DEFINITIONS: 1. FACTORIAL: n! = n (n 1) (n 2) ! = 1. n! = n (n 1)! = n (n 1) (n 2)! (n + 1)! = (n + 1) n! = (n + 1) n (n 1)!


 Sara Green
 1 years ago
 Views:
Transcription
1 MATH 2000 NOTES ON INDUCTION DEFINITIONS: 1. FACTORIAL: n! = n (n 1) (n 2) ! = 1 n! = n (n 1)! = n (n 1) (n 2)! (n + 1)! = (n + 1) n! = (n + 1) n (n 1)! 2. SUMMATION NOTATION: f(i) = f(1) + f(2) + f(3) f(n 1) + f(n) f(i) = f(1) and 2 f(i) = f(2) and i=2 f(i) = f(n) i=n c b 1 f(i) = f(i) + i=a i=a c f(i) i=b n+1 f(i) = f(i) + n+1 i=n+1 f(i) Examples: (3i + 1) = = 4 2 (3i + 1) = ( ) + ( ) = = 11 (3i + 1) = ( ) + ( ) + ( ) (3n + 1) = (3n + 1) n+1 (3i + 1) = ( ) + ( ) + ( ) (3n + 1) + (3 (n + 1) + 1) = (3n + 1) + (3n + 4) 1
2 3. WELL ORDERED : A set S of real numbers is well ordered if every nonempty subset of S has a minimum element. Examples: S = { 6, 0, 15, 102} S is well ordered. S = (0, 1) is not well ordered since it has no minimum element. S = [0, 1] is not well ordered since (0, 1) [0, 1] Q, Z and R are not well ordered. 4. Well ordering axiom: The set N is well ordered. 5. The Principle of Mathematical Induction, abbreviated to PMI. THE INDUCTION PRINCIPLE (PMI): For each n N, let P (n) be a statement. If a) P (1) is true and b) k N, P (k) P (k + 1) is true, then n N, P (n) is true. Condition a), that P (1) is true, is called the basis step or base case of the induction. Condition b) is called the inductive step. Some analogies to describe induction:  dominoes: setting up a line of dominoes so that each one will trigger the next is the inductive step; then tipping over the first domino is the base step: together, the two steps result in all the dominoes being knocked over.  climbing an infinite ladder: you need to be able to step onto the bottom rung (the base step) and then be able to climb from any one rung to the next (the inductive step). Two main uses of induction:  to make inductive definitions  to prove statements about N. 2
3 INDUCTIVE DEFINITIONS: We can use the induction property to define a function on the set N of all natural numbers. Example: The factorial function can be defined inductively by giving a base case and an inductive step: a) 1! = 1, b) n! = n (n 1)!. Example: The odd natural numbers can be inductively defined by: a) 1 is odd; b) for all n, if n is odd then n + 2 is odd. Example: The sum of the numbers from 1 to n can be defined inductively by: a) i = 1 b) n 1 i = n + i Note: induction is closely related to the idea of RECURSION used in computer science. The main difference is the point of view: in induction we start with a base case (usually n = 1) and work our way inductively up the chain of natural numbers; in a recursive calculation of something like 20!, we recursively calculate 19! then 18! then... until we reach a stopping case of 1!. STRONG INDUCTION: There is a variation of the basic principle called the Principle of Strong Induction. In this version we use not just the claim for n, but the claim for all numbers from 1 to n, to prove the claim for n + 1. Example: The fibonacci numbers 1, 2, 3, 5, 8, 13, 21,... can be defined by the rule that each number is the sum of the two previous ones. Inductively: f(n + 1) = f(n) + f(n 1). 3
