Oh Yeah? Well, Prove It.


 Neal King
 1 years ago
 Views:
Transcription
1 Oh Yeah? Well, Prove It. MT 43A  Abstract Algebra Fall 009 A large part of mathematics consists of building up a theoretical framework that allows us to solve problems. This theoretical framework is built upon a set of axioms. Axioms are unproven assumptions that are the foundation of all mathematics. (Not everyone agrees on what axioms should be used, but that s a whole other story.) These basic axioms are combined using the rules of logic into more complicated statements called theorems. If the axioms are true and the logic is sound, then the theorems must also be true. New theorems are created by combining axioms and known theorems. A proof of a theorem is simply an explanation of why the theorem is true. Different people will require different amounts of explanation. It will take some practice to get used to how much explanation I expect. In this class we will not attempt to build all of mathematics from a set of axioms (an extremely difficult proposition!). Instead, we will start with some known mathematical objects (like the integers), and see how some basic definitions, axioms and known theorems can be built into a useful theoretical framework. In order to do this, we need to learn how to prove new theorems. One of the first definitions we will use is that of an even integer. I m sure you all think you know what an even integer is, but many of you will hesitate if asked whether zero is even or odd. The following definition makes this obvious. Definition: An integer is called even if it is divisible by, and called odd otherwise. In other words, the even numbers are those that can be written as k for some integer k, and the odd numbers are those that can be written as k + 1 for some integer k. It should be clear upon a moment s reflection that zero is an even number. A statement that has not been proved is called a conjecture. Conjectures can usually be written in the form If P, then Q. This can also be written as P = Q, which is read P implies Q. P consists of one or more statements called the hypotheses and Q is a statement called the conclusion. Example 1: The statement The sum of two even integers is even. can be written as the statement If a and b are even integers, then a + b is an even integer. The hypotheses are a and b are even integers and the conclusion is a + b is an even integer. Some conjectures are written in the form P if and only if Q. This is also written as P Q, or P iff Q. In this case it is said that Q is a necessary and sufficient condition for P. To prove an if and only if statement, we must prove two statements: If P, then Q (Q is necessary for P ), and If Q, then P (Q is sufficient for P ) One direction is frequently significantly easier to prove than the other.
2 Example : The square of an integer n is even if and only if n is even. P is the statement: The square of an integer n is even, and Q is the statement: n is even. To prove this statement we must prove both If n is even, then n is even. If n is even, then n is even. A proof that P = Q is a logically correct argument that shows that if the hypotheses are true, then the conclusion must also be true. The argument may use axioms, definitions and previously proved statements. Once a statement has been proved, it is called a theorem. A theorem whose main purpose is to prove other more important theorems is called a lemma. A theorem that easily follows from a more important theorem is called a corollary. There is no algorithm, or single list of instructions, for finding proofs, but here are some tips to help you get started. 1. Understand what you are trying to prove. Look up definitions of all the terms. Even though you may know what the terms mean, it is important to have the official definitions in front of you, as these will frequently give you a clue about how you might proceed.. Understand what the hypotheses are. Write down definitions of all the terms involved in the hypotheses. This is almost always the first step in a proof. 3. It is often helpful to look at examples that illustrate the conjecture. This can give you a better understanding of the problem. No collection of specific examples constitues a general proof that a conjecture is true. 4. It is sometimes useful to work backward from the conclusion. Ask yourself, What would I have to know in order to know that the conclusion is true. Then, What would I have to know in order to know that these statements are true. Keep going until you see a way to connect the hypotheses with the conclusion. 5. Once you have discovered your proof, write it up in a clear correct logical form. State the main steps from the hypotheses to the conclusion with some justification for each step. Avoid excessive detail, but provide enough detail so that your argument is convincing. There are several standard methods of proof. Here is a description of some of the most common methods of proof, with examples of each. DIRECT PROOF: Begin with the hypotheses and follow a direct line of reasoning to the conclusion. Theorem: If a and b are even integers, then a + b is an even integer.
