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COMP. 3803 - DISCRETE STRUCTURES: II ASSIGNMENT I DUE: FRIDAY JAN 25, 2013

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Assignment Policy: Late assignments will not be accepted. You are expected to work on the assignments on your own. Past experience has shown conclusively that those who do not put adequate effort into the assignments do not learn the material and have a probability near 1 of doing poorly on the exams. Important note: When writing your solutions, you must follow the guidelines below.

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The answers should be concise, clear and neat. Make sure that your TA can read your solution. Please submit the solutions in the order of the problems, the solution to Problem 1, then to Problem 2 and so on. When presenting proofs, every step should be justified. Assignments should be stapled or placed in an unsealed envelope with your name and student number.

Substantial departures from the above guidelines will not be graded.

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Prove that the sum of n real rational numbers is rational if all of them are rational. Does the reverse hold true? What can you say about the rationality of the product of n rational numbers? Prove that if n is a positive integer, then n is odd if and only if 5n+6 is odd. Show by induction that n5 – n is divisible by 5 for all n ≥ 0. Show by induction that n2 – 1 is divisible by 8 whenever n is an odd positive integer. If possible, give examples of: a) A graph with 4 vertices whose degrees are 1, 2, 3 and 3. b) A graph with 6 vertices whose degrees are 2, 4, 3, 3, 4 and 5. If it is not possible, explain why. Let A = {0,1} and B be a countable set. Prove that the set of all functions from A to B is countable. Let A be the set of all even natural numbers, and B be the set of natural numbers divisible by 3. Prove…...

...Course Design Guide MTH/221 Version 1 1 Course Design Guide College of Information Systems & Technology MTH/221 Version 1 Discrete Math for Information Technology Copyright © 2010 by University of Phoenix. All rights reserved. Course Description Discrete (as opposed to continuous) mathematics is of direct importance to the fields of Computer Science and Information Technology. This branch of mathematics includes studying areas such as set theory, logic, relations, graph theory, and analysis of algorithms. This course is intended to provide students with an understanding of these areas and their use in the field of Information Technology. Policies Faculty and students/learners will be held responsible for understanding and adhering to all policies contained within the following two documents: University policies: You must be logged into the student website to view this document. Instructor policies: This document is posted in the Course Materials forum. University policies are subject to change. Be sure to read the policies at the beginning of each class. Policies may be slightly different depending on the modality in which you attend class. If you have recently changed modalities, read the policies governing your current class modality. Course Materials Grimaldi, R. P. (2004). Discrete and combinatorial mathematics: An applied introduction. (5th ed.). Boston, MA: Pearson Addison Wesley. Article References Albert, I. Thakar, J., Li, S., Zhang, R., & Albert, R...

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...Introduction to Discrete Structures --- Whats and Whys What is Discrete Mathematics ? Discrete mathematics is mathematics that deals with discrete objects. Discrete objects are those which are separated from (not connected to/distinct from) each other. Integers (aka whole numbers), rational numbers (ones that can be expressed as the quotient of two integers), automobiles, houses, people etc. are all discrete objects. On the other hand real numbers which include irrational as well as rational numbers are not discrete. As you know between any two different real numbers there is another real number different from either of them. So they are packed without any gaps and can not be separated from their immediate neighbors. In that sense they are not discrete. In this course we will be concerned with objects such as integers, propositions, sets, relations and functions, which are all discrete. We are going to learn concepts associated with them, their properties, and relationships among them among others. Why Discrete Mathematics ? Let us first see why we want to be interested in the formal/theoretical approaches in computer science. Some of the major reasons that we adopt formal approaches are 1) we can handle infinity or large quantity and indefiniteness with them, and 2) results from formal approaches are reusable. As an example, let us consider a simple problem of investment. Suppose that we invest $1,000 every year with expected return of 10% a year. How much...