4 PROOFS BY INDUCTION: Standard method to prove a statement about all natural numbers:  show that P (1) is true  usually very simple to do!  show that k N; P (k) P k + 1) is true  this is a forallifthen proof!  conclude that P (n) is true n N. We will look at proofs by induction of 3 basic kinds: summation formulas; divisibility statements; order relationships. EXAMPLE: Prove that n N, (3n 2) = n(3n 1)/2. OR (3i 2) = n(3n 1)/2. PROOF BY INDUCTION: a) Base case: Check that P (1) is true. For n = 1, (3i 2) = = 1 and n(3n 1)/2 = 1(3(1) 1)/2 = 1(2)/2 = 1. So P (1) is true. b) Inductive Step: Show that for any k N, P (k) P (k + 1) is true. ASSUME: that P (k) is true, i.e. that (3k 2) = k(3k 1)/2. GOAL: Show that P (k + 1) is true, i.e. that (3k 2) + [3(k + 1) 2] = (k + 1)[3(k + 1) 1]/ (3k 2) + [3(k + 1) 2] = k(3k 1)/2 + (3k + 1), by the induction hypothesis, = k(3k 1)/2 + 2(3k + 1)/2, common denominator, = (3k 2 k + 6k + 2)/2 = (3k 2 + 5k + 2)/2 = (k + 1)(3k + 2)/2 = (k + 1)[3(k + 1) 1]/2. Therefore by induction, (3n 2) = n(3n 1)/2, n N. 4
5 EXAMPLE: Prove n N, 4 5 n 1. PROOF BY INDUCTION: a) Base case: Check that P (1) is true. For n = 1, 5 n 1 = 4, which is divisible by 4. So P (1) is true. b) Inductive Step: Show that for any k N, P (k) P (k + 1) is true. ASSUME: that P (k) is true, i.e. that 4 5 k 1 OR 5 k 1 = 4m, m Z GOAL: Show that P (k + 1)is true, i.e. that 4 5 k k+1 1 = 5(5 k ) 1 = (4 + 1)(5 k ) 1 = 4(5 k ) + 5 k 1 = 4(5 k ) + (5 k 1) = 4(5 k ) + 4m by induction hypothesis = 4(5 k + m) where 5 k + m Z 4 5 k+1 1 Therefore by induction, 4 5 n 1, n N. 5
6 EXAMPLE: Prove that n N, (3i 1) = n(3n + 1)/2. PROOF BY INDUCTION: a) Base case: Check that P (1) is true. For n = 1, So P (1) is true. (3i 1) = 2 and n(3n+1)/2 = (1 4)/2 = 2. b) Inductive Step: Show that for any k N, P (k) P (k + 1) is true. ASSUME: that P (k) is true, i.e. that k (3i 1) = k(3k + 1)/2. k+1 GOAL: Show that P (k + 1)is true, i.e. that (3i 1) = (k + 1)(3(k + 1) + 1)/2. k+1 (3i 1) = k (3i 1) + k+1 i=k+1 (3i 1) = k(3k + 1)/2 + (3(k + 1) 1), by the induction hypothesis, = k(3k + 1)/2 + (3k + 2), = k(3k + 1)/2 + 2(3k + 2)/2, common denominator, = (3k 2 + k + 6k + 4)/2 = (3k 2 + 7k + 4)/2 = (k + 1)(3k + 4)/2 = (k + 1)(3(k + 1) + 1)/2. Therefore by induction, (3i 1) = n(3n + 1)/2, n N. 6
7 EXAMPLE: Prove that n N, 3 5 n 2 n. PROOF BY INDUCTION: a) Base case: Check that P (1) is true. For n = 1, 5 n 2 n = 5 2 = 3, which is divisible by 3. So P (1) is true. b) Inductive Step: Show that for any k N, P (k) P (k + 1) is true. ASSUME: that P (k) is true, i.e. that 3 5 k 2 k OR 5 k 2 k = 3m, m Z GOAL: Show that P (k + 1) is true, i.e. that 3 5 k+1 2 k+1. 5 k+1 2 k+1 = 5(5 k ) 2(2 k ) = (3 + 2)(5 k ) 2(2 k ) = 3(5 k ) + 2(5 k ) 2(2 k ) = 3(5 k ) + 2(5 k 2 k ) = 3(5 k ) + 2(3m) by induction hypothesis = 3(5 k + 2m) where 5 k + 2m Z 3 5 k+1 2 k+1. Therefore by induction, 3 5 n 2 n, n N. 7
8 EXAMPLE: Prove that n 2, n N, n > n 2 + n. PROOF BY EXTENDED INDUCTION: a) Base case: NOTE THE BASE CASE HERE IS n = 2 Check that P (2) is true. For n = 2, n = 9 and n 2 + n = 6, and 9 > 6. So P (2) is true. b) Inductive Step: Show that for any k N, k 2, P (k) P (k + 1) is true. ASSUME: that P (k) is true, i.e. that k > k 2 + k. GOAL: Show that P (k + 1) is true, i.e. that (k + 1) > (k + 1) 2 + (k + 1). (k + 1) = k 3 + 3k 2 + 3k , = (k 3 + 1) + (3k 2 + 3k + 1), regrouping > (k 2 + k) + (3k 2 + 3k + 1), by induction hypothesis, = 4k 2 + 3k + (k + 1), regrouping > 4k 2 + 3k + 2 since k 2, k + 1 > 2, > k 2 + 3k + 2, since 4k 2 > k 2, = (k + 1)(k + 2), = (k + 1)((k + 1) + 1), = (k + 1) 2 + (k + 1). So (k + 1) > (k + 1) 2 + (k + 1). Therefore by induction, n > n 2 + n, n N, n 2. 8
9 EXAMPLE: Prove that n 2, n N, PROOF BY EXTENDED INDUCTION: n A i = n A i. a) Base case: NOTE THE BASE CASE HERE IS n = 2 Check that P (2) is true. For n = 2, So P (2) is true. 2 A i = A 1 A 2 = A 1 A 2, by DeMorgan s 2 = A i b) Inductive Step: Show that for any k N, k 2, P (k) P (k + 1) is true. k ASSUME: that P (k) is true, i.e. that A i = k A i. GOAL: Show that P (k + 1) is true, i.e. that k+1 A i = k+1 A i. k+1 A i = = = = k A i A k+1 k A i A k+1, by DeMorgan s k A i A k+1, by induction hypothesis, k+1 A i. n Therefore by induction, A i = n A i, n N, n 2. 9