3 Proof: If a is an even integer, then a can be written as an integer multiple of (this is the definition of an even integer). Thus, we can write a = m for some integer m. Similarly, since b is even, we can write b = n for some integer n (not necessarily the same integer as we used to write a, of course). Now look at a + b. We can write a + b as m + n = (m + n), and we see that a + b is an integer multiple of. Therefore, a + b is an even integer. It is important to mark the end of every proof. The box,, is commonly used for this. Another common way to mark the end of a proof is with the letters QED. (In fact, the box is often called a QEDbox.) QED is an abbreviation of the latin phrase Quod Erat Demonstrandum  which was to be demonstrated. COUNTEREXAMPLE: To prove that a statement is false in general, it is enough to find one specific case for which the statement is false. Such a statement is called a counterexample. Find a counterexample to the following conjecture: Conjecture: If a, b and d are integers and d divides the product ab, then d divides a or d divides b. CONTRADICTION: Assume the hypothesis is true and the conclusion is false, and show that this contradicts something known to be true. Theorem: There are infinitely many prime numbers. Proof: This theorem is unusual in that there are no hypotheses explicitly stated! We only have basic definitions and previous theorems to go on. We will prove this by contradiction, using the Fundamental Theorem of Arithmetic (Theorem 0.3 on page 6). Every integer greater than one is prime or a product of primes. In order to arrive at a contradiction, assume that there is a finite number of prime numbers. If the number of prime numbers is finite (say there are n of them), then we can list them. p 1, p,... p n Consider the (very large) number we get if we multiply all these primes together and add 1. V = p 1 p... p n + 1 Notice that p k does not divide V for any k from 1 to n (since no primes divide 1). In other words, none of the primes on the list divide V. The Fundamental Theorem of Arithmetic, however, tells us that V can be written uniquely as a product of prime numbers. In other words, some prime numbers have to divide V. But all the primes are on the list, and none of them divide V. This is a contradiction! Contradictions are sometimes denoted with the symbol.
4 This contradiction means that our initial assumption that there is a finite number of prime numbers must be false. Therefore there must be infinitely many prime numbers. Some (uptight) mathematicians don t like to use proof by contradiction. They insist that any statement proved by contradiction can be proved directly. This may or may not be true, but it is true that proof by contradiction is often a much simpler and more elegant way to prove a statement that is difficult to prove directly. Always try a direct proof first, but if you get stuck, a proof by contradiction will sometimes work. CONTRAPOSITIVE: Sometimes the contrapositive of a statement is easier to prove than the statement. The contrapositive of the statement is the statement If P, then Q If not Q, then not P. A statement and its contrapositive are logically equivalent. If one is true the other is necessarily true, and vice versa. For example, the statement If I roll a 7, then I win. is logically equivalent to the statement If I don t win, then I don t roll a 7. Either they are both true or both false. Theorem: If n is even, then n is even. Proof: We prove the contrapositive: If n is odd, then n is odd. If n is odd, then n can be written as k + 1 for some integer k. Then n = (k + 1) = 4k + 4k + 1 = (k + k) + 1. Since n is of the form x + 1 for the integer x = k + k, n is an odd integer. INDUCTION: Induction is a technique for proving that a statement is true for all positive integers. Proof by induction is like knocking over dominos  prove that you can knock over the first domino, then prove that if one domino falls then it knocks over the next domino. If you can prove both those things, then all the dominos will fall. More formally, if you wish use induction to prove a statement S n true for all integers n n 0, you must Prove a base case: prove the statement S n is true for n = n 0.
5 Prove the inductive step: assume the statement is true for case n, and use this to prove the statement is true for case n + 1. In other words prove that S n = S n+1 S n is called the inductive hypothesis. You assume S n is true, and use it to show S n+1 must also be true.
6 Theorem: The sum of the the first n positive integers is n(n+1). In other words, we wish to prove the statement S n : n = for all n 1. (Note that S n is simply a name for the above equation.) Proof: (By induction) Base case: S 1 is the statement 1 = 1(1+1). This is true. The inductive step: Assume S n is true. That is, We wish to use this to prove S n+1 : n = n + (n + 1) = (n + 1)(n + ) To see this, we add (n + 1) to both sides of equation S n and get n + (n + 1) = + (n + 1) Getting a common denominator and simplifying the right side produces We see that if S n is true then + (n + 1) = = n + 3n + (n + 1)(n + ) = + (n + 1) (n + 1)(n + ) n + (n + 1) = is also true, but this is statement S n+1. So S n implies S n+1. Therefore, since we have shown a base case and proved the inductive step, the statement S n is true for all n 1. In other words, the sum of the the first n positive integers is n(n+1). There you have it  a quick and dirty introduction to methods of proof. I hope you find it helpful. Of course the only way to really learn how to write proofs is to grit your teeth, try to write some, get stuck, get frustrated (but don t stay frustrated!), ask for help, make mistakes, learn from them, and repeat all of these steps as often as necessary. Many of you will need to use office hours more than you are used to. I ll be able to answer some questions in class, but the best help will come one on one, or in small groups. Don t be afraid to ask for help. As long as you keep trying, I ll be willing to help as best as I can.
