MI 4 Mathematical Induction Name. Mathematical Induction

Size: px
Start display at page:

Download "MI 4 Mathematical Induction Name. Mathematical Induction"

Transcription

1 Mathematical Induction It turns out that the most efficient solution to the Towers of Hanoi problem with n disks takes n 1 moves. If this isn t the formula you determined, make sure to check your data if you took more moves than this, see if you can find a more efficient solution; if you took fewer make sure you didn t accidentally cheat! How do we know that the Tower of Hanoi can really be solved in n 1 moves, even if n is very large? Sure, the formula worked for small choices of n like 1,, 3, and perhaps you even tried 4 or 5. But there is no way you could try 64 disks, much less a billion. What if some difficulty sets in when the problem becomes complicated enough, and it stops following the pattern? Or conversely, what if there is some shortcut for larger numbers? Then the simple formula n 1 might not work out any more. Math is full of patterns that don t work out. For instance, into how many separate regions can n lines split the plane? n = 0, 1 region n = 1, regions n =, 4 regions n = 3, 7 regions? (1 region) (region 1) (region ) (1) () (3) (4) It looked like we had a nice pattern of doubling each time, but after a while it broke down. And that is the problem with statements about all n they might break down after a while. Worse, they might break down for some large number where we didn t bother to check that high, so we don t even know they broke down. Sometimes, it can be very important to know that some fact is true about every possible integer. For instance, some forms of Internet security depend on the fact that every positive integer can be factored into primes in exactly one way. Now most 50-digit numbers have never been factored, much less double-checked to see if there isn t maybe a second, different way to factor it. There are just too many 50-digit numbers to check them all! But if the 50-digit number that happens to be guarding your credit card can be factored in two different ways, it could compromise your card s security. So it is important to be able to prove that every number can be factored in a unique way, even though it is impossible to actually check them all. There are a number of techniques mathematicians use to prove statements are true about all numbers, without having to check that every possible number works. Induction.1

2 1. The most common type of proof in mathematics is the direct proof. This proceeds from known facts to deduce new facts. For instance: prove that if n is even, then so is n (hint: start by noticing that if n is even, then n = k for some other number k. Note this is not true about odd numbers! Now what is n?) If n is even, then n = k, so n = (k) = 4k = k is also even. So the square of every even number is even. Now since n is a variable in your proof, you could substitute any even number for n, and you know that n is also even. So your have proven that the square of an even number is even, without having to check every even number (or, by a different way of thinking, since n is variable you checked every even number simultaneously).. Another common proof technique is indirect proof, sometimes called proof by contradiction or proof by contraposition (they are slightly different, but let s not worry about that for now). Use proof by contradiction to prove that if n is odd, then n must be odd. (Hint: what do you know if n is even?) Assume that n is even. Then, from the previous problem, we know that n is even, not odd. This is a contradiction, so n must be odd. Again, since n is a variable, you could substitute any n and the statement remains true. These techniques aren t always sufficient to prove everything we want to prove. We would like to prove that Towers of Hanoi can be solved in n 1 moves if you start with a stack of n disks. The problem with direct or indirect proof is that there is no easy way to take a variable n and say which n 1 moves you are going to make, and why no fewer will work. Unlike the even and odd numbers we played with above, it s difficult to see what property of n makes the Towers work in the right number of moves. Remember the notation from the last handout: H(n) is the minimum number of moves needed to move a stack of n disks. Induction.

3 Recursion and Induction 3. Pretend you are faced with a tower of sixteen disks to move. Don t fret! The evil Dr. Mike Fogg just stepped into the room, and he happens to know how to solve a 15-disk puzzle in the most efficient way possible! (In other words, he can do it using exactly H(15) moves.) Explain how you could use his knowledge to solve the 16-disk puzzle. First, use Dr. Fogg s method to move the top 15 disks onto another peg. Then move the biggest disk onto the third peg. Then, use Dr. Fogg s method to put the 15 disks back on top of the biggest disk, thus moving all 16 disks to a new peg. 4. You should be able to use your strategy to explain why H(16) H(15) + 1. Can you think of why this is and not just =? Can you make your argument slightly more precise to explain why it can t be < and therefore must actually be =? Our solution used Dr. Fogg s solution twice, plus one additional move of the biggest disk. So H(16) is at most H(15) + 1. It might be less, because there might be a shortcut not involving Dr. Fogg s method. But not really. Before we can move the biggest disk, all the disks must be moved off of it, and onto a single other peg to leave room to move the biggest disk. This takes at least H(15) moves, since H(15) is the most efficient way possible. Then we must move the biggest disk, then put those smaller ones back on top, taking at least H(15) more moves. So H(16) H(15) + 1. If it is and, is must be =. 5. If H(15) really does equal 15 1, use this new strategy to find and simplify H(16). H(16) = H(15) + 1 = ( 15 1) + 1 = = Is there anything special about the numbers 15 and 16 to make this work? Or could your strategy work for any number, k, so long as Dr. Fogg knows the solution for k 1 disks? Nothing special. Everything works the same with any two consecutive numbers. Induction.3

