More Examples of Proofs
|
|
- Rosaline McBride
- 6 years ago
- Views:
Transcription
1 More Examples of Proofs
2 Contradiction Proofs Definition: A prime number is an integer greater than 1 which is divisible only by 1 and itself. Ex: 2, 5, 11 are primes; 6, 15, 100 are not primes. There are an infinite number of primes. Pf: BWOC suppose that there are only a finite number of primes. Let p 1, p 2,..., p k be the complete list of all prime numbers. Consider the integer N = p 1 p 2...p k + 1. N is not a prime because it is larger than every element in the list. Suppose the integer d, with 1 < d < N, divides N. Let p be a prime divisor of d. p divides N since d divides N. Thus, p divides N p 1 p 2...p k since p is in the list of primes. Therefore, p divides 1. Thus, there are an infinite number of primes.
3 Quiz #2 If all the integers of A are either even or squares, then either there is an even integer in A or all the integers of A are squares. ( x A)(P(x) Q(x)) ( x A)P(x) ( x A)Q(x) ~ (A B) A ~B ( x A)(P(x) Q(x)) ( x A) ~P(x) ( x A) ~Q(x) All the integers of A are either even or squares, and (but) none of the integers in A are even and there is a non-square in A.
4 unless When used in an English sentence the meaning can be ambiguous. The usual interpretation of A unless B is : ~B A ~(~B ~A) B A. Thus, Tuition will increase unless Referenda C & D pass does not usually imply that tuition will not go up if C & D pass, i.e., B ~A ~(B ~~A) ~B ~A. However, I will fix your car unless you don't pay me. Would have the stronger interpretation of ~B A.
5 Contradiction Proofs There do not exist three consecutive natural numbers such that the cube of the largest is the sum of the cubes of the smaller numbers. Pf. Look at (n+2) 3 = (n+1) 3 + n 3 n 3 + 6n n + 8 = n 3 + 3n 2 + 3n n 3 0 = n 3-3n 2-9n - 7 = f(n) f'(n) = 3n 2-6n - 9 = 3(n-3)(n+1) f(-1) = -2, f(3) = f(5) = -2, f(6) = 47 the only root of f(n) is non-integral (between 5 and 6).
6 Contradiction Proofs There are no prime numbers a,b,c such that c 3 = a 3 + b 3. Pf. At least one of a, b is even. May assume b = 2. 8 = c 3 - a 3 = (c - a)(c 2 + ca + a 2 ) but each term in the sum is greater than or equal to 4.
7 Contradiction Proofs If a,b,c are integers such that a 2 + b 2 = c 2, show that at least one of a or b is even. Pf: Suppose that both a and b are odd. There exist integers k and m so that a = 2k + 1 and b = 2m + 1. a 2 + b 2 = (2k+1) 2 + (2m+1) 2 = 4(k 2 +m 2 ) + 4(k+m) + 2. hmmmmmmmm??????? Try again. Pf: Suppose that both a and b are odd. a 2 and b 2 are odd, so c 2 is even. Since c 2 is even, c is even. There is an integer n so that c = 2n and so c 2 = 4n 2. hmmmmmmmmmmm???????
8 Contradiction Proofs If a,b,c are integers such that a 2 + b 2 = c 2, show that at least one of a or b is even. Pf: Suppose that both a and b are odd. There exist integers k and m so that a = 2k + 1 and b = 2m + 1. a 2 + b 2 = (2k+1) 2 + (2m+1) 2 = 4(k 2 +m 2 ) + 4(k+m) + 2. Thus, c 2 = 2[2(k 2 +m 2 + k+m) + 1], which, since k 2 +m 2 + k+m is an integer, is two times an odd number. Since a 2 and b 2 are odd, c 2 is even. Since c 2 is even, c is even. There is an integer n so that c = 2n and so c 2 = 4n 2. So, c 2 = 2(2n 2 ) which is two times an even number.
