Writing Assignment 2 Student Sample Questions
|
|
- Brenda Stokes
- 5 years ago
- Views:
Transcription
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 non-negative 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
MATH 215 Final. M4. For all a, b in Z, a b = b a.
MATH 215 Final We will assume the existence of a set Z, whose elements are called integers, along with a well-defined binary operation + on Z (called addition), a second well-defined binary operation on
More informationLecture Notes 1 Basic Concepts of Mathematics MATH 352
Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,
More informationChapter 5: The Integers
c Dr Oksana Shatalov, Fall 2014 1 Chapter 5: The Integers 5.1: Axioms and Basic Properties Operations on the set of integers, Z: addition and multiplication with the following properties: A1. Addition
More informationFinal Exam Review. 2. Let A = {, { }}. What is the cardinality of A? Is
1. Describe the elements of the set (Z Q) R N. Is this set countable or uncountable? Solution: The set is equal to {(x, y) x Z, y N} = Z N. Since the Cartesian product of two denumerable sets is denumerable,
More information5: The Integers (An introduction to Number Theory)
c Oksana Shatalov, Spring 2017 1 5: The Integers (An introduction to Number Theory) The Well Ordering Principle: Every nonempty subset on Z + has a smallest element; that is, if S is a nonempty subset
More informationa the relation arb is defined if and only if = 2 k, k
DISCRETE MATHEMATICS Past Paper Questions in Number Theory 1. Prove that 3k + 2 and 5k + 3, k are relatively prime. (Total 6 marks) 2. (a) Given that the integers m and n are such that 3 (m 2 + n 2 ),
More informationInduction. Induction. Induction. Induction. Induction. Induction 2/22/2018
The principle of mathematical induction is a useful tool for proving that a certain predicate is true for all natural numbers. It cannot be used to discover theorems, but only to prove them. If we have
More informationRED. Name: Instructor: Pace Nielsen Math 290 Section 1: Winter 2014 Final Exam
RED Name: Instructor: Pace Nielsen Math 290 Section 1: Winter 2014 Final Exam Note that the first 10 questions are true-false. Mark A for true, B for false. Questions 11 through 20 are multiple choice
More information8. Given a rational number r, prove that there exist coprime integers p and q, with q 0, so that r = p q. . For all n N, f n = an b n 2
MATH 135: Randomized Exam Practice Problems These are the warm-up exercises and recommended problems taken from all the extra practice sets presented in random order. The challenge problems have not been
More information4 Powers of an Element; Cyclic Groups
4 Powers of an Element; Cyclic Groups Notation When considering an abstract group (G, ), we will often simplify notation as follows x y will be expressed as xy (x y) z will be expressed as xyz x (y z)
More informationNOTES ON INTEGERS. 1. Integers
NOTES ON INTEGERS STEVEN DALE CUTKOSKY The integers 1. Integers Z = {, 3, 2, 1, 0, 1, 2, 3, } have addition and multiplication which satisfy familar rules. They are ordered (m < n if m is less than n).
More informationIntegers and Division
Integers and Division Notations Z: set of integers N : set of natural numbers R: set of real numbers Z + : set of positive integers Some elements of number theory are needed in: Data structures, Random
More informationCSC Discrete Math I, Spring Relations
CSC 125 - Discrete Math I, Spring 2017 Relations Binary Relations Definition: A binary relation R from a set A to a set B is a subset of A B Note that a relation is more general than a function Example:
More informationElementary Properties of the Integers
Elementary Properties of the Integers 1 1. Basis Representation Theorem (Thm 1-3) 2. Euclid s Division Lemma (Thm 2-1) 3. Greatest Common Divisor 4. Properties of Prime Numbers 5. Fundamental Theorem of
More informationWORKSHEET MATH 215, FALL 15, WHYTE. We begin our course with the natural numbers:
WORKSHEET MATH 215, FALL 15, WHYTE We begin our course with the natural numbers: N = {1, 2, 3,...} which are a subset of the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } We will assume familiarity with their
More informationALGEBRA. 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers
ALGEBRA CHRISTIAN REMLING 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers by Z = {..., 2, 1, 0, 1,...}. Given a, b Z, we write a b if b = ac for some
More informationReview Problems for Midterm Exam II MTH 299 Spring n(n + 1) 2. = 1. So assume there is some k 1 for which
Review Problems for Midterm Exam II MTH 99 Spring 014 1. Use induction to prove that for all n N. 1 + 3 + + + n(n + 1) = n(n + 1)(n + ) Solution: This statement is obviously true for n = 1 since 1()(3)
More informationChapter 1. Greatest common divisor. 1.1 The division theorem. In the beginning, there are the natural numbers 0, 1, 2, 3, 4,...,
Chapter 1 Greatest common divisor 1.1 The division theorem In the beginning, there are the natural numbers 0, 1, 2, 3, 4,..., which constitute the set N. Addition and multiplication are binary operations
More information1.4 Equivalence Relations and Partitions
24 CHAPTER 1. REVIEW 1.4 Equivalence Relations and Partitions 1.4.1 Equivalence Relations Definition 1.4.1 (Relation) A binary relation or a relation on a set S is a set R of ordered pairs. This is a very
More information1. Given the public RSA encryption key (e, n) = (5, 35), find the corresponding decryption key (d, n).