4 7. If Dr. Fogg s solution really does take k 1 1 steps for the k 1-disk tower, compute how many steps it will take you to solve the k-disk tower. H(k) = H(k 1) + 1 = ( k 1 1) + 1 = k = k 1. This is an example of a common strategy in mathematics. If you have a sequence of related problems, figure out how to use the solution to each one as part of the solution to the next more complicated one. This is called recursion. If you could then figure out how to solve the first problem, you could use that as part of the solution to the second problem. Then you could use that to solve the third one. Then you could use this triply-compounded solution to solve the fourth one, and so on. Since, eventually, you will get to any one of those problems if you work for long enough, you can say that (in theory at least it might take you a long time to get to the trillionth problem!) you can solve every one of those problems. This is actually an axiom of the integers: if you can solve problem number one, and whenever you know the solution to problem number k 1 you have a strategy for turning that into a solution for problem number k, then you can solve all the problems! This technique relied on two things: 1) the ability to solve the first problem, and ) having a strategy to use a solution for one problem to solve the next one. Without the first you have no way to get started, and without the second you have no way of continuing once you get started. Let s put it in more math-y terms. Let s say you have some statement about a number n. It might be n is odd or Towers of Hanoi with n disks can be solved, in exactly n 1 moves. We will call this statement S(n) and it can be either true or false when you substitute some particular number for n. For example, if S(n) is n is odd then S(3) is true while S(94) is false. Principle of Mathematical Induction: Let s say you want to prove the S(n) is true no matter what positive integer you plug in for n. (1) if S(1) is true, and () whenever a particular S(k 1) happens to be true, you know how to use that fact to prove that S(k) is also true, then (conclusion) S(n) is true for all positive integers n by the principle of mathematical induction. As noted, the principle of mathematical induction is an axiom of the integers. We can t prove it s true. It is part of the mathematical definition of the integers. It is believable, because we know a way to get from the truth of S(k 1) to the truth of S(k), and we also know that S(1) is true. But then setting k =, we know how to use the fact that S(k 1) = S(1) is true to prove that S() is true. So now we know that S() is true, and we change k to 3. Then we know S(k 1) = S() is true, and we know how to get from S() to S(3), so we now learn that S(3) is true. And we just keep going forever. Notice there are two variables n and k. It s better not to think of k as a variable, but as being a definite, but unknown, number. It doesn t change during your proof; it s some particular number, and it happens not to matter what that number is in your strategy from going from one Induction.4

5 step to the next. That is, your strategy tells how to solve problem number k given that you (or Dr. Fogg) knows how to solve problem number k 1, without knowing what k actually is. It s like climbing a long staircase it doesn t matter which stair you are on in the process of climbing from one stair to the next. Whichever stair you are on (which just happens to be number k 1) you can climb onto the next stair (which is stair number k), and it doesn t matter what k is. On the other hand, n is a variable like in the examples of direct and indirect proof earlier you can substitute any value for n and the statement is true, so the statement is true for all positive integers n. Part (1), which gets you started, is called the base case or basis step. Part (), which keeps you going once you ve started, is called the induction step. Let s write out the whole inductive proof of the Tower of Hanoi problem now: Theorem: the n-disk Tower of Hanoi problem can be solved, and the best solution takes exactly n 1 steps. 8. Base case: Explain how to solve the 1-disk Tower of Hanoi: Just move the disk to a new peg How many moves did you take? Is this equal to 1 1? Yes 10. Induction step: Let s say you (or Dr. Fogg) has a solution to the k 1-disk puzzle, and that the best possible solution takes H(k 1) moves. Explain why you can now solve the k-disk puzzle, needing no more nor less than H(k 1) + 1 moves. Let s say we have a solution to the k 1 disk puzzle, taking the minimum number of H(k 1) moves. Then to solve the k- disk puzzle, we can move the top k 1 disks onto a new peg in H(k 1) moves, then move the largest disk onto the last vacant peg, then use another H(k 1) moves to put the smaller disks back on top of the largest. So we can move the tower in no more than H(k 1) + 1 moves. But we can t do any better than that, because before we move the biggest disk, we must move all the smaller disks onto a single other peg, which takes at least H(k 1) moves. Then one move to move the biggest disk, then at least H(k 1) more moves to put the small ones back on top. 11. So H(k) = H(k 1) + 1. If we knew that H(k 1) = k 1 1, simplify to find H(k). H(k) = H(k 1) + 1 = ( k 1 1) + 1 = k = k 1. By the principle of induction, we are done! We have proven that n-disk Tower of Hanoi can always be solved, taking n 1 moves. Induction.5

