Excersises concerning different fields of mathematics first year university students, Exercises of Mathematics

Download Excersises concerning different fields of mathematics first year university students and more Exercises Mathematics in PDF only on Docsity! 464 Test Bank Questions and Answers TEST BANK Questions for Chapter 1 What is the negation of the propositions in 1–3? 1. Abby has more than 300 friends on facebook. 2. A messaging package for a cell phone costs less than $20 per month. 3. 4.5 + 2.5 = 6 In questions 4–8, determine whether the proposition is TRUE or FALSE. 4. 1 + 1 = 3 if and only if 2 + 2 = 3. 5. If it is raining, then it is raining. 6. If 1 < 0, then 3 = 4. 7. If 2 + 1 = 3, then 2 = 3 ! 1. 8. If 1 + 1 = 2 or 1 + 1 = 3, then 2 + 2 = 3 and 2 + 2 = 4. 9. Write the truth table for the proposition ¬(r " ¬q) # (p $ ¬r). 10. (a) Find a proposition with the truth table at the right. p q ? T T F T F F F T T F F F (b) Find a proposition using only p, q,¬, and the connective # that has this truth table. 11. Find a proposition with three variables p, q, and r that is true when p and r are true and q is false, and false otherwise. 12. Find a proposition with three variables p, q, and r that is true when at most one of the three variables is true, and false otherwise. 13. Find a proposition with three variables p, q, and r that is never true. 14. Find a proposition using only p, q,¬, and the connective # with the truth table at the right. p q ? T T F T F T F T T F F F15. Determine whether p " (q " r) and p " (q $ r) are equivalent. 16. Determine whether p " (q " r) is equivalent to (p " q) " r. 17. Determine whether (p " q) $ (¬p " q) % q. Test Bank Questions and Answers 465 18. Write a proposition equivalent to p # ¬q that uses only p, q,¬, and the connective $. 19. Write a proposition equivalent to ¬p $ ¬q using only p, q,¬, and the connective #. 20. Prove that the proposition “if it is not hot, then it is hot” is equivalent to “it is hot”. 21. Write a proposition equivalent to p " q using only p, q,¬, and the connective #. 22. Write a proposition equivalent to p " q using only p, q,¬, and the connective $. 23. Prove that p " q and its converse are not logically equivalent. 24. Prove that ¬p " ¬q and its inverse are not logically equivalent. 25. Determine whether the following two propositions are logically equivalent: p # (q $ r), (p $ q) # (p $ r). 26. Determine whether the following two propositions are logically equivalent: p " (¬q $ r), ¬p # ¬(r " q). 27. Prove that (q $ (p " ¬q)) " ¬p is a tautology using propositional equivalence and the laws of logic. 28. Determine whether this proposition is a tautology: ((p " q) $ ¬p) " ¬q. 29. Determine whether this proposition is a tautology: ((p " ¬q) $ q) " ¬p. In 30–36, write the statement in the form “If . . . , then . . . .” 30. x is even only if y is odd. 31. A implies B. 32. It is hot whenever it is sunny. 33. To get a good grade it is necessary that you study. 34. Studying is su!cient for passing. 35. The team wins if the quarterback can pass. 36. You need to be registered in order to check out library books. 37. Write the contrapositive, converse, and inverse of the following: If you try hard, then you will win. 38. Write the contrapositive, converse, and inverse of the following: You sleep late if it is Saturday. In 39–41 write the negation of the statement. (Don’t write “It is not true that . . . .”) 39. It is Thursday and it is cold. 40. I will go to the play or read a book, but not both. 41. If it is rainy, then we go to the movies. 42. Explain why the negation of “Al and Bill are absent” is not “Al and Bill are present”. 43. Using c for “it is cold” and d for “it is dry”, write “It is neither cold nor dry” in symbols. 44. Using c for “it is cold” and r for “it is rainy”, write “It is rainy if it is not cold” in symbols. 45. Using c for “it is cold” and w for “it is windy”, write “To be windy it is necessary that it be cold” in symbols. 46. Using c for “it is cold”, r for “it is rainy”, and w for “it is windy”, write “It is rainy only if it is windy and cold” in symbols. 468 Test Bank Questions and Answers 83. All students are freshmen. 84. Every freshman is a full-time student. 85. No math course is upper-level. In 86–88 suppose the variable x represents students and y represents courses, and: U(y): y is an upper-level course M(y): y is a math course F (x): x is a freshman A(x): x is a part-time student T (x, y): student x is taking course y. Write the statement using these predicates and any needed quantifiers. 86. Every student is taking at least one course. 87. There is a part-time student who is not taking any math course. 88. Every part-time freshman is taking some upper-level course. In 89–91 suppose the variable x represents students and y represents courses, and: F (x): x is a freshman A(x): x is a part-time student T (x, y): x is taking y. Write the statement in good English without using variables in your answers. 89. F (Mikko). 90. ¬'y T (Joe, y). 91. 'x (A(x) $ ¬F (x)). In 92–94 suppose the variable x represents students and y represents courses, and: M(y): y is a math course F (x): x is a freshman B(x): x is a full-time student T (x, y): x is taking y. Write the statement in good English without using variables in your answers. 92. &x'y T (x, y). 93. 'x&y T (x, y). 94. &x'y [(B(x) $ F (x)) " (M(y) $ T (x, y))]. In 95–97 suppose the variables x and y represent real numbers, and L(x, y) : x < y G(x) : x > 0 P (x) : x is a prime number. Write the statement in good English without using any variables in your answer. 95. L(7, 3). 96. &x'y L(x, y). 97. &x'y [G(x) " (P (y) $ L(x, y))]. In 98–100 suppose the variables x and y represent real numbers, and L(x, y) : x < y Q(x, y) : x = y E(x) : x is even I(x) : x is an integer. Write the statement using these predicates and any needed quantifiers. 98. Every integer is even. 99. If x < y, then x is not equal to y. 100. There is no largest real number. In 101–102 suppose the variables x and y represent real numbers, and Test Bank Questions and Answers 469 E(x) : x is even G(x) : x > 0 I(x) : x is an integer. Write the statement using these predicates and any needed quantifiers. 101. Some real numbers are not positive. 102. No even integers are odd. In 103–105 suppose the variable x represents people, and F (x): x is friendly T (x): x is tall A(x): x is angry. Write the statement using these predicates and any needed quantifiers. 103. Some people are not angry. 104. All tall people are friendly. 105. No friendly people are angry. In 106–107 suppose the variable x represents people, and F (x): x is friendly T (x): x is tall A(x): x is angry. Write the statement using these predicates and any needed quantifiers. 106. Some tall angry people are friendly. 107. If a person is friendly, then that person is not angry. In 108–110 suppose the variable x represents people, and F (x): x is friendly T (x): x is tall A(x): x is angry. Write the statement in good English. Do not use variables in your answer. 108. A(Bill). 109. ¬'x (A(x) $ T (x)). 110. ¬&x (F (x) " A(x)). In 111–113 suppose the variable x represents students and the variable y represents courses, and A(y): y is an advanced course S(x): x is a sophomore F (x): x is a freshman T (x, y): x is taking y. Write the statement using these predicates and any needed quantifiers. 111. There is a course that every freshman is taking. 112. No freshman is a sophomore. 113. Some freshman is taking an advanced course. In 114–115 suppose the variable x represents students and the variable y represents courses, and A(y): y is an advanced course F (x): x is a freshman T (x, y): x is taking y P (x, y): x passed y. Write the statement using the above predicates and any needed quantifiers. 114. No one is taking every advanced course. 115. Every freshman passed calculus. In 116–118 suppose the variable x represents students and the variable y represents courses, and T (x, y): x is taking y P (x, y): x passed y. Write the statement in good English. Do not use variables in your answers. 116. ¬P (Wisteria, MAT 100). 117. 'y&x T (x, y). 470 Test Bank Questions and Answers 118. &x'y T (x, y). In 119–123 assume that the universe for x is all people and the universe for y is the set of all movies. Write the English statement using the following predicates and any needed quantifiers: S(x, y): x saw y L(x, y): x liked y A(y): y won an award C(y): y is a comedy. 119. No comedy won an award. 120. Lois saw Casablanca, but didn’t like it. 121. Some people have seen every comedy. 122. No one liked every movie he has seen. 123. Ben has never seen a movie that won an award. In 124–126 assume that the universe for x is all people and the universe for y is the set of all movies. Write the statement in good English, using the predicates S(x, y): x saw y L(x, y): x liked y. Do not use variables in your answer. 124. 'y ¬S(Margaret, y). 125. 'y&x L(x, y). 126. &x'y L(x, y). In 127–136 suppose the variable x represents students, y represents courses, and T (x, y) means “x is taking y”. Match the English statement with all its equivalent symbolic statements in this list: 1. 'x&y T (x, y) 2. 'y&x T (x, y) 3. &x'y T (x, y) 4. ¬'x'y T (x, y) 5. 'x&y ¬T (x, y) 6. &y'x T (x, y) 7. 'y&x ¬T (x, y) 8. ¬&x'y T (x, y) 9. ¬'y&x T (x, y) 10. ¬&x'y ¬T (x, y) 11. ¬&x¬&y ¬T (x, y) 12. &x'y ¬T (x, y) 127. Every course is being taken by at least one student. 128. Some student is taking every course. 129. No student is taking all courses. 130. There is a course that all students are taking. 131. Every student is taking at least one course. 132. There is a course that no students are taking. 133. Some students are taking no courses. 134. No course is being taken by all students. 135. Some courses are being taken by no students. 136. No student is taking any course. In 137–147 suppose the variable x represents students, F (x) means “x is a freshman’,’ and M(x) means “x is a math major”. Match the statement in symbols with one of the English statements in this list: 1. Some freshmen are math majors. 2. Every math major is a freshman. 3. No math major is a freshman. Test Bank Questions and Answers 473 172. Determine whether the premises “Some math majors left the campus for the weekend” and “All seniors left the campus for the weekend” imply the conclusion “Some seniors are math majors.” 173. Show that the premises “Everyone who read the textbook passed the exam”, and “Ed read the textbook” imply the conclusion “Ed passed the exam”. 174. Determine whether the premises “No juniors left campus for the weekend” and “Some math majors are not juniors” imply the conclusion “Some math majors left campus for the weekend.” 175. Show that the premise “My daughter visited Europe last week” implies the conclusion “Someone visited Europe last week”. 176. Suppose you wish to prove a theorem of the form “if p then q”. (a) If you give a direct proof, what do you assume and what do you prove? (b) If you give a proof by contraposition, what do you assume and what do you prove? (c) If you give a proof by contradiction, what do you assume and what do you prove? 177. Suppose that you had to prove a theorem of the form “if p then q”. Explain the di"erence between a direct proof and a proof by contraposition. 178. Give a direct proof of the following: “If x is an odd integer and y is an even integer, then x + y is odd”. 179. Give a proof by contradiction of the following: “If n is an odd integer, then n2 is odd”. 180. Consider the following theorem: “if x and y are odd integers, then x + y is even”. Give a direct proof of this theorem. 181. Consider the following theorem: “if x and y are odd integers, then x+y is even”. Give a proof by contradiction of this theorem. 182. Give a proof by contradiction of the following: If x and y are even integers, then xy is even. 183. Consider the following theorem: If x is an odd integer, then x + 2 is odd. Give a direct proof of this theorem 184. Consider the following theorem: If x is an odd integer, then x + 2 is odd. Give a proof by contraposition of this theorem. 185. Consider the following theorem: If x is an odd integer, then x + 2 is odd. Give a proof by contradiction of this theorem. 186. Consider the following theorem: If n is an even integer, then n+1 is odd. Give a direct proof of this theorem. 187. Consider the following theorem: If n is an even integer, then n + 1 is odd. Give a proof by contraposition of this theorem. 188. Consider the following theorem: If n is an even integer, then n + 1 is odd. Give a proof by contradiction of this theorem. 189. Prove that the following is true for all positive integers n: n is even if and only if 3n2 + 8 is even. 190. Prove the following theorem: n is even if and only if n2 is even. 191. Prove: if m and n are even integers, then mn is a multiple of 4. 192. Prove or disprove: For all real numbers x and y, *x! y+ = *x+ ! *y+. 193. Prove or disprove: For all real numbers x and y, *x + *x++ = *2x+. 194. Prove or disprove: For all real numbers x and y, *xy+ = *x+ · *y+. 195. Give a proof by cases that x ( |x| for all real numbers x. 474 Test Bank Questions and Answers 196. Suppose you are allowed to give either a direct proof or a proof by contraposition of the following: if 3n + 5 is even, then n is odd. Which type of proof would be easier to give? Explain why. 197. Prove that the following three statements about positive integers n are equivalent: (a) n is even; (b) n3 + 1 is odd; (c) n2 ! 