9 Case Method The case method is based on the tautology: [(P Q) R ] [(P R) (Q R)] If the hypothesis of an implication can be broken up into special cases (which cover all possibilities) and each special case implies the conclusion, then the implication is true. The special cases may arise naturally from the form of the hypothesis or a definition used in the hypothesis. Other times, the special cases may be more "forced" imposed because you see how to easily prove the statement under additional conditions.
10 Case Method Prove that every natural number n is either a prime, a perfect square or divides (n-1)! Pf: If n = 1, then it is a perfect square, so we may assume n > 1. Any natural number is either a prime or not a prime. If n > 1 is not a prime, then there are integers a and b with n = ab and 1 < a,b < n. Case I: a = b If a = b then n = a 2, so n is a perfect square. Case II: a b Since a < n, a (n-1)!. Since b < n and b a, b (n-1)!/a So, ab (n-1)! Thus, n is either a prime, a perfect square or divides (n-1)!
11 Pigeon-Hole Principle The Pigeon-Hole Principle is the formal statement of a common sense idea that we are all aware of. If there are 7 pigeons and 6 pigeon holes and all the pigeons are in a pigeon hole, then some pigeon hole must have at least 2 pigeons in it! - or - If there are 13 pieces of paper all of which are in one of 3 drawers in a desk, then some drawer must have at least 5 pieces of paper.
12 Pigeon-Hole Principle Pigeon-Hole Principle : If kn + 1 objects are distributed amongst n sets, one of the sets must contain at least k + 1 objects. Pf: If no set contains at least k+1 objects, then each of the n sets has at most k objects. Thus, the total number of objects is at most nk.
13 Problem Let there be 9 points in 3-space with integer coordinates. Show that there is a pair of these points whose line segment contains an interior point whose coordinates are integers. Outline of Proof: Points in 3-space have 3 coordinates, (a,b,c). Integer coordinates are either odd or even. There are 8 odd-even patterns of integer coordinates in 3-space. Since there are 9 points, at least two must have the same pattern by the Pigeon-Hole Principle. The midpoint of the line segment joining two points with the same pattern has integer coordinates. Special Assignment 1 : Fill in the missing details of this proof.
CSCE 222 Discrete Structures for Computing. Proofs. Dr. Hyunyoung Lee. !!!!! Based on slides by Andreas Klappenecker
CSCE 222 Discrete Structures for Computing Proofs Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 What is a Proof? A proof is a sequence of statements, each of which is either assumed, or follows
More informationCOT 2104 Homework Assignment 1 (Answers)
1) Classify true or false COT 2104 Homework Assignment 1 (Answers) a) 4 2 + 2 and 7 < 50. False because one of the two statements is false. b) 4 = 2 + 2 7 < 50. True because both statements are true. c)
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 informationTry the assignment f(1) = 2; f(2) = 1; f(3) = 4; f(4) = 3;.
I. Precisely complete the following definitions: 1. A natural number n is composite whenever... See class notes for the precise definitions 2. Fix n in N. The number s(n) represents... 3. For A and B sets,
More informationBasic Principles of Algebra
Basic Principles of Algebra Algebra is the part of mathematics dealing with discovering unknown numbers in an equation. It involves the use of different types of numbers: natural (1, 2, 100, 763 etc.),
More informationA Level. A Level Mathematics. Proof by Contradiction (Answers) AQA, Edexcel, OCR. Name: Total Marks:
AQA, Edexcel, OCR A Level A Level Mathematics Proof by Contradiction (Answers) Name: Total Marks: A1 Proof Answers AQA, Edexcel, OCR 1) Prove that there is an infinite amount of prime numbers. Assume there
More informationHomework 3: Solutions
Homework 3: Solutions ECS 20 (Fall 2014) Patrice Koehl koehl@cs.ucdavis.edu October 16, 2014 Exercise 1 Show that this implication is a tautology, by using a table of truth: [(p q) (p r) (q r)] r. p q
More informationMath 319 Problem Set #2 Solution 14 February 2002
Math 39 Problem Set # Solution 4 February 00. (.3, problem 8) Let n be a positive integer, and let r be the integer obtained by removing the last digit from n and then subtracting two times the digit ust
More informationMathematical Reasoning Rules of Inference & Mathematical Induction. 1. Assign propositional variables to the component propositional argument.
