have the same type? Medical research. Because of the limited funds. 5 research centers are to be chosen out of 8 suitable ones for a study on heart disease. How many choices are possible? ‘7 H351. All from A 69. Politics. A nominating convention is to select a president (0»). 4 people regardless of department and vice-president from among 4 candidates Campaign but- tons. listing a president and a vice-president, are to be de- signed for each possible outcome before the convention. 67. ,édedicinc. There are 8 standard classifications of blood type. How many different kinds of buttons should be designed? “if” exglfnrianon far prafrftlzesabolatfizy tfchnticlafilcsgé 70. Politics. In how many different ways can 6 candidates for an i sets 0 avmg cat: can 1 a e e ermine e ype or office be “3‘ e d on a balm? (.53) At least 3 from department A cHi‘V‘i'ER ZREVIEW... .. _ Important Terms, Symbols. and Concepts 74 Examples o .\ proposition is a statement (not a question or command) that is either true or false. 0 H gt :nid q are propositions, then the compound propositions. Iis. I_p 3V) «wp. p \/ q. p /\ q. and p—H] L‘.’l‘ be formed using the negation symbol -u and the connectives V. A. and -+ .These propositions are L‘Lll led not p, p or q, p and q, and if p then q, respectively (or negation. disjunction, conjunction, and \ conditional, respectively). Each of these compound propositions is specified by a truth table (see pine (Mill). - \i llh any conditional proposition p -> q we associate the proposition q -’ p. called the converse of l’\. p r» q. and the proposition no —> op.called the contrapositive of p ~—> q. l .2 .ii, 353 0 ,\ truth table for a compound proposition specifies whether it is true or false for any assignment of l~' \. ,‘x. i n ulh values to its variables. A proposition is a tautology if each entry in its Column of the truth table l‘\. 4 p. ‘t' - l‘. a contradiction if each entry is F. and a contingency if at l Inst one entry is T and at least one l'\ S, p ‘w t» :lil‘V is F. 0 ( outsider the rows of the truth tables for the compound propositions P and Q. If whenever P is true. li U also true. we say that P logically implies Q and write P = Q. We call P => Q a logical implication. l“, 1 It the compound propositions P and Q have identical truth tables. we say that P and Q are logically thriivalent and write P E Q. We call P E Q a logical equivalence. } I‘/ \l T. 4444 3 Several logical equivalences are given in Table 2 on page 357. The last of these implies that any condi- tioual proposition is logically equivalent to its contrapositive. l ‘t, set is a collection of objects specified in such a way that we can tell whether any given object is or is i not in the collection. I och object in a set is called a member. or element. of the set. If a is an element of the set A. we write \ ii fl. .\ set without any elements is called the empty. or null, set. denoted t3. -\ set can be described by listing its elements, or by giving a rule that determines the elements of the l"\ l p. itt'fi‘ set. [i P(.r) is a statement about .t. then {.rlP(.r)| denotes the set of all .r such that P(.r) is true. .\ set is finite if its elements ‘an be counted and there is an end; a set such as the positive integers. in which there is no end in counting its elements. is infinite. ‘ We write A C B. and say that A is a subset of B,ifet\cl1 element of A is an element of B.We write l \ ‘, p in ,l 2 B. and say that sets A and B are equal. if they have exactly the same elements. The empty set (25 is 21 subset of every set. 5 [it in; in ...
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MTH101_Chap7 - 372 C H A P T E R 7 Logic, Sets, and...

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