# 1 Introduction to Counting

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 1 Introduction to Counting 1.1 Introduction In this chapter you will learn the fundamentals of enumerative combinatorics, the branch of mathematics concerned with counting. While enumeration problems can be arbitrarily difficult, the fundamentals are easy to master and will prepare you for the more difficult problems ahead. You will be surprised at the degree of difficulty of the problems that you can solve using only these simple tools. Our goal is that this chapter be sufficient to meet the needs of students in MA1025, but some of you might want additional reading material or additional exercises. There are currently two copies of Discrete Mathematics and Its Applications, by Kenneth Rosen, on two-hour reserve in the library for the studetns in MA2025. By now, all of those students have their own copies, so it should be available. If you want more to read, and more problems to work, it is a good choice. The material in this chapter corresponds to Section 5.1, and Section 5.2 through page 335. A number of problems can be found at the end of this chapter. You will also find a list of suggested exercises from the Rosen text. Upon completion of this chapter, you will have mastered the following: The Sum Rule The Product Rule The basic Inclusion-Exclusion Principle You will also encounter a device called a tree diagram and a tool colloquially known as the Pigeonhole Principle, but the first is infrequently used and the second shows up in some very difficult problems, so I don t think mastery of those tools is the goal. 1.2 Overview and Definitions The sum rule tells us in how many ways one can make a single choice from two disjoint sets of alternatives. The product rule tells us in how many ways one can make one choice from each of two sets of alternatives. Both rules generalize to larger families of sets. The basic Principle of Inclusion-Exclusion extends the sum rule to situations in which the two sets of alternatives are not disjoint. This, too, generalizes to larger collections of sets. A finite sequence of choices can be represented by a tree diagram, in which the root represents the initial state, leaves represent outcomes, internal vertices represent intermediate states, and edges represent choices. The Pigeonhole Principle, in its picturesque form, says that if k + 1 pigeons fly into k pigeonholes, at least one pigeonhole must contain at least two pigeons.

2 1.3 The Sum and Product Rules The most fundamental rules are the Sum Rule and the Product Rule. Theorem 1 (Sum Rule) If A B =, then A B = A + B. Although the sum rule tells us that the cardinality of the union of two disjoint sets is the sum of the cardinalities of the two sets, it is typically applied to problems that do not immediately remind us of sets. Example 1: Suppose that you are in a restaurant, and are going to have either soup or salad but not both. There are two soups and four salads on the menu. How many choices do you have? By the Sum Rule, you have = 6 choices. Theorem 2 (Product Rule) For any choice of sets A and B, A B = A B. The product rule tells us that the cardinality of the Cartesian Product of two sets is the product of the cardinality of the two sets. Back in that same restaurant, there are 2 4 = 8 ways to have both soup and salad. Both rules generalize to larger numbers of sets, although the generalization of the sum rule requires that the sets in question are pairwise disjoint. Example 2: The first two of these problems apply the generalized Product Rule. We begin with a couple of definitions: An alphabet is a finite set of symbols. A string of length k over an alphabet A is a finite sequence a 1 a 2...a k of symbols from A, with repetition allowed. (a) How many distinguishable strings of length 3 over the alphabet {A, B,..., Z} exist? Solution: Since there are 26 choices for the first symbol, 26 for the second, and 26 for the third, then by the Product Rule there are 26 3 = such strings of length 3. (b) How many strings of length 4 over the alphabet {0, 1,..., 9} do not begin with 0? Solution: There are nine choices for the first symbol, and ten for each of the second, third, and fourth. By the Product Rule, there are = 9000 strings of length 4 that do not begin with 0. (Note: this is also the number of 4-digit positive integers with no leading zeros.) (c) The standard California license-plate number begins with a nonzero decimal digit that is followed by three uppercase alpha characters, which are in turn followed by three decimal digits. How many of these license numbers exist? Solution: By the preceding problems and the basic Product Rule, there are = such numbers. The fact that we are inserting the string of alpha characters between the first and second digits of the string of decimal digits has no effect on the count. 2