MATH 135: Randomized Exam Practice Problems These are the warm-up exercises and recommended problems taken from all the extra practice sets presented in random order. The challenge problems have not been
More informationFall 2017 Test II review problems
Fall 2017 Test II review problems Dr. Holmes October 18, 2017 This is a quite miscellaneous grab bag of relevant problems from old tests. Some are certainly repeated. 1. Give the complete addition and
More informationDo not open this exam until you are told to begin. You will have 75 minutes for the exam.
Math 2603 Midterm 1 Spring 2018 Your Name Student ID # Section Do not open this exam until you are told to begin. You will have 75 minutes for the exam. Check that you have a complete exam. There are 5
More information2 Elementary number theory
2 Elementary number theory 2.1 Introduction Elementary number theory is concerned with properties of the integers. Hence we shall be interested in the following sets: The set if integers {... 2, 1,0,1,2,3,...},
More informationChapter 1 : The language of mathematics.
MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :
More informationHomework #2 solutions Due: June 15, 2012
All of the following exercises are based on the material in the handout on integers found on the class website. 1. Find d = gcd(475, 385) and express it as a linear combination of 475 and 385. That is
More informationIntroduction to Cryptography CS 355 Lecture 3
Introduction to Cryptography CS 355 Lecture 3 Elementary Number Theory (1) CS 355 Fall 2005/Lecture 3 1 Review of Last Lecture Ciphertext-only attack: Known-plaintext attack: Chosen-plaintext: Chosen-ciphertext:
More informationCh 4.2 Divisibility Properties
Ch 4.2 Divisibility Properties - Prime numbers and composite numbers - Procedure for determining whether or not a positive integer is a prime - GCF: procedure for finding gcf (Euclidean Algorithm) - Definition:
More informationMATH 420 FINAL EXAM J. Beachy, 5/7/97
MATH 420 FINAL EXAM J. Beachy, 5/7/97 1. (a) For positive integers a and b, define gcd(a, b). (b) Compute gcd(1776, 1492). (c) Show that if a, b, c are positive integers, then gcd(a, bc) = 1 if and only
More informationMath 109 September 1, 2016
Math 109 September 1, 2016 Question 1 Given that the proposition P Q is true. Which of the following must also be true? A. (not P ) or Q. B. (not Q) implies (not P ). C. Q implies P. D. A and B E. A, B,
More informationProblem 1: Suppose A, B, C and D are finite sets such that A B = C D and C = D. Prove or disprove: A = B.
Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination III (Spring 2007) Problem 1: Suppose A, B, C and D are finite sets
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationRecitation 7: Existence Proofs and Mathematical Induction
Math 299 Recitation 7: Existence Proofs and Mathematical Induction Existence proofs: To prove a statement of the form x S, P (x), we give either a constructive or a non-contructive proof. In a constructive
More information(e) Commutativity: a b = b a. (f) Distributivity of times over plus: a (b + c) = a b + a c and (b + c) a = b a + c a.
