2.4 Mathematical Induction
|
|
- Marsha Gilmore
- 7 years ago
- Views:
Transcription
1 2.4 Mathematical Induction
2 What Is (Weak) Induction? The Principle of Mathematical Induction works like this:
3 What Is (Weak) Induction? The Principle of Mathematical Induction works like this: We want to show some statement P(n) is true for all n n 0.
4 What Is (Weak) Induction? The Principle of Mathematical Induction works like this: We want to show some statement P(n) is true for all n n 0. We show some base case P(n 0 ) is true.
5 What Is (Weak) Induction? The Principle of Mathematical Induction works like this: We want to show some statement P(n) is true for all n n 0. We show some base case P(n 0 ) is true. We assume P(k) is true for some k n 0.
6 What Is (Weak) Induction? The Principle of Mathematical Induction works like this: We want to show some statement P(n) is true for all n n 0. We show some base case P(n 0 ) is true. We assume P(k) is true for some k n 0. We show P(k + 1) is true.
7 What Is (Weak) Induction? The Principle of Mathematical Induction works like this: We want to show some statement P(n) is true for all n n 0. We show some base case P(n 0 ) is true. We assume P(k) is true for some k n 0. We show P(k + 1) is true. So, basically, we are trying to show P(k) P(k + 1) is a tautology for any choice k n 0.
8 The Idea Behind Induction How to think of induction...
9 The Idea Behind Induction How to think of induction... If a baby can take the first step...
10 The Idea Behind Induction How to think of induction... If a baby can take the first step... If you can step on the first rung of a ladder...
11 When To Use (Weak) Induction We use mathematical induction when... we have a series and we are trying to prove the general formula.
12 When To Use (Weak) Induction We use mathematical induction when... we have a series and we are trying to prove the general formula. we are trying to prove a counting problem.
13 When To Use (Weak) Induction We use mathematical induction when... we have a series and we are trying to prove the general formula. we are trying to prove a counting problem. there seems to be a pattern that seems to change in a fixed manner.
14 First (Famous) Example Example Prove that if n is a positive integer, then n = n i = n(n + 1) 2
15 First (Famous) Example Example Prove that if n is a positive integer, then n = n i = n(n + 1) 2 Let P(n) be the proposition that the sum of the first n positive integers is n(n+1) 2. We will show this is true by mathematical induction.
16 First (Famous) Example Example Prove that if n is a positive integer, then n = n i = n(n + 1) 2 Let P(n) be the proposition that the sum of the first n positive integers is n(n+1) 2. We will show this is true by mathematical induction. Base Case:
17 First (Famous) Example Example Prove that if n is a positive integer, then n = n i = n(n + 1) 2 Let P(n) be the proposition that the sum of the first n positive integers is n(n+1) 2. We will show this is true by mathematical induction. Base Case: P(1)
18 First (Famous) Example Example Prove that if n is a positive integer, then n = n i = n(n + 1) 2 Let P(n) be the proposition that the sum of the first n positive integers is n(n+1) 2. We will show this is true by mathematical induction. Base Case: P(1) P(1) is true because 1 = 1(1+1) 2.
19 First (Famous) Example Example Prove that if n is a positive integer, then n = n i = n(n + 1) 2 Let P(n) be the proposition that the sum of the first n positive integers is n(n+1) 2. We will show this is true by mathematical induction. Base Case: P(1) P(1) is true because 1 = 1(1+1) 2.