4 once in B. This applies to every element in A B, so we must subtract A B to correct the overcount. The principle, then, tells us that A B = A + B A B. Example 5: The following four problems apply the principle of inclusion-exclusion. (a) Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}. Then A B = {1, 2, 3, 4, 5, 6, 7}. By inspection, A B = 7, but you can also verify that A B = 7 = = A + B A B. (b) How many positive integers not bigger than 20 are divisible by either 2 or 3? Solution: There are 20/2 = 10 that are divisible by 2, and 20/3 = 6 that are divisible by 3. But there are also 20/6 = 3 that are divisible by both 2 and 3, so the total is = 13. (c) How many bitstrings of length eight either begin with 00 or end with 101? Solution: There are 2 6 that begin with 00, 2 5 that end with 101, and 2 3 that start with 00 and end with 101. So the number of bitstrings with at least one of the two properties is = 88. You can probably convince yourself that the inclusion-exclusion formula for the cardinality of the union of three sets is A B C = A + B + C A B A C B C + A B C. The principle generalizes to more than three sets, and there is an inductive proof of that generalization. 1.5 Tree Diagrams A tree diagram offers a way to enumerate outcomes resulting from a finite (and preferably small) sequence of choices. Each vertex represents a state, with the initial state represented by the root and the outcomes represented by the leaves. Edges tending downward from a vertex represent choices. Figure 1 shows a decision tree used to enumerate bitstrings of length three that do not contain 11. The leaves, from left to right, represent the strings 101, 100, 010, 001, and 000. You can see that any branch containing two consecutive 1s has been pruned out, leaving only those that do not contain 11. And you can probably guess that the utility of tree diagrams, like that of truth tables and Venn diagrams, is limited to small problems. On the other hand, the tree structure lends itself to computation, so you will probably see this again. 4

5 root Figure 1: A decision tree for counting 11-free strings 1.6 The Pigeonhole Principle The Pigeonhole Principle has as picturesque a statement as any in combinatorics, and in its most basic form the proof is so intuitive as to be obvious. But there are many forms to this principle, and putting it to use can be arduous. Our focus is on relatively easy applications of the principle. Here it is in its simple form: Theorem 3 (The Simplified Pigeonhole Principle) If k pigeons occupy k 1 pigeonholes, then at least one pigeonhole contains at least two pigeons. A more useful presentation is the following: Theorem 4 (The Pigeonhole Principle) If k pigeons occupy j < k pigeonholes, then at least one pigeonhole contains at least two pigeons. This can be restated in terms of functions: If A and B are sets, and if A > B, then there can be no one-to-one mapping f : A B. The proof is straightforward. In applying the principle, the difficulty (when there is one) is in deciding what constitutes a pigeonhole and what constitutes a pigeon. Example 6: (a) The hello, world problem for the pigeonhole principle is the sock problem : In your dresser drawer you have a jumble of socks in two colors, say blue and gray. It s dark, and you don t want to wake your spouse. How many socks must you grab to guarantee that you have a pair of the same color? Solution: Three socks suffice. You might end up with three blue, or three gray, but with only two colors you re guaranteed to have at least two blue or at least two gray. That is, the pigeonholes are the colors, the the pigeons are the socks. As you draw one sock from the drawer, you put it in the respective pigeonhole, and one pigeonhole will have at least two socks. 5

6 (b) Show that in a group of eight people there must be two whose birthdays fall on the same day of the week. Solution: The pigeons are now the people in the group, and the pigeonholes are the days of the week. (c) Show that in a group of ten people there must be two whose birthdays fall on the same day of the week. Solution: The pigeons are now the people in the group, and the pigeonholes are the days of the week. (d) Show that if five integers are selected from the set A = {1, 2, 3, 4, 5, 6, 7, 8}, there must be two whose sum is 9. Solution: Consider the four pairs {1, 8}, {2, 7}, {3, 6}, and {4, 5}. Those are the pigeonholes. The pigeons are the five selected integers. Two must be elements of the same pigeonhole. A generalization of the principle is this: Theorem 5 If n pigeons occupy k pigeonholes, then at least one pigeonhole contains at least n/k pigeons. We can use this version to answer more difficult questions. What is the smallest n such that at least one of k boxes must contain at least r of n objects? By Theorem 4, we need n/k r. So the smallest integer n that forces some box to contain r of n objects is n = k(r 1) + 1. How can you avoid having at least r objects in some box? By setting n < r, of course. But by setting n k(r 1), you stand a chance at having fewer than r objects in every box. Example 7: (a) In your dresser drawer you have a jumble of socks in two colors, say blue and gray. It s dark, and you don t want to wake your spouse. How many socks must you grab to guarantee that you have 3 socks of the same color? Solution: Five socks suffice. You might end up with more than three blue, or three gray, but with only two colors you re guaranteed to have at least three blue or at least three gray. That is, the pigeonholes are the colors, the the pigeons are the socks. As you draw one sock from the drawer, you put it in the respective pigeonhole, and one pigeonhole will have at least three socks. (b) Show that in a group of 25 people there must be four whose birthdays fall on the same day of the week. Solution: The pigeons are now the people in the group, and the pigeonholes are the days of the week. And so 25 = 4 people share the same day of the week. 7 6

