...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 Edition...
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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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...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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...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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...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 exam...
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...* ------------------------------------------------- Homework problems: Section 1.1, pages 12–16: #2, #11, #31 Section 1.2, pages 22–24: #3, #8a,b,c, #24 Section 1.4, pages 53–55: #6, #11, #32 Section 2.3, pages 152–153: #2, #12, #13 Section 2.6, pages 183–184: #2a, #4b * ------------------------------------------------- * ------------------------------------------------- * ------------------------------------------------- * ------------------------------------------------- 2. Which of these are propositions?What are the truth values * ------------------------------------------------- of those that are propositions? * ------------------------------------------------- a) Do not pass go. -No * ------------------------------------------------- b) What time is it? - No * ------------------------------------------------- c) There are no black flies in Maine. - Yes, FALSE * ------------------------------------------------- d) 4 + x = 5. - No * ------------------------------------------------- e) The moon is made of green cheese. Yes, FALSE * ------------------------------------------------- f ) 2n ≥ 100. , No * ------------------------------------------------- * ------------------------------------------------- * ------------------------------------------------- 11. Let p and q be the propositions * ------------------------------------------------- p : It is below freezing. * ------------------------------------------------- ...
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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 neither...
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...* MTH/221 Week Four Individual problems: * * Ch. 11 of Discrete and Combinatorial Mathematics * Exercise 11.1, problems 8, 11 , text-pg:519 Exercise 11.2, problems 1, 6, text-pg:528 Exercise 11.3, problems 5, 20 , text-pg:537 Exercise 11.4, problems 14 , text-pg:553 Exercise 11.5, problems 7 , text-pg:563 * Ch. 12 of Discrete and Combinatorial Mathematics * Exercise 12.1, problems 11 , text-pg:585 Exercise 12.2, problems 6 , text-pg:604 Exercise 12.3, problems 2 , text-pg:609 Exercise 12.5, problems 3 , text-pg:621 Chapter 11 Exercise 11.1 Problem 8: Figure 11.10 shows an undirected graph representing a section of a department store. The vertices indicate where cashiers are located; the edges denote unblocked aisles between cashiers. The department store wants to set up a security system where (plainclothes) guards are placed at certain cashier locations so that each cashier either has a guard at his or her location or is only one aisle away from a cashier who has a guard. What is the smallest number of guards needed? Figure 11.10 Problem 11: Let G be a graph that satisfies the condition in Exercise 10. (a) Must G be loop-free? (b) Could G be a multigraph? (c) If G has n vertices, can we determine how many edges it has? Exercise 11.2 Problem 1: Let G be the undirected graph in Fig. 11.27(a). a) How many connected subgraphs ofGhave four vertices and include a cycle? b) Describe the...
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...Textbook Exercises (UOP Course) For more course tutorials visit www.tutorialrank.com Tutorial Purchased: 3 Times, Rating: A+ Mathematics - Discrete Mathematics Complete 12 questions below by choosing at least four from each section. • Ch. 1 of Discrete and Combinatorial Mathematics o Supplementary Exercises 1, 2, 7, 8, 9, 10, 15(a), 18, 24, & 25(a & b) • Ch. 2 of Discrete and Combinatorial Mathematics o Exercise 2.1, problems 2, 3, 10, & 13, o Exercise 2.2, problems 3, 4, & 17 o Exercise 2.3, problems 1 & 4 o Exercise 2.4, problems 1, 2, & 6 o Exercise 2.5, problems 1, 2, & 4 • Ch. 3 of Discrete and Combinatorial Mathematics o Exercise 3.1, problems 1, 2, 18, & 21 o Exercise 3.2, problems 3 & 8 ----------------------------------------------- MTH 221 Week 1 Individual Assignment Selected Textbook Exercises (UOP Course) For more course tutorials visit www.tutorialrank.com Tutorial Purchased:2 Times, Rating: No rating Complete the six questions listed below: • Ch. 1 of Discrete and Combinatorial Mathematics o Supplementary Exercise 2 • Ch. 2 of Discrete and Combinatorial Mathematics o Exercise 2.1, problem 10; p 54 o Exercise 2.2, problem 4; p 66 o Exercise 2.3, problem 4; p 84 • Ch. 3 of Discrete and Combinatorial Mathematics o Exercise 3.1, problem 18; p 135 ----------------------------------------------- MTH 221 Week 2 Individual and Team Assignment Selected Textbook...
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...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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...• Ch. 1 of Discrete and Combinatorial Mathematics o Supplementary Exercises 1, 2, 7, & 8 1. In the manufacture of a certain type of automobile, four kinds of major defects and seven kinds of minor defects can occur. For those situations in which defects do occur, in how many ways can there be twice as many minor defects as there are major ones? 2. A machine has nine different dials, each with five settings labeled 0, 1, 2, 3, and 4. a) In how many ways can all the dials on the machine be set? b) If the nine dials are arranged in a line at the top of the machine, how many of the machine settings have no two adjacent dials with the same setting? 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 dance with eight of these 12 men? 8. In how many ways can the letters in WONDERING be arranged with exactly two consecutive vowels? • Ch. 2 of Discrete and Combinatorial Mathematics o Exercise 2.1, problems 2 o Exercise 2.2, problems 3 o Exercise 2.4, problems 1 o Exercise 2.5, problems 1 2. Identify the primitive statements in Exercise 1 below: Exercise 1. Determine whether each of the following sentences is a statement. a) In 2003 GeorgeW. Bush was the president of the United States. b) x + 3 is a positive integer. c) Fifteen is an even number. d) If Jennifer is late for the party, then her cousin Zachary will be quite angry...
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...Discrete geometry Apolinario G. Sanger III Submitted to : Professor Rody G. Balete MT-31 Chapter I The Problem and It’s Background Introduction: Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory,toric geometry, and combinatorial topology. Although polyhedra and tessellations have been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied were: the density of circle packings by Thue,projective configurations by Reye and Steinitz, the geometry of numbers by Minkowski, and map colourings by Tait, Heawood, and Hadwiger. László Fejes Tóth, H.S.M. Coxeter and Paul Erdős, laid the foundations of discrete geometry. "This is an introduction to the field of discrete geometry understood as the investigation...
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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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...Signal may be either continuous-time or discrete-time, with either analog or digital values [1]. The signals which are represented by a continuous function are called continuous signals and those which are described by number sequences are called discrete signals [2]. We have seen about a signal in brief. The second component in signal processing is a system which is a process whose input and output are signals. Signal processing is a vast area comprising the concepts of electrical engineering, systems engineering and applied mathematics that deals with both the analog and discrete time signals, represented by variation in time or spatial physical quantities. Precise statistical depiction is required for the development of improved signal processing algorithms of natural signals [3]. The major operations of Signal processing includes 1) signal acquisition and reconstruction, 2) Quality improvement including filtering, smoothing and digitization, 3) feature extraction 4) signal compression and 5) prediction [4] [5]. Analog signal processing, Discrete-time signal processing, Non-linear signal processing and Digital signal processing are the four major categories of signal processing. The signal processing performed over analog signals for the purpose of any of the major operations of signal processing is known to be analog signal processing and the same concept is applied for discrete-time signal processing, where the only difference is discrete signal is employed. An analog signal...
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... | | |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...
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