1 is odd. 198. Given any 40 people, prove that at least four of them were born in the same month of the year. 199. Prove that the equation 2x2 + y2 = 14 has no positive integer solutions. 200. What is wrong with the following “proof” that !3 = 3, using backward reasoning? Assume that !3 = 3. Squaring both sides yields (!3)2 = 32, or 9 = 9. Therefore !3 = 3. Answers for Chapter 1 1. Abby has fewer than 301 friends on facebook. 2. A messaging package for a cell phone costs at least $20 per month. 3. 4.5 + 2.5 )= 6 4. True. 5. True. 6. True. 7. True. 8. False. 9. p q r ¬(r " ¬q) # (p $ ¬r) T T T T T T F T T F T F T F F T F T T T F T F F F F T F F F F F 10. (a) ¬p $ q, (b) ¬(p # ¬q). 11. p $ ¬q $ r. 12. (p $ ¬q $ ¬r) # (¬p $ q $ ¬r) # (¬p $ ¬q $ r) # (¬p $ ¬q $ ¬r). 13. (p $ ¬p) # (q $ ¬q) # (r $ ¬r). 14. ¬(¬p # q) # ¬(p # ¬q). 15. Not equivalent. Let q be false and p and r be true. 16. Not equivalent. Let p, q, and r be false. 17. Both truth tables are identical: p q (p " q) $ (¬p " q) q T T T T T F F F F T T T F F F F Test Bank Questions and Answers 475 18. ¬(¬p $ q). 19. ¬(p # q). 20. Both propositions are true when “it is hot” is true and both are false when “it is hot” is false. 21. ¬p # q. 22. ¬(p $ ¬q). 23. Truth values di"er when p is true and q is false. 24. Truth values di"er when p is false and q is true. 25. No. 26. Yes. 27. (q $ (p " ¬q)) " ¬p ,- (q $ (¬p # ¬q)) " ¬p ,- ((q $ ¬p) # (q $ ¬q)) " ¬p ,- (q $ ¬p) " ¬p ,- ¬(q $ ¬p) # ¬p ,- (¬q # p) # ¬p ,- ¬q # (p # ¬p), which is always true. 28. No. 29. Yes. 30. If x is even, then y is odd. 31. If A, then B. 32. If it is sunny, then it is hot. 33. If you don’t study, then you don’t get a good grade (equivalently, if you get a good grade, then you study). 34. If you study, then you pass. 35. If the quarterback can pass, then the team wins. 36. If you are not registered, then you cannot check out library books (equivalently, if you check out library books, then you are registered). 37. Contrapositive: If you will not win, then you do not try hard. Converse: If you will win, then you try hard. Inverse: If you do not try hard, then you will not win. 38. Contrapositive: If you do not sleep late, then it is not Saturday. Converse: If you sleep late, then it is Saturday. Inverse: If it is not Saturday, then you do not sleep late. 39. It is not Thursday or it is not cold. 40. I will go to the play and read a book, or I will not go to the play and not read a book. 41. It is rainy and we do not go to the movies. 42. Both propositions can be false at the same time. For example, Al could be present and Bill absent. 43. ¬c $ ¬d. 44. ¬c " r. 45. w " c. 46. r " (w $ c). 47. (r $ t) " h 48. Using m, n, k, and i, there are three rows of the truth table that have all five propositions true: the rows TTTT, FFTT, FFFT for m,n, k, i. 49. A is a knight, B is a knave. 50. A is a knave, B is a knight. 51. A is the spy, B is the knight, and C is the knave. 52. A is the knave, B is the spy, and C is the knight. 478 Test Bank Questions and Answers 119. &y (C(y)"¬A(y)). 120. S(Lois,Casablanca) $ ¬L(Lois,Casablanca). 121. 'x&y [C(y)"S(x, y)]. 122. ¬'x&y [S(x, y)"L(x, y)]. 123. ¬'y [A(y) $ S(Ben, y)]. 124. There is a movie that Margaret did not see. 125. There is a movie that everyone liked. 126. Everyone liked at least one movie. 127. 6. 128. 1, 10. 129. 12. 130. 2. 131. 3. 132. 7. 133. 5, 8, 11. 134. 9. 135. 7. 136. 4. 137. 3. 138. 2. 139. 3. 140. 2. 141. 1. 142. 1. 143. 2. 144. 3. 145. 2. 146. 3. 147. 1. 148. 'A ¬F (A). 149. &A&B [(F (B) $ S(A,B))"F (A)]. 150. ¬'A'B (¬F (A) $ F (B) $ S(A,B)). 151. &A (F (A)"S(Ø, A)). 152. No bananas are yellow. 153. Some integers ending in the digit 7 are not odd. 154. Some tests are easy. 155. Roses are not red or violets are not blue. 156. All skiers speak Swedish. Test Bank Questions and Answers 479 157. (a) Depending on which word is emphasized, the sentence can be interpreted as “all bananas are non-ripe fruit” (i.e., no bananas are ripe) or as “not all bananas are ripe” (i.e., some bananas are not ripe). (b) Both statements can be false at the same time. (c) Both statements can be true at the same time. (d) Some bananas are not ripe. 158. Both statements can be true at the same time. 159. Modus tollens. 160. Hypothetical syllogism. 161. p false and q true yield true hypotheses but a false conclusion. 162. Not valid: p false, q false, r true. 163. Not valid: p true, q true, r true. 164. Use resolution on l # f and ¬ l # r to conclude f # r. 165. Not valid: fallacy of a!rming the conclusion. 166. Not valid: fallacy of denying the hypothesis. 167. Valid. 168. Valid. 169. Not valid. 170. Universal instantiation. 171. Universal instantiation. 172. The two premises do not imply the conclusion. 173. Let R(x) be the predicate “x has read the textbook” and P (x) be the predicate “x passed the exam”. The following is the proof: 1. &x (R(x) " P (x)) hypothesis 2. R(Ed) " P (Ed) universal instantiation on 1 3. R(Ed) hypothesis 4. P (Ed) modus ponens on 2 and 3 174. The two premises do not imply the conclusion. 175. Existential generalization. 176. (a) Assume p, prove q. (b) Assume ¬q, prove ¬p. (c) Assume p $ ¬q, show that this leads to a contradiction. 177. Direct proof: Assume p, show q. Indirect proof: Assume ¬q, show ¬p. 178. Suppose x = 2k + 1, y = 2l. Therefore x + y = 2k + 1 + 2l = 2(k + l) + 1, which is odd. 179. Suppose n = 2k + 1 but n2 = 2l. Therefore (2k + 1)2 = 2l, or 4k2 + 4k + 1 = 2l. Hence 2(2k2 + 2k ! l) = !1 (even = odd), a contradiction. Therefore n2 is odd. 180. Let x = 2k + 1, y = 2l + 1. Therefore x + y = 2k + 1 + 2l + 1 = 2(k + l + 1) , which is even. 181. Suppose x = 2k + 1 and y = 2l + 1, but x + y = 2m + 1. Therefore (2k + 1) + (2l + 1) = 2m + 1. Hence 2(k + l !m + 1) = 1 (even = odd), which is a contradiction. Therefore x + y is even. 182. Suppose x = 2k and y = 2l, but xy = 2m + 1. Therefore 2k · 2l = 2m + 1. Hence 2(2kl !m) = 1 (even = odd), which is a contradiction. Therefore xy is even. 183. Let x = 2k + 1. Therefore x + 2 = 2k + 1 + 2 = 2(k + 1) + 1, which is odd. 184. Suppose x + 2 = 2k. Therefore x = 2k ! 2 = 2(k ! 1), which is even. 480 Test Bank Questions and Answers 185. Suppose x is odd but x + 2 is even. Therefore x = 2k + 1 and x + 2 = 2l. Hence (2k + 1) + 2 = 2l. Therefore 2(k + 1 ! l) = !1 (even = odd), a contradiction. 186. Let n = 2k. Therefore n + 1 = 2k + 1, which is odd. 187. Suppose n + 1 is even. Therefore n + 1 = 2k. Therefore n = 2k ! 1 = 2(k ! 1) + 1, which is odd. 188. Suppose n = 2k but n + 1 = 2l. Therefore 2k + 1 = 2l (even = odd), which is a contradiction. 189. If n is even, then n = 2k. Therefore 3n2 +8 = 3(2k)2 +8 = 12k2 +8 = 2(6k2 +4), which is even. If n is odd, then n = 2k + 1. Therefore 3n2 + 8 = 3(2k + 1)2 + 8 = 12k2 + 12k + 11 = 2(6k2 + 6k + 5) + 1, which is odd. 190. If n is even, then n2 = (2k)2 = 2(2k2), which is even. If n is odd, then n2 = (2k + 1)2 = 2(2k2 + 2k) + 1, which is odd. 191. If m = 2k and n = 2l, then mn = 4kl. Hence mn is a multiple of 4. 192. False: x = 2 y = 1/2. 193. False: x = 1/2. 194. False: x = 3/2, y = 3/2. 195. Case 1, x . 0: then x = |x|, so x ( |x|. Case 2, x < 0: here x < 0 and 0 < |x|, so x < |x|. 196. It is easier to give a contraposition proof; it is usually easier to proceed from a simple expression (such as n) to a more complex expression (such as 3n + 5 is even). Begin by supposing that n is not odd. Therefore n is even and hence n = 2k for some integer k. Therefore 3n + 5 = 3(2k) + 5 = 6k + 5 = 2(3k + 2) + 1, which is not even. If we try a direct proof, we assume that 3n + 5 is even; that is, 3n + 5 = 2k for some integer k. From this we obtain n = (2k ! 5)/3, and it it not obvious from this form that n is even. 197. Prove that (a) and (b) are equivalent and that (a) and (c) are equivalent. 198. If at most three people were born in each of the 12 months of the year, there would be at most 36 people. 199. Give a proof by cases. There are only six cases that need to be considered: x = y = 1; x = 1, y = 2; x = 1, y = 3; x = 2, y = 1; x = y = 2; x = 2, y = 3. 200. The steps in the “proof” cannot be reversed. Knowing that the squares of two numbers, !3 and 3, are equal does not allow us to infer that the two numbers are equal. Questions for Chapter 2 For each of the pairs of sets in 1-3 determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other. 1. The set of people who were born in the U.S., the set of people who are U.S. citizens. 2. The set of students studying a programming language, the set of students studying Java. 3. The set of animals living in the ocean, the set of fish. 4. Prove or disprove: A! (B / C) = (A!B) 0 (A! C). 5. Prove that A /B = A 0B by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side). 6. Prove that A /B = A 0B by giving an element table proof. 7. Prove that A /B = A 0B by giving a proof using logical equivalence. Test Bank Questions and Answers 483 61. {Ø} 5 P(A). 62. (1, 1) 5 A6A. In questions 63–65 suppose the following are fuzzy sets: F = {0.7 Ann, 0.1 Bill, 0.8 Fran, 0.3 Olive, 0.5 Tom}, R = {0.4 Ann, 0.9 Bill, 0.9 Fran, 0.6 Olive, 0.7 Tom}. 63. Find F and R. 64. Find F 0R. 65. Find F /R. In questions 66–75, suppose A = {a, b, c} and B = {b, {c}}. Mark the statement TRUE or FALSE. 66. c 5 A!B. 67. |P(A6B)| = 64. 68. Ø 5 P(B). 69. B 2 A. 70. {c} 2 B. 71. {a, b} 5 A6A. 72. {b, c} 5 P(A). 73. {b, {c}} 5 P(B). 74. Ø 2 A6A. 75. {{{c}}} 2 P(B). 76. Find A2 if A = {1, a}. In questions 77–89 determine whether the set is finite or infinite. If the set is finite, find its size. 77. {x | x 5 Z and x2 < 10 }. 78. P({a, b, c, d}), where P denotes the power set. 79. {1, 3, 5, 7, . . .}. 80. A6B, where A = {1, 2, 3, 4, 5} and B = {1, 2, 3}. 81. {x | x 5 N and 9x2 ! 1 = 0 }. 82. P(A), where A is the power set of {a, b, c}. 83. A6B, where A = {a, b, c} and B = Ø. 84. {x | x 5 N and 4x2 ! 8 = 0 }. 85. {x | x 5 Z and x2 = 2 }. 86. P(A), where A = P({1, 2}). 87. {1, 10, 100, 1000, . . .}. 88. S 6 T , where S = {a, b, c} and T = {1, 2, 3, 4, 5}. 484 Test Bank Questions and Answers 89. {x | x 5 Z and x2 < 8 }. 90. Prove that between every two rational numbers a/b and c/d (a) there is a rational number. (b) there are an infinite number of rational numbers. 91. Prove that there is no smallest positive rational number. 92. Consider these functions from the set of licensed drivers in the state of New York. Is a function one-to-one if it assigns to a licensed driver his or her (a) birthdate (b) mother’s first name (c) drivers license number? In 93–94 determine whether each of the following sets is countable or uncountable. For those that are countably infinite exhibit a one-to-one correspondence between the set of positive integers and that set. 93. The set of positive rational numbers that can be written with denominators less than 3. 94. The set of irrational numbers between 7 2 and !/2. 95. Adapt the Cantor diagonalization argument to show that the set of positive real numbers less than 1 with decimal representations consisting only of 0s and 1s is uncountable. 96. Show that (0, 1) has the same cardinality as (0, 2). 97. Show that (0, 1] and R have the same cardinality. In questions 98–106 determine whether the rule describes a function with the given domain and codomain. 98. f :N " N where f(n) = 7 n. 99. h:R " R where h(x) = 7 x. 100. g:N " N where g(n) = any integer > n. 101. F :R " R where F (x) = 1 x! 5 . 102. F :Z " R where F (x) = 1 x2 ! 5 . 103. F :Z " Z where F (x) = 1 x2 ! 5 . 104. G:R " R where G(x) = # x + 2 if x . 0 x! 1 if x ( 4. 105. f :R " R where f(x) = # x2 if x ( 2 x! 1 if x . 4. 106. G:Q " Q where G(p/q) = q. 107. Give an example of a function f :Z " Z that is 1-1 and not onto Z. 108. Give an example of a function f :Z " Z that is onto Z but not 1-1. 109. Give an example of a function f :Z " N that is both 1-1 and onto N. 110. Give an example of a function f :N " Z that is both 1-1 and onto Z. 111. Give an example of a function f :Z " N that is 1-1 and not onto N. Test Bank Questions and Answers 485 112. Give an example of a function f :N " Z that is onto Z and not 1-1. 113. Suppose f :N " N has the rule f(n) = 4n + 1. Determine whether f is 1-1. 114. Suppose f :N " N has the rule f(n) = 4n + 1. Determine whether f is onto N. 115. Suppose f :Z " Z has the rule f(n) = 3n2 ! 1. Determine whether f is 1-1. 116. Suppose f :Z " Z has the rule f(n) = 3n! 1. Determine whether f is onto Z. 117. Suppose f :N " N has the rule f(n) = 3n2 ! 1. Determine whether f is 1-1. 118. Suppose f :N " N has the rule f(n) = 4n2 + 1. Determine whether f is onto N. 119. Suppose f :R " R where f(x) = *x/2+. (a) Draw the graph of f . (b) Is f 1-1? (c) Is f onto R? 120. Suppose f :R " R where f(x) = *x/2+. (a) If S = {x | 1 ( x ( 6 }, find f(S). (b) If T = {3, 4, 5}, find f"1(T ). 121. Determine whether f is a function from the set of all bit strings to the set of integers if f(S) is the position of a 1 bit in the bit string S. 122. Determine whether f is a function from the set of all bit strings to the set of integers if f(S) is the number of 0 bits in S. 123. Determine whether f is a function from the set of all bit strings to the set of integers if f(S) is the largest integer i such that the ith bit of S is 0 and f(S) = 1 when S is the empty string (the string with no bits). 124. Let f(x) = *x3/3+. Find f(S) if S is: (a) {!2,!1, 0, 1, 2, 3}. (b) {0, 1, 2, 3, 4, 5}. (c) {1, 5, 7, 11}. (d) {2, 6, 10, 14}. 125. Suppose f :R " Z where f(x) = 82x! 19. (a) Draw the graph of f . (b) Is f 1-1? (Explain) (c) Is f onto Z? (Explain) 126. Suppose f :R " Z where f(x) = 82x! 19. (a) If A = {x | 1 ( x ( 4}, find f(A). (b) If B = {3, 4, 5, 6, 7}, find f(B). (c) If C = {!9,!8}, find f"1(C). (d) If D = {0.4, 0.5, 0.6}, find f"1(D). 