7 1.6.1 Exercises Here are some exercises that apply the techniques presented in this chapter. 1. A store has t-shirts in 6 different styles and 4 different sizes. How many different kinds of t-shirts does the store have? 2. In Example 2, we described the standard California license-plate number. How many such numbers have (a) no repeated digit? (b) no repeated letter? (c) no repeated symbol? 3. In how many ways can a ballot be validly marked if a person is to vote on three questions, and at least one question must be answered (i.e. one or two questions may be skipped, but not all three of them), if for question 1 there are four options to choose from, for question 2 there are two options to choose from, and two more options to chose from for question 3? 4. How many integers n, with 1 n 200, are not divisible by 2, 3, or 5? 5. Given five points in the plane, with integer coordinates, prove that there are two with the property that the midpoint of the line segment joining them has integer coordinates. 6. Let S be a six-element subset of {1, 2,..., 14}. Show that there are two proper nonempty subsets of S, say A and B such that the sum of the values of the elements of A is the same as the sum of the values of the elements of B. Recall that A is a proper subset of B if A B. 7. Use the principle of mathematical induction to generalize the product rule to arbitrarily many tasks. **************************************************************** Solutions to Exercises, Chapter 1 of Combinatorics Notes for MA1025 **************************************************************** 1. A store has t-shirts in 6 different styles and 4 different sizes. How many different kinds of t-shirts does the store have? Solution: There are = 6 4 = 24 (6 choices for each size). 2. In Example 2, we described the standard California license-plate number. How many such numbers have 7

8 (a) no repeated digit? (b) no repeated letter? (c) no repeated symbol? Solution: (a) There are still 26 3 ways to choose the alpha characters, but now there are 9 choices for the first digit, 9 for the second, 8 for the third, and 7 for the fourth. In all, there are such numbers. (b) There are ways to choose the digits, but now there are ways to choose the letters. In all, there are such numbers. (c) Combining the results from (a) and (b), there are license numbers with no repeated symbols. 3. In how many ways can a ballot be validly marked if a person is to vote on three questions, and at least one question must be answered (i.e. one or two questions may be skipped, but not all three of them), if for question 1 there are four options to choose from, for question 2 there are two options to choose from, and two more options to chose from for question 3? Solution: There are 5 ways to answer question 1 (one of the 4 options, or skipping that question). Then there are 3 ways to answer each of the questions 2 and 3. However we counted the option that no question was answered wich is not a valid ballot. Thus the are = 53 different possible valid ballots. 4. How many integers n, with 1 n 200, are not divisible by 2, 3, or 5? Solution: The number of integers n, with 1 n 200, that are divisible by at least one of 2, 3, or 5 is = = 146, so the number of integers in {1, 2,..., 200} divisible by none of them is = Given five points in the plane, with integer coordinates, prove that there are two with the property that the midpoint of the line segment joining them has integer coordinates. Proof: Given two integer coordinate pairs (x 1, y 1 ) and (x 2, y 2 ), the midpoint of the line segment joining them has coordinates ( x 1+y 1, y 1+y 2 ), which is an integer pair only if both 2 2 x 1 + x 2 and y 1 + y 2 are even, which requires that the parity of x 1 agrees with that of x 2 and that the parity of y 1 agrees with that of y 2. There are four parity patterns that an ordered pair might have. These are (even,even), (odd,odd), (even,odd), and (odd,even). Given five ordered pairs, two must share one of these patterns, but then the midpoint of the line segment joining them has integer coordinates. 8