Mathematical Reasoning Rules of Inference & Mathematical Induction Example. If I take the day off it either rains or snows 2. When It rains, my basement floods 3. When the basement floods or it snows,
More informationCS 340: Discrete Structures for Engineers
CS 340: Discrete Structures for Engineers Instructor: Prof. Harry Porter Office: FAB 115-06 harry@cs.pdx.edu Hours: Mon 3-4, Wed 3-4, or by appointment Website: web.cecs.pdx.edu/~harry/discrete Class Mailing
More informationNumber Theory and Graph Theory
1 Number Theory and Graph Theory Chapter 1 Introduction and Divisibility By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com 1 DIVISION
More informationThe theory of numbers
1 AXIOMS FOR THE INTEGERS 1 The theory of numbers UCU Foundations of Mathematics course 2017 Author: F. Beukers 1 Axioms for the integers Roughly speaking, number theory is the mathematics of the integers.
More informationMATH10040: Numbers and Functions Homework 1: Solutions
MATH10040: Numbers and Functions Homework 1: Solutions 1. Prove that a Z and if 3 divides into a then 3 divides a. Solution: The statement to be proved is equivalent to the statement: For any a N, if 3
More informationWhat is a proof? Proofing as a social process, a communication art.
Proof Methods What is a proof? Proofing as a social process, a communication art. Theoretically, a proof of a mathematical statement is no different than a logically valid argument starting with some premises
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 informationProof by Contradiction
Proof by Contradiction MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Proof by Contradiction Fall 2014 1 / 12 Outline 1 Proving Statements with Contradiction 2 Proving
More information13 = m m = (C) The symmetry of the figure implies that ABH, BCE, CDF, and DAG are congruent right triangles. So
Solutions 2005 56 th AMC 2 A 2. (D It is given that 0.x = 2 and 0.2y = 2, so x = 20 and y = 0. Thus x y = 0. 2. (B Since 2x + 7 = 3 we have x = 2. Hence 2 = bx 0 = 2b 0, so 2b = 8, and b = 4. 3. (B Let
More informationCSC165. Larry Zhang, October 7, 2014
CSC165 Larry Zhang, October 7, 2014 If you did bad, then it is not bad. Proof: assume you left all questions blank # that s pretty bad! then you get 20% # rule on test paper assume class average is 70%
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 information1. Prove that the number cannot be represented as a 2 +3b 2 for any integers a and b. (Hint: Consider the remainder mod 3).
1. Prove that the number 123456782 cannot be represented as a 2 +3b 2 for any integers a and b. (Hint: Consider the remainder mod 3). Solution: First, note that 123456782 2 mod 3. How did we find out?
More informationExercise Set 1 Solutions Math 2020 Due: January 30, Find the truth tables of each of the following compound statements.