127. Suppose g:R " R where g(x) = $ x! 1 2 % . (a) Draw the graph of g. (b) Is g 1-1? (c) Is g onto R? 128. Suppose g:R " R where g(x) = $ x! 1 2 % . (a) If S = {x | 1 ( x ( 6}, find g(S). (b) If T = {2}, find g"1(T ). 129. Show that 8x9 = !*!x+. 130. Prove or disprove: For all positive real numbers x and y, *x · y+ ( *x+ · *y+. 131. Prove or disprove: For all positive real numbers x and y, 8x · y9 ( 8x9 · 8y9. 132. Suppose g:A " B and f :B " C where A = {1, 2, 3, 4}, B = {a, b, c}, C = {2, 7, 10}, and f and g are defined by g = {(1, b), (2, a), (3, a), (4, b)} and f = {(a, 10), (b, 7), (c, 2)}. Find f : g. 133. Suppose g:A " B and f :B " C where A = {1, 2, 3, 4}, B = {a, b, c}, C = {2, 7, 10}, and f and g are defined by g = {(1, b), (2, a), (3, a), (4, b)} and f = {(a, 10), (b, 7), (c, 2)}. Find f"1. 488 Test Bank Questions and Answers 184. an = 2n. 185. an = 2n + 1. 186. an = (!1)n. 187. an = 3n! 1. 188. an = 7 2. 189. You take a job that pays $25,000 annually. (a) How much do you earn n years from now if you receive a three percent raise each year? (b) How much do you earn n years from now if you receive a five percent raise each year? (c) How much do you earn n years from now if each year you receive a raise of $1000 plus two percent of your previous year’s salary. 190. Suppose inflation continues at three percent annually. (That is, an item that costs $1.00 now will cost $1.03 next year.) Let an = the value (that is, the purchasing power) of one dollar after n years. (a) Find a recurrence relation for an. (b) What is the value of $1.00 after 20 years? (c) What is the value of $1.00 after 80 years? (d) If inflation were to continue at ten percent annually, find the value of $1.00 after 20 years. (e) If inflation were to continue at ten percent annually, find the value of $1.00 after 80 years. 191. Find the sum 1/4 + 1/8 + 1/16 + 1/32 + · · · . 192. Find the sum 2 + 4 + 8 + 16 + 32 + · · · + 228. 193. Find the sum 2 ! 4 + 8 ! 16 + 32 ! · · ·! 228 . 194. Find the sum 1 ! 1/2 + 1/4 ! 1/8 + 1/16 ! · · · . 195. Find the sum 2 + 1/2 + 1/8 + 1/32 + 1/128 + · · · . 196. Find the sum 112 + 113 + 114 + · · · + 673. 197. Find 6& i=1 ((!2)i ! 2i). 198. Find 3& j=1 j& i=1 ij. 199. Rewrite 4& i="3 (i2 + 1) so that the index of summation has lower limit 0 and upper limit 7. 200. Find a 2 6 2 matrix A )= ' 0 0 0 0 ( such that A2 = ' 0 0 0 0 ( . 201. Suppose A is a 6 6 8 matrix, B is an 8 6 5 matrix, and C is a 5 6 9 matrix. Find the number of rows, the number of columns, and the number of entries in A(BC). 202. Let A = ' 1 m 0 1 ( . Find An where n is a positive integer. 203. Suppose A = ' 3 5 2 4 ( and C = ' 2 1 0 6 ( . Find a matrix B such that AB = C or prove that no such matrix exists. Test Bank Questions and Answers 489 204. Suppose B = ' 3 5 2 4 ( and C = ' 2 1 0 6 ( . Find a matrix A such that AB = C or prove that no such matrix exists. 205. Suppose B = ' 6 2 3 1 ( and C = ' 2 1 0 6 ( . Find a matrix A such that AB = C or prove that no such matrix exists. In questions 206–212 determine whether the statement is true or false. 206. If AB = AC, then B = C. 207. If A = ' 3 5 1 2 ( , then A"1 = ' 2 5 1 !3 ( . 208. If A = ' 1 3 !5 2 ( , then A2 = ' 1 9 25 4 ( . 209. If A is a 6 6 4 matrix and B is a 4 6 5 matrix, then AB has 16 entries. 210. If A and B are 2 6 2 matrices such that AB= ' 0 0 0 0 ( , then A= ' 0 0 0 0 ( or B= ' 0 0 0 0 ( . 211. If A and B are 2 6 2 matrices, then A+B=B+A. 212. AB=BA for all 2 6 2 matrices A and B. 213. Suppose A = ) * 1 0 1 0 1 1 1 1 0 + , and B = ) * 0 1 0 0 1 1 1 0 0 + ,. Find (a) the join of A and B. (b) the meet of A and B. (c) the Boolean product of A and B. 214. Suppose A is a 2 6 2 matrix with real number entries such that AB=BA for all 2 6 2 matrices. What relationships must exist among the entries of A? Answers for Chapter 2 1. The first is a subset of the second, but the second is not a subset of the first. 2. The second is a subset of the first, but the first is not a subset of the second. 3. Neither is a subset of the other. 4. True, since A! (B / C) = A /B / C = A / (B 0 C) = (A /B) 0 (A / C) = (A!B) 0 (A! C). 5. A /B 2 A0B: Let x 5 A /B. ...x /5 A/B, ...x /5 A or x /5 B, ...x 5 A or x 5 B, ...x 5 A0B. Reversing the steps shows that A 0B 2 A /B. 6. The columns for A /B and A 0B match: each entry is 0 if and only if A and B have the value 1. 7. A /B = {x | x 5 A /B} = {x | x /5 A / B} = {x | ¬(x 5 A / B)} = {x | ¬(x 5 A $ x 5 B)} = {x | ¬(x 5 A) # ¬(x 5 B)} = {x | x /5 A # x /5 B} = {x | x 5 A # x 5 B} = {x | x 5 A 0B} = A 0B. 490 Test Bank Questions and Answers 8. 9. A/ (B 0C) 2 (A/B)0 (A/C): Let x 5 A/ (B 0C). ...x 5 A and x 5 B 0C, ...x 5 A and x 5 B, or x 5 A and x 5 C, ...x 5 (A /B) 0 (A / C). Reversing the steps gives the opposite containment. 10. Each set has the same values in the element table: the value is 1 if and only if A has the value 1 and either B or C has the value 1. 11. A/(B0C) = {x | x 5 A/(B0C)} = {x | x 5 A $ x 5 (B0C)} = {x | x 5 A $ (x 5 B # x 5 C)} = {x | (x 5 A $ x 5 B) # (x 5 A $ x 5 C)} = {x | x 5 A/B # x 5 A/C} = {x | x 5 (A/B) 0 (A/C)} = (A/B) 0 (A/C). 12. 13. False. For example, let A = {1, 2}, B = {1}, C = {2}. 14. True, using either a membership table or a containment proof, for example. 15. =. 16. 3. 17. =. 18. 2. 19. Yes {Ø, a, {a}, {{a}}}. 20. Yes, {a}. 21. No, it lacks {Ø}. 22. Yes, {{a,Ø}}. 23. No, it lacks {a} and {Ø}. 24. Since S 0 T = S / T (De Morgan’s law), the complements are equal. 25. False. 26. False. 27. True. 28. False. 29. True. 30. False. Test Bank Questions and Answers 493 110. f(n) = / 01 02 !n 2 , n even n + 1 2 , n odd. 111. f(n) = # !2n, n ( 0 2n + 1, n > 0. 112. f(n) = / 01 02 !n 2 , n even n! 1 2 , n odd. 113. Yes. 114. No. 115. No. 116. No. 117. Yes. 118. No. 119. (a) (b) No. (c) No. 120. (a) {0, 1, 2, 3} (b) [6, 12). 121. No; there may be no 1 bits or more than one 1 bit. 122. Yes. 123. No; f not defined for the string of all 1’s, for example S = 11111. 124. (a) {!3,!1, 0, 2, 9}. (b) {0, 2, 9, 21, 41}. (c) {0, 41, 114, 443}. (d) {2, 72, 333, 914}. 125. (a) (b) No. (c) Yes. 126. (a) {1, 2, 3, 4, 5, 6, 7}. (b) {5, 7, 9, 11, 13}. (c) (!9/2,!7/2]. (d) Ø. 127. (a) 494 Test Bank Questions and Answers (b) No. (c) No. 128. (a) {0, 1, 2}. (b) [5, 7). 129. Let n = 8x9, so that n ! 1 < x ( n. Multiplying by !1 yields !n + 1 > !x . !n, which means that !n = *!x+. 130. False: x = y = 1.5. 131. True: x ( 8x9, y ( 8y9; therefore xy ( 8x98y9; since 8x98y9 is an integer at least as great as xy, then 8xy9 ( 8x98y9. 132. {(1, 7), (2, 10), (3, 10), (4, 7)}. 133. {(2, c), (7, b), (10, a)}. 134. {(1, 2), (2, 3), (3, 3), (4, 2)}. 135. {(1, 1), (2, 1), (3, 2), (4, 1)}. 136. {(1, 2), (2, 4), (3, 4), (4, 1)}. 137. {(1, 1), (2, 2), (3, 2), (4, 4)}. 138. {(1, 10), (2, 8), (3, 10), (4, 8)}. 139. {(2, c), (8, a), (10, b)}. 140. {(2, 2), (8, 8), (10, 10)}. 141. g"1(a) is equal to both 2 and 4. 142. {(a, 3), (b, 8), (c, 2), (d, 3)}. 143. {(2, 3), (3, 2), (8, 1)}. 144. Suppose f is onto. Let T 5 P(B) and let S = {x 5 A | f(x) 5 T }. Then g(S) = T , and g is onto. If f is not onto B, let y 5 B ! f(A). Then there is no subset S of A such that g(S) = {y}. 145. f"1(10) does not exist. 146. {(1, b), (2, a), (3, c)}. 147. f"1(x) = 5 + x 3 . 148. f"1( 1 2 ) does not exist. 149. f"1(5) is not a single value. 150. No. 151. Yes. 152. f(x) = x + 5. 153. Z, R, Z ! {0}, {0}. 154. Z, Z, Z, total function. 155. Z 6 Z, Q, Z 6 (Z ! {0}), Z 6 {0}. 156. Z 6 Z, Z, Z 6 Z, total function. 157. Z 6 Z, Z, { (m,n) | m > n }, { (m,n) | m ( n }. 158. Z 6 Z, R, { (m,n) | m )= n or m )= !n }, { (m,n) | m = n or m = !n }. 159. (a) 65 = 7,776. (b) 6 · 5 · 4 · 3 · 2 = 720. 160. an = 4n + 1. 161. an = 3. 162. an = 5(n + 2). 163. an = 1 ! (n! 1)/10. Test Bank Questions and Answers 495 164. an = 1/(2n! 1). 165. an = 1 + (!1)n+1. 166. an = 1 + (!1)n. 167. an = 5an"1, a1 = 5. 168. an = an"1 + an"2, a1 = a2 = 1. 169. an = an"2, a1 = 0, a2 = 1. 170. an = an"1 + n, a1 = 1. 171. an = an"1 ! 1, a1 = 3. 172. an = nan"1, a1 = 1. 173. an = an"1 1 + an"1 , a1 = 1/2. 174. an = an"1 + 1/10n, a1 = 0.1. 175. an = an"1 + 2n! 1, a1 = 1. 176. an = 100an"1 + 11. 177. an = 2 · an"1, a1 = 2. 178. an = 100an"1 + 1, a1 = 1. 179. 4 · 6 ! 3 · 6 = 1 · 6 = 6. 180. 4 · 3n"1 ! 3 · 3n"2 = 4 · 3n"1 ! 3n"1 = 3 · 3n"1 = 3n. 181. 4 · 3n+3 ! 3 · 3n+2 = 4 · 3n+3 ! 3n+3 = 3 · 3n+3 = 3n+4. 182. 4(3n"1 + 1) ! 3(3n"2 + 1) = 4 · 3n"1 ! 3n"1 + 4 ! 3 = 3n"1(4 ! 1) + 1 = 3n + 1. 183. 4(7 · 3n"1 ! !) ! 3(7 · 3n"2 ! !) = 28 · 3n"1 ! 7 · 3n"1 ! 4! + 3! = 7 · 3n ! !. 184. an = 2an"1, a0 = 1. 185. an = 2an"1 ! 1, a0 = 2. 186. an = !an"1, a0 = 1. 187. an = an"1 + 3, a0 = !1. 188. an = an"1, a0 = 7 2. 189. (a) 25, 000 · 1.03n. (b) 25, 000 · 1.05n. (c) 25, 000 · 1.02n + 1, 000 3 1.02n"1 0.02 4 . 190. (a) an = an"1/1.03. (b) a20 = 1/1.0320 ; 0.55. (c) a80 = 1/1.0380 ; 0.09. (d) 1/1.120 ; 0.15. (e) 1/1.180 ; 0.00. 191. 1/2. 192. 229 ! 2. 193. 2 3 + 2 3 (229). 194. 2/3. 195. 8/3. 196. 220,585. 197. !84. 198. 25. 199. 7& i=0 ((i! 3)2 + 1). 200. A matrix of the form ' !2a a !4a 2a ( where a )= 0. 201. A(BC) has 6 rows, 9 columns, and 54 entries. 202. An = ' 1 mn 0 1 ( . 498 Test Bank Questions and Answers In questions 36–46 find the “best” big-O notation to describe the complexity of the algorithm. Choose your answers from the following: 1, log2 n, n, n log2 n, n2, n3, . . . , 2n, n! . 36. A binary search of n elements. 37. A linear search to find the smallest number in a list of n numbers. 38. An algorithm that lists all ways to put the numbers 1, 2, 3, . . . , n in a row. 39. An algorithm that prints all bit strings of length n. 40. The number of print statements in the following: i := 1, j := 1 while i ( n while j ( i print “hello”; j := j + 1 i := i + 1 41. The number of print statements in the following: while n > 1 print “hello”; n := *n/2+ 42. An iterative algorithm to compute n!, (counting the number of multiplications). 43. An algorithm that finds the average of n numbers by adding them and dividing by n. 44. An algorithm that prints all subsets of size three of the set {1, 2, 3, . . . , n}. 45. The best-case analysis of a linear search of a list of size n (counting the number of comparisons). 46. The worst-case analysis of a linear search of a list of size n (counting the number of comparisons). 47. Give a big-O estimate for the number of operations (where an operation is an addition or a multiplication) used in this segment of an algorithm: t := 1 for i = n to n2 t := t + 2it 48. Give a big-O estimate for the number of operations (where an operation is an addition or a multiplication) used in this segment of an algorithm: t := 0 for i = 1 to n for j = 1 to n t := (it + jt + 1)2 In 49–52 assume that the number of multiplications of entries used to multiply a p6 q and a q 6 r matrix is pqr. 49. What is the most e!cient way to multiply the matrices A1, A2, A3 of sizes 20 6 5, 5 6 50, 50 6 5? 50. What is the most e!cient way to multiply the matrices A1, A2, A3 of sizes 10 6 50, 50 6 10, 10 6 40? 51. What is the best order to form the product ABC if A, B and C are matrices with dimensions 2 6 5, 5 6 7 and 7 6 3, respectively? 52. What is the best order to form the product ABC if A, B and C are matrices with dimensions 8 6 3, 3 6 6 and 6 6 12, respectively? Test Bank Questions and Answers 499 Answers for Chapter 3 1. procedure greaterthanfive(a1, . . . , an: integers) answer := 0 for i := 1 to n if ai > 5 then answer := answer +1 return answer 2. procedure secondlargest(a1, . . . , an: integers) largest := a1 secondlargest := a2 if a2 > a1 then secondlargest := a1 largest := a2 if n = 2 then return secondlargest for i := 3 to n if ai > largest then secondlargest := largest largest := ai if (ai > secondlargest and ai ( largest) then secondlargest := ai return secondlargest 3. procedure lasteven(a1, . . . , an: integers) location := 0 for i := 1 to n if 2 | ai then location := i return location 4. procedure avgmaxmin(a1, . . . , an: integers) max := a1 min := a1 for i := 2 to n if ai > max then max := ai if ai < min then min := ai return (max + min)/2 5. procedure secondmax(a1, a2, . . . , an : integers) for i = 2 to n if a1 < ai then exchange a1 and ai secondmax := a2 for j = 3 to n if secondmax < aj then secondmax := aj return secondmax {secondmax is the second largest element} 6. procedure largestproduct(a1, a2, . . . , an, N : real numbers) largestproduct := !4 for i = 2 to n for j = 1 to i! 1 if ai · aj < N then if ai · aj > largestproduct then largestproduct := ai · aj return largestproduct {largestproduct is the largest product of two numbers in the list that is less than N , or !4 if all products are greater than or equal to N} 7. To search for x in an ordered list a1, . . . , an, find the “midpoint” of the list and choose the appropriate half of the list. Continue until the list consists of one element. Either this element is x, or else x is not in the list. 8. The consecutive choices of sublists of the original list are: 15 21 25 31, 25 31, and 25. Since 27 )= 25, the integer 25 is not in the list. 500 Test Bank Questions and Answers 9. The greedy algorithm first chooses a 12-foot-long board, and then three one-foot-long boards. This requires four boards. But only three boards are needed: each five feet long. 10. True. Note that each denomination divides the next largest one. 11. False. The algorithm gives change of 25 using 20, 1, 1, 1, 1, 1 (a total of six coins), but it can be done using 8, 8, 8, 1 (a total of only four coins). 12. 12 + 22 + · · · + n2 ( n2 + n2 + · · · + n2 = n · n2 = n3. 