6 As you were filling in the steps of the proof, it probably felt like you were just repeating the arguments from before. You were you really had completed the proof before, and were just putting all the pieces together in one place. Let s try another one. Theorem: if n (non-parallel) lines are drawn in a plane, the resulting picture has n(n +1) L( n) = +1 regions. (You might want to check that this formula works for n = 0, 1,, and 3 from the pictures on the first page!) 1. Base case: does the formula work for one line? How many regions are there? What does the formula give when you substitute n = 1? Are these equal? With one line, we get two regions. 1 (1 + 1) / + 1 =, so the formula works in this case. 13. Inductive step: Let s say that for some number k, Dr. Fogg counted the number of regions determined by k 1 lines and knows that this really does fit the formula there really are (k 1)(k) L( k 1) = +1 regions. How many regions will there be when you draw that kth line? If the new line crosses through an old region, it splits that region into two pieces one region becomes two. So you add one new region for each old region that your new line crosses. Explain why your new line crosses exactly k regions. Hint: when does your new line go from one old region to a different one? When we add the kth line, it starts in one region, and crosses into a new region each time it crosses one of the k 1 old lines. So it crosses into k 1 other regions. Thus the new line crosses a total of k regions, splitting each in two. Thus, when we add a new line to the picture, we add k new regions. So now we know that L(k) = L(k 1) + k. Substitute the formula for L(k 1) above and simplify. Do this match the hoped-for formula for L(k)? (k 1)k L(k) = L(k 1) + k = = k + k k(k +1) +1 = +1+ k = k k +1. Yes, it matches! Then we re done, by the principle of induction! +1+ k k = k k + k +1 Induction.6

Chapter 2. Mathematical Reasoning. 2.1 Mathematical Models

Chapter 2. Mathematical Reasoning. 2.1 Mathematical Models Contents Mathematical Reasoning 3.1 Mathematical Models........................... 3. Mathematical Proof............................ 4..1 Structure of Proofs........................ 4.. Direct Method..........................

More information

Induction 1 = 1(1+1) = 2(2+1) = 3(3+1) 2

Induction 1 = 1(1+1) = 2(2+1) = 3(3+1) 2 Induction 0-8-08 Induction is used to prove a sequence of statements P(), P(), P(3),... There may be finitely many statements, but often there are infinitely many. For example, consider the statement ++3+

More information

CHAPTER 4 SOME METHODS OF PROOF

CHAPTER 4 SOME METHODS OF PROOF CHAPTER 4 SOME METHODS OF PROOF In all sciences, general theories usually arise from a number of observations. In the experimental sciences, the validity of the theories can only be tested by carefully

More information

What can you prove by induction?

What can you prove by induction? MEI CONFERENCE 013 What can you prove by induction? Martyn Parker M.J.Parker@keele.ac.uk Contents Contents iii 1 Splitting Coins.................................................. 1 Convex Polygons................................................

More information

cse547, math547 DISCRETE MATHEMATICS Professor Anita Wasilewska

cse547, math547 DISCRETE MATHEMATICS Professor Anita Wasilewska cse547, math547 DISCRETE MATHEMATICS Professor Anita Wasilewska LECTURE 1 INTRODUCTION Course Web Page www.cs.stonybrook.edu/ cse547 The webpage contains: detailed lectures notes slides; very detailed

More information

ASSIGNMENT 1 SOLUTIONS

ASSIGNMENT 1 SOLUTIONS MATH 271 ASSIGNMENT 1 SOLUTIONS 1. (a) Let S be the statement For all integers n, if n is even then 3n 11 is odd. Is S true? Give a proof or counterexample. (b) Write out the contrapositive of statement

More information

CS1800: Mathematical Induction. Professor Kevin Gold

CS1800: Mathematical Induction. Professor Kevin Gold CS1800: Mathematical Induction Professor Kevin Gold Induction: Used to Prove Patterns Just Keep Going For an algorithm, we may want to prove that it just keeps working, no matter how big the input size

More information

Math 300: Final Exam Practice Solutions

Math 300: Final Exam Practice Solutions Math 300: Final Exam Practice Solutions 1 Let A be the set of all real numbers which are zeros of polynomials with integer coefficients: A := {α R there exists p(x) = a n x n + + a 1 x + a 0 with all a

More information

Cool Results on Primes

Cool Results on Primes Cool Results on Primes LA Math Circle (Advanced) January 24, 2016 Recall that last week we learned an algorithm that seemed to magically spit out greatest common divisors, but we weren t quite sure why

More information

Mathematical Induction

Mathematical Induction Mathematical Induction Let s motivate our discussion by considering an example first. What happens when we add the first n positive odd integers? The table below shows what results for the first few values

More information

Introducing Proof 1. hsn.uk.net. Contents

Introducing Proof 1. hsn.uk.net. Contents Contents 1 1 Introduction 1 What is proof? 1 Statements, Definitions and Euler Diagrams 1 Statements 1 Definitions Our first proof Euler diagrams 4 3 Logical Connectives 5 Negation 6 Conjunction 7 Disjunction

More information

What is proof? Lesson 1

What is proof? Lesson 1 What is proof? Lesson The topic for this Math Explorer Club is mathematical proof. In this post we will go over what was covered in the first session. The word proof is a normal English word that you might

More information

WORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...}

WORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...} WORKSHEET ON NUMBERS, MATH 215 FALL 18(WHYTE) We start our study of numbers with the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } and their subset of natural numbers: N = {1, 2, 3,...} For now we will not

More information

MATH CSE20 Homework 5 Due Monday November 4

MATH CSE20 Homework 5 Due Monday November 4 MATH CSE20 Homework 5 Due Monday November 4 Assigned reading: NT Section 1 (1) Prove the statement if true, otherwise find a counterexample. (a) For all natural numbers x and y, x + y is odd if one of

More information

1 Sequences and Summation

1 Sequences and Summation 1 Sequences and Summation A sequence is a function whose domain is either all the integers between two given integers or all the integers greater than or equal to a given integer. For example, a m, a m+1,...,

More information

, p 1 < p 2 < < p l primes.