20 First (Famous) Example Inductive Step: We assume that P(k) holds for any positive integer k. That is, we assume k = k i = k(k + 1) 2
21 First (Famous) Example Inductive Step: We assume that P(k) holds for any positive integer k. That is, we assume k = We want to show P(k + 1) is true. k i = k(k + 1) 2
22 First (Famous) Example Inductive Step: We assume that P(k) holds for any positive integer k. That is, we assume k = We want to show P(k + 1) is true. k i = k(k + 1) 2 Aside: P(k + 1) is the same thing as the formula we have but we replace k with k + 1. So our goal is to arrive at k+1 = (k + 1)(k + 2) 2
23 Consider k+1 i = k i + (k + 1)
24 Consider k+1 i = = k i + (k + 1) k(k + 1) 2 + k + 1
25 Consider k+1 i = = k i + (k + 1) k(k + 1) = + k k(k + 1) 2(k + 1) + 2 2
26 Consider k+1 i = = k i + (k + 1) k(k + 1) = + k k(k + 1) 2(k + 1) = k2 + 3k + 2 2
27 Consider k+1 i = = k i + (k + 1) k(k + 1) = + k k(k + 1) 2(k + 1) = k2 + 3k (k + 1)(k + 2) = 2
28 Consider k+1 i = = k i + (k + 1) k(k + 1) = + k k(k + 1) 2(k + 1) = k2 + 3k (k + 1)(k + 2) = 2 This last equation shows that P(k + 1) is true under the assumption that P(k) is true. So, by mathematical induction, we have shown that that n n(n + 1) i = n Z + 2
29 Union/Intersection Example Example Let A 1, A 2,..., A n be any n sets. Show that ( n ) A i = n A i
30 Union/Intersection Example Example Let A 1, A 2,..., A n be any n sets. Show that Note: Does this look familiar? ( n ) A i = n A i
31 Union/Intersection Example Example Let A 1, A 2,..., A n be any n sets. Show that ( n ) A i = n Note: Does this look familiar? This is just an extension of DeMorgan s law. A i
32 Union/Intersection Example Example Let A 1, A 2,..., A n be any n sets. Show that ( n ) A i = n Note: Does this look familiar? This is just an extension of DeMorgan s law. A i Let P(n) be the predicate that the equality holds for any n sets. We will prove by mathematical induction that for all n 1, P(n) is true.
33 Union/Intersection Example Base Case:
34 Union/Intersection Example Base Case: P(1):
35 Union/Intersection Example Base Case: P(1): Certainly A 1 = A 1 is true.
36 Union/Intersection Example Base Case: P(1): Certainly A 1 = A 1 is true. Inductive Step: We assume P(k) ( k ) A i = k is true. We want to show P(k + 1) is true. A i
37 Union/Intersection Example Consider ( k+1 ) A i = A 1 A 2... A k A k+1
38 Union/Intersection Example Consider ( k+1 A i ) = A 1 A 2... A k A k+1 = (A 1 A 2... A k ) A k+1
39 Union/Intersection Example Consider ( k+1 A i ) = A 1 A 2... A k A k+1 = (A 1 A 2... A k ) A k+1 = (A 1 A 2... A k ) A k+1
40 Union/Intersection Example Consider ( k+1 A i ) = A 1 A 2... A k A k+1 = (A 1 A 2... A k ) A k+1 = (A 1 A 2... A k A k+1 ( k ) = A i A k+1
41 Union/Intersection Example Consider ( k+1 A i ) = A 1 A 2... A k A k+1 = (A 1 A 2... A k ) A k+1 = (A 1 A 2... A k A k+1 ( k ) = A i A k+1 = ( k+1 ) A i
42 Union/Intersection Example Consider ( k+1 A i ) = A 1 A 2... A k A k+1 = (A 1 A 2... A k ) A k+1 = (A 1 A 2... A k A k+1 ( k ) = A i A k+1 = ( k+1 A i ) So, we have shown that P(k + 1) is true when we assume P(k) is true, and so by mathematical induction we have proven our statement.
43 Inequalities Example Use mathematical induction to prove the inequality n < 2 n for all positive integers n.
44 Inequalities Example Use mathematical induction to prove the inequality n < 2 n for all positive integers n. Let P(n) be the proposition that n < 2 n.