7 EXERCISES: 1. Use the following outline to prove by induction that for all integers greater than or equal to 1, the sum of the first n odd integers is n. Since the n th odd integer is n 1, the symbolic form for this statement is S n : (n 1) = n (a) Check that the statement is true in the case n = 1. (b) Write the inductive hypothesis, S n. This is what you assume to be true. (c) Write the expression S n+1. This is what you hope to show is true. (d) What simple algebraic operation would you have to do to the left side of S n to get the left side of S n+1? Of course, if you do this to the left side of equation S n, you must do it to the right side of S n, as well. Do it! (e) Does the left side of the expression from part (d) equal the left side of S n+1? (It better!) (f) Do some simple algebra to see if the right side of part (d) equals the right side of S n+1. If it does, you have shown that S n = S n+1, and thus completed the proof by induction.. Prove that if a, b and c are integers and a divides b and a divides c, then a divides b + c and a divides b c. 3. Prove that if a, b and c are integers and a divides b and b divides c, then a divides c. 4. Prove that the product of two even integers is divisible by Prove that the product of two odd integers has remainder 1 when divided by Prove that the square of any integer is congruent to 0 or 1 mod Prove that if a, b and c are integers such that a + b = c, then either a or b is even. (This is an inclusive or  either a or b is even, or possibly both are even.) 8. Use induction to prove that n = n+1 1 for n The Fibonacci numbers are defined recursively by the formulas F 1 = 1 F = 1 F n = F n 1 + F n calculate the first ten Fibonacci numbers. Use induction to prove that for all n 1. F 1 + F + F F n = F n F n+1
Basic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
More informationInduction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition
Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More informationWRITING PROOFS. Christopher Heil Georgia Institute of Technology
WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationThe Foundations: Logic and Proofs. Chapter 1, Part III: Proofs
The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments
More informationSECTION 102 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
More information2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.
2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then
More informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
More informationSection 62 Mathematical Induction
6 Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that
More informationIntroduction. Appendix D Mathematical Induction D1
Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationIf n is odd, then 3n + 7 is even.
Proof: Proof: We suppose... that 3n + 7 is even. that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. that 3n + 7 is even. Since n is odd, there exists an integer k so that
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More informationCHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a
More informationDiscrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University
Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called
More informationAppendix F: Mathematical Induction
Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More informationProof of Infinite Number of Fibonacci Primes. Stephen Marshall. 22 May Abstract
Proof of Infinite Number of Fibonacci Primes Stephen Marshall 22 May 2014 Abstract This paper presents a complete and exhaustive proof of that an infinite number of Fibonacci Primes exist. The approach
More informationThe Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.
The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,
More informationMathematical Induction. Mary Barnes Sue Gordon
Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by
More informationMathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
More informationSolutions for Practice problems on proofs
Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
More informationWe now explore a third method of proof: proof by contradiction.
CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement
More informationMathematical Induction. Lecture 1011
Mathematical Induction Lecture 1011 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
More informationMODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.
MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on
More informationCongruent Numbers, the Rank of Elliptic Curves and the Birch and SwinnertonDyer Conjecture. Brad Groff
Congruent Numbers, the Rank of Elliptic Curves and the Birch and SwinnertonDyer Conjecture Brad Groff Contents 1 Congruent Numbers... 1.1 Basic Facts............................... and Elliptic Curves.1
More informationFractions and Decimals
Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationHomework 5 Solutions
Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which
More informationCS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On wellordering and induction: (a) Prove the induction principle from the wellordering principle. (b) Prove the wellordering
More information1.2. Successive Differences
1. An Application of Inductive Reasoning: Number Patterns In the previous section we introduced inductive reasoning, and we showed how it can be applied in predicting what comes next in a list of numbers
More informationMath Workshop October 2010 Fractions and Repeating Decimals
Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,
More informationAPPLICATIONS OF THE ORDER FUNCTION
APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and
More informationApplications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
More informationCHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
More informationPythagorean Triples. Chapter 2. a 2 + b 2 = c 2
Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More informationElementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
More informationAn Innocent Investigation
An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number
More informationPythagorean Triples Pythagorean triple similar primitive
Pythagorean Triples One of the most farreaching problems to appear in Diophantus Arithmetica was his Problem II8: To divide a given square into two squares. Namely, find integers x, y, z, so that x 2
More informationMATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
More informationCS 441 Discrete Mathematics for CS Lecture 5. Predicate logic. CS 441 Discrete mathematics for CS. Negation of quantifiers
CS 441 Discrete Mathematics for CS Lecture 5 Predicate logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Negation of quantifiers English statement: Nothing is perfect. Translation: x Perfect(x)
More informationNumber Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may
Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition
More informationIntroduction to Proofs
Chapter 1 Introduction to Proofs 1.1 Preview of Proof This section previews many of the key ideas of proof and cites [in brackets] the sections where they are discussed thoroughly. All of these ideas are
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More informationStyle Guide For Writing Mathematical Proofs
Style Guide For Writing Mathematical Proofs Adapted by Lindsey Shorser from materials by Adrian Butscher and Charles Shepherd A solution to a math problem is an argument. Therefore, it should be phrased
More informationIntegers and division
CS 441 Discrete Mathematics for CS Lecture 12 Integers and division Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Symmetric matrix Definition: A square matrix A is called symmetric if A = A T.
