# Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014

Save this PDF as:

Size: px
Start display at page:

Download "Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014"

## Transcription

1 Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring : 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the product of two consecutive integers. The results of the preview activities verify that this product is even by examining the two possibilities: m is odd, or m is even. In both cases, the product of m with m + 1 is divisible by We prove the following proposition. Proposition. Suppose that u is an odd integer; then the equation x 2 + x u = 0 has no integer solutions. Proof. Suppose for contradiction that there exists a integer solution m of the equation x 2 + x u = 0, where u is an odd integer. We will examine the two cases: m is odd and m is even. If m is odd, then there exists an integer k such that m = 2k + 1. Since m is a root of our equation, it must be that This implies that (2k + 1) 2 + (2k + 1) u = 0. 4k 2 + 4k k + 1 u = 0. This is clearly not possible, for 4k k + 2 is even. Now suppose that m is even. If so, then there exists an integer k such that m = 2k; then 4k 2 + 2k u = 0. This is also impossible, for 4k 2 + 2k is even. 3. We prove the following. Proposition. If n is an odd integer, the n = 4k + 1 for some integer k or n = 4k + 3 for some integer k.

2 Proof. Suppose that n is odd; then there is an integer j such that n = 2j + 1. There are two possibilities for j. Either j is even, or j is odd. If j is even, then there is an integer k such that j = 2k. Thus n = 4k + 1. On the other hand, if j is odd, there is an integer k such that j = 2k + 1. This means that n = 2(2k + 1) + 1. This is equivalent to n = 4k (a) We prove the following. Proposition. If m and n are consecutive integers, then 4 divides m 2 + n 2 1. Proof. Suppose that m and n are consecutive integers; we can assume that n = m + 1 in this situation. Then m 2 +n 2 1 = m 2 +(m+1) 2 1 = m 2 +m 2 +2m+1 1 = 2m 2 +2m = 2m(m+1). Now there are two cases: either m is even, or m is odd. If m is even, then 4 will divide 2m, and therefore 2m(m + 1). If m is odd, then m + 1 is even. 2 will divide 2m, and 2 will also divide m + 1. It follows that 4 will divide 2m(m + 1). On the other hand, the converse is untrue. Let m = 1 and let n = 4. (b) We prove the following. Proposition. For all integers m, n, if m and n are both even or if m and n are both odd, then 4 divides m 2 n 2. Proof. First suppose that both m and n are even. If so, then there exist integers j and k such that m = 2j and n = 2j. So m 2 n 2 = 4k 2 4j 2 = 4(k 2 j 2 ). This implies that 4 divides m 2 n 2. Now suppose that m and n are both odd. If so, then there exist integers k and j such that m = 2k + 1 and n = 2j + 1. So m 2 n 2 = 4k 2 + 4k + 1 4j 2 4j 1 = 4(k 2 + k j 2 j) This means that 4 divides m 2 n 2 in this case also. Proposition. For all integers m, n, if 4 divides m 2 n 2, then either both m and n are even, or m and n are odd.

3 7. We prove Proof. Consider the contrapositive. Suppose that m is odd and n is even. If so, then there are two integers j and k such that m = 2j + 1 and n = 2k. So m 2 n 2 = (2j + 1) 2 4k 2 = 4j 2 + 4j + 1 4k 2. This is an odd number, so is not divisible by 4. Suppose that m is even and n is odd. If so, then there are two integers j and k such that m = 2j and n = 2k + 1. So m 2 n 2 = 4j 2 4k 2 4k 1. This number is also odd, so it is not divisible by 4. Proposition. If n is an odd integer, then 8 n 2 1. Proof. If n is odd, then there is an integer k such that n = 2k + 1. Then n 2 1 = (2k + 1) 2 1 = 4k 2 + 4k = 4(k 2 + k). By a result proved in class (and appearing elsewhere in this homework assignment), k 2 + k must be even, i.e., k 2 + k = 2j for some integer j. Thus n 2 1 = 4(k 2 + k) = 4(2j) = 8j, so 8 n (b) We prove the following. Proposition. For all real numbers x and y, xy = x y. Proof. There are several cases to consider. Clearly, if either x or y is equal to 0, then xy = x y. The equation is clearly true if x > 0 and y > 0, for xy = xy = x y in this case. Suppose that x > 0 and y < 0. Then, xy = xy. On ther other hand, x = x and y = y, so x y = xy as well. Similarly, the equation is true if x < 0 and y > 0. Finally, consider what happens when x < 0 and y < 0. In this case, the product xy > 0, so xy = xy. On the other hand, x = x and y = y. This implies that x y = xy as well. 13. (a) This proof is incorrect, but the statement is true. It correctly deduces that an 3 + 2bn = 3, but then it incorrectly factors the left side as n(an 2 + b). Also, since n and an 2 + b are both integers, n > 0, and 3 is a prime number, it does follow (as in the proof) that one of n, an 2 + b has to equal 3 and the other has to equal 1. This proof only considers the case where n = 3 and an 2 + b = 1. The n = 1 needs to be considered, and doing so is analogous to the n = 3 case. The proof should proceed as follows. Rewrite an 3 + 2bn = 3 as n(an 2 + 2b) = 3. Since n and an 2 + 2b are both integers whose product is 3, and since n is positive an 2 + 2b must be a positive. Since 3 is prime, there are only two possibilities: n = 3 or n = 1. If n = 3, then we have a(3) 3 + 2b(3) = 3,