, p 1 < p 2 < < p l primes. Solutions Math 347 Homework 1 9/6/17 Exercise 1. When we take a composite number n and factor it into primes, that means we write it as a product of prime numbers, usually in increasing order, using exponents

More information

Appendix G: Mathematical Induction

Appendix G: Mathematical Induction Appendix G: 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

6.042/18.062J Mathematics for Computer Science March 17, 2005 Srini Devadas and Eric Lehman. Recurrences

6.042/18.062J Mathematics for Computer Science March 17, 2005 Srini Devadas and Eric Lehman. Recurrences 6.04/8.06J Mathematics for Computer Science March 7, 00 Srini Devadas and Eric Lehman Lecture Notes Recurrences Recursion breaking an object down into smaller objects of the same type is a major theme

More information

You separate binary numbers into columns in a similar fashion. 2 5 = 32

You separate binary numbers into columns in a similar fashion. 2 5 = 32 RSA Encryption 2 At the end of Part I of this article, we stated that RSA encryption works because it s impractical to factor n, which determines P 1 and P 2, which determines our private key, d, which

More information

Winter Camp 2009 Number Theory Tips and Tricks

Winter Camp 2009 Number Theory Tips and Tricks Winter Camp 2009 Number Theory Tips and Tricks David Arthur darthur@gmail.com 1 Introduction This handout is about some of the key techniques for solving number theory problems, especially Diophantine

More information

Basics of Proofs. 1 The Basics. 2 Proof Strategies. 2.1 Understand What s Going On

Basics of Proofs. 1 The Basics. 2 Proof Strategies. 2.1 Understand What s Going On Basics of Proofs The Putnam is a proof based exam and will expect you to write proofs in your solutions Similarly, Math 96 will also require you to write proofs in your homework solutions If you ve seen

More information

Writing proofs. Tim Hsu, San José State University. May 31, Definitions and theorems 3. 2 What is a proof? 3. 3 A word about definitions 4

Writing proofs. Tim Hsu, San José State University. May 31, Definitions and theorems 3. 2 What is a proof? 3. 3 A word about definitions 4 Writing proofs Tim Hsu, San José State University May 31, 2006 Contents I Fundamentals 3 1 Definitions and theorems 3 2 What is a proof? 3 3 A word about definitions 4 II The structure of proofs 6 4 Assumptions

More information

The Basics COPYRIGHTED MATERIAL. chapter. Algebra is a very logical way to solve

The Basics COPYRIGHTED MATERIAL. chapter. Algebra is a very logical way to solve chapter 1 The Basics Algebra is a very logical way to solve problems both theoretically and practically. You need to know a number of things. You already know arithmetic of whole numbers. You will review

More information

Matrix Multiplication

Matrix Multiplication Matrix Multiplication Given two vectors a = [a 1,..., a k ] and b = [b 1,..., b k ], their inner product (or dot product) is a b = k i=1 a ib i [1, 2, 3] [ 2, 4, 6] = (1 2) + (2 4) + (3 6) = 24. We can

More information

Math Circle: Recursion and Induction

Math Circle: Recursion and Induction Math Circle: Recursion and Induction Prof. Wickerhauser 1 Recursion What can we compute, using only simple formulas and rules that everyone can understand? 1. Let us use N to denote the set of counting

More information

MATHEMATICAL INDUCTION

MATHEMATICAL INDUCTION MATHEMATICAL INDUCTION MATH 3A SECTION HANDOUT BY GERARDO CON DIAZ Imagine a bunch of dominoes on a table. They are set up in a straight line, and you are about to push the first piece to set off the chain

More information

Inference and Proofs (1.6 & 1.7)

Inference and Proofs (1.6 & 1.7) EECS 203 Spring 2016 Lecture 4 Page 1 of 9 Introductory problem: Inference and Proofs (1.6 & 1.7) As is commonly the case in mathematics, it is often best to start with some definitions. An argument for

More information

Chapter12. Relations, Functions, and Induction

Chapter12. Relations, Functions, and Induction . Relations. Functions. Seuences.4 Mathematical Induction.5 Recurrence Relations Chapter Review Chapter Relations, Functions, and Induction In this chapter we introduce the concepts of a relation and a

More information

Solving with Absolute Value

Solving with Absolute Value Solving with Absolute Value Who knew two little lines could cause so much trouble? Ask someone to solve the equation 3x 2 = 7 and they ll say No problem! Add just two little lines, and ask them to solve

More information

QUADRATICS 3.2 Breaking Symmetry: Factoring

QUADRATICS 3.2 Breaking Symmetry: Factoring QUADRATICS 3. Breaking Symmetry: Factoring James Tanton SETTING THE SCENE Recall that we started our story of symmetry with a rectangle of area 36. Then you would say that the picture is wrong and that