13. 666 3n! 8 ! 4n3 2n! 1 666 ( 666 3n3 + 8n3 + 4n3 2n! n 666 = 666 15n3 n 666 = 15n2 if n . 1. 14. 13 + 23 + · · · + n3 ( n3 + n3 + · · · + n3 = n · n3 = n4. 15. 666 6n + 4n5 ! 4 7n2 ! 3 666 ( 666 6n5 + 4n5 7n2 ! n2 666 = 666 10n5 6n2 666 = 5 3 |n3|, if n . 2. 16. 1 · 2 + 2 · 3 + · · · + (n! 1) · n ( (n! 1) · n + (n! 1) · n + · · · + (n! 1) · n = (n! 1)2 · n ( n3. 17. f(n) ( 3n2 + 8n2 + 7n2 = 18n2 if n . 1; therefore C = 18 and k = 1 can be used. 18. n2. 19. n2. 20. n. 21. n4. 22. n2. 23. n4. 24. n& j=1 (j3 + j) ( n& j=1 (n3 + n3) = n · 2n3 = 2n4. 25. log2(x2 + 1) and log2(x3 + 1) are each O(log2 x). Thus each term is O(x log2 x), and hence so is the sum. 26. n3. 27. x2. 28. 5x4 + 2x3 ! 1 is O(x4) since |5x4 + 2x3 ! 1| ( |5x4 + 2x4| ( 7|x4| (if x . 1). Also, x4 is O(5x4 + 2x3 ! 1) since |x4| ( |5x4 + x3| ( |5x4 + 2x3 ! 1| (if x . 1). 29. x3 + 7x2 + 3 2x + 1 is O(x2) since x3 + 7x2 + 3 2x + 1 ( x3 + 7x3 + 3x3 2x = 11x3 2x = 11 2 x2 (if x . 1). Also, x2 is O ' x3 + 7x2 + 3 2x + 1 ( since x2 = x3 x ( x3 + 7x 2x ( x3 + 7x + 3 2x + 1 ( x3 + 7x2 + 3 2x + 1 (if x . 1). 30. x3 + 7x + 2 . 1 · x3 (if x . 1). 31. log(n!), log(nn), n3/2, (n100)n 32. 10000, logn4, n2 log n, n3 + 88n2 + 3, n · 2n, 3n, 33. log logn, log n2, log(n2 + 1), log 2n, n log n, 34. (100n3 + n2,n2 + n3), (3n3 + 2n, n2 + 2n). 35. The first algorithm uses fewer operations as n grows. 36. log2 n. 37. n. 38. n!. 39. 2n. 40. n2. 41. log2 n. 42. n. 43. n. 44. n3. Test Bank Questions and Answers 503 56. (54 mod 7)3 mod 13. 57. Show that if a, b, k and m are integers such that k . 1, m . 2, and a % b (mod m), then ka % kb (mod m). In questions 58–64 determine whether each of the following “theorems” is true or false. Assume that a, b, c, d, and m are integers with m > 1. 58. If a % b (mod m), and a % c (mod m), then a % b + c (mod m). 59. If a % b (mod m) and c % d (mod m), then ac % b + d (mod m). 60. If a % b (mod m), then 2a % 2b (mod m). 61. If a % b (mod m), then 2a % 2b (mod 2m). 62. If a % b (mod m), then a % b (mod 2m). 63. If a % b (mod 2m), then a % b (mod m). 64. If a % b (mod m2), then a % b (mod m). 65. Either find an integer x such that x % 2 (mod 6) and x % 3 (mod 9) are both true, or else prove that there is no such integer. 66. What sequence of pseudorandom numbers is generated using the pure multiplicative generator xn+1 = 3xn mod 11 with seed x0 = 2? 67. Explain in words the di"erence between a|b and b a . 68. Prove or disprove: if p and q are prime numbers, then pq + 1 is prime. 69. (a) Find two positive integers, each with exactly three positive integer factors greater than 1. (b) Prove that there are an infinite number of positive integers, each with exactly three positive integer factors greater than 1. 70. Convert (204)10 to base 2. 71. Convert (1 1101)2 to base 16. 72. Convert (1 1101)2 to base 10. 73. Convert (2AC)16 to base 10. 74. Convert (10,000)10 to base 2. 75. Convert (8091)10 to base 2. 76. Convert (BC1)16 to base 2. 77. Convert (100 1100 0011)2 to base 16. 78. Convert (271)8 to base 2. 79. Convert (6253)8 to base 2. 80. Convert (10 1011)2 to base 8. 81. Convert (110 1011 1100)2 to base 8. In 82–83 find the sum and product of each of these pairs of numbers. Express your answer as a binary expansion. 82. (10 1011)2, (110 1011)2 83. (110 1011 1100)2, (111 0111 0111)2 In 84–85 find the sum and product of each of these pairs of numbers. Express your answer as a base 3 504 Test Bank Questions and Answers expansion. 84. (202)3, (122)3 85. (21202)3, (12212)3 In 86–87 find the sum and product of each of these pairs of numbers. Express your answer as an octal expansion. 86. (371)8, (624)8 87. (4274)8, (5366)8 In 88–89 find the sum and product of each of these pairs of numbers. Express your answer as a hexadecimal expansion. 88. (2A)16, (BF)16 89. (E3A)16, (B5F8)16 90. Take any three-digit integer, reverse its digits, and subtract. For example, 742 ! 247 = 495. The di"erence is divisible by 9. Prove that this must happen for all three-digit numbers abc. 91. Prove or disprove that 30! ends in exactly seven 0’s. 92. Here is a sample proof that contains an error. Explain why the proof is not correct. Theorem: If a|b and b|c, then a|c. Proof: Since a|b, b = ak. Since b|c, c = bk. Therefore c = bk = (ak)k = ak2. Therefore a|c. 93. Prove: if n is an integer that is not a multiple of 3, then n2 % 1 mod 3. 94. Prove: if n is an integer that is not a multiple of 4, then n2 % 0 mod 4 or n2 % 1 mod 4. 95. Use the Euclidean algorithm to find gcd(44, 52). 96. Use the Euclidean algorithm to find gcd(144, 233). 97. Use the Euclidean algorithm to find gcd(203, 101). 98. Use the Euclidean algorithm to find gcd(300, 700). 99. Use the Euclidean algorithm to find gcd(34, 21). 100. Use the Euclidean Algorithm to find gcd(900, 140). 101. Use the Euclidean Algorithm to find gcd(580, 50). 102. Use the Euclidean Algorithm to find gcd(390, 72). 103. Use the Euclidean Algorithm to find gcd(400, 0). 104. Use the Euclidean Algorithm to find gcd(128, 729). 105. Find the two’s complement of 12. 106. Find the two’s complement of !13. 107. Find the two’s complement of 9. 108. Given that gcd(620, 140) = 20, write 20 as a linear combination of 620 and 140. 109. Given that gcd(662, 414) = 2, write 2 as a linear combination of 662 and 414. 110. Express gcd(84, 18) as a linear combination of 18 and 84. 111. Express gcd(450, 120) as a linear combination of 120 and 450. Test Bank Questions and Answers 505 112. Find an inverse of 5 modulo 12. 113. Find an inverse of 17 modulo 19. 114. Solve the linear congruence 2x % 5 (mod 9). 115. Solve the linear congruence 5x % 3 (mod 11). 116. Find an inverse of 6 modulo 7. 117. Find an inverse of 5 modulo 17. 118. Find an inverse of 2 modulo 31. 119. Solve the linear congruence 15x % 31 (mod 47) given that the inverse of 15 modulo 47 is 22. 120. Solve the linear congruence 54x % 12 (mod 73) given that the inverse of 54 modulo 73 is 23. 121. Solve the linear congruence 31x % 57 (mod 61). 122. Use Fermat’s little theorem to find 945 mod 23. 123. Use Fermat’s little theorem to find 251202 mod 61. 124. Show that 7 is a primitive root of 13. 125. Find the discrete logarithms of 5 and 8 to the base 7 modulo 13. 126. Find the first five terms of the sequence of four-digit pseudorandom numbers generated by the middle square method starting with 1357. 127. Find the first five terms of the sequence of four-digit pseudorandom numbers generated by the middle square method starting with 9361. 128. Find the sequence of pseudorandom numbers generated by the power generator xn+1 = x2 n mod 17, and seed x0 = 5. 129. Find the sequence of pseudorandom numbers generated by the power generator xn+1 = x3 n mod 23, and seed x0 = 3. The numbers in question 130–133 refer to an 8-digit student id at a large university. The eighth digit is a check digit equal to the sum of the first seven digits modulo 7. 130. Find the check digit of the student id starting with 2365 415. 131. Find the check digit of the student id starting with 3179 822. 132. Suppose the first digit of the student id X123 4566 is illegible (indicated by X). Can you tell what the first digit has to be? 133. Suppose the first digit of the student id X923 4562 is illegible (indicated by X). Can you tell what the first digit has to be? 134. Encrypt the message NEED HELP by translating the letters into numbers (A=0, B=1, . . ., Z=25), applying the encryption function f(p) = (p + 3) mod 26, and then translating the numbers back into letters. 135. Encrypt the message NEED HELP by translating the letters into numbers (A=0, B=1, . . ., Z=25), applying the encryption function f(p) = (3p + 7) mod 26, and then translating the numbers back into letters. 136. Suppose that a computer has only the memory locations 0, 1, 2, . . . , 19. Use the hashing function h where h(x) = (x + 5) mod 20 to determine the memory locations in which 57, 32, and 97 are stored. 137. A message has been encrypted using the function f(x) = (x + 5) mod 26. If the message in coded form is JCFHY, decode the message. 138. Explain why f(x) = (2x + 3) mod 26 would not be a good coding function. 508 Test Bank Questions and Answers 51. -6 52. 194 53. 9 54. 0 55. 6 56. 8 57. The hypothesis a % b (mod m) means that m|(a ! b). Therefore m|(k · (a ! b)), which means precisely that ka % kb (mod m). 58. False. 59. False. 60. True. 61. True. 62. False. 63. True. 64. True. 65. There is no such x; if there were, then there would be integers k and l such that x! 2 = 6k and x! 3 = 9l. Hence 1 = 6k ! 9l = 3(2k ! 3l), which is not possible. 66. The sequence 2, 6, 7, 10, 8 repeats. 67. a | b is a statement; b a represents a number. 68. False: p = q = 3. 69. (a) 8, 27. (b) Any integer of the form p3 where p is prime. 70. 1100 1100. 71. 1D. 72. 29. 73. 684. 74. 10 0111 0001 0000. 75. 1 1111 1001 1011. 76. 1011 1100 0001. 77. 4C3. 78. (1011 1001)2 79. (1100 1010 1011)2 80. (53)8 81. (3274)8 82. (1001 0110)2, (1 0001 1111 1001)2 83. (1110 0011 0011)2, (11 0010 0100 0101 0110 0100)2 84. (1101)3, (110121)3 85. (111121)3, (1200022001)3 86. (1215)8, (304364)8 87. (11662)8, (27736250)8 88. (E9)16, (1F56)16 89. (C432)16, (A1CCA30)16 90. abc! cba = 100a + 10b + c! (100c + 10b + a) = 99a! 99c = 9(11a! 11c). Therefore 9 | abc! cba. Test Bank Questions and Answers 509 91. True. When the factors 5, 10, 15, 20, and 30 are multiplied by the factor 2, each contributes one zero; when the factor 25 is multiplied by two factors 2, it contributes two zeros. 92. The proof is not correct since there is no guarantee that the multiple k will be the same in both cases. 93. Proof by cases. Suppose n is not a multiple of 3. Then n = 3k + 1 or n = 3k + 2 for some integer k. Case 1, n = 3k + 1: therefore n2 = (3k + 1)2 = 9k2 + 6k + 1 = 3(3k2 + 2k) + 1, and hence n2 % 1 mod 3. Case 2, n = 3k + 2: therefore n2 = (3k + 2)2 = 9k2 + 12k + 4 = 3(3k2 + 4k + 1) + 1, and hence n2 % 1 mod 3. 94. Proof by cases. Suppose n is not a multiple of 4. Then there is an integer k such that n = 4k + 1, n = 4k + 2, or n = 4k + 3. Case 1, n = 4k + 1: therefore n2 = (4k + 1)2 = 16k2 + 8k + 1 = 4(4k2 + 2k) + 1, and hence n2 % 1 mod 4. Case 2, n = 4k + 2: therefore n2 = (4k + 2)2 = 16k2 + 16k + 4 = 4(4k2 + 4k + 1), and hence n2 % 0 mod 4. Case 3, n = 4k+3: therefore n2 = (4k+3)2 = 16k2 +24k+9 = 4(4k2 +6k+2)+1, and hence n2 % 1 mod 4. 95. 4. 96. 1. 97. 1. 98. 100. 99. 1. 100. 20. 101. 10. 102. 6. 103. 400. 104. 1. 105. 0 1100. 106. 1 0011. 107. 0 1001. 108. 620 · (!2) + 140 · 9. 109. 662 · (!5) + 414 · 8. 110. 18 · (!9) + 84 · 2. 111. 120 · 4 + 450 · (!1). 112. 5. 113. 9. 114. 7 + 9k. 115. 5 + 11k. 116. 6 117. 7 118. 16 119. 24 120. 57 121. 53 122. 9 123. 15 124. The powers of 7 modulo 13 are 7, 10, 5, 9, 11, 12, 6, 3, 8, 4, 2, 1. 125. 3, 9 126. 1357, 8414, 7953, 2502, 2600 127. 9361, 6283, 4760, 6576, 2437 510 Test Bank Questions and Answers 128. 8, 13, 16, 1, 1, 1, . . . 129. 4, 18, 13, 12, 3, 4, . . . 130. 5 131. 4 132. 6 133. 1 or 8 134. Encrypted form: QHHG KHOS. 135. Encrypted form: UTTQ CTOA. 136. 2, 17, 3. 137. EXACT. 138. f is not 1 ! 1 (f(0) = f(13)), and hence f"1 is not a function. 139. YZUV GZ TUUT. 140. WZPU UXPURQH 141. FLLQ FL BQ OXXO 142. HOMECOMING 143. f(x) = 9x + 15 144. CTWA TUHO 145. CLATCNE REERHOD 146. REA LLY 147. YSZD QONV 148. FIIZSN 149. SPLASHED 150. 1740 2314 151. 1506 0075 152. (712)25 mod 67 = 59 153. (79236)334 mod 431 = 353 154. 1519 2124 2227 1111 Questions for Chapter 5 1. Suppose you wish to prove that the following is true for all positive integers n by using the Principle of Mathematical Induction: 1 + 3 + 5 + · · · + (2n! 1) = n2. (a) Write P (1) (b) Write P (72) (c) Write P (73) (d) Use P (72) to prove P (73) (e) Write P (k) (f) Write P (k + 1) (g) Use the Principle of Mathematical Induction to prove that P (n) is true for all positive integers n 2. Suppose you wish to use the Principle of Mathematical Induction to prove that 1·1!+2·2!+3·3!+· · ·+n·n! = (n + 1)! ! 1 for all n . 1. (a) Write P (1) (b) Write P (5) (c) Write P (k) (d) Write P (k + 1) (e) Use the Principle of Mathematical Induction to prove that P (n) is true for all n . 1 Test Bank Questions and Answers 513 invariant. 53. Verify that the following program segment is correct with respect to the initial assertion T and the final assertion (x ( y $ max = y) # (x > y $ max = x): if x ( y then max := y else max := x Answers for Chapter 5 1. (a) 1 = 12. (b) 1 + 3 + 5 + · · · + 143 = 722. (c) 1 + 3 + 5 + · · · + 145 = 732. (d) 1 + 3 + 5 + · · · + 145 = (1 + 3 + 5 + · · · + 143) + 145 = 722 + 145 = 722 + 2 · 72 + 1 = (72 + 1)2 = 732. (e) 1 + 3 + · · · + (2k ! 1) = k2. (f) 1 + 3 + · · · + (2k + 1) = (k + 1)2. (g) P (1) is true since 1 = 12. P (k) " P (k + 1): 1 + 3 + · · · + (2k + 1) = k2 + (2k + 1) = (k + 1)2. 2. (a) 1 · 1! = 2! ! 1. (b) 1 · 1! + 2 · 2! + · · · + 5 · 5! = 6! ! 1. (c) 1 · 1! + 2 · 2! + · · · + k · k! = (k + 1)! ! 1. (d) 1 · 1! + 2 · 2! + · · · + (k + 1)(k + 1)! = (k + 2)! ! 1. (e) P (1) is true since 1 · 1! = 1 and 2! ! 1 = 1. P (k) " P (k + 1): 1 · 1! + 2 · 2! + · · · + (k + 1)(k + 1)! = (k + 1)! ! 1 + (k + 1)(k + 1)! = (k + 1)![1 + (k + 1)] ! 1 = (k + 1)!(k + 2) ! 1 = (k + 2)! ! 1. 3. P (1): 1 ! 2 = 22(!1) + 1 3 , which is true since both sides are equal to !1. P (k) " P (k + 1): 1 ! 2 + 22 + · · · + (!1)k+12k+1 = 2k+1(!1)k + 1 3 + (!1)k+12k+1 = 2k+1(!1)k + 1 + 3(!1)k+12k+1 3 = 2k+1(!1)k(1 + 3(!1)) + 1 3 = 2k+1(!1)k(!2) + 1 3 = 2k+2(!1)k+1 + 1 3 . 4. P (1): 1+21 ( 31, which is true since both sides are equal to 3. P (k) " P (k+1): 1+2k+1 = (1+2k)+2k ( 3k + 2k ( 3k + 3k = 2 · 3k < 3 · 3k = 3k+1. 