4 and dividing both sides by 3, we obtain If n = 1, then we have This completes the proof. 9a + 2b = 1. a + 2b = 3. (b) This proof is incorrect, since it assumes the negation of the hypothesis of the statement. A correct proof by contradiction assumes the negation of the conclusion. See (a) for a correct proof. 5.1: 1. (a) A = B. It is evident that A B. If x B, then x = 3, x = 3, x = 2, or x = 2. So, B A. (b) Yes. (c) No, the set C is not equal to the set D. The set C is empty. The set D contains many, many real numbers. (d) Yes; the empty set is a subset of any set. (e) No, A contains negative numbers, e.g. 3, but D doesn t. 3. The following statements are true. A B, A B, A B. 5 C. A C, A C, A C. {1, 2} A, {1, 2} = A. 4 B. card(a) = card(d). A P(A). A, A, = A. {5} C, {5} C, {5} = C. {1, 2} B, {1, 2} B, {1, 2} = B. {3, 2, 1} D, {3, 2, 1} D, {3, 2, 1} D. D, D. card(a) < card(b), card(a) card(b). A P(B). 5. (a) It is not the case that {a, b} {a, c, d, e}. This is because there is an element b {a, b} which is not an element of {a, c, d, e}. (b) This is true. The set {x Z x 2 < 5} is the set { 2, 1, 0, 1, 2}. The even members of this set are { 2, 0, 2}. (c) This is true. The empty set is a subset of any set. (d) This is not true. The set {a} is a subset of {a, b}, and so {a} is an element of the power set of A. 6. (a) x A B if and only if x A or x B. (b) x A B if and only if x A and x B. (c) x A B if and only if x A c or x B. 8. (a) A B = {7, 9, 11, 13,...}. (b) A B = {1, 3, 5, 7, 8, 9, 10, 11,...}. (c) (A B) c = {2, 4, 6}. (d) A c B c = {2, 4, 6}. (e) (A B) C = {3, 9, 12, 15, 18, 21,... }.

5 (f) (A C) (B C) = {3, 9, 12, 15, 18, 21,... }. (g) B D = { }. (h) (B D) c = {1, 2, 3,...}. (i) A D = {7, 9, 11, 13, 15,...}. (j) B D = {1, 3, 5, 7, 9,...}. (k) (A D) (B D) = {1, 3, 5, 7, 9,...}. (l) (A B) D = {1, 3, 5, 7, 9,...}. 9. (a) For all x U, if x P Q then x R S. (b) There is an x such that x P Q and x R S. (c) For all x U, if x R S, then x P Q. 11. Picture rectangles with lots of circles inside them!

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### Handout #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

### Mathematical Induction

Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

### The set consisting of all natural numbers that are in A and are in B is the set f1; 3; 5g;

Chapter 5 Set Theory 5.1 Sets and Operations on Sets Preview Activity 1 (Set Operations) Before beginning this section, it would be a good idea to review sets and set notation, including the roster method

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

Section 5.1 Climbing an Infinite Ladder Suppose we have an infinite ladder and the following capabilities: 1. We can reach the first rung of the ladder. 2. If we can reach a particular rung of the ladder,

### Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction.

Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction. February 21, 2006 1 Proof by Induction Definition 1.1. A subset S of the natural numbers is said to be inductive if n S we have

### Solutions 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

### SOLUTIONS FOR PROBLEM SET 2

SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such

### 3. 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)

### Logic, Sets, and Proofs

Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical

### 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 Mathematical Induction. October 27, 2016

More Mathematical Induction October 7, 016 In these slides... Review of ordinary induction. Remark about exponential and polynomial growth. Example a second proof that P(A) = A. Strong induction. Least

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### We 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

### Quantifiers are used to describe variables in statements. - The universal quantifier means for all. - The existential quantifier means there exists.

