Math 213 F1 Final Exam Solutions Page 1 Prof. A.J. Hildebrand

Save this PDF as:

Size: px
Start display at page:

Download "Math 213 F1 Final Exam Solutions Page 1 Prof. A.J. Hildebrand"

Transcription

1 Math 213 F1 Final Exam Solutions Page 1 Prof. A.J. Hildebrand 1. For the questions below, just provide the requested answer no explanation or justification required. As always, answers should be left in raw, unevaluated form, involving binomials, factorials, powers, etc. (such as 3 ) 2 3 ). Do not use notations C(n, k) or P (n, k) in your answers. All of the questions have a simple answer and require virtually no hand computations. Write legibly and circle or box your answer. (a) How many elements does the power set of {1, 2,..., 10} have? 2 10 (b) What is the number of reflexive relations on the set A = {1, 2,..., 10}? = 2 90 (c) What is the number of bijections from the set {1, 2,..., 10} to the set {1, 2,..., 10}? 10! (d) What is the number of one-to-one functions from the set {1, 2, 3} to the set {1, 2,..., 10}? = 10!/7! (e) What is the constant term (i.e., the coefficient of x 0 ) in the expansion of (2x 3 1 x )100. ( ) (f) Evaluate the sum x x 99 (1 x) 1 + x 98 (1 x) 2 + x 97 (1 x) x 1 (1 x) 99 + (1 x) 100 for 0 < x < 1. x 101 (1 x) 101 x (1 x) 2. Suppose you roll an ordinary die 10 times. Determine the following probabilities. As always, leave your answers in raw form, in terms of fractions, factorials, or binomial coefficients. (a) The probability that at least 2 sixes come up. By the complement trick and the binomial distribution, this is 1 0 ) ( 5 ) 10 (b) The probability that exactly 4 sixes come up. p = 1/): 1 ) 1 ( ) 9 5 This is a standard binomial probability (with n = 10, k = 4, and 4 ) 4 ( ) 5 (c) The probability that the 4th six occurs at the last (i.e., 10-th) roll: The given event is equivalent to getting (i) a six in trial 10 (which as prob. 1/) and (ii) exactly 3 sixes in trials 1 9 (which has probability ( 9 3) (1/) 3 (5/). Multiplying these two probabilities gives the answer: ( 9 3 ) 4 ( ) 5 (d) The probability that a six comes up in rolls 1, 4, and 8, but in none of the other rolls. This corresponds to getting the specific success/failure sequence SFFSFFFSFF, which has probability (1/) 3 (5/) 7. (e) The probability that exactly two distinct numbers show up in the ten rolls. This is similar to a wordcounting example worked out in class. First pick the two numbers ( ( 2) ways), then multiply by the number of 10-letter words ( ) consisting of exactly these two numbers ( ): 10 (2 10 2) 2

2 Math 213 F1 Final Exam Solutions Page 2 Prof. A.J. Hildebrand (f) The probability that exactly three 2 s, exactly three 4 s and exactly four s show up in the ten rolls. The numerator in this probability is the number of 10-letter words consisting of three 2 s, three 4 s, and four s. There are 10!/(3!3!4!) of these, and dividing by 10 10! gives the probability 10 3!3!4! 3. The questions below are independent of each other. Provide the requested answer, along with a brief explanation/justification (e.g., by citing an appropriate theorem). (a) Does there exist a graph with 9 vertices, with degree sequence 1, 1, 2, 2, 3, 3, 4, 4, 5? Justify your answer! NO By the Handshake Theorem, the number of odd-degree vertices must be even. The given graph has an odd number of vertices with an odd degree. (b) If a planar graph has 12 vertices, all of the same degree d, and divides the plane into 20 regions, what is the degree d of the vertices? Justify your answer. By Euler s formula, the graph has e = r + v 2 = = 30 edges. By the Handshake Theorem, 2e = vd = 12d, so d = 2e/12 = 2 30/12 = 5. (c) Given a positive integer n, consider the graph Q n whose vertices are bit strings of length n, and such that two bitstrings are connected by an edge if and only if they differ in exactly one bit. Answer the following questions and give a brief justification/explanation (one sentence is enough). A. How many vertices does this graph have? 2 n (number of binary strings of length n) B. How many edges does this graph have? n 2 n 1. A vertex is connected to exactly n other vertices, namely those in which exactly one of the n bits of this of the bitstring representing this vertex is flipped. Hence the degree of the vertices is n. Since there are 2 n vertices, by the Handshake Theorem, the number of edges is e = (1/2)2 n n = n2 n 1. C. For which values of n does this graph have an Euler circuit? An Euler circuit exists if and only if all degrees are even. Since in the graph Q n all degrees are n, an Euler circuit exists if and only if n is even (d) Complete the following definition by filling in the blanks with the appropriate graph-theoretic term (consisting of exactly two words). A Gray code is a in the graph Q n. A Gray code is a Hamilton circuit in Q n For which values of n does there exist a Gray code? (Just state the answer. No justification required for this part.) For all n. (e) The Königsberg bridge problem (see picture) can be modeled by a certain graph. Draw this graph (in the usual way, with dots as vertices and lines/curves as edges), and find its adjacency matrix. The regions A,B,C,D in the graph correspond to the vertices, and the bridges to the edges. The adjacency matrix, with rows and columns labeled A,B,C,D, ) is. (