10 6. At a party, there are n people for some n 2. Show that there must be two people at the party who know precisely the same number of other people at the party. Self-Quiz Solutions: Fundamentals of Counting 1. Bud s Grill offers an earlybird dinner special, which includes one of four entrees, either a soup (two kinds) or a salad (four choices), and a dessert (three choices). In how many different ways can one order all three courses from the special menu? Solution: A hypothetical diner can order an entree in four ways, one of six first courses, and one of three desserts. By the product rule, there are = 72 ways to order. 2. How many functions are there from a 5-element set A to a 6-element set B? How many are one-to-one? Solution: Let A = {a 1, a 2,..., a 5 }. In the first case, there are 6 choices for the image of each element a i A, so by the product rule there are 6 5 = 7776 such functions. If the function must be one-to-one, there are six choices for the image of a 1, then five choices for the image of a 2, etc. So there are = 6! = 720 one-to-one functions from A to B. 3. How many subsets of {1, 2,..., 8} contain more than one element? Solution: All except the empty set and the eight singletons, so altogether = 247 subsets contain more than one element. 4. How many bitstrings of length 8 either start with 01 or end with 01? Solution: There are 2 6 that start with 01, 2 6 that end with 01, and 2 4 with both properties, so the number in question is = How many numbers must be selected from {1, 2,..., 10} to guarantee that there is a pair whose sum is 11? How many, if the sum is to be 13? Solution: There are five pairs ({1, 10}, {2, 9}, {3, 8}, {4, 7}, and {5, 6}) that add up to 11, so we must choose six numbers to guarantee that we have both elements in one of the five sets. There are four pairs ({3, 10}, {4, 9}, {5, 8}, and {6, 7}) that add up to 13, and there are also the numbers 1 and 2, neither of which combines with another element of the set to make 13. So we must choose seven numbers to guarantee that two of them add up to At a party, there are n people for some n 2. Show that there must be two people at the party who know precisely the same number of other people at the party. Proof: The number of people known by each celebrant (these are the pigeons) is an integer between 0 and n 1 (these are the labels on the pigeonholes). So it appears that we have ten pigeonholes for ten pigeons. But now there are two cases. If any 10

11 individual knows nobody, then the pigeonhole labeled n 1 must be empty. Otherwise, the pigeonhole labeled 0 must be empty. In either case, we have n pigeons but only n 1 pigeonholes. One of them must contain at least two pigeons. 11

### Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting

Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Colin Stirling Informatics Slides originally by Kousha Etessami Colin Stirling (Informatics) Discrete Mathematics (Chapter 6) Today 1 /

### CS 441 Discrete Mathematics for CS Lecture 16. Counting. CS 441 Discrete mathematics for CS. Course administration

CS 44 Discrete Mathematics for CS Lecture 6 Counting Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Course administration Midterm exam: Thursday, March 6, 24 Homework assignment 8: due after Spring

### 1 Introduction. 2 Basic Principles. 2.1 Multiplication Rule. [Ch 9] Counting Methods. 400 lecture note #9

400 lecture note #9 [Ch 9] Counting Methods 1 Introduction In many discrete problems, we are confronted with the problem of counting. Here we develop tools which help us counting. Examples: o [9.1.2 (p.

### Section Summary. The Product Rule The Sum Rule The Subtraction Rule The Division Rule Examples, Examples, and Examples Tree Diagrams

Chapter 6 Chapter Summary The Basics of Counting The Pigeonhole Principle Permutations and Combinations Binomial Coefficients and Identities Generalized Permutations and Combinations Generating Permutations

### The Basics of Counting. Niloufar Shafiei

The Basics of Counting Niloufar Shafiei Counting applications Counting has many applications in computer science and mathematics. For example, Counting the number of operations used by an algorithm to

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

### 94 Counting Solutions for Chapter 3. Section 3.2

94 Counting 3.11 Solutions for Chapter 3 Section 3.2 1. Consider lists made from the letters T, H, E, O, R, Y, with repetition allowed. (a How many length-4 lists are there? Answer: 6 6 6 6 = 1296. (b

### Pigeonhole Principle Solutions

Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such

### Arrangements of Objects

Section 6.1 Arrangements of Objects The study of arrangements of objects is called combinatorics Counting objects in these arrangements is an important part of combinatorics as well as in other disciplines

