Writing Assignment 2 Student Sample Questions


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1 Writing Assignment 2 Student Sample Questions 1. Let P and Q be statements. Then the statement (P = Q) ( P Q) is a tautology. 2. The statement If the sun rises from the west, then I ll get out of the bed. is trivially true. 3. Let A and B be the any two sets. Consider the following statement: (A B) B = B (A B) Which of the following would make the statement true? A. B A, and B. B. A B, and A. C. B =. D. A =. E. More than one of the above. F. None of the above. 4. Let P be the statement For all integer x greater than 9, x 2 is greater than 9x + 5. Which of the following has the same meaning as P? A. x Z 9, x 2 9x + 5. B. x Z >9, x 2 > 9x + 5. C. x Z >9, x 2 9x + 5. D. x / Z 9, x 2 < 9x Prove or disprove the following: Proposition: For every integer n > 5, n can be written as n = 4x + 3y, for some nonnegative x, y Z. 6. For all integers n > 9, 2 n > n Every subset of the natural numbers has a smallest element. 8. The Greatest Common Denominator of 592 and 566 is A. x = 16. B. x = 2. C. x = 3. D. x = 8. E. More than one of the above. F. None of the above. 9. When proving using the method of induction we prove that following is (are) true. A. P (0)
2 B. P (k) C. P (k + 1) D. All of the above E. A and C 10. Prove or disprove the following: Proposition: Let n N, then n ( ) n ( 1) k = 0 k 11. The following relation R is antisymmetric: R = {(a, b) R R : a = b}. k=0 12. For some finite set of primes {p 1, p 2,..., p n }, the result p 1 p 2 p n + 1 must be a prime number not contained within the set. 13. What is the coefficient of the term x 2 y 3 in the expansion of (8x + 7y) 5? A. 10. B C D E. None of the above. 14. Determine which of the following is the Greatest Common Divisor of 1332 and 612 using the Euclidean Algorithm. A. 18. B. 12. C. 24. D. 21. E. None of the above. 15. Prove or disprove the following: Proposition: For all n N, 8 n 3 n is divisible by The definition of n! is Given n Z and n 0, n! = n (n 1)! 17. Since a bc and a b, a c 18. Which of the following are true statements about the GCD? (mark multiple) A. GCD(a, 0) = a B. GCD(a, b) = GCD( a, b ) C. The smallest linear combination of two integers a, b is GCD(a, b) Page 2
3 D. Let a, b, c Z. If a bc and GCD(a, b) = 1, then a c. 19. Evaluate the following proposition and disproof. Proposition: Let a N. Then a is a product of primes Disproof. Let a = 1, so a N, but since prime numbers are defined as an integer p > 1, there is no prime factor of 1. A. The disproof is correct and the proposition is true. B. The disproof is correct and the proposition is false. C. The proposition is true but the disproof isn t mathematically rigorous. D. The proposition is true but the disproof incorrectly uses the definition of prime numbers. E. The proposition is true but the disproof makes an arithmetic mistake. 20. Prove or disprove the following: Proposition: Let a, b Z with b 0. If d is a common divisor of a and b, and d = ax + by for some x, y Z, then GCD(a, b) = d. 21. The GCD(1386, 408) = R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (2, 3), (3, 2), (1, 3), (3, 1)} is an equivalence relation on the set A = {1, 2, 3, 4, 5}. 23. Let R be a equivalence relation on a set A = {1, 4, 10}. Let R be defined as R = {(1, 1), (4, 4), (10, 10), (1, 10), (10, 1)}. How many equivalence classes does R have? A. 1 B. 2 C. 3 D. 4 E The coefficient of x 4 y 2 in (3x + y) 6 is: A B. 538 C D Prove the following by induction: Proposition: For all integers, n 21, the Fibonacci number F n is greater than n A prime number is an integer p > 0 such that the only positive divisors of p are 1 and p. 27. The GCD(a, 0) = a. 28. Let R be the relation on N given by arb if a b. R is: A. Reflexive only. Page 3