Worksheet 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 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 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 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 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 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 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 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 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 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 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 informationMITES Physics III Summer Introduction 1. 3 Π = Product 2. 4 Proofs by Induction 3. 5 Problems 5
MITES Physics III Summer 010 Sums Products and Proofs Contents 1 Introduction 1 Sum 1 3 Π Product 4 Proofs by Induction 3 5 Problems 5 1 Introduction These notes will introduce two topics: A notation which
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 informationMathematical 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
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 informationBasic Set Theory. Chapter Set Theory. can be written: A set is a Many that allows itself to be thought of as a One.
Chapter Basic Set Theory A set is a Many that allows itself to be thought of as a One.  Georg Cantor This chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical
More informationGod created the integers and the rest is the work of man. (Leopold Kronecker, in an afterdinner 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 afterdinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work
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 informationThis asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.
3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but
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 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 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 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 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 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 informationGROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.
Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the
More informationCS473  Algorithms I
CS473  Algorithms I Lecture 4 The DivideandConquer Design Paradigm View in slideshow mode 1 Reminder: Merge Sort Input array A sort this half sort this half Divide Conquer merge two sorted halves Combine
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
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 informationGraph Theory Problems and Solutions
raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is
More informationRecursive Algorithms. Recursion. Motivating Example Factorial Recall the factorial function. { 1 if n = 1 n! = n (n 1)! if n > 1
Recursion Slides by Christopher M Bourke Instructor: Berthe Y Choueiry Fall 007 Computer Science & Engineering 35 Introduction to Discrete Mathematics Sections 717 of Rosen cse35@cseunledu Recursive Algorithms
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 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 informationNotes on counting finite sets
Notes on counting finite sets Murray Eisenberg February 26, 2009 Contents 0 Introduction 2 1 What is a finite set? 2 2 Counting unions and cartesian products 4 2.1 Sum rules......................................