1. Find the truth tables of each of the following compound statements. (a) ( (p q)) (p q), p q p q (p q) q p q ( (p q)) (p q) 0 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 (b) [p ( p q)] [( (p
More informationIntroduction to Decision Sciences Lecture 10
Introduction to Decision Sciences Lecture 10 Andrew Nobel October 17, 2017 Mathematical Induction Given: Propositional function P (n) with domain N + Basis step: Show that P (1) is true Inductive step:
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 information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More informationDiscrete Mathematics & Mathematical Reasoning Predicates, Quantifiers and Proof Techniques
Discrete Mathematics & Mathematical Reasoning Predicates, Quantifiers and Proof Techniques Colin Stirling Informatics Some slides based on ones by Myrto Arapinis Colin Stirling (Informatics) Discrete Mathematics
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 informationInference and Proofs (1.6 & 1.7)
EECS 203 Spring 2016 Lecture 4 Page 1 of 9 Introductory problem: Inference and Proofs (1.6 & 1.7) As is commonly the case in mathematics, it is often best to start with some definitions. An argument for
More informationMCS-236: Graph Theory Handout #A4 San Skulrattanakulchai Gustavus Adolphus College Sep 15, Methods of Proof
MCS-36: Graph Theory Handout #A4 San Skulrattanakulchai Gustavus Adolphus College Sep 15, 010 Methods of Proof Consider a set of mathematical objects having a certain number of operations and relations
More informationMATH 135 Fall 2006 Proofs, Part IV
MATH 135 Fall 006 s, Part IV We ve spent a couple of days looking at one particular technique of proof: induction. Let s look at a few more. Direct Here we start with what we re given and proceed in a
More informationQ 1 Find the square root of 729. 6. Squares and Square Roots Q 2 Fill in the blank using the given pattern. 7 2 = 49 67 2 = 4489 667 2 = 444889 6667 2 = Q 3 Without adding find the sum of 1 + 3 + 5 + 7
More informationOn the polynomial x(x + 1)(x + 2)(x + 3)
On the polynomial x(x + 1)(x + 2)(x + 3) Warren Sinnott, Steven J Miller, Cosmin Roman February 27th, 2004 Abstract We show that x(x + 1)(x + 2)(x + 3) is never a perfect square or cube for x a positive
More informationPutnam Pigeonhole Principle October 25, 2005
Putnam Pigeonhole Principle October 5, 005 Introduction 1. If n > m pigeons are placed into m boxes, then there exists (at least) one box with at least two pigeons.. If n > m, then any function f : [n]
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 informationSolutions Manual. Selected odd-numbers problems from. Chapter 3. Proof: Introduction to Higher Mathematics. Seventh Edition
Solutions Manual Selected odd-numbers problems from Chapter 3 of Proof: Introduction to Higher Mathematics Seventh Edition Warren W. Esty and Norah C. Esty 5 4 3 2 1 2 Section 3.1. Inequalities Chapter
More informationFor all For every For each For any There exists at least one There exists There is Some
Section 1.3 Predicates and Quantifiers Assume universe of discourse is all the people who are participating in this course. Also let us assume that we know each person in the course. Consider the following
More informationHomework 3, solutions
Homework 3, solutions Problem 1. Read the proof of Proposition 1.22 (page 32) in the book. Using simialr method prove that there are infinitely many prime numbers of the form 3n 2. Solution. Note that
More informationWORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...}
WORKSHEET ON NUMBERS, MATH 215 FALL 18(WHYTE) We start our study of numbers with the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } and their subset of natural numbers: N = {1, 2, 3,...} For now we will not
More informationSection 3.1: Direct Proof and Counterexample 1
Section 3.1: Direct Proof and Counterexample 1 In this chapter, we introduce the notion of proof in mathematics. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion
More informationSection 2.1: Introduction to the Logic of Quantified Statements
Section 2.1: Introduction to the Logic of Quantified Statements In the previous chapter, we studied a branch of logic called propositional logic or propositional calculus. Loosely speaking, propositional
More informationAN INTRODUCTION TO MATHEMATICAL PROOFS NOTES FOR MATH Jimmy T. Arnold
AN INTRODUCTION TO MATHEMATICAL PROOFS NOTES FOR MATH 3034 Jimmy T. Arnold i TABLE OF CONTENTS CHAPTER 1: The Structure of Mathematical Statements.............................1 1.1. Statements..................................................................