3 Math 213 F1 Final Exam Solutions Page 3 Prof. A.J. Hildebrand 4. Evaluate the following sums and products (where n denotes a (general) positive integer). The answer should be a simple function of n (e.g., a polynomial, factorial, binomial coefficients, or a power, or some simple combination of these). (a) (b) (c) n (n i) n n n + i i n n i n(n 1) 2 (2n)! n! = ( ) 2n 2 n n n(n 1)/2 (d) ( 1) n 3 n 1 ( 3) n+1 1 ( 3) = 1 4 (1 + ( 1)n+1 3 n+1 ) (geom. series formula) (e) n + 1 4n n n 4 + (Note that this is an infinite sum.) 1 1 (1/2n) = 5. The following problems are independent of each other. (a) Solve the recurrence a n = 2n 2 a n 1 with initial condition a 0 = 1. By iteration, a n = 2n 2 a n 1 = 2 2 n 2 (n 1) 2 a n 2 = = 2 n n 2 (n 1) a 0 = 2 n (n!) 2. 2n 2n 1 (b) Find the generating function of the sequence defined by a 0 = 1 and a n = 3a n for n 1. (No need to solve the recurrence. Just find the generating function.) (Cf. 7.4:33) G(x) = 1 1 3x +. The following questions are independent of each other. 2x (1 x)(1 3x). (a) Consider the relation R on the set of positive integers defined by (x, y) R if x and y have no common prime factor. Determine whether this relation is (a) reflexive, (b) symmetric, (c) transitive. If the property holds, just say so; if it does not hold, give a specific counterexample. Reflexive: NO. Counterexample: x = 2. 2 has a common prime factor 2 with itself, so (2, 2) R. Symmetric: YES. (Obvious.) Transitive: NO. Let x = 2, y = 1, and z = 2. Then x and y have no common prime factor, so (x, y) R, y and z have no common prime factor, so (y, z) R, but x and z have common prime factor 2, so (x, y) R. (b) Write down a formula for the number of positive integers less than or equal to 1000 that are divisible by none of the numbers 5, 7, and 11? The answer should be in raw form and involve expressions like By inclusion/exclusion, the number sought is For the following questions, either give a specific example with the requested properties, or explain (in one or two sentences) why no such example exists. (a) A function from Z + to Z that is a bijection. (b) A function from R to Z that is onto. Example: the mapping 1 0, 2 1, 3 1, 4 2, 5 2, etc.

4 Math 213 F1 Final Exam Solutions Page 4 Prof. A.J. Hildebrand Example: The floor function f(x) = x. (c) An infinite set of real numbers in the interval [0, 1] that is countable. Example: The rational numbers in this interval.. 8. The following questions are independent of each other. (a) A partition of a set A is a collection of nonempty subsets A i of A satisfying certain properties. State these properties. The properties defining a partition of S are: (i) Pairwise disjoint: A i A j = for all i j. (ii) Union is A: A 1 A 2 A 3 = A. (b) The equivalence class of 213 in the congruence relation modulo 7 is a certain set. Write out this set explicitly (by listing its elements), using proper set-theoretic notation. You can use ellipses (... ), provided the pattern is completely clear. {3, 10, 17, 24,..., 4, 11, 18,... } (c) Complete the following definition: A function f(x) is said to be of order O(g(x)) with witnesses C and k if f(x) C g(x) for all x k 9. A test has 5 questions, each with possible scores of 0, 1, 2, or 3 points. The scores are recorded on a score sheet that shows the number of points the student obtained on each question just like the table on the cover page of this exam. Two score sheets are considered equal if and only if, on each of the questions, both sheets show the same score. (Thus, for example, two students receiving scores of 2,1,1,3,0 and 1,2,1,3,0, respectively, on Questions 1 5 would have different score sheets since the first got 2 points on Question 1 and 1 point on Question 2 and the second got 1 point on Question 1 and 2 points on Question 2. The following questions depend on these assumptions, but are otherwise independent of each other. (a) What is the probability that, in a class of 40, there are at least two students with the same score sheets? A score sheet can be represented by a tuple (s 1, s 2, s 3, s 4, s 5 ), with s i {0, 1, 2, 3}. There are 4 5 = 2 10 = 1024 such tuples, so there are 1024 possible score sheets. The given problem is a birthday type problem with the 1024 possible score sheets corresponding to the 35 possible birthdays. The probability that all score sheets are distinct is ( )/ , and the probability that at least two score sheets are identical is 1 minus the above, i.e., Remark: The numerical value of the above quantity is ). Thus, in a class of 40 there is already a greater than chance that two score sheets are identical. That only 40 students should be needed for this is surprising, but consistent with the birthday paradox. (b) What is the minimal number of students in a class needed in order to guarantee that there are two students with identical score sheets? Clearly explain/justify your reasoning, citing appropriate theorems if necessary.

