CSC 125 :: Final Exam December 15, 2010

Transcription

1 1-5. Complete the truth tables below: CSC 125 :: Final Exam December 15, 2010 p q p q p q p q p q p q T T F F T F T F Fill in the missing portion of each of the rules of inference named below: 6. Modus ponens 7. Modus tollens 8. Hypothetical syllogism 9. Disjunctive syllogism 10. Addition p q p.? p q?. p p q q r? p q?. q?. p q 1

2 11. Simplification 12. Conjunction 13. Resolution?. p p q? p q?. q r (14 16) Let S(x) be the statement x studied hard last week, where the domain for x consists of all students in your school. Match each of the quantified expressions below with the sentences found in the Answer Bank. Write the letter of the matching sentence just to the right of the expression. 14. xs(x) 15. xs(x) 16. xs(x) (17 24) Let B(x,y) be the statement student x from this class attended basketball game y, where the domain for x consists of all students in this class and the domain for y consists of all basketball games this season. Match each of the quantified expressions below with the sentences found in the Answer Bank. Write the letter of the matching sentence just to the right of the expression. 17. xyb(x,y) 18. xyb(x,y) 19. x yb(x,y) 20. x yb(x,y) 21. xyb(x,y) 22. xyb(x,y) 2

3 23. x yb(x,y) 24. x yb(x,y) (25 30) Let F(x,y) be the statement x can fool y, where the domain for both x and y consists of all people in the world. Use quantifiers to express these sentences. 25. Everyone can fool Fred. 26. Evelyn can fool everybody. 27. No one can fool everybody. 28. Someone can fool everybody. 29. Everybody can fool someone. 30. No one can fool him- or herself. (31 42) Determine whether the arguments below are valid or invalid, then circle either or. If valid, using the list of valid argument forms found in the Answer Bank, write the letter of the rule of inference; if invalid write the letter of the logical fallacy, again using the list of invalid argument forms found in the Answer Bank. 31. If pigs can fly, elephants can sing. Pigs cannot fly Elephants cannot sing. 32. If pigs can fly, elephants can sing. Pigs can fly. Elephants can sing. 3

4 33. Freedom is precious and fragile. Freedom is precious. 34. If we pigs can fly, Santa Claus is real. Santa Claus is not real. Pigs cannot fly. 35. Either it will rain or snow today. It did not rain. It snowed. 36. Freedom is a precious commodity. Freedom must be safeguarded by every generation. Freedom is precious and must be safeguarded by every generation. 37. If the Eagles win this game, they will make the playoffs. The Eagles made the playoffs. The Eagles won this game. 38. The crocuses are blooming. Either the crocuses are blooming or robins are singing. 39. The crocuses are blooming. The crocuses are blooming and robins are singing. 40. If freedom is fragile, it can be lost to tyranny. If freedom can be lost to tyranny, it must be safeguarded carefully. If freedom is fragile, it must be safeguarded carefully. 4

5 41. Logic is either hard or it is nutty. Either logic is easy or it is impossible. Logic is either nutty or it is impossible. 42. We are having ham or eggs for breakfast. We are having eggs for breakfast. (43 46) Let A = {1, 3, 5, 6} B = {1, 2, 3, 6, 7} 43. A B = 44. A B = 45. A B = 46. A B = 47. Let A = {1, 2, 3} Give the power set of A, P(A) = (48 49) Let A = {a,b,c} B = {x,y} C = {0,1}. Find 48. A B = 49. A B C = 5

6 (50 57) Determine whether each sentence below is true or false. Circle either True or False. 50. {a, b, c} {a, b, c} True False 51. {a, b} {a, b, c} True False 52. {0} {0} True False 53. {0} True False 54. {a, b, c} {a, b, c} True False 55. { True False 56. { { True False 57. {{ {{ True False (58 64) Let the universal set U = {a, b, c, d, e, f, g, h} 58. If the set A is represented by the bit string , then A = 59. If the set B is represented by the bit string , then B = 60. Give the bit string for A B. 61. Give the bit string for A B. 62. Give the bit string for A B. 63. Give the bit string for the complement of A. 64. Give the bit string for the empty set,. 6

7 (65 67) Which region(s) in the Venn diagrams above represent: 65. (A B) C 66. cmp(a B C) 67. A B (68 69) Compute the value of these expressions involving the floor and ceiling functions (70 71) What are the terms a 0, a 1, a 2, and a 3, of the sequence {a n }, where a n equals: n + ( 2) n 71. n/3 7

8 (72 75) For the questions below, the notation n j=0 x represents the summation of x as j goes from 0 to n. What is the value of this sum? k=0 (k+2) Use this formula: n k=0 ar k = a[(r n+1 1)/(r 1)] to compute the value of each of this geometric progression k=0 2 k (74 75) Compute each of these double sums i=0 3 j=1 (i+j) i=0 2 j=0 (2i+3j) 76. Let S = {1, 3, 4, 7} What is j S (j 2 2j)? 77. Let S be a set. Then P(S) = 78. Arrange these complexity classes in ascending order of complexity: O(n n ), O(n b ), O(n), O(1), O(n!), O(n log n), O(log n), O(b n ) Answer:: 79. P(9,3) = 8

9 80. C(10,4) = 81. How many different bit strings of length 8 are there? mod 17 = mod 13 = 84. Give the prime factorization of Give the prime factorization of Give the prime factorization of 7! 87. How many bit strings of length 8 contain exactly 3 1's? 88. What is the minimum number of students, each of whom comes from one of the 50 states, that must be enrolled in a university to guarantee that there are at least 100 that come from the same state? 89. How many bit strings of length 10 contain at least 4 1's? 90. How many bit strings of length 10 have an equal number of 1's and 0's? 91. How many bit strings of length 8 either begin with 111 or end with 00? 92. What is the coefficient of x 5 y 8 in (x + y) 13? 93. What is the coefficient of x 7 in (x + 1) 11? 9

10 94. A jar contains 5 red balls and 5 blue ones. If you pick balls at random, how many must you select to guarantee that you have drawn 3 red balls? (95 96) Let S = {1, 2, 3, 4, 5} 95. How many 3-permutations of S are there? 96. List all the 3-combinations of S. 97. In a 15-horse race, how many different possibilities are there for win, place and show? Extra Credit EC-1. Give the recursive definition of factorial: EC-2. What is the coefficient of y 3 x 5 in (3y + 2x) 8? EC-3. What is the coefficient of x 9 in (2 x) 19? EC-4. Prove or disprove: If a bc, with a,b & c positive integers, then either a b or a c. EC-5. At a KU awards ceremony, 7 women and 5 men are to be seated in a row of 12 chairs. If the seating arrangement is chosen at random, what is the probability that all of the men will be seated next to each other in 5 consecutive positions? EC-6. A fair coin is tossed 100 times, with each toss resulting in a head or a tail. Which probability is greater? Justify your answer. a. the total number of heads is exactly 50? b. the total number of heads is 70? EC-7. Prove: For all b > 0, c b such that n! > b n, for all n > c b. 10