45 Inequalities Example Use mathematical induction to prove the inequality n < 2 n for all positive integers n. Let P(n) be the proposition that n < 2 n. Base Case:
46 Inequalities Example Use mathematical induction to prove the inequality n < 2 n for all positive integers n. Let P(n) be the proposition that n < 2 n. Base Case: P(1)
47 Inequalities Example Use mathematical induction to prove the inequality for all positive integers n. n < 2 n Let P(n) be the proposition that n < 2 n. Base Case: P(1) Since 1 < 2 1 = 2, P(1) is true.
48 Inequalities Example Use mathematical induction to prove the inequality for all positive integers n. n < 2 n Let P(n) be the proposition that n < 2 n. Base Case: P(1) Since 1 < 2 1 = 2, P(1) is true. Inductive Step: Assume P(k) is true. That is, assume k < 2 k for k Z +. We want to show that P(k + 1) is true.
49 Inequalities Consider k + 1 < 2 k + 1
50 Inequalities Consider k + 1 < 2 k k + 2 k
51 Inequalities Consider k + 1 < 2 k k + 2 k = 2 2 k
52 Inequalities Consider k + 1 < 2 k k + 2 k = 2 2 k = 2 k+1
53 Inequalities Consider k + 1 < 2 k k + 2 k = 2 2 k = 2 k+1 This shows that P(k + 1) is true, based on the assumption that P(k) is true. Therefore, by mathematical induction, we have shown that for all n Z +, n < 2 n.
54 Harmonic Series Example Definition The harmonic series, H j, is the series j 1 i =
55 Harmonic Series Example Definition The harmonic series, H j, is the series j 1 i = Example Prove for any nonnegative integer n, H 2 n 1 + n 2
56 Harmonic Series Example Let P(n) be the proposition that H 2 n 1 + n 2.
57 Harmonic Series Example Let P(n) be the proposition that H 2 n 1 + n 2. Base Case:
58 Harmonic Series Example Let P(n) be the proposition that H 2 n 1 + n 2. Base Case: P(0)
59 Harmonic Series Example Let P(n) be the proposition that H 2 n 1 + n 2. Base Case: P(0) P(0) is true because H 2 0 = H 1 =
60 Harmonic Series Example Let P(n) be the proposition that H 2 n 1 + n 2. Base Case: P(0) P(0) is true because H 2 0 = H 1 = Inductive Step: Assume P(k) is true. That is, assume H 2 k 1 + k 2, whenever k is a nonnegative integer.
61 Harmonic Series Example Let P(n) be the proposition that H 2 n 1 + n 2. Base Case: P(0) P(0) is true because H 2 0 = H 1 = Inductive Step: Assume P(k) is true. That is, assume H 2 k 1 + k 2, whenever k is a nonnegative integer. We want to show that if P(k) is true, then P(k + 1), which states that H 2 k k+1 2, is also true.
62 Harmonic Series Example Consider H 2 k+1 = k k k+1
63 Harmonic Series Example Consider H 2 k+1 = k k k+1 = H 2 k k k+1
64 Harmonic Series Example Consider H 2 k+1 = k k k+1 = H 2 k k k+1 ( 1 + k 2 ) k k+1
65 Harmonic Series Example Consider H 2 k+1 = k k k+1 = H 2 k k k+1 ( 1 + k 2 ) k ( 1 + k ) + 2 k 2 2 k k+1
66 Harmonic Series Example ( 1 + k )
67 Harmonic Series Example ( 1 + k ) = k 2
68 Harmonic Series Example ( 1 + k ) = k 2 = 2 + k + 1 2
69 Harmonic Series Example ( 1 + k ) = k 2 = 2 + k = 1 + k + 1 2
70 Harmonic Series Example ( 1 + k ) This establishes the proof. = k 2 = 2 + k = 1 + k + 1 2
71 Divisibility Example Example Prove that n 3 n is divisible by 3 whenever n Z +.
72 Divisibility Example Example Prove that n 3 n is divisible by 3 whenever n Z +. Let P(n) denote the proposition n 3 n is divisible by 3.