5. P (2): 23 > 22 +3 is true since 8 > 7. P (k) " P (k+1): (k+1)2 +3 = k2 +2k+1+3 = (k2 +3)+2k+1 < k3 + 2k + 1 ( k3 + 3k ( k3 + 3k2 + 3k + 1 = (k + 1)3. 6. P (0): 2 | 02 + 0, which is true since 2 | 0. P (k) " P (k + 1): (k + 1)2 + (k + 1) = (k2 + k) + 2(k + 1), which is divisible by 2 since 2 | k2 + k and 2 | 2(k + 1). 7. P (0): 1 = 31 ! 1 2 , which is true since 1 = 1. P (k) " P (k + 1): 1 + 3 + · · · + 3k+1 = 3k+1 ! 1 2 + 3k+1 = 3k+1 ! 1 + 2 · 3k+1 2 = 3k+2 ! 1 2 . 8. P (1): 1 = 1 · 2 2 , which is true since 1 = 1. P (k) " P (k+1): 1+4+· · ·+(3(k+1)!2) = k(3k ! 1) 2 +(3k+1) = k(3k ! 1) + 2(3k + 1) 2 = 3k2 + 5k + 2 2 = (3k + 2)(k + 1) 2 = (k + 1)(3(k + 1) ! 1) 2 . 9. P (1): 2 | 12 + 3 · 1, which is true since 2 | 4. P (k) " P (k + 1): (k + 1)2 + 3(k + 1) = (k2 + 3k) + 2(k + 2), which is divisible by 2 since 2 | k2 + 3k and 2 | 2(k + 2). 10. P (4): 2 · 4 + 3 ( 24, which is true since 11 ( 16. P (k) " P (k + 1): 2(k + 1) + 3 = (2k + 3) + 2 ( 2k + 2 ( 2k + 2k = 2k+1. 11. P (1): 3 | 13 + 3 · 12 + 2 · 1, which is true since 3 | 6. P (k) " P (k + 1): (k + 1)3 + 3(k + 1)2 + 2(k + 1) = (k3 + 3k2 + 2k) + 3(k2 + 3k + 2), which is divisible by 3 since each of the two terms is divisible by 3. 514 Test Bank Questions and Answers 12. P (18): use one 4-cent stamp and two 7-cent stamps. P (k) " P (k + 1): if a pile of stamps for k cents postage has a 7-cent stamp, replace one 7-cent stamp with two 4-cent stamps; if the pile contains only 4-cent stamps (there must be at least five of them), replace five 4-cent stamps with three 7-cent stamps. 13. P (18): Eighteen dollars can be made using six 3-dollar bills. P (k) " P (k + 1): Suppose that k dollars can be formed, for some k . 18. If at least two 10-dollar bills are used, replace them by seven 3-dollar bills to form k+1 dollars. Otherwise (that is, at most one 10-dollar bill is used), at least three 3-dollar bills are being used, and three of them can be replaced by one 10-dollar bill to form k + 1 dollars. 14. P (6): Six cents postage can be made from two 3-cent stamps. P (k) " P (k + 1): either replace a 3-cent stamp by a 4-cent stamp or else (if there are only 4-cent stamps in the pile of stamps making k cents postage) replace two 4-cent stamps by three 3-cent stamps. 15. The basis case holds since 1& j=1 (2j +1) = 3 = 3 · 12. Now assume that 2k"1& j=k (2j +1) = 3k2 for some k. It follows that 2(k+1)"1& j=k+1 (2j + 1) = 2k"1& j=k (2j + 1) ! (2k + 1) + (4k + 1) + (4k + 3) = 3k2 + 6k + 3 = 3(k + 1)2. 16. The basis step follows since one line divides the plane into 2 regions. Now assume that k lines passing through the same point divide the plane into 2k regions. Adding the (k + 1)st line splits exactly two of these regions into two parts each. Hence, the k + 1 lines split the plane into 2k + 2 = 2(k + 1) regions. 17. Let P (n) be the proposition that an ( 3n. The proof uses the Principle of Strong Induction. The basis step follows since a1 = 2 ( 3 = 31 and a2 = 9 ( 9 = 32. Now assume that P (k) is true for all k such that 1 ( k < n (where n . 3). Then ak ( 3k for 1 ( k < n. Hence an = 2an"1 + 3an"2 ( 2 · 3n"1 + 3 · 3n"2 = 2 · 3n"1 + 3n"1 = 3 · 3n"1 = 3n. 18. P (8): use one of each type. P (k) " P (k + 1): If a 1# 6 5# tile is used as part of the covering of a 1# 6 k# strip, replace a 1#6 5# tile with two 1#6 3# tiles to cover a 1#6 (k+1)# strip. Otherwise, the tiles for the 1#6k# strip must include three 1# 6 3# tiles; replace three of these with two 1# 6 5# tiles to cover a 1# 6 (k + 1)# strip. 19. P (2): The figure at the right shows a tiling of a 4 6 4 board. P (k) " P (k + 1): Divide the 2k+1 6 2k+1 board into four quarters, each of which is a 2k 6 2k board. P (k) guarantees that each of these four 2k 6 2k boards can be tiled. Put these four tiled boards together to obtain a tiling for the 2k+1 6 2k+1 board. 20. P (0): 4 | 1 ! 1 is true since 4 | 0. P (k) " P (k + 1): 9k+1 ! 5k+1 = 9(9k ! 5k) + 5k(9 ! 5). Each term is divisible by 4: 4 | 9k ! 5k (by P (k)) and 4 | 9 ! 5. 21. P (1): 5 | 7 ! 2 is true since 5 | 5. P (k) " P (k + 1): 7k+1 ! 2k+1 = 7(7k ! 2k) + 2k(7 ! 2). Each term is divisible by 5: 5 | 7k ! 2k (by P (k)) and 5 | 7 ! 2. 22. The second form of mathematical induction is used. P (3) is true since it is the ordinary distributive law for intersection over union. P (3)$· · ·$P (n) " P (n+1): A1/(A20· · ·0An+1) = A1/((A20· · ·0An)0An+1) = [A1/(A20 · · ·0An)]0(A1/An+1) = [(A1/A2)0 · · ·0(A1/An)]0(A1/An+1) = (A1/A2)0 · · ·0(A1/An+1). 23. P (1): 1 2 = (22"2"1) 21 , which is true since the right side is equal to 1/2. P (k) " P (k+1): 1 2 + 2 4 + 3 8 + · · ·+ k+1 2k+1 = 2k+1"2"k 2k + k+1 2k+1 = 2k+2"4"2k+k+1 2k+1 = 2k+2"3"k 2k+1 = 2k+2"2"(k+1) 2k+1 . 24. No basis case has been shown. 25. f(n) = 2f(n! 1), f(1) = 2. 26. f(n) = nf(n! 1), f(0) = 1. 27. f(n) = f(n! 1) + 5, f(1) = 7. 28. an = an"1 ! 3, a1 = 16. 29. an = an"1 + an"2, a1 = 1, a2 = 1. 30. 0 5 S; x 5 S " x + 3 5 S. 31. 1 5 S; x 5 S " x + 4 5 S. Test Bank Questions and Answers 515 32. 1 5 S; x 5 S " x/3 5 S. 33. 0 5 S; x 5 S " x± 2 5 S. 34. an = 2an"1, a1 = 2. 35. an = an"1 + 3, a1 = !2. 36. an = an"1 + 1/3, a1 = 2/3. 37. an = an"1, a1 = 7 2. 38. an = 7 an"1, a1 = 7 2. 39. an = an"1 + 2n, a1 = 2. 40. 3 5 S; x 5 S " x + 4 5 S. 41. 5 5 S; x 5 S " x + 5 5 S. 42. 1 5 S; x 5 S " x± 2 5 S. 43. 0.1 5 S; x 5 S " x/10 5 S. 44. 1 5 S; x 5 S " x11 5 S (or x 5 S " 100x + 11 5 S). 45. f(2) = 30, f(3) = 66. 46. f(2) = 5, f(3) = 21. 47. f(2) = 5/2, f(3) = 1/2. 48. a3 = 63 and a4 = 3,968. 49. The following procedure computes na: procedure mult(a: real number, n: positive integer) if n = 1 then mult(a, n) := a else mult(a, n) := a + mult(a, n! 1). 50. The following procedure computes 32n : procedure power(n: nonnegative integer) if n = 0 then power(n) := 3 else power(n) := power(n! 1) · power(n! 1). 51. Suppose c = 3. The program segment assigns 2 to a and then assigns a + c = 2 + 3 = 5 to b. 52. Before the loop is entered p is true since total = 1·2 2 and i ( n. Suppose p is true and i < n after an execution of the loop. Suppose that the while loop is executed again. The variable i is incremented by 1, and hence i ( n. The variable total was (i"1)i 2 , which now becomes (i"1)i 2 + i = i(i+1) 2 . Hence p is a loop invariant. 53. If x < y initially, max is set equal to y, so (x ( y $ max = y) is true. If x = y initially, max is set equal to y, so (x ( y $ max = y) is again true. If x > y, max is set equal to x, so (x > y $ max = x) is true. Questions for Chapter 6 In questions 1–12 suppose that a “word” is any string of seven letters of the alphabet, with repeated letters allowed. 1. How many words are there? 2. How many words end with the letter T? 3. How many words begin with R and end with T? 4. How many words begin with A or B? 5. How many words begin with A or end with B? 518 Test Bank Questions and Answers 59. Find |A|. 60. How many strings in A begin with 774? 61. How many strings in A have exactly one 5? 62. How many strings in A have exactly three 5s? 63. Make up a word problem in good English whose answer is 15!/10!. 64. Make up a word problem in good English whose answer is 315 4 4 · 37 3 4 . 65. How many subsets with an odd number of elements does a set with 10 elements have? 66. How many subsets with more than two elements does a set with 100 elements have? 67. Each user has a password 6 characters long where each character is an uppercase letter, a lowercase letter, or a digit. Each password must contain at least one digit. How long will it take to check every possible character combination, if each check takes one unit of time. In questions 68–71 suppose you have a class with 30 students — 10 freshmen, 12 sophomores, and 8 juniors. 68. In how many ways can you put all 30 in a line? 69. In how many ways can you put all students in a line so that the freshmen are first, the sophomores are in the middle, and the juniors are at the end? 70. In how many ways can you get a committee of 7? 71. In how many ways can you get a committee of 4 freshmen and 3 sophomores? 72. Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8. 73. Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that begin and end with T. 74. Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that begin and end with the same letter. 75. Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that have exactly one B. 76. Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that have at least one C. 77. Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that begin with L or end with R. 78. How many ways are there to select 6 students from a class of 25 to serve on a committee? 79. How many ways are there to select 6 students from a class of 25 to hold six di"erent executive positions on a committee? 80. Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain the number 5. 81. Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain neither 5 nor 6. 82. Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain both 5 and 6. 83. Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain no odd numbers. 84. Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements. 85. Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements, one of which is 3. 86. Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly five elements, all of them even. Test Bank Questions and Answers 519 87. Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements, all of them even. 88. Find the number of subsets of S = {1, 2, 3, . . . , 10} with exactly five elements, two of which are 3 and 4. 89. Find the number of subsets of S = {1, 2, 3, . . . , 10} with exactly five elements, including 3 or 4 but not both. 90. Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly five elements, but neither 3 nor 4. 91. Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly five elements, the sum of which is even. 92. Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly four elements, the sum of which is odd. 93. Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly four elements, the sum of which is even. 94. Suppose a restaurant serves a “special dinner” consisting of soup, salad, entree, dessert, and beverage. The restaurant has five kinds of soup, three kinds of salad, ten entrees, five desserts, and four beverages. How many di"erent special dinners are possible? (Two special dinners are di"erent if they di"er in at least one selection.) 95. The figure at the right shows a 4-block by 5-block grid of streets. Find the number of ways in which you can go from point A to point B, where at each stage you can only go right or up. (You are not allowed to go left or down.) For example, one allowable route from A to B is: Right, Right, Up, Right, Up, Up, Right, Right, Up. 96. Here is an incorrect solution to a problem. Find the error, explain why it is not correct, and give the correct answer. “Problem: Find the number of ways to get two pairs of two di"erent ranks (such as 2 jacks and 2 fives) in a 4-card hand from an ordinary deck of 52 cards.” “Solution: There are 13 ways to get a rank (such as “kings”) for the first pair and 34 2 4 ways to get a pair of that rank. Similarly, there are 12 ways to get a rank (such as “sevens”) for the second pair and 34 2 4 ways to get a pair of that rank. Therefore there are 13 · 34 2 4 · 12 · 34 2 4 ways to get 2 pairs.” 97. A game consisting of flipping a coin ends when the player gets two heads in a row, two tails in a row, or flips the coin four times. (a) Draw a tree diagram to show the ways in which the game can end. (b) In how many ways can the game end? 98. A factory makes automobile parts. Each part has a code consisting of a letter and three digits, such as C117, O076, or Z920. Last week the factory made 60,000 parts. Prove that there are at least three parts that have the same serial number. 99. A factory makes automobile parts. Each part has a code consisting of a digit, a letter, and a digit, with the digits distinct, such as 5C7, 1O6, or 3Z0. Last week the factory made 5,000 parts. Find the minimum number of parts that must have the same serial number. 100. Show that if five points are picked on or in the interior of a square of side length 2, then there are at least two of these points no farther than 7 2 apart. 101. A professor teaching a Discrete Math course gives a multiple choice quiz that has ten questions, each with four possible responses: a, b, c, d. What is the minimum number of students that must be in the professor’s class in order to guarantee that at least three answer sheets must be identical? (Assume that no answers are left blank.) 102. Show that in a group of ten people (where any two people are either friends or enemies) there are either three mutual friends or four mutual enemies. 103. A computer network consists of six computers. Each computer is directly connected to zero or more of the 520 Test Bank Questions and Answers other computers. Show that there are at least two computers in the network that are directly connected to the same number of computers. 104. A computer is programmed to print subsets of {1, 2, 3, 4, 5} at random. If the computer prints 40 subsets, prove that some subset must have been printed at least twice. 105. A computer randomly prints three-digit codes, with no repeated digits in any code (for example, 387, 072, 760). What is the minimum number of codes that must be printed in order to guarantee that at least six of the codes are identical? 106. Explain how the Pigeonhole Principle can be used to show that among any 11 integers, at least two must have the same last digit. 