More information

Lecture 3: Sizes of Infinity

Lecture 3: Sizes of Infinity Math/CS 20: Intro. to Math Professor: Padraic Bartlett Lecture 3: Sizes of Infinity Week 2 UCSB 204 Sizes of Infinity On one hand, we know that the real numbers contain more elements than the rational

More information

base 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation.

base 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation. EXPONENTIALS Exponential is a number written with an exponent. The rules for exponents make computing with very large or very small numbers easier. Students will come across exponentials in geometric sequences

More information

MA103 STATEMENTS, PROOF, LOGIC

MA103 STATEMENTS, PROOF, LOGIC MA103 STATEMENTS, PROOF, LOGIC Abstract Mathematics is about making precise mathematical statements and establishing, by proof or disproof, whether these statements are true or false. We start by looking

More information

3 The language of proof

3 The language of proof 3 The language of proof After working through this section, you should be able to: (a) understand what is asserted by various types of mathematical statements, in particular implications and equivalences;

More information

Row Reduction

Row Reduction Row Reduction 1-12-2015 Row reduction (or Gaussian elimination) is the process of using row operations to reduce a matrix to row reduced echelon form This procedure is used to solve systems of linear equations,

More information

WORKSHEET MATH 215, FALL 15, WHYTE. We begin our course with the natural numbers:

WORKSHEET MATH 215, FALL 15, WHYTE. We begin our course with the natural numbers: WORKSHEET MATH 215, FALL 15, WHYTE We begin our course with the natural numbers: N = {1, 2, 3,...} which are a subset of the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } We will assume familiarity with their

More information

Solution Set 2. Problem 1. [a] + [b] = [a + b] = [b + a] = [b] + [a] ([a] + [b]) + [c] = [a + b] + [c] = [a + b + c] = [a] + [b + c] = [a] + ([b + c])

Solution Set 2. Problem 1. [a] + [b] = [a + b] = [b + a] = [b] + [a] ([a] + [b]) + [c] = [a + b] + [c] = [a + b + c] = [a] + [b + c] = [a] + ([b + c]) Solution Set Problem 1 (1) Z/nZ is the set of equivalence classes of Z mod n. Equivalence is determined by the following rule: [a] = [b] if and only if b a = k n for some k Z. The operations + and are

More information

Another Proof By Contradiction: 2 is Irrational

Another Proof By Contradiction: 2 is Irrational Another Proof By Contradiction: 2 is Irrational Theorem: 2 is irrational. Proof: By contradiction. Suppose 2 is rational. Then 2 = a/b for some a, b N +. We can assume that a/b is in lowest terms. Therefore,

More information

An analogy from Calculus: limits

An analogy from Calculus: limits COMP 250 Fall 2018 35 - big O Nov. 30, 2018 We have seen several algorithms in the course, and we have loosely characterized their runtimes in terms of the size n of the input. We say that the algorithm

More information

Introduction to Basic Proof Techniques Mathew A. Johnson

Introduction to Basic Proof Techniques Mathew A. Johnson Introduction to Basic Proof Techniques Mathew A. Johnson Throughout this class, you will be asked to rigorously prove various mathematical statements. Since there is no prerequisite of a formal proof class,

More information

means is a subset of. So we say A B for sets A and B if x A we have x B holds. BY CONTRAST, a S means that a is a member of S.

means is a subset of. So we say A B for sets A and B if x A we have x B holds. BY CONTRAST, a S means that a is a member of S. 1 Notation For those unfamiliar, we have := means equal by definition, N := {0, 1,... } or {1, 2,... } depending on context. (i.e. N is the set or collection of counting numbers.) In addition, means for

More information

Section 20: Arrow Diagrams on the Integers

Section 20: Arrow Diagrams on the Integers Section 0: Arrow Diagrams on the Integers Most of the material we have discussed so far concerns the idea and representations of functions. A function is a relationship between a set of inputs (the leave

More information

Fibonacci mod k. In this section, we examine the question of which terms of the Fibonacci sequence have a given divisor k.

Fibonacci mod k. In this section, we examine the question of which terms of the Fibonacci sequence have a given divisor k. Fibonacci mod k I start by giving out a table of the first 0 Fibonacci numbers actually the first, because we begin with u 0 =0 (and I have a reason for that which will soon become apparent). Okay, which

More information

Preptests 59 Answers and Explanations (By Ivy Global) Section 1 Analytical Reasoning

Preptests 59 Answers and Explanations (By Ivy Global) Section 1 Analytical Reasoning Preptests 59 Answers and Explanations (By ) Section 1 Analytical Reasoning Questions 1 5 Since L occupies its own floor, the remaining two must have H in the upper and I in the lower. P and T also need

More information

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions. Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,

More information

Lecture 12 : Recurrences DRAFT

Lecture 12 : Recurrences DRAFT CS/Math 240: Introduction to Discrete Mathematics 3/1/2011 Lecture 12 : Recurrences Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last few classes we talked about program correctness. We