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### The Pigeonhole Principle

The Pigeonhole Principle 1 Pigeonhole Principle: Simple form Theorem 1.1. If n + 1 objects are put into n boxes, then at least one box contains two or more objects. Proof. Trivial. Example 1.1. Among 13

### A set is a Many that allows itself to be thought of as a One. (Georg Cantor)

Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains

### 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......................................

### Answer: (a) Since we cannot repeat men on the committee, and the order we select them in does not matter, ( )

1. (Chapter 1 supplementary, problem 7): There are 12 men at a dance. (a) In how many ways can eight of them be selected to form a cleanup crew? (b) How many ways are there to pair off eight women at the

### Applications of Methods of Proof

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

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### 136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### 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

### 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

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Sets. A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object.

Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: real numbers, complex numbers, C integers, All students in

### Sets, Relations and Functions

Sets, Relations and Functions Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org epacuit@umd.edu ugust 26, 2014 These notes provide a very brief background in discrete

### 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,

### Reading 13 : Finite State Automata and Regular Expressions

CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model

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

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

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

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

### Lecture 17 : Equivalence and Order Relations DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

### Discrete Mathematics: Homework 6 Due:

Discrete Mathematics: Homework 6 Due: 2011.05.20 1. (3%) How many bit strings are there of length six or less? We use the sum rule, adding the number of bit strings of each length to 6. If we include the

### (Refer Slide Time: 1:41)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must

### Notes. Sets. Notes. Introduction II. Notes. Definition. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry.

Sets Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 1.6 1.7 of Rosen cse235@cse.unl.edu Introduction

### 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

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

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

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

### (Refer Slide Time: 01.26)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 27 Pigeonhole Principle In the next few lectures

### (Refer Slide Time 1.50)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Module -2 Lecture #11 Induction Today we shall consider proof

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

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

### 1(a). How many ways are there to rearrange the letters in the word COMPUTER?

CS 280 Solution Guide Homework 5 by Tze Kiat Tan 1(a). How many ways are there to rearrange the letters in the word COMPUTER? There are 8 distinct letters in the word COMPUTER. Therefore, the number of

### Automata and Languages

Automata and Languages Computational Models: An idealized mathematical model of a computer. A computational model may be accurate in some ways but not in others. We shall be defining increasingly powerful

### Sets and set operations

CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used

### Just the Factors, Ma am

1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive

### NOTES ON COUNTING KARL PETERSEN

NOTES ON COUNTING KARL PETERSEN It is important to be able to count exactly the number of elements in any finite set. We will see many applications of counting as we proceed (number of Enigma plugboard

### The Pigeonhole Principle

Section 6.2 The Pigeonhole Principle If a flock of 20 pigeons roosts in a set of 19 pigeonholes, one of the pigeonholes must have more than 1 pigeon. It is an important proof principle The Pigeonhole Principle

### 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 Mathematical Induction. October 27, 2016

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

### 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

### 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,

Some Definitions about Sets Definition: Two sets are equal if they contain the same elements. I.e., sets A and B are equal if x[x A x B]. Notation: A = B. Recall: Sets are unordered and we do not distinguish

### 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

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### Automata and Formal Languages

Automata and Formal Languages Winter 2009-2010 Yacov Hel-Or 1 What this course is all about This course is about mathematical models of computation We ll study different machine models (finite automata,

### 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

### Lecture 16 : Relations and Functions DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

### PROBLEM SET 7: PIGEON HOLE PRINCIPLE

PROBLEM SET 7: PIGEON HOLE PRINCIPLE The pigeonhole principle is the following observation: Theorem. Suppose that > kn marbles are distributed over n jars, then one jar will contain at least k + marbles.

### SETS, RELATIONS, AND FUNCTIONS

September 27, 2009 and notations Common Universal Subset and Power Set Cardinality Operations A set is a collection or group of objects or elements or members (Cantor 1895). the collection of the four

### This chapter describes set theory, a mathematical theory that underlies all of modern mathematics.

Appendix A Set Theory This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. A.1 Basic Definitions Definition A.1.1. A set is an unordered collection of elements.