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...PART ONE Introduction to Discrete-Event System Simulation 1 Introduction to Simulation A simulation is the imitation of the operation of a real-world process or system over time. Whether done by hand or on a computer, simulation involves the generation of an artiﬁcial history of a system, and the observation of that artiﬁcial history to draw inferences concerning the operating characteristics of the real system. The behavior of a system as it evolves over time is studied by developing a simulation model. This model usually takes the form of a set of assumptions concerning the operation of the system. These assumptions are expressed in mathematical, logical, and symbolic relationships between the entities, or objects of interest, of the system. Once developed and validated, a model can be used to investigate a wide variety of “what-if” questions about the realworld system. Potential changes to the system can ﬁrst be simulated in order to predict their impact on system performance. Simulation can also be used to study systems in the design stage, before such systems are built. Thus, simulation modeling can be used both as an analysis tool for predicting the effect of changes to existing systems, and as a design tool to predict the performance of new systems under varying sets of circumstances. In some instances, a model can be developed which is simple enough to be “solved” by mathematical methods. Such solutions may be found by the use of differential calculus,......

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...Continued on ﬁrst page of back endpapers. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. DISCRETE MATHEMATICS Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. DISCRETE MATHEMATICS WITH APPLICATIONS FOURTH EDITION SUSANNA S. EPP DePaul University Australia · Brazil · Japan · Korea · Mexico · Singapore · Spain · United Kingdom · United States Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect......

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...ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth......

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...decrypt the message.This system doesn’t require secure key transmission.So, it resolves the one of the problem faced by symmetric key cryptosystem. If someone is able to compute respective private key from a given public key, then this system is no more secure. So, Public key cryptosystem requires that calculation of respective private key is computationally impossible from given public key. In most of the Public key cryptosystem, private key is related to public key via Discrete Logarithm. Examples are Diﬃe-Hellman Key Exchange, Digital Signature Algorithm (DSA), Elgamal which are based on DLP in ﬁnite multiplicative group. 1 2. Discrete logarithm problem The Discrete Logarithm Problem (DLP)is the problem of ﬁnding an exponent x such that g x ≡ h (mod p) where, g is a primitive root for Fp and h is a non-zero element of Fp . Let, n be the order of g. Then solution x is unique up to multiples of n and x is called discrete logarithm of h to the base g (i.e.) x = logg h. In cryptosystem based on Discrete Logarithm , x is used as private key and (Fp , g , h) is used as public key. So, one way to break such cryptosystem is to solve DLP. If we consider calculation of g a as base step, then trial and error method takes O(p) steps to solve DLP. If 2k ≤ p ≤ 2k+1 , then it takes O(2k ) steps (i.e) it takes exponential time. Pollard ρ method and Pollard kangaroo method are some of the good algorithm to solve DLP. These are generic methods to solve DLP because they do not......

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...operators , and , as is done in various programming languages. Information is often represented using bit strings, which are sequences of zeros and ones. When this is done, operations on the bit strings can be used to manipulate this information. A bit string is a sequence of zero or more bits. The length of this string is the number of bits in the string. We can extend bit operations to bit strings. We define the bitwise OR, bitwise AND and bitwise XOR of two strings of the same length to be the strings that have as their bits the OR, AND and XOR of the corresponding bits in the two strings, respectively. We use the symbols , and to represent the bitwise OR, bitwise AND and bitwise XOR operations respectively. Literature: Kenneth H. Rosen, Discrete Mathematics and Its Applications, fourth edition, 1999. Chapter 1, paragraph 1-4, pp. 6-18. Control questions: 1. Proposition. 2. Logical operations over propositions 3. Bit operations. Lecture 2. Propositional Equivalences An important type of step used in a mathematical argument is the replacement of a statement with another statement with the same truth value. Because of this, methods that produce propositions with the same truth value as a given compound proposition are used extensively in the construction of mathematical arguments. We begin our discussion with a classification of compound propositions according to their possible truth values. A compound proposition that is always true, no matter what the truth......

