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)!

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "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)!"

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 non-empty 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 for-all-if-then 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 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 information

Mathematical Induction. Lecture 10-11

Mathematical 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 information

Math 55: Discrete Mathematics

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 information

Sample Induction Proofs

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

More information

Introduction. Appendix D Mathematical Induction D1

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

More information

3. Mathematical Induction

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)

More information

Appendix F: Mathematical Induction

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

More information

Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction. 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 information

Section 6-2 Mathematical Induction

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

More information

SECTION 10-2 Mathematical Induction

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

More information

MATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.

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

More information

MITES Physics III Summer Introduction 1. 3 Π = Product 2. 4 Proofs by Induction 3. 5 Problems 5

MITES 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 information

Mathematical Induction

Mathematical 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 information

Mathematical induction. Niloufar Shafiei

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

More information

Mathematical Induction. Mary Barnes Sue Gordon

Mathematical 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 information

Basic Set Theory. Chapter Set Theory. can be written: A set is a Many that allows itself to be thought of as a One.

Basic 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 information

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 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 information

Basic Proof Techniques

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 information

This asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.

This 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 information

Cartesian Products and Relations

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

More information

x if x 0, x if x < 0.

x 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 information

Math 115 Spring 2011 Written Homework 5 Solutions

Math 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 information

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 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 information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

Discrete 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 information

Section IV.1: Recursive Algorithms and Recursion Trees

Section 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 information

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

GROUPS 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 information

CS473 - Algorithms I

CS473 - Algorithms I CS473 - Algorithms I Lecture 4 The Divide-and-Conquer Design Paradigm View in slide-show mode 1 Reminder: Merge Sort Input array A sort this half sort this half Divide Conquer merge two sorted halves Combine

More information

Basic 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 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 information

Collatz Sequence. Fibbonacci Sequence. n is even; Recurrence Relation: a n+1 = a n + a n 1.

Collatz 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 information

Graph Theory Problems and Solutions

Graph 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 information

Recursive Algorithms. Recursion. Motivating Example Factorial Recall the factorial function. { 1 if n = 1 n! = n (n 1)! if n > 1

Recursive 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 71-7 of Rosen cse35@cseunledu Recursive Algorithms

More information

9.2 Summation Notation

9.2 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 information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate 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 information

Notes on counting finite sets

Notes 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 information

Solutions for Practice problems on proofs

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

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Series 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 information

1.2. Successive Differences

1.2. 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 information

The thing that started it 8.6 THE BINOMIAL THEOREM

The 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 information

10.2 Series and Convergence

10.2 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 information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX 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 information

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80)

Discrete 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 information

WRITING PROOFS. Christopher Heil Georgia Institute of Technology

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

More information

Functions 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. 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 information

CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 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 information

Chapter 3 Nuclear Fission and Neutron Multiplication. Ryan Schow

Chapter 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

#1-12: Write the first 4 terms of the sequence. (Assume n begins with 1.)

#1-12: Write the first 4 terms of the sequence. (Assume n begins with 1.) Section 9.1: Sequences #1-12: 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 information

STUDENT S SOLUTIONS MANUAL ELEMENTARY NUMBER THEORY. Bart Goddard. Kenneth H. Rosen AND ITS APPLICATIONS FIFTH EDITION. to accompany.

STUDENT 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 Addison-Wesley from electronic files supplied

More information

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas 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 information

Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities

Creating, 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 information

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

This 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 information

Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.

Cardinality. 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 information

Iteration, Induction, and Recursion

Iteration, 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 information

Matrix 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. 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 information

Induction Problems. Tom Davis November 7, 2005

Induction 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 information

The Language of Mathematics

The 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 information

WOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology

WOLLONGONG 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 information

Core Maths C1. Revision Notes

Core 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 information

Linear 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. 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 information

8 Primes and Modular Arithmetic

8 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 information

Taylor Polynomials and Taylor Series Math 126

Taylor 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 information

4/1/2017. 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

4/1/2017. 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 information

IB Maths SL Sequence and Series Practice Problems Mr. W Name

IB 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 information

Deterministic Finite Automata

Deterministic 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 information

Geometric Series and Annuities

Geometric 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 information

Sequence of Numbers. Mun Chou, Fong QED Education Scientific Malaysia

Sequence 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 information

Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014

Solutions 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 information

CmSc 175 Discrete Mathematics Lesson 10: SETS A B, A B

CmSc 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 information

If n is odd, then 3n + 7 is even.

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

More information

Overview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series

Overview. 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 information

Course Notes for Math 162: Mathematical Statistics Approximation Methods in Statistics

Course 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 information

Assignment 5 - Due Friday March 6

Assignment 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 information

Recursion Theory in Set Theory

Recursion 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 information

Proseminar on Semantic Theory Fall 2013 Ling 720. Problem Set on the Formal Preliminaries : Answers and Notes

Proseminar 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 information

4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.

4.1. 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 information

Discrete Mathematics Problems

Discrete Mathematics Problems Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 E-mail: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................

More information

Continued Fractions and the Euclidean Algorithm

Continued 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 information

1.1 Logical Form and Logical Equivalence 1

1.1 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 information

CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS 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 information

L - Standard Letter Grade P - Pass/No Pass Repeatability: N - Course may not be repeated

L - 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 information

MATRIX 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. + + 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 information

Homework 5 Solutions

Homework 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 information

Maths Workshop for Parents 2. Fractions and Algebra

Maths 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 information

RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES

RECURSIVE 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 information

Catalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni- versity.

Catalan 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 information

Full and Complete Binary Trees

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

More information

Handout #1: Mathematical Reasoning

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

More information

A Course in Discrete Structures. Rafael Pass Wei-Lung Dustin Tseng

A Course in Discrete Structures. Rafael Pass Wei-Lung Dustin Tseng A Course in Discrete Structures Rafael Pass Wei-Lung Dustin Tseng Preface Discrete mathematics deals with objects that come in discrete bundles, e.g., 1 or 2 babies. In contrast, continuous mathematics

More information

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

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:

More information

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

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

More information

Math Circle Beginners Group October 18, 2015

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

More information

Removing Partial Inconsistency in Valuation- Based Systems*

Removing 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 information

MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 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 information

To define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions

To 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 information

Set operations and Venn Diagrams. COPYRIGHT 2006 by LAVON B. PAGE

Set 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 information

Math 4310 Handout - Quotient Vector Spaces

Math 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 information

Problem Set I: Preferences, W.A.R.P., consumer choice

Problem 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 information

Jacobi s four squares identity Martin Klazar

Jacobi s four squares identity Martin Klazar Jacobi s four squares identity Martin Klazar (lecture on the 7-th 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 information

8 Divisibility and prime numbers

8 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 information

An Innocent Investigation

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

More information