More informationCOMP 182 Algorithmic Thinking. Proofs. Luay Nakhleh Computer Science Rice University
COMP 182 Algorithmic Thinking Proofs Luay Nakhleh Computer Science Rice University 1 Reading Material Chapter 1, Section 3, 6, 7, 8 Propositional Equivalences The compound propositions p and q are called
More informationIn Exercises 1 12, list the all of the elements of the given set. 2. The set of all positive integers whose square roots are less than or equal to 3
APPENDIX A EXERCISES In Exercises 1 12, list the all of the elements of the given set. 1. The set of all prime numbers less than 20 2. The set of all positive integers whose square roots are less than
More informationPRODUCTS THAT ARE POWERS. A mathematical vignette Ed Barbeau, University of Toronto
PRODUCTS THAT ARE POWERS. A mathematical vignette Ed Barbeau, University of Toronto This investigation was originally designed for school students to gain fluency with the factoring of positive integers
More informationChapter 3: Section 3.1: Factors & Multiples of Whole Numbers
Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers Prime Factor: a prime number that is a factor of a number. The first 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
More informationSection 3.4 Library of Functions; Piecewise-Defined Functions
Section. Library of Functions; Piecewise-Defined Functions Objective #: Building the Library of Basic Functions. Graph the following: Ex. f(x) = b; constant function Since there is no variable x in the
More informationPreparing for the CS 173 (A) Fall 2018 Midterm 1
Preparing for the CS 173 (A) Fall 2018 Midterm 1 1 Basic information Midterm 1 is scheduled from 7:15-8:30 PM. We recommend you arrive early so that you can start exactly at 7:15. Exams will be collected
More informationBC Exam Solutions Texas A&M High School Math Contest October 24, p(1) = b + 2 = 3 = b = 5.
C Exam Solutions Texas &M High School Math Contest October 4, 01 ll answers must be simplified, and If units are involved, be sure to include them. 1. p(x) = x + ax + bx + c has three roots, λ i, with
More informationMath 38: Graph Theory Spring 2004 Dartmouth College. On Writing Proofs. 1 Introduction. 2 Finding A Solution
Math 38: Graph Theory Spring 2004 Dartmouth College 1 Introduction On Writing Proofs What constitutes a well-written proof? A simple but rather vague answer is that a well-written proof is both clear and
More informationDiscrete Mathematics and Probability Theory Fall 2013 Vazirani Note 1
CS 70 Discrete Mathematics and Probability Theory Fall 013 Vazirani Note 1 Induction Induction is a basic, powerful and widely used proof technique. It is one of the most common techniques for analyzing
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 informationnot to be republished NCERT REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results
REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results Euclid s Division Lemma : Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Euclid s Division
More informationBefore you get started, make sure you ve read Chapter 1, which sets the tone for the work we will begin doing here.
Chapter 2 Mathematics and Logic Before you get started, make sure you ve read Chapter 1, which sets the tone for the work we will begin doing here. 2.1 A Taste of Number Theory In this section, we will
More informationBaltic Way 2003 Riga, November 2, 2003
altic Way 2003 Riga, November 2, 2003 Problems and solutions. Let Q + be the set of positive rational numbers. Find all functions f : Q + Q + which for all x Q + fulfil () f ( x ) = f (x) (2) ( + x ) f
More informationSample Problems for all sections of CMSC250, Midterm 1 Fall 2014
Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014 1. Translate each of the following English sentences into formal statements using the logical operators (,,,,, and ). You may also use mathematical
More informationx P(x) x P(x) CSE 311: Foundations of Computing announcements last time: quantifiers, review: logical Inference Fall 2013 Lecture 7: Proofs