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...Phase One Individual Project: BeT Proposal Natalie Braggs IT106-1401A-03: Introduction to Programming Logic Colorado Technical University 01/12/2014 Table of Contents Introduction 3 Problem Solving Techniques (Week 1) 4 Data Dictionary 5 Equations 6 Expressions 7 8 Sequential Logic Structures (Week 2) 9 PAC (Problem Analysis Chart)-Transfers 10 IPO (Input, Processing, Output)-Viewing Balances 11 Structure Chart (Hierarchical Chart)-Remote Deposit 12 12 Problem Solving with Decisions (Week 3) 14 Problem Solving with Loops (Week 4) 15 Case Logic Structure (Week 5) 16 Introduction This Design Proposal (BET-Banking e-Teller) is going to show a banking application that allows customers to perform many of the needed transactions from the mobile phones. The Banking e-Teller will allow customers to check balances, make remote capture deposits, and perform transfers to their checking and/or savings accounts. Problem Solving Techniques (Week 1) Data Dictionary ------------------------------------------------- Problem Solving Techniques A data dictionary allows you see what data (items) you are going to use in your program, and lets you see what type of data type you will be using. Providing these important details allows the programmers to collectively get together and brainstorm on what is needed and not needed. Data Item | Data Item Name | Data Type | First name of account owner | firstName | String | Last name of account......

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...1. Current scenario As an established auto parts shop, the company are looking forward to extend their business reach to its customers and also further increase their reputation in the automotive industry. As of now, the company only provides their services within their shop. Customers can either call or come to their shop to order and buy or make an inquiries regarding a specific products or parts. Most of the customers are the local resident of Brunei and are yet to cater customers from other countries. In this era of e-commerce, the company are well aware of the advantages of setting up an online auto shop to gain a competitive advantage over its competitor in the industry as there are currently no local auto parts shop have set up an online auto shop. Customers can browse and buy specific parts with detailed information on the parts according to the customer’s car model and year. This makes it easier and less time consuming for the customers. Setting up an online auto shop also enables the company to cater customers from other countries and enter the international markets. In terms of marketing, the company carry out their advertisement through newspapers, brochures and banners. This methods of advertisement does work for the company however with the increase use of smart phones, tablets and computers, going online will help them more to reach out to more customers locally and internationally where they can provide news, updates on their new products and services...

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...elements, and so on, until the last iteration of a loop, when it processes all elements. Thus, if n = 4, the runtime would be 1 + 2 + 3 + 4 = 10. * Create a table that depicts the runtime for arrays of length 1 to 10. Would you expect the general runtime to be O(n), O(n2), O(n3), or some other function of n? Explain. * 1= 1= 1 * 2 = 1 + 2= 3 * 3 = 1 + 2 +3 = 6 * 4 = 1 + 2 + 3 + 4= 10 * 5 = 1 + 2 + 3 + 4 + 5 = 15 * 6 = 1 + 2 + 3 + 4 + 5 + 6 = 21 * 7 = 1 + 2 + 3 + 4 + 5 + 6 + 7= 28 * 8 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8= 36 * 9 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9= 45 * 10 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1= 55 * Run time equals O ( ( n * (n + 1)) / 2 References Johnsborough, R. (2009) Discrete Mathematics 7th Edition. (N.D.) Binary search algorithm. Retrieved on March 24, 2014 from http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Binary_search_algorithm.html (November 24, 2011). Computer Algorithms: Sequential Search. Retrieved on March 24, 2014 from http://www.stoimen.com/blog/2011/11/24/computer-algorithms-sequential-search/...

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...Task Name: Phase 4 Individual Project Deliverable Length: 4 Parts: See Assignment Details Details: Weekly tasks or assignments (Individual or Group Projects) will be due by Monday and late submissions will be assigned a late penalty in accordance with the late penalty policy found in the syllabus. NOTE: All submission posting times are based on midnight Central Time. Task Background: This assignment involves solving problems by using various discrete techniques to model the problems at hand. Quite often, these models form the foundations for writing computer programming code that automate the tasks. To carry out these tasks effectively, a working knowledge of sets, relations, graphs, finite automata structures and Grammars is necessary. Part I: Set Theory Look up a roulette wheel diagram. The following sets are defined: A = the set of red numbers B = the set of black numbers C = the set of green numbers D = the set of even numbers E = the set of odd numbers F = {1,2,3,4,5,6,7,8,9,10,11,12} From these, determine each of the following: A∪B A∩D B∩C C∪E B∩F E∩F Part II: Relations, Functions, and Sequences The implementation of the program that runs the game involves testing. One of the necessary tests is to see if the simulated spins are random. Create an n-ary relation, in table form, that depicts possible results of 10 trials of the game. Include the following results of the game: Number Color Odd or even (note: 0 and 00 are considered......

