Name: Mathematics 1C03

Transcription

1 Name: Student ID Number: Mathematics 1C03 Day Class Instructor: M. Harada Duration: 2.5 hours April 2018 McMaster University PRACTICE Final Examination This is a PRACTICE final exam. The actual final exam will look almost exactly like this one. In particular, the number of problems on this practice exam is the same as what will be on the real exam. THIS EXAM IS PRINTED DOUBLE-SIDED AND INCLUDES 20 PAGES AND 12 QUESTIONS. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE PAPER IS COMPLETE. BRING ANY DISCREPANCIES TO THE ATTENTION OF THE INVIGILATOR. Instructions: You are not permitted to use books, notes, or other course aids, including calculators. The test contains 12 questions. YOU ARE TO CHOOSE 10 OUT OF THE 12 QUESTIONS. The exam is out of 100 marks total. Pages 15 to 18 are blank for additional rough work. For full credit you must show all your work and justify your statements. You are to select 10 out of 12 of the problems. You MUST CROSS OUT, with a clear X, the boxes for the questions which you do NOT wish to have marked, in the chart below. Problem Marks Problem Marks TOTAL Page 1 of 18 continued...

2 Page 2 of 18 continued... Problem 1. [10 marks total] True or False. Decide whether the statement is True or False and give a one-sentence justification of your answer. There is some space for scratch work, but write your answer in the space provided. a. (2 marks) For integers a and b, if 2 a and 2 b, then 4 ab. (Circle one) True False Justification: b. (2 marks) For an equivalence relation R on a set S, if two equivalence classes [a] and [b] satisfy [a] [b], then [a] = [b]. (Circle one) True False Justification: c. (2 marks) The statement P Q is logically equivalent to (P = Q) AND NOT Q. (Circle one) True False Justification: d. (2 marks) If S is the set {1, 2, 3}, then {2, 3} S. (Circle one) True False Justification: e. (2 marks) If S is the set {1, 2, 3}, then {1} S. (Circle one) True False Justification: Page 2 of 18 continued...

3 Page 3 of 18 continued... Problem 2. [10 marks total] a. (1 mark) Fill in the blank. A Diophantine equation ax + by = c has a solution iff c. b. (1 mark) Fill in the blank. An integer x is invertible modulo m iff x and m are. c. (1 mark) Fill in the blank. The of P = Q is Q = P. d. (1 mark) Fill in the blank. A function f : X Y is if distinct elements x 1 and x 2 of X must be mapped to distinct elements f(x 1 ) and f(x 2 ) of Y. e. (2 marks) Let Z 5 be the set of integers modulo 5. Using mathematical notation, give a precise description of the congruence class of 3 in Z 5 as a subset of Z. f. (2 marks) Let m > 0 be a positive integer and a, x, y be integers. Under what circumstances does ax ay modulo m imply that x y modulo m? g. (2 marks) What is the difference between the principle of mathematical induction (POMI) and the principle of strong induction (POSI)? (You do not have to give the full statment of POMI and POSI. Just say what the difference is.) Page 3 of 18 continued...

4 Page 4 of 18 continued... Problem 3. [10 marks total] a. (5 marks) Use the Euclidean algorithm to find gcd (43, 17). Show your work. Page 4 of 18 continued...

5 Page 5 of 18 continued... Problem 3, cont d. b. (5 marks) Find all solutions to the Diophantine equation: 43x+17y = gcd(43, 17). Show your work and justify your answer. Page 5 of 18 continued...

6 Page 6 of 18 continued... Problem 4. [10 marks total] a. (5 marks) Find all solutions to the simultaneous linear congruences: x 6 mod 7 and x 9 mod 12. Show your work. b. (5 marks) Give an example of a relation on the set X = {1, 2, 3} which is reflexive and transitive, but not symmetric. Justify your answer. Page 6 of 18 continued...

7 Page 7 of 18 continued... Problem 5. [10 marks total] a. (5 marks) Write the contrapositive and the negation of the following statement: If x Z such that x 2 = y then y is a positive integer. b. (5 marks) Give an example of two infinite sets X and Y of the same cardinality and a function f : X Y which is not a bijection. Page 7 of 18 continued...

8 Page 8 of 18 continued... Problem 6. [10 marks] Let a, b, c 1, c 2 Z. Suppose that the Diophantine equations ax + by = c 1 and ax + by = c 2 have integer solutions (x 1, y 1 ) and (x 2, y 2 ) respectively. Prove that ax + by = kc 1 + lc 2 also has an integer solution (x, y) for all k, l Z. Page 8 of 18 continued...

9 Page 9 of 18 continued... Problem 7. [10 marks] Does x 3 a modulo p always have a solution for every value of a and any prime p? Fully justify your answer. Page 9 of 18 continued...

10 Page 10 of 18 continued... Problem 8. [10 marks] Let f 1, f 2, f 3,..., be the Fibonacci sequence, which is the sequence of integers defined (inductively!) by: f 1 = 1, f 2 = 1, and for integers n with n 3, we define f n = f n 1 + f n 2. Prove, by using the principle of mathematical induction, that gcd(f n, f n 1 ) = 1, n Z, n 2. Page 10 of 18 continued...

11 Page 11 of 18 continued... Problem 9. [10 marks] Suppose that X and Y are sets and f : X Y is injective. Prove: for any set T and any functions g : T X and h : T X, f g = f h implies that g = h. Page 11 of 18 continued...

12 Page 12 of 18 continued... Problem 10. [10 marks] For any set X, let P(X) denote the set of all subsets (including and X) of the set X. If # X = n, a finite number, find a formula for # P(X). Prove your answer. (Hint: try some examples with sets of small size in order to make an educated guess for what the formula should be. Then see if you can prove your guess.) Page 12 of 18 continued...

13 Page 13 of 18 continued... Problem 11. [10 marks] If φ(m) is the Euler φ-function, prove that φ(m) = φ(2m) if and only if m is odd. Page 13 of 18 continued...

14 Page 14 of 18 continued... Problem 12. [10 marks] Let p and q be distinct odd primes and let n = pq. Let r be the least common multiple of p 1 and q 1, i.e., the smallest positive integer which is a multiple of both p 1 and q 1. Let e be an encryption exponent for the RSA cryptosystem based on n. This means that encryption is given by m m e mod n. Show that any d with de 1 mod r can be used as a decryption exponent. (Hint 1: use the Chinese Remainder Theorem. Hint 2: this problem is harder than the others. Generous partial marks will be awarded, so even if you do not have a complete solution, you should write down your thoughts.) END OF PRACTICE EXAM QUESTIONS Page 14 of 18 continued...

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