73 Divisibility Example Example Prove that n 3 n is divisible by 3 whenever n Z +. Let P(n) denote the proposition n 3 n is divisible by 3. Base Case:
74 Divisibility Example Example Prove that n 3 n is divisible by 3 whenever n Z +. Let P(n) denote the proposition n 3 n is divisible by 3. Base Case: P(1) P(1) is true because = 0 and 0 is divisible by 3.
75 Divisibility Example Example Prove that n 3 n is divisible by 3 whenever n Z +. Let P(n) denote the proposition n 3 n is divisible by 3. Base Case: P(1) P(1) is true because = 0 and 0 is divisible by 3. Inductive Step: Assume P(k) is true, which is to say that if k Z +. k 3 k is divisible by 3. We want to show that P(k + 1) is true, where P(k + 1) is (k + 1) 3 (k + 1).
76 Divisibility Example Now, (k + 1) 3 (k + 1) =
77 Divisibility Example Now, (k + 1) 3 (k + 1) = k 3 + 3k 2 + 3k + 1 k 1
78 Divisibility Example Now, (k + 1) 3 (k + 1) = k 3 + 3k 2 + 3k + 1 k 1 (k 3 k) + 3(k 2 + k)
79 Divisibility Example Now, (k + 1) 3 (k + 1) = k 3 + 3k 2 + 3k + 1 k 1 (k 3 k) + 3(k 2 + k) k 3 k is divisible by 3 by our inductive hypothesis. And, since k 2 + k is an integer for any integer k, we have that 3(k 2 + k) is also divisible by 3. It follows that (k 3 k) + 3(k 2 + k) is divisible by 3, completing the proof.
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
More informationMathematical Induction. Lecture 10-11
Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
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 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 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 informationSECTION 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
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 informationBasic 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 informationFull 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
More informationSCORE SETS IN ORIENTED GRAPHS
Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in
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 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 informationSample 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
More informationSection IV.1: Recursive Algorithms and Recursion Trees
Section IV.1: Recursive Algorithms and Recursion Trees Definition IV.1.1: A recursive algorithm is an algorithm that solves a problem by (1) reducing it to an instance of the same problem with smaller
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 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 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 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 informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More informationLikewise, we have contradictions: formulas that can only be false, e.g. (p p).
CHAPTER 4. STATEMENT LOGIC 59 The rightmost column of this truth table contains instances of T and instances of F. Notice that there are no degrees of contingency. If both values are possible, the formula
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 informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
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 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 informationCSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis. Linda Shapiro Winter 2015
CSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis Linda Shapiro Today Registration should be done. Homework 1 due 11:59 pm next Wednesday, January 14 Review math essential
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
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 informationWOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology
First Name: Family Name: Student Number: Class/Tutorial: WOLLONGONG COLLEGE AUSTRALIA A College of the University of Wollongong Diploma in Information Technology Mid-Session Test Summer Session 008-00
More informationExamination paper for MA0301 Elementær diskret matematikk
Department of Mathematical Sciences Examination paper for MA0301 Elementær diskret matematikk Academic contact during examination: Iris Marjan Smit a, Sverre Olaf Smalø b Phone: a 9285 0781, b 7359 1750
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 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 informationDiscrete Math in Computer Science Homework 7 Solutions (Max Points: 80)
Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you
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 informationDIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents
DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition
More informationMISS. INDUCTION SEQUENCES and SERIES. J J O'Connor MT1002 2009/10
MISS MATHEMATICAL INDUCTION SEQUENCES and SERIES J J O'Connor MT002 2009/0 Contents This booklet contains eleven lectures on the topics: Mathematical Induction 2 Sequences 9 Series 3 Power Series 22 Taylor
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
More informationRemoving Partial Inconsistency in Valuation- Based Systems*
Removing Partial Inconsistency in Valuation- Based Systems* Luis M. de Campos and Serafín Moral Departamento de Ciencias de la Computación e I.A., Universidad de Granada, 18071 Granada, Spain This paper
More informationSolutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014
Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 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
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 informationLinear Equations and Inequalities
Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................