More information

At the start of the term, we saw the following formula for computing the sum of the first n integers:

At the start of the term, we saw the following formula for computing the sum of the first n integers: Chapter 11 Induction This chapter covers mathematical induction. 11.1 Introduction to induction At the start of the term, we saw the following formula for computing the sum of the first n integers: Claim

More information

MCS-236: Graph Theory Handout #A4 San Skulrattanakulchai Gustavus Adolphus College Sep 15, Methods of Proof

MCS-236: Graph Theory Handout #A4 San Skulrattanakulchai Gustavus Adolphus College Sep 15, Methods of Proof MCS-36: Graph Theory Handout #A4 San Skulrattanakulchai Gustavus Adolphus College Sep 15, 010 Methods of Proof Consider a set of mathematical objects having a certain number of operations and relations

More information

PHIL 422 Advanced Logic Inductive Proof

PHIL 422 Advanced Logic Inductive Proof PHIL 422 Advanced Logic Inductive Proof 1. Preamble: One of the most powerful tools in your meta-logical toolkit will be proof by induction. Just about every significant meta-logical result relies upon

More information

Solution to Proof Questions from September 1st

Solution to Proof Questions from September 1st Solution to Proof Questions from September 1st Olena Bormashenko September 4, 2011 What is a proof? A proof is an airtight logical argument that proves a certain statement in general. In a sense, it s

More information

Mathematical Statements

Mathematical Statements Mathematical Statements Joseph R. Mileti January 26, 2015 1 Mathematical Statements and Mathematical Truth Unfortunately, many people view mathematics only as complicated equations and elaborate computational

More information

Writing proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases

Writing proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases Writing proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases September 22, 2018 Recall from last week that the purpose of a proof

More information

Solving Quadratic & Higher Degree Equations

Solving Quadratic & Higher Degree Equations Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,

More information

Algebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Algebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is

More information

THE SIMPLE PROOF OF GOLDBACH'S CONJECTURE. by Miles Mathis

THE SIMPLE PROOF OF GOLDBACH'S CONJECTURE. by Miles Mathis THE SIMPLE PROOF OF GOLDBACH'S CONJECTURE by Miles Mathis miles@mileswmathis.com Abstract Here I solve Goldbach's Conjecture by the simplest method possible. I do this by first calculating probabilites

More information

35 Chapter CHAPTER 4: Mathematical Proof

35 Chapter CHAPTER 4: Mathematical Proof 35 Chapter 4 35 CHAPTER 4: Mathematical Proof Faith is different from proof; the one is human, the other is a gift of God. Justus ex fide vivit. It is this faith that God Himself puts into the heart. 21

More information

Name Solutions Linear Algebra; Test 3. Throughout the test simplify all answers except where stated otherwise.

Name Solutions Linear Algebra; Test 3. Throughout the test simplify all answers except where stated otherwise. Name Solutions Linear Algebra; Test 3 Throughout the test simplify all answers except where stated otherwise. 1) Find the following: (10 points) ( ) Or note that so the rows are linearly independent, so

More information

CS 124 Math Review Section January 29, 2018

CS 124 Math Review Section January 29, 2018 CS 124 Math Review Section CS 124 is more math intensive than most of the introductory courses in the department. You re going to need to be able to do two things: 1. Perform some clever calculations to

More information

PROBLEM SET 3: PROOF TECHNIQUES

PROBLEM SET 3: PROOF TECHNIQUES PROBLEM SET 3: PROOF TECHNIQUES CS 198-087: INTRODUCTION TO MATHEMATICAL THINKING UC BERKELEY EECS FALL 2018 This homework is due on Monday, September 24th, at 6:30PM, on Gradescope. As usual, this homework

More information

HOW TO WRITE PROOFS. Dr. Min Ru, University of Houston

HOW TO WRITE PROOFS. Dr. Min Ru, University of Houston HOW TO WRITE PROOFS Dr. Min Ru, University of Houston One of the most difficult things you will attempt in this course is to write proofs. A proof is to give a legal (logical) argument or justification

More information

Sequences and infinite series

Sequences and infinite series Sequences and infinite series D. DeTurck University of Pennsylvania March 29, 208 D. DeTurck Math 04 002 208A: Sequence and series / 54 Sequences The lists of numbers you generate using a numerical method

More information

Lecture 6: Finite Fields

Lecture 6: Finite Fields CCS Discrete Math I Professor: Padraic Bartlett Lecture 6: Finite Fields Week 6 UCSB 2014 It ain t what they call you, it s what you answer to. W. C. Fields 1 Fields In the next two weeks, we re going

More information

Examples: P: it is not the case that P. P Q: P or Q P Q: P implies Q (if P then Q) Typical formula:

Examples: P: it is not the case that P. P Q: P or Q P Q: P implies Q (if P then Q) Typical formula: Logic: The Big Picture Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about time (and

More information

Discrete Mathematics and Probability Theory Fall 2013 Vazirani Note 1

Discrete Mathematics and Probability Theory Fall 2013 Vazirani Note 1 CS 70 Discrete Mathematics and Probability Theory Fall 013 Vazirani Note 1 Induction Induction is a basic, powerful and widely used proof technique. It is one of the most common techniques for analyzing