CSE 311: Foundations of Computing Fall 2013 Lecture 7: Proofs announcements Reading assignment Logical inference 1.6-1.7 7 th Edition 1.5-1.7 6 th Edition Homework #2 due today last time: quantifiers,
More informationCommutative Rings and Fields
Commutative Rings and Fields 1-22-2017 Different algebraic systems are used in linear algebra. The most important are commutative rings with identity and fields. Definition. A ring is a set R with two
More informationStatements, Implication, Equivalence
Part 1: Formal Logic Statements, Implication, Equivalence Martin Licht, Ph.D. January 10, 2018 UC San Diego Department of Mathematics Math 109 A statement is either true or false. We also call true or
More informationSect 2.6 Graphs of Basic Functions
Sect. Graphs of Basic Functions Objective : Understanding Continuity. Continuity is an extremely important idea in mathematics. When we say that a function is continuous, it means that its graph has no
More informationMath Homework # 4
Math 446 - Homework # 4 1. Are the following statements true or false? (a) 3 5(mod 2) Solution: 3 5 = 2 = 2 ( 1) is divisible by 2. Hence 2 5(mod 2). (b) 11 5(mod 5) Solution: 11 ( 5) = 16 is NOT divisible
More informationProof worksheet solutions
Proof worksheet solutions These are brief, sketched solutions. Comments in blue can be ignored, but they provide further explanation and outline common misconceptions Question 1 (a) x 2 + 4x +12 = (x +
More informationBoolean Algebra and Proof. Notes. Proving Propositions. Propositional Equivalences. Notes. Notes. Notes. Notes. March 5, 2012
March 5, 2012 Webwork Homework. The handout on Logic is Chapter 4 from Mary Attenborough s book Mathematics for Electrical Engineering and Computing. Proving Propositions We combine basic propositions
More informationNumbers and their divisors
Chapter 1 Numbers and their divisors 1.1 Some number theoretic functions Theorem 1.1 (Fundamental Theorem of Arithmetic). Every positive integer > 1 is uniquely the product of distinct prime powers: n
More informationUNC Charlotte 2005 Comprehensive March 7, 2005
March 7, 2005 1 The numbers x and y satisfy 2 x = 15 and 15 y = 32 What is the value xy? (A) 3 (B) 4 (C) 5 (D) 6 (E) none of A, B, C or D 2 Suppose x, y, z, and w are real numbers satisfying x/y = 4/7,
More informationMath 13, Spring 2013, Lecture B: Midterm
Math 13, Spring 2013, Lecture B: Midterm Name Signature UCI ID # E-mail address Each numbered problem is worth 12 points, for a total of 84 points. Present your work, especially proofs, as clearly as possible.
More informationMore AMC 8 Problems. Dr. Titu Andreescu. November 9, 2013
More AMC 8 Problems Dr. Titu Andreescu November 9, 2013 Problems 1. Complete 12 + 12 6 2 3 with one set of brackets ( ) in order to obtain 12. 2. Evaluate 10000000 100000 90. 3. I am thinking of two numbers.
More informationSun Life Financial Canadian Open Mathematics Challenge Section A 4 marks each. Official Solutions
Sun Life Financial Canadian Open Mathematics Challenge 2015 Official Solutions COMC exams from other years, with or without the solutions included, are free to download online. Please visit http://comc.math.ca/2015/practice.html
More informationCounting Methods. CSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo
Counting Methods CSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 48 Need for Counting The problem of counting
More informationand LCM (a, b, c) LCM ( a, b) LCM ( b, c) LCM ( a, c)
CHAPTER 1 Points to Remember : REAL NUMBERS 1. Euclid s division lemma : Given positive integers a and b, there exists whole numbers q and r satisfying a = bq + r, 0 r < b.. Euclid s division algorithm
More informationYavapai County Math Contest College Bowl Competition. January 28, 2010
Yavapai County Math Contest College Bowl Competition January 28, 2010 Is your adrenalin engaged? What is 1 2 + 3 4? 82 Solve for x in: 2x + 7 = 1 3x. x=-6/5 (or x=-1.2) If a fair die is rolled once, what
More informationProof by contrapositive, contradiction
Proof by contrapositive, contradiction Margaret M. Fleck 9 September 2009 This lecture covers proof by contradiction and proof by contrapositive (section 1.6 of Rosen). 1 Announcements The first quiz will