More informationCONTRIBUTIONS TO ZERO SUM PROBLEMS
CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg
More informationMathematics 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
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 informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationWHERE DOES THE 10% CONDITION COME FROM?
1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay
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 non-negative integers. We say that A divides B, denoted
More informationMath 115 Spring 2011 Written Homework 5 Solutions
. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence
More informationChapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way
Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)
More informationBasics of Counting. The product rule. Product rule example. 22C:19, Chapter 6 Hantao Zhang. Sample question. Total is 18 * 325 = 5850
Basics of Counting 22C:19, Chapter 6 Hantao Zhang 1 The product rule Also called the multiplication rule If there are n 1 ways to do task 1, and n 2 ways to do task 2 Then there are n 1 n 2 ways to do
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
More informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional
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 informationMAT-71506 Program Verication: Exercises
MAT-71506 Program Verication: Exercises Antero Kangas Tampere University of Technology Department of Mathematics September 11, 2014 Accomplishment Exercises are obligatory and probably the grades will
More informationPredicate Logic. Example: All men are mortal. Socrates is a man. Socrates is mortal.
Predicate Logic Example: All men are mortal. Socrates is a man. Socrates is mortal. Note: We need logic laws that work for statements involving quantities like some and all. In English, the predicate is
More informationLecture 16 : Relations and Functions DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence
More informationPUTNAM 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
More informationCollatz Sequence. Fibbonacci Sequence. n is even; Recurrence Relation: a n+1 = a n + a n 1.
Fibonacci Roulette In this game you will be constructing a recurrence relation, that is, a sequence of numbers where you find the next number by looking at the previous numbers in the sequence. Your job
More informationMTH6120 Further Topics in Mathematical Finance Lesson 2
MTH6120 Further Topics in Mathematical Finance Lesson 2 Contents 1.2.3 Non-constant interest rates....................... 15 1.3 Arbitrage and Black-Scholes Theory....................... 16 1.3.1 Informal
More informationarxiv:math/0202219v1 [math.co] 21 Feb 2002
RESTRICTED PERMUTATIONS BY PATTERNS OF TYPE (2, 1) arxiv:math/0202219v1 [math.co] 21 Feb 2002 TOUFIK MANSOUR LaBRI (UMR 5800), Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France
More informationMathematical Induction
Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How
More information4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION
4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More informationCHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS
CHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS CENTRAL LIMIT THEOREM (SECTION 7.2 OF UNDERSTANDABLE STATISTICS) The Central Limit Theorem says that if x is a random variable with any distribution having
More informationObjective. Materials. TI-73 Calculator
0. Objective To explore subtraction of integers using a number line. Activity 2 To develop strategies for subtracting integers. Materials TI-73 Calculator Integer Subtraction What s the Difference? Teacher
More informationTheorem3.1.1 Thedivisionalgorithm;theorem2.2.1insection2.2 If m, n Z and n is a positive
Chapter 3 Number Theory 159 3.1 Prime Numbers Prime numbers serve as the basic building blocs in the multiplicative structure of the integers. As you may recall, an integer n greater than one is prime
More informationSEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov
Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices
More informationWald s Identity. by Jeffery Hein. Dartmouth College, Math 100
Wald s Identity by Jeffery Hein Dartmouth College, Math 100 1. Introduction Given random variables X 1, X 2, X 3,... with common finite mean and a stopping rule τ which may depend upon the given sequence,
More informationWarm up. Connect these nine dots with only four straight lines without lifting your pencil from the paper.