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More informationChapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications
More information3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.
SOLUTIONS TO HOMEWORK 2  MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid
More informationdef: An axiom is a statement that is assumed to be true, or in the case of a mathematical system, is used to specify the system.
Section 1.5 Methods of Proof 1.5.1 1.5 METHODS OF PROOF Some forms of argument ( valid ) never lead from correct statements to an incorrect. Some other forms of argument ( fallacies ) can lead from true
More informationComputing exponents modulo a number: Repeated squaring
Computing exponents modulo a number: Repeated squaring How do you compute (1415) 13 mod 2537 = 2182 using just a calculator? Or how do you check that 2 340 mod 341 = 1? You can do this using the method
More informationSYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me
SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBINCAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According
More informationPRINCIPLES OF PROBLEM SOLVING
PRINCIPLES OF PROBLEM SOLVING There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problemsolving process and to
More information13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcsftl 2010/9/8 0:40 page 379 #385
mcsftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite
More informationAn Interesting Way to Combine Numbers
An Interesting Way to Combine Numbers Joshua Zucker and Tom Davis November 28, 2007 Abstract This exercise can be used for middle school students and older. The original problem seems almost impossibly
More informationInteger Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions
Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationFoundations of Geometry 1: Points, Lines, Segments, Angles
Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationToday s Topics. Primes & Greatest Common Divisors
Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime
More informationMATH 289 PROBLEM SET 4: NUMBER THEORY
MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides
More information3.4 Complex Zeros and the Fundamental Theorem of Algebra
86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and
More informationLecture 13  Basic Number Theory.
Lecture 13  Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are nonnegative integers. We say that A divides B, denoted
More informationTheory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras
Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationFactoring Polynomials
Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationMath 3000 Section 003 Intro to Abstract Math Homework 2
Math 3000 Section 003 Intro to Abstract Math Homework 2 Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Solutions (February 13, 2012) Please note that these
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationHandout NUMBER THEORY
Handout of NUMBER THEORY by Kus Prihantoso Krisnawan MATHEMATICS DEPARTMENT FACULTY OF MATHEMATICS AND NATURAL SCIENCES YOGYAKARTA STATE UNIVERSITY 2012 Contents Contents i 1 Some Preliminary Considerations
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
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 informationCHAPTER 2. Logic. 1. Logic Definitions. Notation: Variables are used to represent propositions. The most common variables used are p, q, and r.
CHAPTER 2 Logic 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A proposition is a declarative sentence that is either true (denoted either T or 1) or false (denoted either F or 0). Notation:
More informationPrimes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov
Lee 1 Primes in Sequences By: Jae Young Lee Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 2 Jae Young Lee MA341 Number Theory PRIMES IN SEQUENCES
More informationDay 2: Logic and Proof
Day 2: Logic and Proof George E. Hrabovsky MAST Introduction This is the second installment of the series. Here I intend to present the ideas and methods of proof. Logic and proof To begin with, I will
More informationMath 313 Lecture #10 2.2: The Inverse of a Matrix
Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationWorksheet on induction Calculus I Fall 2006 First, let us explain the use of for summation. The notation
Worksheet on induction MA113 Calculus I Fall 2006 First, let us explain the use of for summation. The notation f(k) means to evaluate the function f(k) at k = 1, 2,..., n and add up the results. In other
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationSUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by
SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples
More information12 Greatest Common Divisors. The Euclidean Algorithm
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to
More informationContinued Fractions. Darren C. Collins
Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history
More informationIntroduction to Diophantine Equations
Introduction to Diophantine Equations Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles September, 2006 Abstract In this article we will only touch on a few tiny parts of the field
More informationElementary Algebra. Section 0.4 Factors
Section 0.4 Contents: Definitions: Multiplication Primes and Composites Rules of Composite Prime Factorization Answers Focus Exercises THE MULTIPLICATION TABLE x 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5
More informationNim Games. There are initially n stones on the table.
Nim Games 1 One Pile Nim Games Consider the following twoperson game. There are initially n stones on the table. During a move a player can remove either one, two, or three stones. If a player cannot
More information6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More information