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 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 informationSeries Convergence Tests Math 122 Calculus III D Joyce, Fall 2012
Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it
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 informationThe thing that started it 8.6 THE BINOMIAL THEOREM
476 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers (b) Based on your results for (a), guess the minimum number of moves required if you start with an arbitrary number of n disks.
More information10.2 Series and Convergence
10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and
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 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 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 informationFunctions Recursion. C++ functions. Declare/prototype. Define. Call. int myfunction (int ); int myfunction (int x){ int y = x*x; return y; }
Functions Recursion C++ functions Declare/prototype int myfunction (int ); Define int myfunction (int x){ int y = x*x; return y; Call int a; a = myfunction (7); function call flow types type of function
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 informationChapter 3 Nuclear Fission and Neutron Multiplication. Ryan Schow
Chapter 3 Nuclear Fission and Neutron Multiplication Ryan Schow OBJECTIVES. Describe the fission process, fission product characteristics, and energy release in fission. 2. Define delayed neutron fraction,
More information#112: Write the first 4 terms of the sequence. (Assume n begins with 1.)
Section 9.1: Sequences #112: Write the first 4 terms of the sequence. (Assume n begins with 1.) 1) a n = 3n a 1 = 3*1 = 3 a 2 = 3*2 = 6 a 3 = 3*3 = 9 a 4 = 3*4 = 12 3) a n = 3n 5 Answer: 3,6,9,12 a 1
More informationSTUDENT S SOLUTIONS MANUAL ELEMENTARY NUMBER THEORY. Bart Goddard. Kenneth H. Rosen AND ITS APPLICATIONS FIFTH EDITION. to accompany.
STUDENT S SOLUTIONS MANUAL to accompany ELEMENTARY NUMBER THEORY AND ITS APPLICATIONS FIFTH EDITION Bart Goddard Kenneth H. Rosen AT&T Labs Reproduced by Pearson AddisonWesley from electronic files supplied
More informationArkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 6 Permutation Groups Let S be a nonempty set and M(S be the collection of all mappings from S into S. In this section,
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationCardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.
Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection
More informationIteration, Induction, and Recursion
CHAPTER 2 Iteration, Induction, and Recursion The power of computers comes from their ability to execute the same task, or different versions of the same task, repeatedly. In computing, the theme of iteration
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 informationInduction Problems. Tom Davis November 7, 2005
Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily
More informationThe Language of Mathematics
CHPTER 2 The Language of Mathematics 2.1. Set Theory 2.1.1. Sets. set is a collection of objects, called elements of the set. set can be represented by listing its elements between braces: = {1, 2, 3,
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 Final Examination Spring Session 2008 WUCT121
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationLinear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:
Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),
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 informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More information4/1/2017. PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY
PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY 1 Oh the things you should learn How to recognize and write arithmetic sequences
More informationIB Maths SL Sequence and Series Practice Problems Mr. W Name
IB Maths SL Sequence and Series Practice Problems Mr. W Name Remember to show all necessary reasoning! Separate paper is probably best. 3b 3d is optional! 1. In an arithmetic sequence, u 1 = and u 3 =
More informationDeterministic Finite Automata
1 Deterministic Finite Automata Definition: A deterministic finite automaton (DFA) consists of 1. a finite set of states (often denoted Q) 2. a finite set Σ of symbols (alphabet) 3. a transition function
More informationGeometric Series and Annuities
Geometric Series and Annuities Our goal here is to calculate annuities. For example, how much money do you need to have saved for retirement so that you can withdraw a fixed amount of money each year for
More informationSequence of Numbers. Mun Chou, Fong QED Education Scientific Malaysia
Sequence of Numbers Mun Chou, Fong QED Education Scientific Malaysia LEVEL High school after students have learned sequence. OBJECTIVES To review sequences and generate sequences using scientific calculator.