Warm up Connect these nine dots with only four straight lines without lifting your pencil from the paper. Sometimes we need to think outside the box! Warm up Solution Warm up Insert the Numbers 1 8 into
More informationMultiplying and Dividing Listen & Learn PRESENTED BY MATHMANIAC Mathematics, Grade 8
Number Sense and Numeration Integers Multiplying and Dividing PRESENTED BY MATHMANIAC Mathematics, Grade 8 Integers Multiplying and Dividing Introduction Welcome to today s topic Parts of Presentation,
More information7: The CRR Market Model
Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The Cox-Ross-Rubinstein
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 informationInteger roots of quadratic and cubic polynomials with integer coefficients
Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street
More informationThe sum of digits of polynomial values in arithmetic progressions
The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France E-mail: stoll@iml.univ-mrs.fr
More informationBX in ( u, v) basis in two ways. On the one hand, AN = u+
1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x
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 informationExam Introduction Mathematical Finance and Insurance
Exam Introduction Mathematical Finance and Insurance Date: January 8, 2013. Duration: 3 hours. This is a closed-book exam. The exam does not use scrap cards. Simple calculators are allowed. The questions
More informationHow To Know If A Domain Is Unique In An Octempo (Euclidean) Or Not (Ecl)
Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness
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 informationSet operations and Venn Diagrams. COPYRIGHT 2006 by LAVON B. PAGE
Set operations and Venn Diagrams Set operations and Venn diagrams! = { x x " and x " } This is the intersection of and. # = { x x " or x " } This is the union of and. n element of! belongs to both and,
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationSection 6.4: Counting Subsets of a Set: Combinations
Section 6.4: Counting Subsets of a Set: Combinations In section 6.2, we learnt how to count the number of r-permutations from an n-element set (recall that an r-permutation is an ordered selection of r
More informationGod created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886)
Chapter 2 Numbers God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work
More informationZachary Monaco Georgia College Olympic Coloring: Go For The Gold
Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various
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 informationSome strong sufficient conditions for cyclic homogeneous polynomial inequalities of degree four in nonnegative variables
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 6 (013), 74 85 Research Article Some strong sufficient conditions for cyclic homogeneous polynomial inequalities of degree four in nonnegative
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 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationConnectivity and cuts
Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every
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 informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationLogic is a systematic way of thinking that allows us to deduce new information
CHAPTER 2 Logic Logic is a systematic way of thinking that allows us to deduce new information from old information and to parse the meanings of sentences. You use logic informally in everyday life and
More informationSection 4.4 Inner Product Spaces
Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer
More informationNote on some explicit formulae for twin prime counting function
Notes on Number Theory and Discrete Mathematics Vol. 9, 03, No., 43 48 Note on some explicit formulae for twin prime counting function Mladen Vassilev-Missana 5 V. Hugo Str., 4 Sofia, Bulgaria e-mail:
More information1 Propositions. mcs-ftl 2010/9/8 0:40 page 5 #11. Definition. A proposition is a statement that is either true or false.
mcs-ftl 2010/9/8 0:40 page 5 #11 1 Propositions Definition. A proposition is a statement that is either true or false. For example, both of the following statements are propositions. The first is true
More informationStrictly speaking, all our knowledge outside mathematics consists of conjectures.
1 Strictly speaking, all our knowledge outside mathematics consists of conjectures. There are, of course, conjectures and conjectures. There are highly respectable and reliable conjectures as those expressed
More informationON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS
ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS YANN BUGEAUD, FLORIAN LUCA, MAURICE MIGNOTTE, SAMIR SIKSEK Abstract If n is a positive integer, write F n for the nth Fibonacci number, and ω(n) for the number
More informationFrom Binomial Trees to the Black-Scholes Option Pricing Formulas
Lecture 4 From Binomial Trees to the Black-Scholes Option Pricing Formulas In this lecture, we will extend the example in Lecture 2 to a general setting of binomial trees, as an important model for a single
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 information