107. Let s1, s2 . . . , s101 be 101 bit strings of length at most 9. Prove that there exist two strings, si and sj , where i )= j, that contain the same number of 0’s and the same number of 1’s. (For example, strings 001001 and 101000 contain the same number of 0’s and the same number of 1’s.) 108. You pick cards one at a time without replacement from an ordinary deck of 52 playing cards. What is the minimum number of cards you must pick in order to guarantee that you get (a) a pair (for example, two kings or two 5s). (b) three of a kind (for example, three 7s). 109. Use the binomial theorem to expand (2a + b)4. 110. Use the binomial theorem to expand (x + y)5. 111. Use the binomial theorem to expand (a + 2)6. 112. Use the binomial theorem to expand (2c! 3d)4. 113. Use the binomial theorem to expand 3 x! 3 x 45. 114. Use the binomial theorem to expand 3 x2 + 1 x 47. 115. Use the binomial theorem to prove the following: 36 0 4 + 36 1 4 + · · · + 36 6 4 = 26. 116. Use the binomial theorem to prove the following: 3100 0 4 + 3100 2 4 + 3100 4 4 + 3100 6 4 + · · · + 3100 98 4 + 3100 100 4 = 3100 1 4 + 3100 3 4 + 3100 5 4 + · · · + 3100 97 4 + 3100 99 4 . 117. Use the binomial theorem to prove the following: 3100 = 3100 0 4 + 3100 1 4 · 2 + 3100 2 4 · 22 + 3100 3 4 · 23 + · · · + 3100 99 4 · 299 + 3100 100 4 · 2100. 118. Find the coe!cient of x7y5 in the expansion of (3x! y)12. 119. Find the coe!cient of x5y6 in the expansion of (2x! y)11. 120. Find the coe!cient of x8 in the expansion of (x2 + 2)13. 121. Find the coe!cient of x9 in the expansion of (2 + x3)10. 122. Find the coe!cient of x5 in (2 + x2)12. 123. Find the number of terms in the expansion of (5a + 8b)15. 124. Find the largest coe!cient in the expansion of (x + 1)6. 125. Find the largest coe!cient in the expansion of (x + 3)5. 126. List the derangements of 1, 2, 3, 4. 127. Find the number of positive integers not exceeding 1000 that are not divisible by 4, 6, or 9. 128. How many permutations of 12345 are there that leave 3 in the third position but leave no other integer in its Test Bank Questions and Answers 523 26. 7! ! 2 · 6!. 27. 46 28. 5670 29. 4 · 5 · 4 · 3 = 240 30. 1524 31. 5P (8, 4). 32. 5 · 4 · P (7, 3). 33. P (7, 5). 34. P (7, 3). 35. 2 · 5 · P (7, 4). 36. 2 · 4 · P (7, 3). 37. 5 · 4 · P (7, 3) ! 2 · 4 · P (7, 3). 38. 388 · 12. 39. 225 . 40. 210. 41. 26. 42. 310 6 4 . 43. 310 5 4 . 44. 310 6 4 + 310 7 4 + · · · + 310 10 4 = 210"(10 5 ) 2 . 45. 30!. 46. 330 4 4 . 47. 310 3 4315 7 4 . 48. 15! · 5! · 10!. 49. (a) 335 5 4 . (b) 320 3 4315 2 4 . 50. 337 6 4 . 51. 320 3 4317 3 4 . 52. 317 2 4320 4 4 + 317 3 4320 3 4 + 317 4 4320 2 4 + 317 5 4320 1 4 + 317 6 4320 0 4 . 53. 320 6 4 + 317 6 4 . 54. (a) 37 · 36 · 35. (b) 20 · 19 · 17. 55. (a) 35 · 34 · 33 · 32. (b) 20 · 19 · 15 · 14. 56. 104. 57. P (10, 4). 58. 0. 59. 105. 60. 102. 61. 5 · 94. 62. 35 3 4 · 92. 63. In how many ways can 5 out of 15 people be put in a row for a picture? 64. A class has 15 women and 7 men. In how many ways can a committee of 4 women and 3 men be formed? 65. C(10, 1) + C(10, 3) + C(10, 5) + C(10, 7) + C(10, 9). 66. 2100 ! C(100, 0) ! C(100, 1) ! C(100, 2). 67. (26 + 26 + 10)6 ! (26 + 26)6 units of time. 524 Test Bank Questions and Answers 68. 30!. 69. 10! · 12! · 8!. 70. 330 7 4 . 71. 310 4 4312 3 4 . 72. 268. 73. 266. 74. 26 · 266. 75. 8 · 257. 76. 268 ! 258. 77. 267 + 267 ! 266. 78. C(25, 6). 79. P (25, 6). 80. 29. 81. 28. 82. 28. 83. 25. 84. C(10, 3). 85. C(9, 2). 86. 1. 87. C(5, 3). 88. C(8, 3). 89. 2C(8, 4). 90. C(8, 5). 91. C(5, 1)C(5, 4) + C(5, 3)C(5, 2) + 1. 92. 2C(5, 3)C(5, 1). 93. 2C(5, 4) + C(5, 2)2. 94. 5 · 3 · 10 · 5 · 4. 95. C(9, 5). 96. The same hand is counted twice. (For example, getting the kings of hearts and diamonds first and the sevens of clubs and hearts second is the same as getting the pair of sevens first and the pair of kings second.) To obtain the correct answer, divide the given answer by two. 97. (a) Test Bank Questions and Answers 525 (b) 8. 98. The number of codes is 26 · 103 = 26,000. Since 860,000/26,0009 = 3, at least three parts have the same code number. 99. The number of codes is 10 · 26 · 9 = 2,340. Since 86,000/2,3409 = 3, at least three parts have the same code number. 100. Divide the square into four congruent 16 1 squares. At least two of the five points lie in or on the edge of one of these 1 6 1 squares. The maximum distance between these two points is 7 2. 101. There are 410 possible answer sheets. Therefore 2 · 410 + 1 is the minimum number that will guarantee three identical answer sheets. 102. Let A be one of the people. A either has at least four friends or else has at least six enemies among the other nine people. Case 1: A has at least four friends, say B,C,D,E. If any two of B,C,D,E are friends, then these two together with A form a group of three mutual friends. If none of B,C,D,E are friends with each other, then B,C,D,E are four mutual enemies. Case 2: A has at least six enemies, say B,C,D,E, F,G. Applying the Pigeonhole Principle to this set of six, there are either three mutual friends or three mutual enemies. If there are three friends, we are done. If there are three mutual enemies, then these three together with A form a group of four mutual enemies. 103. Each computer can be connected to 0, 1, 2, 3, 4, or 5 other computers, but it is not possible in the network to have a computer connected to 0 others and a computer connected to all 5 others. Therefore there are only five possible connection numbers, which is smaller than the number of computers. By the Pigeonhole Principle at least two must have the same number of connections. 104. There are 25 = 32 subsets. If 33 or more subsets are printed, at least one will have been printed twice. 105. There are 10 · 9 · 8 = 720 di"erent codes. Therefore 5 · 720 + 1 = 3,601 is the minimum number of printed codes that guarantees that at least six identical codes will be printed. 106. Use the eleven integers as the pigeons and the ten possible last digits as the pigeonholes. 107. There are ten possible lengths a bit string can have — 0, 1, 2, . . . , 9. Since there are 101 bit strings, there is a length number k such that at least 11 bit strings have length k. The number of 0’s in these 11 bit strings must be one of the ten numbers 0, 1, 2, . . . , 9. Therefore, there are at least two bit strings si and sj , with the same number of 0’s. Since si and sj have the same length, k, they both have the same number of 1’s. 108. (a) 14. (b) 27. 109. 16a4 + 32a3b + 24a2b2 + 8ab3 + b4. 110. x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5. 111. a6 + 12a5 + 60a4 + 160a3 + 240a2 + 192a + 64. 112. 16c4 ! 96c3d + 216c2d2 ! 216cd3 + 81d4. 113. x5 ! 15x3 + 90x! 270/x + 405/x3 ! 243/x5. 114. x14 + 7x11 + 21x8 + 35x5 + 35x2 + 21/x + 7/x4 + 1/x7. 115. In (a + b)n use n = 6, a = b = 1. 116. In (a + b)n use n = 100, a = 1, b = !1. 117. In (a + b)n use n = 100, a = 1, b = 2. 118. ! 312 7 4 37. 119. 311 5 4 25 120. 313 9 4 29. 121. 310 3 4 27. 122. 0. 123. 16. 124. 20. 125. 405. 528 Test Bank Questions and Answers 24. What is the probability that the bit string begins and ends with 0? 25. What is the probability that the bit string has more 0’s than 1’s? 26. What is the probability that the bit string has the sum of its digits equal to seven? 27. What is the probability that the bit string begins with 111? 28. A group of ten women and ten men are in a room. If five of the 20 are selected at random and put in a row for a picture, what is the probability that the five are of the same sex? 29. A group of ten women and ten men are in a room. A committee of four is chosen at random. Find the probability that the committee consists only of women? 30. You pick a word at random from the set of all words of length six of letters of the alphabet with no repeated letters. What is the probability that the word has exactly one vowel? 31. You pick a word at random from the set of all words of length six of letters of the alphabet with no repeated letters. What is the probability that the word begins and ends with a vowel? 32. A red and a green die are rolled. What is the probability of getting a sum of six, given that the number on the red die is even. 33. A red and a green die are rolled. What is the probability of getting a sum of six, given that the number on the green die is odd? In 34–39 an experiment consists of picking at random a bit string of length five. Consider the following events: E1: the bit string chosen begins with 1; E2: the bit string chosen ends with 1; E3: the bit string chosen has exactly three 1’s. 34. Find p(E1|E3). 35. Find p(E3|E2). 36. Find p(E2|E3). 37. Find p(E3|E1 / E2). 38. Determine whether E1 and E2 are independent. 39. Determine whether E2 and E3 are independent. In questions 40–42 you flip a biased coin, where p(heads) = 3/4 and p(tails) = 1/4, ten times. 40. Find p(exactly 9 heads). 41. Find p(exactly 7 heads). 42. Find p(at least 7 heads). 43. Urn 1 contains 2 blue tokens and 8 red tokens; urn 2 contains 12 blue tokens and 3 red tokens. You pick an urn at random and draw out a token at random from that urn. Given that the token is blue, what is the probability that the token came from urn 1? 44. Urn 1 contains 2 blue tokens and 8 red tokens; urn 2 contains 12 blue tokens and 3 red tokens. You roll a die to determine which urn to choose: if you roll a 1 or 2 you choose urn 1; if you roll a 3, 4, 5, or 6 you choose urn 2. Once the urn is chosen, you draw out a token at random from that urn. Given that the token is blue, what is the probability that the token came from urn 1? In questions 45–47 a bowl has eight ping pong balls numbered 1,2,2,3,4,5,5,5. You pick a ball at random. 45. Find p(the number on the ball drawn is . 3). 46. Find p(the number on the ball drawn is even). Test Bank Questions and Answers 529 47. Find E(X), where X = the number on the ball you draw. 48. A die has the numbers 1, 2, 2, 3, 3, 3 on its six sides. If the die is rolled, what is the expected value and variance of the number showing? 49. A pair of dice, each with the numbers 1, 2, 2, 3, 3, 3 on its six sides are rolled. (a) What is the expected value of the sum of the numbers showing? (b) What is the expected value of the product of the numbers showing in part (a)? 50. You have seven cards, numbered 3 through 9, and you pick one at random. If you pick a card with a prime number, you get 1 point; if you pick a card with a composite number, you lose 1 point. Find the expected value of the number of points you get. 51. You flip a coin. If it lands heads, you lose 1 point. If it lands tails, you flip the coin again, and lose 1 point if it lands heads and get 3 points if it lands tails. What is the expected value of the number of points you get when you play this game. 52. Each of 26 cards has a di"erent letter of the alphabet on it. You pick one card at random. A vowel is worth 3 points and a consonant is worth 0 points. Let X = the value of the card picked. Find E(X), V (X), and the standard deviation of X. 53. You have two decks of 26 cards. Each card in each of the two decks has a di"erent letter of the alphabet on it. You pick at random one card from each of the two decks. A vowel is worth 3 points and a consonant is worth 0 points. Let X = the sum of the values of the two cards picked. Find E(X), V (X), and the standard deviation of X. Answers for Chapter 7 1. 4/52. 2. 50/100. 3. 31/366. 4. 18/36. 5. 1/26. 6. C(5, 4)/25 = 5/32. 7. (a) HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. (b) 3/8. 8. (a) (13/52)(12/51). (b) 3/51. 9. (a) 1/64. (b) 3/64. (c) 8/64. (d) 23/64. (e) 28/64. 10. False. Choose one card at random from a deck of 52 cards. Let E = choose a diamond, F = choose a king. Then p(E 0 F ) = 16/52 while p(E) + p(F ) = 17/52. 11. The probabilities of the three outcomes are not equal. Using {HH, HT, TH, TT} as the sample space, the correct answer, 1/4, is obtained. 12. (a) 10/105. (b) 95/105. 13. 15/40. 14. 35/40. 15. (15 · 14)/(40 · 39). 16. 1 ! 20·19+15·14+5·4 40·39 . 17. 22/30. 18. (10 · 9)/(30 · 29). 19. 9/29. 530 Test Bank Questions and Answers 20. 311 3 4 / 316 3 4 . 21. 348 6 4 / 354 6 4 . 22. 310 2 4 /210. 23. 9/210. 24. 28/210. 25. 3310 6 4 + 310 7 4 + 310 8 4 + 310 9 4 + 310 10 44 /210. 26. 310 7 4 /210. 27. 27/210. 28. 10·9·8·7·6 20·19·18·17·16 + 10·9·8·7·6 20·19·18·17·16 . 29. 310 4 4 / 320 4 4 . 30. 5 · 6 · P (21, 5)/P (26, 6). 31. 5 · 4 · P (24, 4)/P (26, 6). 32. 1/9. 33. 1/6. 34. 6/10. 35. 6/16. 36. 6/10. 37. 3/8. 38. Yes. 39. No. 40. 310 9 4 3 3 4 49 1 4 . 41. 310 7 4 3 3 4 47 3 1 4 43. 42. 310 7 4 3 3 4 47 3 1 4 43 + 310 8 4 3 3 4 48 3 1 4 42 + 310 9 4 3 3 4 49 1 4 + 310 10 4 3 3 4 410. 43. ( 2 10 · 1 2 )/( 2 10 · 1 2 + 12 15 · 1 2 ) = 1/5. 44. ( 2 10 · 1 3 )/( 2 10 · 1 3 + 12 15 · 2 3 ) = 1/9. 45. 5/8. 46. 3/8. 47. 27/8. 48. 7/3, 5/9. 49. (a) 14/3. (b) 49/9. 50. !1/7. 51. 0. 52. 15/26, 1.38, 1.18. 53. 2 · 15 26 , 2 · 1.38, 1.66. Questions for Chapter 8 In questions 1–4, describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. Test Bank Questions and Answers 533 39. Suppose f(n) = 2f(n/2) + 3, f(16) = 51. Find f(2). 40. Suppose f(n) = 4f(n/2) + n + 2, f(1) = 2. Find f(8). 41. Use generating functions to solve an = 3an"1 + 2n, a0 = 5. 42. Use generating functions to solve an = 5an"1 + 1, a0 = 1. In questions 43–52 write the first seven terms of the sequence determined by the generating function. 43. (x + 3)2. 44. (1 + x)5. 45. (1 + x)9. 46. 1/(1 ! 3x). 47. x2/(1 ! x). 48. (1 + x)/(1 ! x). 49. 5. 50. ex + e"x. 51. cosx. 52. 1 1 ! x ! x2 ! x3. In questions 53–63 find the coe!cient of x8 in the power series of each of the function. 53. (1 + x2 + x4)3. 54. (1 + x2 + x4 + x6)3. 