More informationCS Foundations of Computing
IIT KGP Dept. of Computer Science & Engineering CS 30053 Foundations of Computing Debdeep Mukhopadhyay Pigeon Hole Principle 1 Pigeonhole Principle If n+1 or more objects (pigeons) are placed into n boxes,
More informationLecture 7 Feb 4, 14. Sections 1.7 and 1.8 Some problems from Sec 1.8
Lecture 7 Feb 4, 14 Sections 1.7 and 1.8 Some problems from Sec 1.8 Section Summary Proof by Cases Existence Proofs Constructive Nonconstructive Disproof by Counterexample Nonexistence Proofs Uniqueness
More informationMersenne and Fermat Numbers
NUMBER THEORY CHARLES LEYTEM Mersenne and Fermat Numbers CONTENTS 1. The Little Fermat theorem 2 2. Mersenne numbers 2 3. Fermat numbers 4 4. An IMO roblem 5 1 2 CHARLES LEYTEM 1. THE LITTLE FERMAT THEOREM
More informationMeritPath.com. Problems and Solutions, INMO-2011
Problems and Solutions, INMO-011 1. Let,, be points on the sides,, respectively of a triangle such that and. Prove that is equilateral. Solution 1: c ka kc b kb a Let ;. Note that +, and hence. Similarly,
More informationCS 210 Foundations of Computer Science
IIT Madras Dept. of Computer Science & Engineering CS 210 Foundations of Computer Science Debdeep Mukhopadhyay Counting-II Pigeonhole Principle If n+1 or more objects (pigeons) are placed into n boxes,
More informationWe say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:
R.4 Polynomials in one variable A monomial: an algebraic expression of the form ax n, where a is a real number, x is a variable and n is a nonnegative integer. : x,, 7 A binomial is the sum (or difference)
More informationChapter 2: The Logic of Quantified Statements. January 22, 2010
Chapter 2: The Logic of Quantified Statements January 22, 2010 Outline 1 2.1- Introduction to Predicates and Quantified Statements I 2 2.2 - Introduction to Predicates and Quantified Statements II 3 2.3
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More informationSimplifying Rational Expressions and Functions
Department of Mathematics Grossmont College October 15, 2012 Recall: The Number Types Definition The set of whole numbers, ={0, 1, 2, 3, 4,...} is the set of natural numbers unioned with zero, written
More informationSteinhardt School of Culture, Education, and Human Development Department of Teaching and Learning. Mathematical Proof and Proving (MPP)
Steinhardt School of Culture, Education, and Human Development Department of Teaching and Learning Terminology, Notations, Definitions, & Principles: Mathematical Proof and Proving (MPP) 1. A statement
More informationIntermediate Math Circles February 14, 2018 Contest Prep: Number Theory
Intermediate Math Circles February 14, 2018 Contest Prep: Number Theory Part 1: Prime Factorization A prime number is an integer greater than 1 whose only positive divisors are 1 and itself. An integer
More informationSTRATEGIES OF PROBLEM SOLVING
STRATEGIES OF PROBLEM SOLVING Second Edition Maria Nogin Department of Mathematics College of Science and Mathematics California State University, Fresno 2014 2 Chapter 1 Introduction Solving mathematical
More informationMath 55 Second Midterm Exam, Prof. Srivastava April 5, 2016, 3:40pm 5:00pm, F295 Haas Auditorium.
Math 55 Second Midterm Exam, Prof Srivastava April 5, 2016, 3:40pm 5:00pm, F295 Haas Auditorium Name: SID: Instructions: Write all answers in the provided space Please write carefully and clearly, in complete
More informationR1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member
Chapter R Review of basic concepts * R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Ex: Write the set of counting numbers
More informationProof techniques (section 2.1)
CHAPTER 1 Proof techniques (section 2.1) What we have seen so far: 1.1. Theorems and Informal proofs Argument: P 1 P n Q Syntax: how it's written Semantic: meaning in a given interpretation Valid argument:
More informationSingapore International Mathematical Olympiad 2008 Senior Team Training. Take Home Test Solutions. 15x 2 7y 2 = 9 y 2 0 (mod 3) x 0 (mod 3).