4 B. Reflexive, transitive, and antisymmetric only. C. Reflexive and symmetric only. D. Reflexive and transitive only. 29. Determine the coefficient of a 3 b 2 in the expansion of (a + b) 5. A. 5! 2! B. 5! 1!(5 1)! C. 5! 2!(5 2)! D. 5! 30. Prove the following by induction: Proposition: For all integers n 1,. n 2 i = 2 n+1 2 i=1 31. If GCD(119, 63) = 119x + 63y for some x, y Z, then the only solution for x, y is x = 1 and y = Let p be a prime number, and let a Z. If p a, then GCD(p, a) = p. 33. Let A = {1, 3, 8, 9}. Define a relation R on A by R = {(1, 3), (1, 9), (3, 9)}. Which of the following statements about R is correct? A. R is symmetric and transitive, but not reflexive. B. R is symmetric, reflexive, and transitive. C. R is transitive, but is not symmetric, and not reflexive. D. R is not symmetric, reflexive, or transitive. 34. Let n, k Z and n 0. Which of the following is true? A. ( ( n n) = n 1). B. ( ) ( n k + n ) ( k+1 = n+1 k+1). C. ( n 0) = 1. D. Both A and C are true. E. Both B and C are true. F. None of the above are true. 35. Prove or disprove the following: Proposition: For all n > 8, we have n! > 4 n. 36. The negation of the definition of antisymmetry, a, b A, (a b b a) a = b is a, b A, (a b) (b a) (a b) where means a and b are related. 37. To write the fraction a b in lowest terms, we find GCD(a, b) and divide a, b by GCD(a, b). Page 4
5 38. Find x, y Z such that GCD(87612, 43651) = 87612x y A. x = 2957 and y = B. x = 5935 and y = C. x = 5935 and y = D. x = 2957 and y = Find the canonical factorization of A B C D Prove or disprove the following: Proposition: For every natural number n > 20, we can write n = 3x+5y +10z with x, y, z nonnegative integers. 41. Let a, b Z. If GCD(a, b) = 1, then a and b are relatively prime. A. True B. False 42. We say that R is symmetric if for all a A, we have ara. A. True B. False 43. Using the Euclidean Algorithm, compute the GCD of 986 and 476. A. 1 B. 3 C. 7 D The binomial coefficient ( 7 4) is equivalent to which of the following? A. 4 B. 21 C. 35 D Prove that for any positive n Z, n 3 + 2n is divisible by Let P (n) be an open sentence, where the domain of n is N. then P (n) is true for all n N. If P (1) is true, and k N, P (k) = P (k + 1), 47. Let n, d Z with d 0. Then there are unique integers q, r such that and 0 r < d. n = qd + r Page 5
6 48. What is the GCD(631, 448)? A. 3. B. 1. C. 7. D. 9. E. None of the above. 49. Let p be a prime number, and let a Z. What are the possibilities of the GCD(p, a)? Mark all that apply. A. p. B. 2a. C. 1. D. 15. E. All of the above. 50. Prove that for every n N: 2n < 2 n Let R be a relation on a set A. We say that R is symmetric if for all a, b, c A, we have 52. GCD(768, 236) = 4 ((arb) (brc)) = (arc). 53. Determine the coefficient of x 2 y 5 in the expansion of (3x + y) 7. A. 243 B. 9 C. 21 D. 189 E. None of the above. 54. Let R be an equivalence relation on the set A = {1, 2, 3, 4, 5, 6, 7}. Assume that 1R5 and 5R6. Given these conditions, which ordered pair of the following must belong to R? A. (3, 4) B. (1, 7) C. (1, 6) D. (1, 2) E. None of the above. 55. Prove that for every n N, n i=1 1 i(i + 1) = n n Let S = {1, 2}. Define a relation R on S by R = {(1, 1), (2, 2), (1, 2)}. Then R is symmetric. Page 6