Math 299 Midterm 2 Review Nov 4, 2013 Midterm Exam 2: Thu Nov 7, in Recitation class 5:00 6:20pm, Wells A-201. Topics 1. Methods of proof (can be combined) (a) Direct proof (b) Proof by cases (c) Proof
More informationBasic Proof Examples
Basic Proof Examples Lisa Oberbroeckling Loyola University Maryland Fall 2015 Note. In this document, we use the symbol as the negation symbol. Thus p means not p. There are four basic proof techniques
More informationThe following is an informal description of Euclid s algorithm for finding the greatest common divisor of a pair of numbers:
Divisibility Euclid s algorithm The following is an informal description of Euclid s algorithm for finding the greatest common divisor of a pair of numbers: Divide the smaller number into the larger, and
More informationMathematical Reasoning & Proofs
Mathematical Reasoning & Proofs MAT 1362 Fall 2018 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0
More informationMath 3000 Section 003 Intro to Abstract Math Final Exam
Math 3000 Section 003 Intro to Abstract Math Final Exam Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Name: Problem 1a-j 2 3a-b 4a-b 5a-c 6a-c 7a-b 8a-j
More informationCSE 1400 Applied Discrete Mathematics Proofs
CSE 1400 Applied Discrete Mathematics Proofs Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Axioms 1 Logical Axioms 2 Models 2 Number Theory 3 Graph Theory 4 Set Theory 4
More informationPRACTICE PROBLEMS: SET 1
PRACTICE PROBLEMS: SET MATH 437/537: PROF. DRAGOS GHIOCA. Problems Problem. Let a, b N. Show that if gcd(a, b) = lcm[a, b], then a = b. Problem. Let n, k N with n. Prove that (n ) (n k ) if and only if
More informationChapter 2. Divisibility. 2.1 Common Divisors
Chapter 2 Divisibility 2.1 Common Divisors Definition 2.1.1. Let a and b be integers. A common divisor of a and b is any integer that divides both a and b. Suppose that a and b are not both zero. By Proposition
More informationGreatest Common Divisor MATH Greatest Common Divisor. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Greatest Common Divisor Benjamin V.C. Collins James A. Swenson The world s least necessary definition Definition Let a, b Z, not both zero. The largest integer d such that d a and d b is called
More informationExercises Exercises. 2. Determine whether each of these integers is prime. a) 21. b) 29. c) 71. d) 97. e) 111. f) 143. a) 19. b) 27. c) 93.
Exercises Exercises 1. Determine whether each of these integers is prime. a) 21 b) 29 c) 71 d) 97 e) 111 f) 143 2. Determine whether each of these integers is prime. a) 19 b) 27 c) 93 d) 101 e) 107 f)
More informationD-MATH Algebra I HS18 Prof. Rahul Pandharipande. Solution 1. Arithmetic, Zorn s Lemma.
D-MATH Algebra I HS18 Prof. Rahul Pandharipande Solution 1 Arithmetic, Zorn s Lemma. 1. (a) Using the Euclidean division, determine gcd(160, 399). (b) Find m 0, n 0 Z such that gcd(160, 399) = 160m 0 +
More informationCOMP239: Mathematics for Computer Science II. Prof. Chadi Assi EV7.635
COMP239: Mathematics for Computer Science II Prof. Chadi Assi assi@ciise.concordia.ca EV7.635 The Euclidean Algorithm The Euclidean Algorithm Finding the GCD of two numbers using prime factorization is
More informationCSE 20 DISCRETE MATH. Winter
CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Today's learning goals Define and use the congruence modulo m equivalence relation Perform computations using modular arithmetic
More information14 Equivalence Relations
14 Equivalence Relations Tom Lewis Fall Term 2010 Tom Lewis () 14 Equivalence Relations Fall Term 2010 1 / 10 Outline 1 The definition 2 Congruence modulo n 3 Has-the-same-size-as 4 Equivalence classes
More informationPart IA Numbers and Sets
Part IA Numbers and Sets Definitions Based on lectures by A. G. Thomason Notes taken by Dexter Chua Michaelmas 2014 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationDirect Proof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Direct Proof Fall / 24
Direct Proof MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Direct Proof Fall 2014 1 / 24 Outline 1 Overview of Proof 2 Theorems 3 Definitions 4 Direct Proof 5 Using
More information2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.
2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say
More informationM381 Number Theory 2004 Page 1
M81 Number Theory 2004 Page 1 [[ Comments are written like this. Please send me (dave@wildd.freeserve.co.uk) details of any errors you find or suggestions for improvements. ]] Question 1 20 = 2 * 10 +
More information2. THE EUCLIDEAN ALGORITHM More ring essentials
2. THE EUCLIDEAN ALGORITHM More ring essentials In this chapter: rings R commutative with 1. An element b R divides a R, or b is a divisor of a, or a is divisible by b, or a is a multiple of b, if there
More informationHomework #2 Solutions Due: September 5, for all n N n 3 = n2 (n + 1) 2 4
Do the following exercises from the text: Chapter (Section 3):, 1, 17(a)-(b), 3 Prove that 1 3 + 3 + + n 3 n (n + 1) for all n N Proof The proof is by induction on n For n N, let S(n) be the statement
More informationWednesday, February 21. Today we will begin Course Notes Chapter 5 (Number Theory).