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 informationCmSc 175 Discrete Mathematics Lesson 10: SETS A B, A B
CmSc 175 Discrete Mathematics Lesson 10: SETS Sets: finite, infinite, : empty set, U : universal set Describing a set: Enumeration = {a, b, c} Predicates = {x P(x)} Recursive definition, e.g. sequences
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 informationOverview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series
Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them
More informationCourse Notes for Math 162: Mathematical Statistics Approximation Methods in Statistics
Course Notes for Math 16: Mathematical Statistics Approximation Methods in Statistics Adam Merberg and Steven J. Miller August 18, 6 Abstract We introduce some of the approximation methods commonly used
More informationAssignment 5  Due Friday March 6
Assignment 5  Due Friday March 6 (1) Discovering Fibonacci Relationships By experimenting with numerous examples in search of a pattern, determine a simple formula for (F n+1 ) 2 + (F n ) 2 that is, a
More informationRecursion Theory in Set Theory
Contemporary Mathematics Recursion Theory in Set Theory Theodore A. Slaman 1. Introduction Our goal is to convince the reader that recursion theoretic knowledge and experience can be successfully applied
More informationProseminar on Semantic Theory Fall 2013 Ling 720. Problem Set on the Formal Preliminaries : Answers and Notes
1. Notes on the Answers Problem Set on the Formal Preliminaries : Answers and Notes In the pages that follow, I have copied some illustrative answers from the problem sets submitted to me. In this section,
More information4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.
Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,
More informationDiscrete Mathematics Problems
Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 Email: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................
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 information1.1 Logical Form and Logical Equivalence 1
Contents Chapter I The Logic of Compound Statements 1.1 Logical Form and Logical Equivalence 1 Identifying logical form; Statements; Logical connectives: not, and, and or; Translation to and from symbolic
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 informationL  Standard Letter Grade P  Pass/No Pass Repeatability: N  Course may not be repeated
Course: MATH 26 Division: 10 Also Listed As: 200930, INACTIVE COURSE Short Title: Full Title: DISCRETE MATHEMATIC Discrete Mathematics Contact Hours/Week Lecture: 4 Lab: 0 Other: 0 Total: 4 4 Number of
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
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 informationMaths Workshop for Parents 2. Fractions and Algebra
Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)
More informationRECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES
RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES DARRYL MCCULLOUGH AND ELIZABETH WADE In [9], P. W. Wade and W. R. Wade (no relation to the second author gave a recursion formula that produces Pythagorean
More informationCatalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni versity.
7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,
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 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 informationA Course in Discrete Structures. Rafael Pass WeiLung Dustin Tseng
A Course in Discrete Structures Rafael Pass WeiLung Dustin Tseng Preface Discrete mathematics deals with objects that come in discrete bundles, e.g., 1 or 2 babies. In contrast, continuous mathematics
More informationClicker 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] 1.11.7 Exercises: Do before next class; not to hand in [J] pp. 1214:
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 informationMath Circle Beginners Group October 18, 2015
Math Circle Beginners Group October 18, 2015 Warmup 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
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 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 informationTo define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions
Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions
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 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 informationProblem Set I: Preferences, W.A.R.P., consumer choice
Problem Set I: Preferences, W.A.R.P., consumer choice Paolo Crosetto paolo.crosetto@unimi.it Exercises solved in class on 18th January 2009 Recap:,, Definition 1. The strict preference relation is x y
More informationJacobi s four squares identity Martin Klazar
Jacobi s four squares identity Martin Klazar (lecture on the 7th PhD conference) Ostrava, September 10, 013 C. Jacobi [] in 189 proved that for any integer n 1, r (n) = #{(x 1, x, x 3, x ) Z ( i=1 x i
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 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 information