55. (1 + x2 + x4 + x6 + x8)3. 56. (1 + x2 + x4 + x6 + x8 + x10)3. 57. (1 + x3)12. 58. (1 + x)(1 + x2)(1 + x3)(1 + x4)(1 + x5). 59. 1/(1 ! 2x). 60. x3/(1 ! 3x). 61. 1/(1 ! x)2. 62. x2/(1 + 2x)2. 63. 1/(1 ! 3x2). In questions 64–76 find a closed form for the generating function for the sequence. 64. 4, 8, 16, 32, 64, . . . . 65. 1, 0, 1, 0, 1, 0, 1, 0, . . . . 66. 2, 0, 0, 2, 0, 0, 2, 0, 0, 2, . . . . 67. 2, 4, 6, 8, 10, 12, . . . . 68. 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, . . . . 69. 2, 3, 4, 5, 6, 7, . . . . 70. 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1 . . . . 534 Test Bank Questions and Answers 71. 1,!1, 1 2! ,! 1 3! , 1 4! ,! 1 5! , . . . . 72. 1, 1 2! , 1 4! , 1 6! , 1 8! . . . . 73. 1,!1, 1,!1, 1,!1, 1,!1, . . . . 74. 1, 0,!1, 0, 1, 0,!1, 0, 1, 0,!1, . . . . 75. 350 50 4 , 350 49 4 , 350 48 4 , . . . , 350 1 4 , 350 0 4 , 0, 0, 0, . . . . 76. 350 1 4 , 2 350 2 4 , 3 350 3 4 , . . . , 50 350 50 4 , 0, 0, 0, . . . . 77. Set up a generating function and use it to find the number of ways in which eleven identical coins can be put in three distinct envelopes if each envelope has at least two coins in it. 78. Set up a generating function and use it to find the number of ways in which eleven identical coins can be put in three distinct envelopes if each envelope has most six coins in it. 79. Set up a generating function and use it to find the number of ways in which eleven identical coins can be put in three distinct envelopes if no envelope is empty. 80. Set up a generating function and use it to find the number of ways in which eleven identical coins can be put in three distinct envelopes if each envelope has an even number of coins in it. 81. Set up a generating function and use it to find the number of ways in which eleven identical coins can be put in three distinct envelopes if each envelope has at least two but no more than five coins in it. 82. Set up a generating function and use it to find the number of ways in which eleven identical coins can be put in three distinct envelopes (labeled A, B, C) if envelope A has at least three coins in it. 83. Set up a generating function and use it to find the number of ways in which eleven identical coins can be put in three distinct envelopes (labeled A, B, C) envelopes A and B have the same number of coins in them. 84. Set up a generating function and use it to find the number of ways in which nine identical blocks can be given to four children if each child gets at least one block. 85. Set up a generating function and use it to find the number of ways in which nine identical blocks can be given to four children, if each child gets at least two blocks. 86. Set up a generating function and use it to find the number of ways in which nine identical blocks can be given to four children, if each child gets at most five blocks. 87. Set up a generating function and use it to find the number of ways in which nine identical blocks can be given to four children, if the oldest child gets three blocks. 88. Set up a generating function and use it to find the number of ways in which nine identical blocks can be given to four children, if the oldest child gets at most three blocks. 89. Set up a generating function and use it to find the number of ways in which nine identical blocks can be given to four children, if the oldest child gets either 2 or 3 blocks. 90. If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a0, a1, a2, . . . . 91. If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a3, a4, a5, . . . . 92. If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for a3, a4, a5, a6, . . . . 93. If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for a0, 0, a1, 0, a2, 0, a3, 0, a4, . . . . 94. If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for Test Bank Questions and Answers 535 a0, 3a1, 9a2, 27a3, 81a4, . . . . 95. If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for a0, 0, 0, a1, 0, 0, a2, 0, 0, a3, . . . . 96. If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for 5, a1, 0, a3, a4, a5, . . . . 97. Use generating functions to solve an = 5an"1 + 3, a0 = 2. 98. Use generating functions to solve an = 7an"1 ! 10an"2, a0 = 1, a1 = 1. 99. Use generating functions to solve an = 3an"1 + 2n + 5, a0 = 1. 100. Find |A1 0 A2 0 A3 0 A4| if each set Ai has 100 elements, each intersection of two sets has 60 elements, each intersection of three sets has 20 elements, and there are 10 elements in all four sets. 101. Find |A1 0 A2 0 A3 0 A4| if each set Ai has 150 elements, each intersection of two sets has 80 elements, each intersection of three sets has 20 elements, and there are no elements in all four sets. 102. Suppose you use the principle of inclusion-exclusion to find the size of the union of four sets. How many terms must be added or subtracted? 103. Find the number of positive integers ( 1000 that are multiples of at least one of 3, 5, 11. 104. Find the number of positive integers ( 1000 that are multiples of at least one of 2, 6, 12. 105. Find the number of positive integers ( 1000 that are multiples of at least one of 3, 4, 12. 106. Suppose |A| = |B| = |C| = 100, |A /B| = 60, |A /C| = 50, |B /C| = 40, and |A 0B 0C| = 175. How many elements are in A /B / C? 107. How many positive integers not exceeding 1000 are not divisible by either 4 or 6? 108. A doughnut shop sells 20 kinds of doughnuts. You want to buy 30 doughnuts. How many possibilities are there if you want at most six of any one kind? 109. A doughnut shop sells 20 kinds of doughnuts. You want to buy 30 doughnuts. How many possibilities are there if you want at most 12 of any one kind? 110. A market sells ten kinds of soda. You want to buy 12 bottles. How many possibilities are there if you want (a) at least one of each kind? (b) at most seven bottles of any kind? 111. A market sells ten kinds of soda. You want to buy 12 bottles. How many possibilities are there? if you want at most three bottles of any kind? 112. Suppose you have 100 identical marbles and five jars (labeled A, B, C, D, E). In how many ways can you put the marbles in the jars if: (a) each jar has at least six marbles in it? (b) each jar has at most forty marbles in it? 113. How many ways are there to choose five donuts if there are eight varieties and only the type of each donut matters? 114. A market sells 40 kinds of candy bars. You want to buy 15 candy bars. (a) How many possibilities are there? (b) How many possibilities are there if you want at least three peanut butter bars and at least five almond bars? (c) How many possibilities are there if you want exactly three peanut butter bars and exactly five almond bars? (d) How many possibilities are there if you want at most four to"ee bars and at most six mint bars? 115. How many permutations of all 26 letters of the alphabet are there that contain at least one of the words DOG, 538 Test Bank Questions and Answers 51. 1, 0, "1 2! , 0, 1 4! , 0, "1 6! . 52. 1, 1, 0, 0, 1, 1, 1. 53. 6. 54. 12. 55. 15. 56. 15. 57. 0. 58. 3. 59. 28. 60. 35. 61. 9. 62. 7 · 26. 63. 34. 64. 4 1 ! 2x . 65. 1 1 ! x2 . 66. 2 1 ! x3 . 67. 2 (1 ! x)2 . 68. x3(1 + x + x2 + x3). 69. 1 (1 ! x)2 + 1 1 ! x = 2 ! x (1 ! x)2 . 70. 1 1 ! x ! 1 1 ! x3 . 71. e"x. 72. ex 2 . 73. 1 1 + x . 74. 1 1 + x2 . 75. (1 + x)50. 76. 50(1 + x)49. 77. (x2 + x3 + x4 + · · ·)3, 21. 78. (1 + x + x2 + · · · + x6)3, 33. 79. (x + x2 + x3 + · · ·)3, 45. 80. (1 + x2 + x4 + x6 + · · ·)3, 0. 81. (x2 + x3 + x4 + x5)3, 12. 82. (x3 + x4 + x5 + x6 + · · ·)(1 + x + x2 + x3 + · · ·)2, 45. 83. (1 + x2 + x4 + x6 + x8 + x10)(1 + x + x2 + x3 + · · ·), 6. 84. (x + x2 + x3 + · · ·)4, 56. 85. (x2 + x3 + x4 + · · ·)4, 4. 86. (1 + x + x2 + x3 + x4 + x5)4, 140. 87. x3(1 + x + x2 + x3 + · · ·)3, 28. 88. (1 + x + x2 + x3)(1 + x + x2 + x3 + · · ·)3, 164. Test Bank Questions and Answers 539 89. (x2 + x3)(1 + x + x2 + x3 + · · ·)3, 64. 90. x3G(x). 91. G(x) ! a0 ! a1x! a2x2. 92. 1 x3 (G(x) ! a0 ! a1x! a2x 2). 93. G(x2). 94. G(3x). 95. G(x3). 96. G(x) ! a0 ! a2x2 + 5. 97. an = ! 3 4 + 11 4 ·5n. 98. an = ! 1 3 ·5 n + 4 3 ·2 n. 99. an = 11 2 ·3n ! 2n+1 ! 5 2 . 100. 110. 101. 200. 102. 15. 103. 515. 104. 500. 105. 542. 106. 25. 107. 667. 108. 349 19 4 ! 320 1 4342 19 4 + 320 2 4335 19 4 ! 320 3 4328 19 4 + 320 4 4321 19 4 . 109. 349 19 4 ! 320 1 4336 19 4 + 320 2 4323 19 4 . 110. (a) 311 2 4 (b) 321 9 4 ! 310 1 4313 9 4 . 111. 321 9 4 ! 310 1 4317 9 4 + 310 2 4313 9 4 ! 310 3 439 9 4 . 112. (a) 374 4 4 . (b) 3104 4 4 ! 35 1 4363 4 4 + 35 2 4322 4 4 . 113. 312 7 4 . 114. (a) 354 39 4 . (b) 346 39 4 . (c) 344 37 4 . (d) 354 39 4 ! 349 39 4 ! 347 39 4 + 342 39 4 . 115. 24! · 3. 116. 3 · 23! ! 3 · 20! + 17!. 117. 3 · 22! ! 18!. 118. 26! ! 3 · 23! + 20!. 119. 24!. 120. 26! ! 24!. 121. D10 = 10! ! 310 1 4 9! + 310 2 4 8! ! 310 3 4 7! + · · · + 310 10 4 0!. 122. 48 ! 34 1 4 38 + 34 2 4 28 ! 34 3 4 18. 123. 49 ! 34 1 4 39 + 34 2 4 29 ! 34 3 4 19. 124. 512 ! 35 1 4 412 + 35 2 4 312 ! 35 3 4 212 + 35 4 4 112. 125. 58 ! 35 1 4 48 + 35 2 4 38 ! 35 3 4 28 + 35 4 4 18. 126. an = an"1 + an"2 + 2n"2, a1 = 0, a2 = 1. Hence a8 = 201. 127. an = an"1 + an"3, a1 = a2 = 1, a3 = 2. Hence a12 = 60. 128. an = 2an"1 + 2an"2, a1 = 3, a2 = 8. Hence a6 = 448. 540 Test Bank Questions and Answers Questions for Chapter 9 1. List all the binary relations on the set {0, 1}. 2. List the reflexive relations on the set {0, 1}. 3. List the irreflexive relations on the set {0, 1}. 4. List the symmetric relations on the set {0, 1}. 5. List the transitive relations on the set {0, 1}. 6. List the antisymmetric relations on the set {0, 1}. 7. List the asymmetric relations on the set {0, 1}. 8. List the relations on the set {0, 1} that are reflexive and symmetric. 9. List the relations on the set {0, 1} that are neither reflexive nor irreflexive. In questions 10–23 determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. 10. The relation R on {1, 2, 3, . . .} where aR b means a|b. 11. The relation R on {w, x, y, z} where R = {(w,w), (w, x), (x,w), (x, x), (x, z), (y, y), (z, y), (z, z)}. 12. The relation R on Z where aR b means |a! b| ( 1. 13. The relation R on Z where aR b means a2 = b2. 14. The relation R on {a, b, c} where R = {(a, a), (b, b), (c, c), (a, b), (a, c), (c, b)}. 15. The relation R on A = {x, y, z} where R = {(x, x), (y, z), (z, y)}. 16. The relation R on Z where aR b means a )= b. 17. The relation R on Z where aR b means that the units digit of a is equal to the units digit of b. 18. The relation R on N where aR b means that a has the same number of digits as b. 19. The relation R on the set of all subsets of {1, 2, 3, 4} where SRT means S 2 T . 20. The relation R on the set of all people where aR b means that a is at least as tall as b. 21. The relation R on the set of all people where aR b means that a is younger than b. 22. The relation R on the set {(a, b) | a, b 5 Z} where (a, b)R (c, d) means a = c or b = d. 23. The relation R on R where aR b means a! b 5 Z. 24. A company makes four kinds of products. Each product has a size code, a weight code, and a shape code. The following table shows these codes: Size Code Weight Code Shape Code #1 42 27 42 #2 27 38 13 #3 13 12 27 #4 42 38 38 Find which of the three codes is a primary key. If none of the three codes is a primary key, explain why. 25. If X =(Fran Williams, 617885197, MTH 202, 248B West), find the projections P1,3(X) and P1,2,4(X). Test Bank Questions and Answers 543 61. Which of the following are partitions of {1, 2, 3, . . . , 10}? (a) {2, 4, 6, 8}, {1, 3, 5, 9}, {7, 10}. (b) {1, 2, 4, 8}, {2, 5, 7, 10}, {3, 6, 9}. (c) {3, 8, 10}, {1, 2, 5, 9}, {4, 7, 8}. (d) {1}, {2}, . . . , {10}. (e) {1, 2, . . . , 10}. 62. Suppose R is the relation on N where aR b means that a ends in the same digit in which b ends. Determine whether R is an equivalence relation on N . 63. Suppose the relation R is defined on the set Z where aR b means that ab ( 0. Determine whether R is an equivalence relation on Z. 64. Suppose A is the set composed of all ordered pairs of positive integers. Let R be the relation defined on A where (a, b)R(c, d) means that a + d = b + c. (a) Prove that R is an equivalence relation. (b) Find [(2, 4)]. 65. Suppose that R and S are equivalence relations on a set A. Prove that the relation R/S is also an equivalence relation on A. 66. Let R be the relation on A = {1, 2, 3, 4, 5} where R = {(1, 1), (1, 3), (1, 4), (2, 2), (3, 1), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4), (5, 5)}. Write the matrix for R. 67. Let R be the relation on A = {1, 2, 3, 4, 5} where R = {(1, 1), (1, 3), (1, 4), (2, 2), (3, 1), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4), (5, 5)}. Draw the directed graph for R. 68. Let R be the relation on A = {1, 2, 3, 4, 5} where R = {(1, 1), (1, 3), (1, 4), (2, 2), (3, 1), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4), (5, 5)}. R is an equivalence relation. Find the equivalence classes for the partition of A given by R. In questions 69–71 give an example or else prove that there are none. 69. A relation on {a, b, c} that is reflexive and transitive, but not antisymmetric. 70. A relation on {1, 2} that is symmetric and transitive, but not reflexive. 71. A relation on {1, 2, 3} that is reflexive and transitive, but not symmetric. 72. Suppose |A| = n. Find the number of binary relations on A. 73. Suppose |A| = n. Find the number of symmetric binary relations on A. 74. Suppose |A| = n. Find the number of reflexive, symmetric binary relations on A. Answers for Chapter 9 1. There are 16 binary relations: (a) { }; (b) {(0, 0)}; (c) {(0, 1)}; (d) {(1, 0)}; (e) {(1, 1)}; (f) {(0, 0), (0, 1)}; (g) {(0, 0), (1, 0)}; (h) {(0, 0), (1, 1)}; (i) {(0, 1), (1, 0)}; (j) {(0, 1), (1, 1)}; (k) {(1, 0), (1, 1)}; (l) {(0, 0), (0, 1), (1, 0)}; (m) {(0, 0), (0, 1), (1, 1)}; (n) {(0, 0), (1, 0), (1, 1)}; (o) {(0, 1), (1, 0), (1, 1)}; (p) {(0, 0), (0, 1), (1, 0), (1, 1)}. 