More information

( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of

( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of Factoring Review for Algebra II The saddest thing about not doing well in Algebra II is that almost any math teacher can tell you going into it what s going to trip you up. One of the first things they

More information

Proof Techniques (Review of Math 271)

Proof Techniques (Review of Math 271) Chapter 2 Proof Techniques (Review of Math 271) 2.1 Overview This chapter reviews proof techniques that were probably introduced in Math 271 and that may also have been used in a different way in Phil

More information

Invariants II. LA Math Circle (Advanced) November 15, 2015

Invariants II. LA Math Circle (Advanced) November 15, 2015 Invariants II LA Math Circle (Advanced) November 15, 2015 Recall that an invariant is some quantity associated with a system that is left unchanged by a specified process. We typically use them to show

More information

[Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty.]

[Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty.] Math 43 Review Notes [Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty Dot Product If v (v, v, v 3 and w (w, w, w 3, then the

More information

Math 425 Fall All About Zero

Math 425 Fall All About Zero Math 425 Fall 2005 All About Zero These notes supplement the discussion of zeros of analytic functions presented in 2.4 of our text, pp. 127 128. Throughout: Unless stated otherwise, f is a function analytic

More information

Mathematical induction

Mathematical induction Mathematical induction Notes and Examples These notes contain subsections on Proof Proof by induction Types of proof by induction Proof You have probably already met the idea of proof in your study of

More information

MAS114: Solutions to Exercises

MAS114: Solutions to Exercises MAS114: s to Exercises Up to week 8 Note that the challenge problems are intended to be difficult! Doing any of them is an achievement. Please hand them in on a separate piece of paper if you attempt them.

More information

MATH 2200 Final Review

MATH 2200 Final Review MATH 00 Final Review Thomas Goller December 7, 01 1 Exam Format The final exam will consist of 8-10 proofs It will take place on Tuesday, December 11, from 10:30 AM - 1:30 PM, in the usual room Topics

More information

Manual of Logical Style (fresh version 2018)

Manual of Logical Style (fresh version 2018) Manual of Logical Style (fresh version 2018) Randall Holmes 9/5/2018 1 Introduction This is a fresh version of a document I have been working on with my classes at various levels for years. The idea that

More information

MITOCW watch?v=y6ma-zn4olk

MITOCW watch?v=y6ma-zn4olk MITOCW watch?v=y6ma-zn4olk PROFESSOR: We have to ask what happens here? This series for h of u doesn't seem to stop. You go a 0, a 2, a 4. Well, it could go on forever. And what would happen if it goes

More information

Preptests 55 Answers and Explanations (By Ivy Global) Section 4 Logic Games

Preptests 55 Answers and Explanations (By Ivy Global) Section 4 Logic Games Section 4 Logic Games Questions 1 6 There aren t too many deductions we can make in this game, and it s best to just note how the rules interact and save your time for answering the questions. 1. Type

More information

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS Basic Concepts Paul Dawkins Table of Contents Preface... Basic Concepts... 1 Introduction... 1 Definitions... Direction Fields... 8 Final Thoughts...19 007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx

More information

No Solution Equations Let s look at the following equation: 2 +3=2 +7

No Solution Equations Let s look at the following equation: 2 +3=2 +7 5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

More information

Fundamentals of Mathematics I

Fundamentals of Mathematics I Fundamentals of Mathematics I Kent State Department of Mathematical Sciences Fall 2008 Available at: http://www.math.kent.edu/ebooks/10031/book.pdf August 4, 2008 Contents 1 Arithmetic 2 1.1 Real Numbers......................................................

More information

NIT #7 CORE ALGE COMMON IALS

NIT #7 CORE ALGE COMMON IALS UN NIT #7 ANSWER KEY POLYNOMIALS Lesson #1 Introduction too Polynomials Lesson # Multiplying Polynomials Lesson # Factoring Polynomials Lesson # Factoring Based on Conjugate Pairs Lesson #5 Factoring Trinomials

More information

3.1 Induction: An informal introduction

3.1 Induction: An informal introduction Chapter 3 Induction and Recursion 3.1 Induction: An informal introduction This section is intended as a somewhat informal introduction to The Principle of Mathematical Induction (PMI): a theorem that establishes

More information

CS280, Spring 2004: Final

CS280, Spring 2004: Final CS280, Spring 2004: Final 1. [4 points] Which of the following relations on {0, 1, 2, 3} is an equivalence relation. (If it is, explain why. If it isn t, explain why not.) Just saying Yes or No with no

More information

Divisibility = 16, = 9, = 2, = 5. (Negative!)