### 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,

### 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

### 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

### CS 341 Homework 9 Languages That Are and Are Not Regular

CS 341 Homework 9 Languages That Are and Are Not Regular 1. Show that the following are not regular. (a) L = {ww R : w {a, b}*} (b) L = {ww : w {a, b}*} (c) L = {ww' : w {a, b}*}, where w' stands for w

### Basics of Counting. The product rule. Product rule example. 22C:19, Chapter 6 Hantao Zhang. Sample question. Total is 18 * 325 = 5850

Basics of Counting 22C:19, Chapter 6 Hantao Zhang 1 The product rule Also called the multiplication rule If there are n 1 ways to do task 1, and n 2 ways to do task 2 Then there are n 1 n 2 ways to do

### . 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

### Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

### 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

### Mathematical induction & Recursion

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

### 6.2 Unions and Intersections

22 CHAPTER 6. PROBABILITY 6.2 Unions and Intersections The probability of a union of events Exercise 6.2- If you roll two dice, what is the probability of an even sum or a sum of 8 or more? Exercise 6.2-2

### 6.2 Permutations continued

6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

### 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

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

### It is not immediately obvious that this should even give an integer. Since 1 < 1 5

Math 163 - Introductory Seminar Lehigh University Spring 8 Notes on Fibonacci numbers, binomial coefficients and mathematical induction These are mostly notes from a previous class and thus include some

### Mathematical Induction

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

### Sets, Venn Diagrams & Counting

MT 142 College Mathematics Sets, Venn Diagrams & Counting Module SC Terri Miller revised January 5, 2011 What is a set? Sets set is a collection of objects. The objects in the set are called elements of

### MA3059 Combinatorics and Graph Theory

MA3059 Combinatorics and Graph Theory Martin Stynes Department of Mathematics, UCC January 23, 2007 Abstract These notes are designed to accompany the textbook A first course in discrete mathematics by

### 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:

### Relations Graphical View

Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary

### The chromatic spectrum of mixed hypergraphs

The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex

### Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.

Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square

### NOTES: Justify your answers to all of the counting problems (give explanation or show work).

Name: Solutions Student Number: CISC 203 Discrete Mathematics for Computing Science Test 3, Fall 2010 Professor Mary McCollam This test is 50 minutes long and there are 40 marks. Please write in pen and

### 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

### HMMT February Saturday 21 February Combinatorics

HMMT February 015 Saturday 1 February 015 1. Evan s analog clock displays the time 1:13; the number of seconds is not shown. After 10 seconds elapse, it is still 1:13. What is the expected number of seconds

### Module 6: Basic Counting

Module 6: Basic Counting Theme 1: Basic Counting Principle We start with two basic counting principles, namely, the sum rule and the multiplication rule. The Sum Rule: If there are n 1 different objects

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

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

### Georg Cantor (1845-1918):

Georg Cantor (845-98): The man who tamed infinity lecture by Eric Schechter Associate Professor of Mathematics Vanderbilt University http://www.math.vanderbilt.edu/ schectex/ In papers of 873 and 874,

### Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

### 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

### Solutions to In-Class Problems Week 4, Mon.

Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics for Computer Science September 26 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised September 26, 2005, 1050 minutes Solutions

### 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...............................

### Principle of (Weak) Mathematical Induction. P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n))

Outline We will cover (over the next few weeks) Mathematical Induction (or Weak Induction) Strong (Mathematical) Induction Constructive Induction Structural Induction Principle of (Weak) 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)

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

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

### We give a basic overview of the mathematical background required for this course.

1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The

### Chapter 4. Trees. 4.1 Basics

Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

### Chapter 5 - Probability

Chapter 5 - Probability 5.1 Basic Ideas An experiment is a process that, when performed, results in exactly one of many observations. These observations are called the outcomes of the experiment. The set

### List of Standard Counting Problems

List of Standard Counting Problems Advanced Problem Solving This list is stolen from Daniel Marcus Combinatorics a Problem Oriented Approach (MAA, 998). This is not a mathematical standard list, just a

### Axiom A.1. Lines, planes and space are sets of points. Space contains all points.

73 Appendix A.1 Basic Notions We take the terms point, line, plane, and space as undefined. We also use the concept of a set and a subset, belongs to or is an element of a set. In a formal axiomatic approach