Wednesday, February 21 Today we will begin Course Notes Chapter 5 (Number Theory). 1 Return to Chapter 5 In discussing Methods of Proof (Chapter 3, Section 2) we introduced the divisibility relation from
More informationNumber Theory Math 420 Silverman Exam #1 February 27, 2018
Name: Number Theory Math 420 Silverman Exam #1 February 27, 2018 INSTRUCTIONS Read Carefully Time: 50 minutes There are 5 problems. Write your name neatly at the top of this page. Write your final answer
More informationNumber Theory Basics Z = {..., 2, 1, 0, 1, 2,...} For, b Z, we say that divides b if z = b for some. Notation: b Fact: for all, b, c Z:
Number Theory Basics Z = {..., 2, 1, 0, 1, 2,...} For, b Z, we say that divides b if z = b for some z Z Notation: b Fact: for all, b, c Z:, 1, and 0 0 = 0 b and b c = c b and c = (b + c) b and b = ±b 1
More informationRED. Fall 2016 Student Submitted Sample Questions
RED Fall 2016 Student Submitted Sample Questions Name: Last Update: November 22, 2016 The questions are divided into three sections: True-false, Multiple Choice, and Written Answer. I will add questions
More informationPUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime.
PUTNAM TRAINING NUMBER THEORY (Last updated: December 11, 2017) Remark. This is a list of exercises on Number Theory. Miguel A. Lerma Exercises 1. Show that the sum of two consecutive primes is never twice
More informationLecture 2. The Euclidean Algorithm and Numbers in Other Bases
Lecture 2. The Euclidean Algorithm and Numbers in Other Bases At the end of Lecture 1, we gave formulas for the greatest common divisor GCD (a, b), and the least common multiple LCM (a, b) of two integers
More informationContribution of Problems
Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions
More informationUNIVERSITY OF VICTORIA DECEMBER EXAMINATIONS MATH 122: Logic and Foundations
UNIVERSITY OF VICTORIA DECEMBER EXAMINATIONS 2013 MATH 122: Logic and Foundations Instructor and section (check one): K. Mynhardt [A01] CRN 12132 G. MacGillivray [A02] CRN 12133 NAME: V00#: Duration: 3
More informationMath 511, Algebraic Systems, Fall 2017 July 20, 2017 Edition. Todd Cochrane
Math 511, Algebraic Systems, Fall 2017 July 20, 2017 Edition Todd Cochrane Department of Mathematics Kansas State University Contents Notation v Chapter 0. Axioms for the set of Integers Z. 1 Chapter 1.
More informationDirect Proof Divisibility
Direct Proof Divisibility Lecture 15 Section 4.3 Robb T. Koether Hampden-Sydney College Fri, Feb 7, 2014 Robb T. Koether (Hampden-Sydney College) Direct Proof Divisibility Fri, Feb 7, 2014 1 / 23 1 Divisibility
More informationSenior Math Circles Cryptography and Number Theory Week 2
Senior Math Circles Cryptography and Number Theory Week 2 Dale Brydon Feb. 9, 2014 1 Divisibility and Inverses At the end of last time, we saw that not all numbers have inverses mod n, but some do. We
More informationMath Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number
More informationASSIGNMENT 1 SOLUTIONS
MATH 271 ASSIGNMENT 1 SOLUTIONS 1. (a) Let S be the statement For all integers n, if n is even then 3n 11 is odd. Is S true? Give a proof or counterexample. (b) Write out the contrapositive of statement
More informationMATH 363: Discrete Mathematics
MATH 363: Discrete Mathematics Learning Objectives by topic The levels of learning for this class are classified as follows. 1. Basic Knowledge: To recall and memorize - Assess by direct questions. The
More informationThe set of integers will be denoted by Z = {, -3, -2, -1, 0, 1, 2, 3, 4, }
Integers and Division 1 The Integers and Division This area of discrete mathematics belongs to the area of Number Theory. Some applications of the concepts in this section include generating pseudorandom
More informationLogic. Facts (with proofs) CHAPTER 1. Definitions
CHAPTER 1 Logic Definitions D1. Statements (propositions), compound statements. D2. Truth values for compound statements p q, p q, p q, p q. Truth tables. D3. Converse and contrapositive. D4. Tautologies
More informationSection Summary. Relations and Functions Properties of Relations. Combining Relations
Chapter 9 Chapter Summary Relations and Their Properties n-ary Relations and Their Applications (not currently included in overheads) Representing Relations Closures of Relations (not currently included
More informationa + b = b + a and a b = b a. (a + b) + c = a + (b + c) and (a b) c = a (b c). a (b + c) = a b + a c and (a + b) c = a c + b c.