2. h, m, n, p (using the letter names in the previous question). 3. a, c, d, i. 4. a, b, e, h, i, l, o, p. 5. All except i, l, o. 6. All except i, l, o, p. 7. a, c, d. 544 Test Bank Questions and Answers 8. h, p. 9. b, e, f, g, j, k, l, o. 10. 1, 3, 4. 11. 1. 12. 1, 2. 13. 1, 2, 4. 14. 1, 3, 4. 15. 2. 16. 2. 17. 1, 2, 4. 18. 1, 2, 4. 19. 1, 3, 4. 20. 1, 4. 21. 3, 4. 22. 1, 2. 23. 1, 2, 4. 24. Shape code. 25. P1,3(X) = (Fran Williams, MTH 202) P1,2,4(X) = (Fran Williams, 617885197, 248B West). 26. {(a, a), (a, c), (b, c), (c, c), (d, b), (d, d)}. 27. {(a, b), (a, c), (a, d), (b, c), (c, c), (d, a), (d, c)}. 28. {(b, a), (d, c)}. 29. {(b, c)}. 30. {(a, c), (b, a), (d, b), (d, d)}. 31. {(a, a), (a, d), (d, c)}. 32. ) 7* 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 + 8,. 33. ) 7* 1 1 0 0 1 1 0 1 0 0 1 0 0 0 1 1 + 8,. 34. ) 777* 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1 + 888, . 35. ) 77777* 1 1 1 1 1 1 0 1 0 1 1 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 1 + 88888, . Test Bank Questions and Answers 545 36. ) 777* 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1 + 888, . 37. ) 777* 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1 + 888, . 38. ) 7* 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 + 8,. 39. ) 7* 1 1 0 1 1 1 1 1 0 0 1 0 0 0 1 1 + 8,. 40. ) 7* 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 + 8,. 41. ) 7* 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 0 + 8,. 42. (a) Yes. (b) No. (c) No. (d) No. 43. (a) Yes. (b) No. (c) Yes. (d) Yes. 44. 45. 46. 548 Test Bank Questions and Answers 7. C4. 8. W5. 9. K4,5. 10. Q3. In questions 11–51 fill in the blanks. 11. Kn has edges and vertices. 12. Km,n has edges and vertices. 13. Wn has edges and vertices. 14. Qn has edges and vertices. 15. The length of the longest simple circuit in K5 is . 16. The length of the longest simple circuit in W10 is . 17. The length of the longest simple circuit in K4,10 is . 18. List all positive integers n such that Cn is bipartite . 19. The adjacency matrix for Km,n has columns. 20. The adjacency matrix for Kn has 1’s and 0’s. 21. There are 0’s and 1’s in the adjacency matrix for Cn. 22. The adjacency matrix for Q4 has entries. 23. The incidence matrix for Wn has rows and columns. 24. The incidence matrix for Q5 has rows and columns. 25. There are non-isomorphic simple undirected graphs with 5 vertices and 3 edges. 26. There are non-isomorphic simple digraphs with 3 vertices and 2 edges. 27. There are non-isomorphic simple graphs with 3 vertices. 28. List all positive integers n such that Kn has an Euler circuit. 29. List all positive integers n such that Qn has an Euler circuit. 30. List all positive integers n such that Wn has an Euler circuit. 31. Every Euler circuit for K9 has length . 32. List all positive integers n such that Kn has a Hamilton circuit. 33. List all positive integers n such that Wn has a Hamilton circuit. 34. List all positive integers n such that Qn has a Hamilton circuit. 35. List all positive integers m and n such that Km,n has a Hamilton circuit. 36. Every Hamilton circuit for Wn has length . 37. List all positive integers n such that Kn has a Hamilton circuit but no Euler circuit. 38. List all positive integers m and n such that Km,n has a Hamilton path but no Hamilton circuit. 39. The largest value of n for which Kn is planar is . 40. The largest value of n for which K6,n is planar is . a b cd e e d c ba f e d cba f e d cb a Test Bank Questions and Answers 549 41. List all the positive integers n such that K2,n is planar. 42. The Euler formula for planar connected graphs states that . 43. If G is a connected graph with 12 regions and 20 edges, then G has vertices. 44. If G is a planar connected graph with 20 vertices, each of degree 3, then G has regions. 45. If a regular graph G has 10 vertices and 45 edges, then each vertex of G has degree . 46. The edge-chromatic number for K2,5 = . 47. The vertex-chromatic number for K7,7 = . 48. The vertex-chromatic number for C15 = . 49. The region-chromatic number for W9 = . 50. The vertex-chromatic number for W9 = . 51. The vertex-chromatic number for Kn = . 52. Determine whether the graph is strongly connected, and if not, whether it is weakly connected. 53. Determine whether the graph is strongly connected, and if not, whether it is weakly connected. 54. Find the strongly connected components of the graph. 55. Find the strongly connected components of the graph. a b c d e a b c de a b c d e f 550 Test Bank Questions and Answers For each of the graphs in 56–58 find %(G),&(G), and min!%V deg('), and determine which of the two inequal- ities in %(G) ( &(G) ( min!%V deg(') are strict. 56. 57. 58. In questions 59–83 either give an example or prove that there are none. 59. A simple graph with 6 vertices, whose degrees are 2, 2, 2, 3, 4, 4. 60. A simple graph with 8 vertices, whose degrees are 0, 1, 2, 3, 4, 5, 6, 7. 61. A simple graph with degrees 1, 2, 2, 3. 62. A simple graph with degrees 2, 3, 4, 4, 4. 63. A simple graph with degrees 1, 1, 2, 4. 64. A simple digraph with indegrees 0, 1, 2 and outdegrees 0, 1, 2. 65. A simple digraph with indegrees 1, 1, 1 and outdegrees 1, 1, 1. 66. A simple digraph with indegrees 0, 1, 2, 2 and outdegrees 0, 1, 1, 3. 67. A simple digraph with indegrees 0, 1, 2, 4, 5 and outdegrees 0, 3, 3, 3, 3. 68. A simple digraph with indegrees 0, 1, 1, 2 and outdegrees 0, 1, 1, 1. 69. A simple digraph with indegrees: 0, 1, 2, 2, 3, 4 and outdegrees: 1, 1, 2, 2, 3, 4. 70. A simple graph with 6 vertices and 16 edges. 71. A graph with 7 vertices that has a Hamilton circuit but no Euler circuit. 72. A graph with 6 vertices that has an Euler circuit but no Hamilton circuit. 73. A graph with a Hamilton path but no Hamilton circuit. 74. A graph with a Hamilton circuit but no Hamilton path. 75. A connected simple planar graph with 5 regions and 8 vertices, each of degree 3. 76. A graph with 4 vertices that is not planar. 77. A planar graph with 10 vertices. 78. A graph with vertex-chromatic number equal to 6. Test Bank Questions and Answers 553 103. How many di"erent channels are needed for six television stations (A,B,C,D,E, F ) whose distances (in miles) from each other are shown in the following table? Assume that two stations cannot use the same channel when they are within 150 miles of each other? A B C D E F A ! 85 175 100 50 100 B 85 ! 125 175 100 130 C 175 125 ! 100 200 250 D 100 175 100 ! 210 220 E 50 100 200 210 ! 100 F 100 130 250 220 100 ! 104. Consider the graph at the right. (a) Does it have an Euler circuit? (b) Does it have an Euler path? (c) Does it have a Hamilton circuit? (d) Does it have a Hamilton path? 105. Consider the graph at the right. (a) Does it have an Euler circuit? (b) Does it have an Euler path? (c) Does it have a Hamilton circuit? (d) Does it have a Hamilton path? 106. Consider the graph at the right. (a) Does it have an Euler circuit? (b) Does it have an Euler path? (c) Does it have a Hamilton circuit? (d) Does it have a Hamilton path? 107. Use Dijkstra’s Algorithm to find the shortest path length between the vertices a and z in this weighted graph. 554 Test Bank Questions and Answers 108. Use Dijkstra’s Algorithm to find the shortest path length between the vertices a and z in this weighted graph. 109. The Math Department has 6 committees that meet once a month. How many di"erent meeting times must be used to guarantee that no one is scheduled to be at 2 meetings at the same time, if committees and their members are: C1 = {Allen, Brooks, Marg}, C2 = {Brooks, Jones, Morton}, C3 = {Allen, Marg, Morton}, C4 = {Jones, Marg, Morton}, C5 = {Allen, Brooks}, C6 = {Brooks, Marg, Morton}. 110. Determine whether this graph is planar. 111. Determine whether this graph is planar. 112. Determine whether this graph is planar. 113. The picture at the right shows the floor plan of an o!ce. Use graph theory ideas to prove that it is impossible to plan a walk that passes through each doorway exactly once, starting and ending at A. Test Bank Questions and Answers 555 114. Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for K3,2. 115. Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for K4. 116. Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for C7. 117. Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for Q3. 118. Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for W5. 119. Give a recurrence relation for en = the number of edges of the graph Kn. 120. Give a recurrence relation for vn = number of vertices of the graph Qn. 121. Give a recurrence relation for en = number of edges of the graph Qn. 122. Give a recurrence relation for en = the number of edges of the graph Wn. 123. Solve the traveling salesman problem for the graph at the right by finding the total weight of all Hamilton circuits and determining a circuit with minimum total weight. 124. Solve the traveling salesman problem for the graph at the right by finding the total weight of all Hamilton circuits and determining a circuit with minimum total weight. In questions 125–134 the grid graph Gm,n refers to the graph obtained by taking an m 6 n rectangular grid of streets (m ( n) with m north/south blocks and n east/west blocks. For example: 125. Find a formula for the number of vertices of Gm,n. 126. Find a formula for the number of edges of Gm,n. 127. Find a formula for the number of regions (including the infinite region) of Gm,n. 128. For which positive integers m and n does Gm,n have an Euler circuit? 129. For which positive integers m and n does Gm,n have an Euler path but no Euler circuit? 130. For which positive integers m and n does Gm,n have a Hamilton circuit? 131. For which positive integers m and n does Gm,n have a Hamilton path but no Hamilton circuit? 132. Find the vertex-chromatic number for Gm,n. 133. Find the edge-chromatic number for Gm,n. 134. Find the region-chromatic number for Gm,n (including the infinite face). 558 Test Bank Questions and Answers 14. n2n"1, 2n. 15. 10. 16. 15. 17. 40. 18. n even. 19. m + n. 20. n(n! 1), n. 21. n2 ! 2n, 2n. 22. 256. 23. n + 1, 2n. 24. 32, 80. 25. 4. 26. 4. 27. 4. 28. n odd. 29. n even. 30. None. 31. 36. 32. All n except n = 2. 33. All n. 34. All n except n = 1. 35. m = n > 1. 36. n + 1. 37. n even ( )= 2). 38. m = n + 1 or n = m + 1. 39. 4. 40. 2. 41. All n. 42. v ! e + r = 2. 43. 10. 44. 12. 45. 9. 46. 5. 47. 2. 48. 3. 49. 4 (if the infinite region is colored). 50. 4. 51. n. 52. The circuits a, e, c, d, a and a, e, b, a show that the graph is strongly connected. 53. Since there are no paths to d, the graph is not strongly connected, but is weakly connected. 54. The circuit a, f, b, c, f, b, a shows that these four vertices are in the same strong component. There are no paths from e to this strong component, and no paths from d to any other vertex. Therefore the strongly connected components are {a, b, c, f}, {d}, and {e}. Test Bank Questions and Answers 559 55. The circuit a, b, f, a shows that these three vertices are in the same strong component. Similarly, the circuit c, d, e, c shows that these three vertices are in the same strong component. There are no paths from {c, d, e} to {a, b, f}. Therefore the strongly connected components are {a, b, f}, {c, d, e}. 56. %(G) = 2,&(G) = 2, min!%V deg(') = 2 57. %(G) = 2,&(G) = 4, min!%V deg(') = 4 58. %(G) = 1,&(G) = 1, min!%V deg(') = 2 59. None. It is not possible to have one vertex of odd degree. 60. None. It is not possible to have a vertex of degree 7 and a vertex of degree 0 in this graph. 61. 62. None. It is not possible to have a graph with one vertex of odd degree. 63. None. In a simple graph with 4 vertices, the largest degree a vertex can have is 3. 64. 65. 66. 67. None. In a simple graph with five vertices, there cannot be a vertex with indegree 5. 68. None. The sum of the outdegrees must equal the sum of the indegrees. 69. None. The sum of the outdegrees must equal the sum of the indegrees. 70. None. The largest number of edges in a simple graph with six vertices is 15. 71. W6. 72. 73. K1,1. 74. None. Every Hamilton circuit is a Hamilton path. 75. None. The graph would have 12 edges, and hence v ! e + r = 8 ! 12 + 5 = 1, which is not possible. 76. None. The largest such graph, K4, is planar. 77. C10. 78. K6. 79. C9 with one edge removed. 80. None. The 4-color theorem rules this out. 81. Q3. 560 Test Bank Questions and Answers 82. 83. None. Any bipartite Hamilton graph must have an even number of vertices. 84. The graphs are isomorphic: A–7, B–4, C–3, D–6, E–5, F–2, G–1. 85. The graphs are not isomorphic: the graph on the left is planar, but the one on the right is isomorphic to K3,3. 86. The digraphs are isomorphic: label the center vertex 4, the top vertex 2, the left vertex 1, the right vertex 3. 87. The graphs are isomorphic: label the graph clockwise from the top with 2, 3, 6, 5, 4, 1. 88. (a) Use the 5,3-entry of A12. (b) Examine the 5,3-entry of A,A2, A3, . . . , A16. The smallest positive integer i such that the 5,3-entry of Ai is not zero is the length of a shortest path from v5 to v3. If the 5,3-entry is always zero, there is no path from v5 to v3. 89. (a) All n . 3, n = 3, all n . 1, all n . 0. (b) m = n. 90. v(v"1) 2 ! e. 91. 92. The numbers on the edges of the graph indicate the multiplicities of the edges. 93. (a) 6. (b) 9. (c) 2, 4, 2, 3, 4, 3.