5 Math 213 F1 Final Exam Solutions Page 5 Prof. A.J. Hildebrand This is pigeonhole problem. Letting the pigeonholes be the score sheets, and the pigeons the students, the pigeonhole principle guarantees that if the number of students is greater than the number of score sheets, then there will be at least two students with the same score sheet. By the first part, there are 1024 different score sheets, so the number of students needed is one more, namely (That this is indeed the minimal such number is clear, since if there are no more students than score sheets, each can obviously have a different score sheet. 10. Let F n = n f i, where f 1 = 1, f 2 = 1, f 3 = 2, f 4 = 3, f 5 = 5, f = 8, f 7 = 13,... are the Fibonacci numbers. Use induction to prove that F n = f n+2 1 for all positive integers n. Your write-up of the induction proof must be mathematically rigorous and complete, be presented in logical order with all necessary steps and explanations where needed (e.g., say things like by the definition of..., by the induction hypothesis, or by algebra at the appropriate places). A list of disconnected formulas does not constitute a proof. Let ( ) denote the formula F n = f n+2 1 we seek to prove.. Base step: For n = 1, we have F 1 = 1 = 2 1 = f 3 1, so ( ) holds in this case. Inductive step: Assume ( ) holds for n = k, where k is a positive integer. Then k+1 F k+1 = f i (by def. of F k+1 ) = k f i + f k+1 (algebra) = F k + f k+1 (by def. of F k ) = f k f k+1 (by ind. hyp.) = f k+3 1 (by the recurrence for f n ). This proves that ( ) holds for n = k + 1, so the induction is complete. Conclusion: By the principle of induction, ( ) holds for all positive integers n.

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

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

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.

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......................................

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

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

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

Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way

Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)

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

Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

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

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

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

MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

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...............................

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

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

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

1 Introduction to Counting

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

Spring 2007 Math 510 Hints for practice problems

Spring 2007 Math 510 Hints for practice problems Section 1 Imagine a prison consisting of 4 cells arranged like the squares of an -chessboard There are doors between all adjacent cells A prisoner in one

Week 5: Binary Relations

1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: 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

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

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

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

k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

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)

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

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

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

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

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

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

, for x = 0, 1, 2, 3,... (4.1) (1 + 1/n) n = 2.71828... b x /x! = e b, x=0

Chapter 4 The Poisson Distribution 4.1 The Fish Distribution? The Poisson distribution is named after Simeon-Denis Poisson (1781 1840). In addition, poisson is French for fish. In this chapter we will

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

Lecture 1: Course overview, circuits, and formulas

Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik

Introduction to Graph Theory

Introduction to Graph Theory Allen Dickson October 2006 1 The Königsberg Bridge Problem The city of Königsberg was located on the Pregel river in Prussia. The river divided the city into four separate

Overview of Math Standards

Algebra 2 Welcome to math curriculum design maps for Manhattan- Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse

Mathematics for Computer Science

Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: patrick.blackburn@loria.fr Course website: http://www.loria.fr/~blackbur/courses/math

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

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

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

Connectivity and cuts

Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every

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,

Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

3 Some Integer Functions

3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple

Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

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

Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

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

. 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

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

Discrete Mathematics. Hans Cuypers. October 11, 2007

Hans Cuypers October 11, 2007 1 Contents 1. Relations 4 1.1. Binary relations................................ 4 1.2. Equivalence relations............................. 6 1.3. Relations and Directed Graphs.......................

POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

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

Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

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

Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

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

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

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

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

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. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

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

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

LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

Finite and discrete probability distributions

8 Finite and discrete probability distributions To understand the algorithmic aspects of number theory and algebra, and applications such as cryptography, a firm grasp of the basics of probability theory

Mathematical Induction

Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How

Homework until Test #2

MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

We can express this in decimal notation (in contrast to the underline notation we have been using) as follows: 9081 + 900b + 90c = 9001 + 100c + 10b

In this session, we ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should

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

Math 223 Abstract Algebra Lecture Notes

Math 223 Abstract Algebra Lecture Notes Steven Tschantz Spring 2001 (Apr. 23 version) Preamble These notes are intended to supplement the lectures and make up for the lack of a textbook for the course

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

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

1. Determine all real numbers a, b, c, d that satisfy the following system of equations.

altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c

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

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,

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,

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

ALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite

ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,

4. How many integers between 2004 and 4002 are perfect squares?

5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started

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

Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.

Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson

Sec 4.1 Vector Spaces and Subspaces

Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)