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
More informationPROBLEMS ON CONGRUENCES AND DIVISIBILITY
PROBLEMS ON CONGRUENCES AND DIVISIBILITY 1. Do there exist 1,000,000 consecutive integers each of which contains a repeated prime factor? 2. A positive integer n is powerful if for every prime p dividing
More informationComplete Induction and the Well- Ordering Principle
Complete Induction and the Well- Ordering Principle Complete Induction as a Rule of Inference In mathematical proofs, complete induction (PCI) is a rule of inference of the form P (a) P (a + 1) P (b) k
More informationProperties of the Integers
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
More informationMATH FINAL EXAM REVIEW HINTS
MATH 109 - FINAL EXAM REVIEW HINTS Answer: Answer: 1. Cardinality (1) Let a < b be two real numbers and define f : (0, 1) (a, b) by f(t) = (1 t)a + tb. (a) Prove that f is a bijection. (b) Prove that any
More informationMathematics for Computer Science Exercises for Week 10
Mathematics for Computer Science Exercises for Week 10 Silvio Capobianco Last update: 7 November 2018 Problems from Section 9.1 Problem 9.1. Prove that a linear combination of linear combinations of integers
More informationName: Mathematics 1C03
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
More informationMath 110 FOUNDATIONS OF THE REAL NUMBER SYSTEM FOR ELEMENTARY AND MIDDLE SCHOOL TEACHERS
4-1Divisibility Divisibility Divisibility Rules Divisibility An integer is if it has a remainder of 0 when divided by 2; it is otherwise. We say that 3 divides 18, written, because the remainder is 0 when
More informationWUCT121. Discrete Mathematics. Logic. Tutorial Exercises
WUCT11 Discrete Mathematics Logic Tutorial Exercises 1 Logic Predicate Logic 3 Proofs 4 Set Theory 5 Relations and Functions WUCT11 Logic Tutorial Exercises 1 Section 1: Logic Question1 For each of the
More informationProofs. Methods of Proof Divisibility Floor and Ceiling Contradiction & Contrapositive Euclidean Algorithm. Reading (Epp s textbook)
Proofs Methods of Proof Divisibility Floor and Ceiling Contradiction & Contrapositive Euclidean Algorithm Reading (Epp s textbook) 4.3 4.8 1 Divisibility The notation d n is read d divides n. Symbolically,
More informationMath 230 Final Exam, Spring 2009
IIT Dept. Applied Mathematics, May 13, 2009 1 PRINT Last name: Signature: First name: Student ID: Math 230 Final Exam, Spring 2009 Conditions. 2 hours. No book, notes, calculator, cell phones, etc. Part
More informationWe want to show P (n) is true for all integers
Generalized Induction Proof: Let P (n) be the proposition 1 + 2 + 2 2 + + 2 n = 2 n+1 1. We want to show P (n) is true for all integers n 0. Generalized Induction Example: Use generalized induction to
More informationThe number of ways to choose r elements (without replacement) from an n-element set is. = r r!(n r)!.