11 Quantifiers are used to describe variables in statements. - The universal quantifier means for all. - The existential quantifier means there exists. The phrases, for all x in R if x is an arbitrary

### Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

### MATH 55: HOMEWORK #2 SOLUTIONS

MATH 55: HOMEWORK # SOLUTIONS ERIC PETERSON * 1. SECTION 1.5: NESTED QUANTIFIERS 1.1. Problem 1.5.8. Determine the truth value of each of these statements if the domain of each variable consists of all

### 2. 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

### MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

### Mathematical induction & Recursion

CS 441 Discrete Mathematics for CS Lecture 15 Mathematical induction & Recursion Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Proofs Basic proof methods: Direct, Indirect, Contradiction, By Cases,

### Finish Set Theory Nested Quantifiers

Finish Set Theory Nested Quantifiers Margaret M. Fleck 18 February 2009 This lecture does a final couple examples of set theory proofs. It then fills in material on quantifiers, especially nested ones,

### SETS. Chapter Overview

Chapter 1 SETS 1.1 Overview This chapter deals with the concept of a set, operations on sets.concept of sets will be useful in studying the relations and functions. 1.1.1 Set and their representations

### Egyptian fraction representations of 1 with odd denominators

Egyptian fraction representations of with odd denominators P. Shiu Department of Mathematical Sciences Loughborough University Leicestershire LE 3TU United Kingdon Email: P.Shiu@lboro.ac.uk Abstract The

### MATHEMATICAL 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

### Math 2602 Finite and Linear Math Fall 14. Homework 9: Core solutions

Math 2602 Finite and Linear Math Fall 14 Homework 9: Core solutions Section 8.2 on page 264 problems 13b, 27a-27b. Section 8.3 on page 275 problems 1b, 8, 10a-10b, 14. Section 8.4 on page 279 problems

### Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.

M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider

### Students in their first advanced mathematics classes are often surprised

CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### DISCRETE MATHEMATICS W W L CHEN

DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### Solutions Manual for How to Read and Do Proofs

Solutions Manual for How to Read and Do Proofs An Introduction to Mathematical Thought Processes Sixth Edition Daniel Solow Department of Operations Weatherhead School of Management Case Western Reserve

### Discrete 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

### CHAPTER 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

### Mathematical Induction

MCS-236: Graph Theory Handout #A5 San Skulrattanakulchai Gustavus Adolphus College Sep 15, 2010 Mathematical Induction The following three principles governing N are equivalent. Ordinary Induction Principle.

### Introduction. 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

### CS 173: Discrete Structures, Fall 2010 Homework 6 Solutions

CS 173: Discrete Structures, Fall 010 Homework 6 Solutions This homework was worth a total of 5 points. 1. Recursive definition [13 points] Give a simple closed-form definition for each of the following

### Elementary 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

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### WRITING 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

### Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.

Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the

### Math Circle Beginners Group October 18, 2015

Math Circle Beginners Group October 18, 2015 Warm-up problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd

### APPLICATIONS 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

### Direct Proofs. CS 19: Discrete Mathematics. Direct Proof: Example. Indirect Proof: Example. Proofs by Contradiction and by Mathematical Induction

Direct Proofs CS 19: Discrete Mathematics Amit Chakrabarti Proofs by Contradiction and by Mathematical Induction At this point, we have seen a few examples of mathematical proofs. These have the following

### Section 3 Sequences and Limits

Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.

### Foundations of Computing Discrete Mathematics Solutions to exercises for week 2

Foundations of Computing Discrete Mathematics Solutions to exercises for week 2 Agata Murawska (agmu@itu.dk) September 16, 2013 Note. The solutions presented here are usually one of many possiblities.

### Proofs are short works of prose and need to be written in complete sentences, with mathematical symbols used where appropriate.

Advice for homework: Proofs are short works of prose and need to be written in complete sentences, with mathematical symbols used where appropriate. Even if a problem is a simple exercise that doesn t

### MATHEMATICAL INDUCTION

MATHEMATICAL INDUCTION MATH 5A SECTION HANDOUT BY GERARDO CON DIAZ Imagine a bunch of dominoes on a table. They are set up in a straight line, and you are about to push the first piece to set off the chain

### CS 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)

### LESSON 1 PRIME NUMBERS AND FACTORISATION

LESSON 1 PRIME NUMBERS AND FACTORISATION 1.1 FACTORS: The natural numbers are the numbers 1,, 3, 4,. The integers are the naturals numbers together with 0 and the negative integers. That is the integers

### Section 6-2 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

### Mathematical induction. Niloufar Shafiei

Mathematical induction Niloufar Shafiei Mathematical induction Mathematical induction is an extremely important proof technique. Mathematical induction can be used to prove results about complexity of

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### If 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

### Sample Induction Proofs

Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### Tons of Free Math Worksheets at:

Topic : Equations of Circles - Worksheet 1 form which has a center at (-6, 4) and a radius of 5. x 2-8x+y 2-8y-12=0 circle with a center at (4,-4) and passes through the point (-7, 3). center located at

### SECTION 10-2 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

### 5-4 Prime and Composite Numbers

5-4 Prime and Composite Numbers Prime and Composite Numbers Prime Factorization Number of Divisorss Determining if a Number is Prime More About Primes Prime and Composite Numbers Students should recognizee

### Math 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

### Clicker Question. Theorems/Proofs and Computational Problems/Algorithms MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES

MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES Tuesday, 1/21/14 General course Information Sets Reading: [J] 1.1 Optional: [H] 1.1-1.7 Exercises: Do before next class; not to hand in [J] pp. 12-14:

### 3.3 MATHEMATICAL INDUCTION

Section 3.3 Mathematical Induction 3.3.1 3.3 MATHEMATICAL INDUCTION From modus ponens: p p q q basis assertion conditional assertion conclusion we can easily derive double modus ponens : p 0 p 0 p 1 p

### Cartesian 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

### This chapter is all about cardinality of sets. At first this looks like a

CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### An 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

### Revised 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.

### Appendix 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

### Mathematical Induction. Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition)

Mathematical Induction Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition) Mathmatical Induction Mathmatical induction can be used to prove statements that assert that P(n) is true for all positive

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### 2 The Euclidean algorithm

2 The Euclidean algorithm Do you understand the number 5? 6? 7? At some point our level of comfort with individual numbers goes down as the numbers get large For some it may be at 43, for others, 4 In

### 3. Recurrence Recursive Definitions. To construct a recursively defined function:

3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a

### Factoring and Applications

Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

### Introduction to mathematical arguments

Introduction to mathematical arguments (background handout for courses requiring proofs) by Michael Hutchings A mathematical proof is an argument which convinces other people that something is true. Math

### APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS

APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic

### Intermediate Math Circles March 7, 2012 Linear Diophantine Equations II

Intermediate Math Circles March 7, 2012 Linear Diophantine Equations II Last week: How to find one solution to a linear Diophantine equation This week: How to find all solutions to a linear Diophantine

### Algebra for Digital Communication

EPFL - Section de Mathématiques Algebra for Digital Communication Fall semester 2008 Solutions for exercise sheet 1 Exercise 1. i) We will do a proof by contradiction. Suppose 2 a 2 but 2 a. We will obtain

### Math 317 HW #7 Solutions

Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

### Mathematical Induction

Mathematical Induction MAT30 Discrete Mathematics Fall 016 MAT30 (Discrete Math) Mathematical Induction Fall 016 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)

### Practice Problems for First Test

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.-

### PRINCIPLE OF MATHEMATICAL INDUCTION

Chapter 4 PRINCIPLE OF MATHEMATICAL INDUCTION Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called induction Newton was indebted to it for his theorem

### Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

### WUCT121. Discrete Mathematics. Logic

WUCT121 Discrete Mathematics Logic 1. Logic 2. Predicate Logic 3. Proofs 4. Set Theory 5. Relations and Functions WUCT121 Logic 1 Section 1. Logic 1.1. Introduction. In developing a mathematical theory,

### Factoring Patterns in the Gaussian Plane

Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood

### Number Theory Homework.

Number Theory Homework. 1. Pythagorean triples and rational points on quadratics and cubics. 1.1. Pythagorean triples. Recall the Pythagorean theorem which is that in a right triangle with legs of length

### WHAT 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

### Mathematical Induction

Mathematical Induction William Cherry February 2011 These notes provide some additional examples to supplement the section of the text on mathematical induction. Inequalities. It happens that often in

### Geometry - Chapter 2 Review

Name: Class: Date: Geometry - Chapter 2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine if the conjecture is valid by the Law of Syllogism.

### Applications 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

### Chapter I Logic and Proofs

MATH 1130 1 Discrete Structures Chapter I Logic and Proofs Propositions A proposition is a statement that is either true (T) or false (F), but or both. s Propositions: 1. I am a man.. I am taller than

### Full and Complete Binary Trees

Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

### SYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me

SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBIN-CAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines

### p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)

.9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other

### Discrete Mathematics, Chapter 5: Induction and Recursion

Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Well-founded

### CHAPTER 1. Logic, Proofs Propositions

CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: Paris is in France (true), London

### THE PROOF IS IN THE PICTURE

THE PROOF IS IN THE PICTURE (an introduction to proofs without words) LAMC INTERMEDIATE GROUP - 11/24/13 WARM UP (PAPER FOLDING) Theorem: The square root of 2 is irrational. We prove this via the mathematical