7 57. A prime number is any integer p such that the only positive divisors of p are 1 and p. 58. Let S = {1, 2, 3, 4, 5} and define a relation R on S by R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (3, 4), (4, 3), (4, 4), (5, 5)}. R is an equivalence relation. How many equivalence classes are there in A? A. 1. B. 2. C. 3. D Find the GCD(48, 120) A. 1. B. 15. C. 24. D Prove or disprove the following: Proposition: 7 n 1 is divisible by 6 for all n 1, where n Z. 61. Let GCD(a, b) = c. Then! z N, with z < c, such that for x, y Z, ax + by = z. 62. Let a, b Z, where both are greater than 2 and have the same parity. Then the GCD(a, b) Assume there exists a world were the only currency is in the form of coins valued at 3, 7, and 5 units. What is the smallest value of x such that c x, c is a value that can be made with the coins? A. x = 3. B. x = 10. C. x = 15. D. x = Given the equation (αx + βy) 10, what is the coefficient on the x 4 y 6 term? A. 210αβ B. 210α 4 β 6 C. 5040αβ D. 5040α 6 β Prove or disprove the following: ( N Proposition: k ) N k k+1 = ( N k+1 ) 66. Let A = {1, 2, 3, 4, 5} and let S = {A 1, A 2,..., A 6 }, where A i = {x A : x i (mod 6)}. Then S is a partition of A. Page 7
8 67. The greatest positive integer linear combination of integers a and b, not both equal to zero, is GCD(a, b). 68. During the inductive step of strong induction, what do we get to assume (inductive hypothesis) and what are we trying to prove? A. Assume P (1) and prove P (k). B. Assume P (k) and prove P (k + 1). C. Assume P (k + 1) and prove P (k + 2). D. Assume P (1)... P (k) and prove P (k + 1). E. More than one of the above. F. None of the above. 69. Let A = {1, 2, 3, 4, 5, 6, 7, 8} and define a relation R on A by. The number of equivalence classes in A is A. 1 B. 2 C. 3 D Prove or disprove the following: R = {(a, b) A A : a b 7} Proposition: Let a, b, c Z. If a bc and GCD(a, b) = 2, then a c. 71. Let A = {1, 2, 3} and define a relation R on A by R is an equivalence relation. R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)}. 72. Let a, b Z, not both zero, and let d = GCD(a, b). Then ( a GCD d, b ) = 1. d 73. Let A = {1, 2, 3, 4} and define a relation R on A as How many equivalence classes are in A? A. 1. B. 2. C. 3. D. 4. E. More than one of the above. F. R is not an equivalence relation. R = {(1, 1), (2, 2), (2, 3), (3, 2), (3, 3), (4, 4)}. Page 8
9 74. Which of the following are true? A. GCD(594, 63) = 3. B. GCD(651, 196) = 7. C. GCD(333, 243) = 3. D. GCD(789, 654) = Prove that for every n N, n i=1 i 3 = n2 (n + 1) Let A be a set and let R be an equivalence relation on A. Then the set of all equivalence classes of R is a partition of A. 77. Let P be a partition of a nonempty set A. Then every possible equivalence relation on A has equivalence classes that are precisely the parts of P. 78. Which of the following cannot be a partition of a nonempty set A? A. {{1}, {2}, {4}, {5}}. B. {{1, 2, 3}, { }}. C. {, {1, 2}}. D. {{a, b, c}}. E. More than one of the above. F. None of the above. 79. An equivalence relation must have all but which one of the following properties? A. Reflexivity. B. Symmetry. C. Transitivity. D. Antisymmetry. 80. Let A be the set {1, 2, 3, 4} and let R be a relation on A given by Prove or disprove the following: R = {(1, 1), (2, 2), (2, 4), (3, 3), (3, 4), (4, 3), (4, 2)(4, 4)}. Proposition: R is an equivalence relation on A. 81. The negation of the definition of antisymmetry, a, b A, (a b b a) a = b is a, b A, (a b) (b a) (a b) where means a and b are related. 82. To write the fraction a b in lowest terms, we find GCD(a, b) and divide a, b by GCD(a, b). 83. Find x, y Z such that GCD(87612, 43651) = 87612x y A. x = 2957 and y = B. x = 5935 and y = Page 9
10 C. x = 5935 and y = D. x = 2957 and y = Find the canonical factorization of A B C D Prove or disprove the following: Proposition: For every natural number n > 20, we can write n = 3x+5y +10z with x, y, z nonnegative integers. You may include my questions and solutions in the file to be distributed to all the students in the class. Page 10
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