The first exam will be on Friday, September 23, 2011. The syllabus will be sections 0.1 through 0.4 and 0.6 in Nagpaul and Jain, and the corresponding parts of the number theory handout found on the class
More information1 Take-home exam and final exam study guide
Math 215 - Introduction to Advanced Mathematics Fall 2013 1 Take-home exam and final exam study guide 1.1 Problems The following are some problems, some of which will appear on the final exam. 1.1.1 Number
More informationMa/CS 6a Class 2: Congruences
Ma/CS 6a Class 2: Congruences 1 + 1 5 (mod 3) By Adam Sheffer Reminder: Public Key Cryptography Idea. Use a public key which is used for encryption and a private key used for decryption. Alice encrypts
More informationChapter VI. Relations. Assumptions are the termites of relationships. Henry Winkler
Chapter VI Relations Assumptions are the termites of relationships. Henry Winkler Studying relationships between objects can yield important information about the objects themselves. In the real numbers,
More informationNOTES ON SIMPLE NUMBER THEORY
NOTES ON SIMPLE NUMBER THEORY DAMIEN PITMAN 1. Definitions & Theorems Definition: We say d divides m iff d is positive integer and m is an integer and there is an integer q such that m = dq. In this case,
More informationModule 1. Integers, Induction, and Recurrences
Module 1. Integers, Induction, and Recurrences This module will look at The integers and the natural numbers. Division and divisors, greatest common divisors Methods of reasoning including proof by contradiction
More informationChapter 5. Number Theory. 5.1 Base b representations
Chapter 5 Number Theory The material in this chapter offers a small glimpse of why a lot of facts that you ve probably nown and used for a long time are true. It also offers some exposure to generalization,
More information10 Problem 1. The following assertions may be true or false, depending on the choice of the integers a, b 0. a "
Math 4161 Dr. Franz Rothe December 9, 2013 13FALL\4161_fall13f.tex Name: Use the back pages for extra space Final 70 70 Problem 1. The following assertions may be true or false, depending on the choice
More information2k n. k=0. 3x 2 7 (mod 11) 5 4x 1 (mod 9) 2 r r +1 = r (2 r )
MATH 135: Randomized Exam Practice Problems These are the warm-up exercises and recommended problems take from the extra practice sets presented in random order. The challenge problems have not been included.
More informationFermat s Little Theorem. Fermat s little theorem is a statement about primes that nearly characterizes them.
Fermat s Little Theorem Fermat s little theorem is a statement about primes that nearly characterizes them. Theorem: Let p be prime and a be an integer that is not a multiple of p. Then a p 1 1 (mod p).
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More informationExam in Discrete Mathematics
Exam in Discrete Mathematics First Year at the Faculty of Engineering and Science and the Technical Faculty of IT and Design June 4th, 018, 9.00-1.00 This exam consists of 11 numbered pages with 14 problems.
More informationThe Euclidean Algorithm and Multiplicative Inverses
1 The Euclidean Algorithm and Multiplicative Inverses Lecture notes for Access 2009 The Euclidean Algorithm is a set of instructions for finding the greatest common divisor of any two positive integers.
More information18 Divisibility. and 0 r < d. Lemma Let n,d Z with d 0. If n = qd+r = q d+r with 0 r,r < d, then q = q and r = r.
118 18. DIVISIBILITY 18 Divisibility Chapter V Theory of the Integers One of the oldest surviving mathematical texts is Euclid s Elements, a collection of 13 books. This book, dating back to several hundred
More information3 Finite continued fractions
MTH628 Number Theory Notes 3 Spring 209 3 Finite continued fractions 3. Introduction Let us return to the calculation of gcd(225, 57) from the preceding chapter. 225 = 57 + 68 57 = 68 2 + 2 68 = 2 3 +
More informationEUCLID S ALGORITHM AND THE FUNDAMENTAL THEOREM OF ARITHMETIC after N. Vasiliev and V. Gutenmacher (Kvant, 1972)
Intro to Math Reasoning Grinshpan EUCLID S ALGORITHM AND THE FUNDAMENTAL THEOREM OF ARITHMETIC after N. Vasiliev and V. Gutenmacher (Kvant, 1972) We all know that every composite natural number is a product
More informationThis is a recursive algorithm. The procedure is guaranteed to terminate, since the second argument decreases each time.
8 Modular Arithmetic We introduce an operator mod. Let d be a positive integer. For c a nonnegative integer, the value c mod d is the remainder when c is divided by d. For example, c mod d = 0 if and only
More informationChapter 5.1: Induction
Chapter.1: Induction Monday, July 1 Fermat s Little Theorem Evaluate the following: 1. 1 (mod ) 1 ( ) 1 1 (mod ). (mod 7) ( ) 8 ) 1 8 1 (mod ). 77 (mod 19). 18 (mod 1) 77 ( 18 ) 1 1 (mod 19) 18 1 (mod
More informationNumber Theory and Graph Theory. Prime numbers and congruences.
1 Number Theory and Graph Theory Chapter 2 Prime numbers and congruences. By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com 2 Module-1:Primes
More information