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...Discrete Mathematics Lecture Notes, Yale University, Spring 1999 L. Lov´sz and K. Vesztergombi a Parts of these lecture notes are based on ´ ´ L. Lovasz – J. Pelikan – K. Vesztergombi: Kombinatorika (Tank¨nyvkiad´, Budapest, 1972); o o Chapter 14 is based on a section in ´ L. Lovasz – M.D. Plummer: Matching theory (Elsevier, Amsterdam, 1979) 1 2 Contents 1 Introduction 2 Let 2.1 2.2 2.3 2.4 2.5 us count! A party . . . . . . . . Sets and the like . . . The number of subsets Sequences . . . . . . . Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 7 7 9 12 16 17 21 21 23 24 27 27 28 29 30 32 33 35 35 38 45 45 46 47 51 51 52 53 55 55 56 58 59 63 64 69 3 Induction 3.1 The sum of odd numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Subset counting revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Counting regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Counting subsets 4.1 The number of ordered subsets . . . . 4.2 The number of subsets of a given size 4.3 The Binomial Theorem . . . . . . . . 4.4 Distributing presents . . . . . . . . . . 4.5 Anagrams . . . . . . . . . . . . . . . . 4.6 Distributing money . . . . . . . . . . ...

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...the definition of problems and the linkage of problems with solutions, and institutions as filters of ideas and regulators of the process of policy choice — and a complete explanation of race policy will consider not only both elements but also the ways in which they interact to reconstruct patterns of race relations." In some instances both countries policies could read the same, yet be interpreted differently. In an effort to keep all entities involved headed in the same direction there has to be a clear communication and regulatory rules to ensure the partnership remains stable. Conversely, they say, companies have to give employees the "freedom" that's essential to innovate. Organizational Structure Raymond Miles and Charles Snow, authors of ,"Organizational Strategy, Structure, and Process" help provide the strategy typology appropriate for outcome desired. The particular typology appropriate for Office Depot and Reliance Retail is the "Analyzer". This is the description of a company that is purposely more innovative in their product marketing initiatives. The Analyzer has a watchful eye yet is willing to take risks in order to succeed. The other three typologies are, " Prospector", "Reactor", and "Defender". The Prospector is described as a company who prospers by stimulating and meeting new product opportunities. The Reactor is the company who vacillate in an approach to their environment . and ultimately fail. The Defender prosper th...

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...Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic[1] – do not vary smoothly in this way, but have distinct, separated values.[2] Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets[3] (sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers). However, there is no exact definition of the term "discrete mathematics."[4] Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions. The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are...

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...MAT 1348B Discrete Mathematics for Computer Science Winter 2011 Professor: Alex Hoﬀnung Dept. of Mathematics & Statistics, 585 King Edward (204B) email: hoffnung@uottawa.ca Important: Please include MAT1348 in the subject line of every email you send me. Otherwise your email may be deleted unread. Please do not use Virtual Campus to send me messages as I may not check them regularly. Course Webpages: This web page will contain detailed and up-to-date information on the course, including a detailed course outline and course policies, homework assignments, handouts to download etc. You are responsible for this information. Consult this page regularly. Timetable: Lectures: Mon. 2:30–4:00 pm, Thurs: 4:00–5:30 pm in STE B0138 Oﬃce hours: Mon. 4:00–5:00 pm, Thurs: 3:00 - 4:00 pm DGD: Wed. 10–11:30 am. Textbook: K. H. Rosen, Discrete Mathematics and Its Applications, 6th Edition, McGrawHill. We’ll be covering most of Chapters 1, 2, and 9, and parts of Chapters 4, 5, and 8. The course may contain a small of amount of material not covered by the textbook. This text has been used in Discrete Math courses at Ottawa U. for many years, so secondhand copies can easily be found. Copies of the book are at the bookstore or available from Amazon. Coursework Evaluation: The ﬁnal grade will be calculated as follows: • 5 homework assignments : 10% • Midterm exam: 30% • Final exam: 60% The midterm test is on February 17 . 1 Note that students must pass the ﬁnal......

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