Singapore International Mathematical Olympiad 2008 Senior Team Training Take Home Test Solutions. Show that the equation 5x 2 7y 2 = 9 has no solution in integers. If the equation has a solution in integer,
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 informationLagrange s polynomial
Lagrange s polynomial Nguyen Trung Tuan November 16, 2016 Abstract In this article, I will use Lagrange polynomial to solve some problems from Mathematical Olympiads. Contents 1 Lagrange s interpolation
More informationFinals: Solutions. ECS 20 (Fall 2009) Patrice Koehl May 20, 2016
Finals: Solutions ECS 20 (Fall 2009) Patrice Koehl koehl@cs.ucdavis.edu May 20, 2016 Part I: Logic Exercise 1 On a distant island, every inhabitant is either a Knight or Knave. Knights only tell the truth.
More informationGRE. Advanced GRE Math Questions
Advanced GRE Math Questions Quantitative Arithmetic 1. What is the sum of all integers x, such that 7 < x 5? 7 7 6 6 7 1. C Quantitative Fractions and Ratios 1. The current ratio of boys to girls at a
More informationCMPSCI 250: Introduction to Computation. Lecture 11: Proof Techniques David Mix Barrington 5 March 2013
CMPSCI 250: Introduction to Computation Lecture 11: Proof Techniques David Mix Barrington 5 March 2013 Proof Techniques Review: The General Setting for Proofs Types of Proof: Direct, Contraposition, Contradiction
More informationCS 173: Induction. Madhusudan Parthasarathy University of Illinois at Urbana-Champaign. February 7, 2016
CS 173: Induction Madhusudan Parthasarathy University of Illinois at Urbana-Champaign 1 Induction February 7, 016 This chapter covers mathematical induction, and is an alternative resource to the one in
More informationChapter 3: Factors, Roots, and Powers
Chapter 3: Factors, Roots, and Powers Section 3.1 Chapter 3: Factors, Roots, and Powers Section 3.1: Factors and Multiples of Whole Numbers Terminology: Prime Numbers: Any natural number that has exactly
More informationEXAMPLES OF MORDELL S EQUATION
EXAMPLES OF MORDELL S EQUATION KEITH CONRAD 1. Introduction The equation y 2 = x 3 +k, for k Z, is called Mordell s equation 1 on account of Mordell s long interest in it throughout his life. A natural
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 3
EECS 70 Discrete Mathematics and Probability Theory Spring 014 Anant Sahai Note 3 Induction Induction is an extremely powerful tool in mathematics. It is a way of proving propositions that hold for all
More informationa 2 + b 2 = (p 2 q 2 ) 2 + 4p 2 q 2 = (p 2 + q 2 ) 2 = c 2,
5.3. Pythagorean triples Definition. A Pythagorean triple is a set (a, b, c) of three integers such that (in order) a 2 + b 2 c 2. We may as well suppose that all of a, b, c are non-zero, and positive.
More informationBasic Logic and Proof Techniques
Chapter 3 Basic Logic and Proof Techniques Now that we have introduced a number of mathematical objects to study and have a few proof techniques at our disposal, we pause to look a little more closely
More informationFoundations of Discrete Mathematics
Foundations of Discrete Mathematics Chapter 0 By Dr. Dalia M. Gil, Ph.D. Statement Statement is an ordinary English statement of fact. It has a subject, a verb, and a predicate. It can be assigned a true
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 informationSection 3.1: Characteristics of Polynomial Functions
Chapter 3: Polynomial Functions Section 3.1: Characteristics of Polynomial Functions pg 107 Polynomial Function: a function of the form f(x) = a n x n + a n 1 x n 1 +a n 2 x n 2 +...+a 2 x 2 +a 1 x+a 0
More information