Divisibility = 16, = 9, = 2, = 5. (Negative!) Divisibility 1-17-2018 You probably know that division can be defined in terms of multiplication. If m and n are integers, m divides n if n = mk for some integer k. In this section, I ll look at properties

More information

Logic. Propositional Logic: Syntax. Wffs

Logic. Propositional Logic: Syntax. Wffs Logic Propositional Logic: Syntax Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about

More information

Chapter 1 Review of Equations and Inequalities

Chapter 1 Review of Equations and Inequalities Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve

More information

CS 4104 Data and Algorithm Analysis. CS4014: What You Need to Already Know. CS4104: What We Will Do. Introduction to Problem Solving (1)

CS 4104 Data and Algorithm Analysis. CS4014: What You Need to Already Know. CS4104: What We Will Do. Introduction to Problem Solving (1) Department of Computer Science Virginia Tech Blacksburg, Virginia Copyright c 010,017 by Clifford A. Shaffer Data and Algorithm Analysis Title page Data and Algorithm Analysis Clifford A. Shaffer Spring

More information

Logic. Propositional Logic: Syntax

Logic. Propositional Logic: Syntax Logic Propositional Logic: Syntax Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about

More information

a. See the textbook for examples of proving logical equivalence using truth tables. b. There is a real number x for which f (x) < 0. (x 1) 2 > 0.

a. See the textbook for examples of proving logical equivalence using truth tables. b. There is a real number x for which f (x) < 0. (x 1) 2 > 0. For some problems, several sample proofs are given here. Problem 1. a. See the textbook for examples of proving logical equivalence using truth tables. b. There is a real number x for which f (x) < 0.

More information

Math /Foundations of Algebra/Fall 2017 Foundations of the Foundations: Proofs

Math /Foundations of Algebra/Fall 2017 Foundations of the Foundations: Proofs Math 4030-001/Foundations of Algebra/Fall 017 Foundations of the Foundations: Proofs A proof is a demonstration of the truth of a mathematical statement. We already know what a mathematical statement is.

More information

Fall 2017 Test II review problems

Fall 2017 Test II review problems Fall 2017 Test II review problems Dr. Holmes October 18, 2017 This is a quite miscellaneous grab bag of relevant problems from old tests. Some are certainly repeated. 1. Give the complete addition and

More information

Egyptian Fractions: Part I

Egyptian Fractions: Part I Egyptian Fractions: Part I Prepared by: Eli Jaffe October 8, 2017 1 Cutting Cakes 1. Imagine you are a teacher. Your class of 10 students is on a field trip to the bakery. At the end of the tour, the baker

More information

Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur

Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Lecture 02 Groups: Subgroups and homomorphism (Refer Slide Time: 00:13) We looked

More information

Mathematics-I Prof. S.K. Ray Department of Mathematics and Statistics Indian Institute of Technology, Kanpur. Lecture 1 Real Numbers

Mathematics-I Prof. S.K. Ray Department of Mathematics and Statistics Indian Institute of Technology, Kanpur. Lecture 1 Real Numbers Mathematics-I Prof. S.K. Ray Department of Mathematics and Statistics Indian Institute of Technology, Kanpur Lecture 1 Real Numbers In these lectures, we are going to study a branch of mathematics called

More information

Seunghee Ye Ma 8: Week 1 Notes September 29, 2016

Seunghee Ye Ma 8: Week 1 Notes September 29, 2016 Week Notes Summary This week, we will learn about mathematical proofs. We start by defining what we mean by a mathematical proof and look at a few important things to avoid/keep in mind when writing mathematical

More information

Chapter III. Basic Proof Techniques. Proving the obvious has never been easy. Marty Rubin

Chapter III. Basic Proof Techniques. Proving the obvious has never been easy. Marty Rubin Chapter III Basic Proof Techniques Proving the obvious has never been easy. Marty Rubin The last two chapters were an introduction to the language of mathematics. Knowing the definitions and concepts of

More information

Mathematical Writing and Methods of Proof

Mathematical Writing and Methods of Proof Mathematical Writing and Methods of Proof January 6, 2015 The bulk of the work for this course will consist of homework problems to be handed in for grading. I cannot emphasize enough that I view homework

More information

ASTRO 114 Lecture Okay. We re now gonna continue discussing and conclude discussing the entire

ASTRO 114 Lecture Okay. We re now gonna continue discussing and conclude discussing the entire ASTRO 114 Lecture 55 1 Okay. We re now gonna continue discussing and conclude discussing the entire universe. So today we re gonna learn about everything, everything that we know of. There s still a lot

More information

Numbers, proof and all that jazz.

Numbers, proof and all that jazz. CHAPTER 1 Numbers, proof and all that jazz. There is a fundamental difference between mathematics and other sciences. In most sciences, one does experiments to determine laws. A law will remain a law,

More information

CS1800: Strong Induction. Professor Kevin Gold

CS1800: Strong Induction. Professor Kevin Gold CS1800: Strong Induction Professor Kevin Gold Mini-Primer/Refresher on Unrelated Topic: Limits This is meant to be a problem about reasoning about quantifiers, with a little practice of other skills, too

More information

Example. How to Guess What to Prove

Example. How to Guess What to Prove How to Guess What to Prove Example Sometimes formulating P (n) is straightforward; sometimes it s not. This is what to do: Compute the result in some specific cases Conjecture a generalization based on

More information

Solving Quadratic & Higher Degree Equations

Solving Quadratic & Higher Degree Equations Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,

More information