...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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...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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...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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...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...
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...CHAPTER 5—DISCRETE PROBABILITY DISTRIBUTIONS MULTIPLE CHOICE 1. A numerical description of the outcome of an experiment is called a a. descriptive statistic b. probability function c. variance d. random variable ANS: D PTS: 1 TOP: Discrete Probability Distributions 2. A random variable that can assume only a finite number of values is referred to as a(n) a. infinite sequence b. finite sequence c. discrete random variable d. discrete probability function ANS: C PTS: 1 TOP: Discrete Probability Distributions 3. A probability distribution showing the probability of x successes in n trials, where the probability of success does not change from trial to trial, is termed a a. uniform probability distribution b. binomial probability distribution c. hypergeometric probability distribution d. normal probability distribution ANS: B PTS: 1 TOP: Discrete Probability Distributions 4. Variance is a. a measure of the average, or central value of a random variable b. a measure of the dispersion of a random variable c. the square root of the standard deviation d. the sum of the squared deviation of data elements from the mean ANS: B PTS: 1 TOP: Discrete Probability Distributions 5. A continuous random variable may assume a. any value in an interval or collection of intervals b. only integer values in an interval or collection of intervals c. only fractional values in an interval or collection of intervals d. only the positive integer values in an interval ANS: A PTS: 1 TOP: Discrete Probability...
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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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...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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...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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...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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...* ------------------------------------------------- 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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...21-110: Problem Solving in Recreational Mathematics Homework assignment 7 solutions Problem 1. An urn contains five red balls and three yellow balls. Two balls are drawn from the urn at random, without replacement. (a) In this scenario, what is the experiment? What is the sample space? (b) What is the probability that the first ball drawn is red? (c) What is the probability that at least one of the two balls drawn is red? (d) What is the (conditional) probability that the second ball drawn is red, given that the first ball drawn is red? Solution. (a) The experiment is the drawing of two balls from the urn without replacement. The sample space is the set of possible outcomes, of which there are four: drawing two red balls; drawing two yellow balls; drawing a red ball first, and then a yellow ball; and drawing a yellow ball first, and then a red ball. One way to denote the sample space is in set notation, abbreviating the colors red and yellow: sample space = {RR, YY, RY, YR}. Note that these four outcomes are not equally likely. We can also represent the experiment and the possible outcomes in a probability tree diagram, as shown below. Note in particular the probabilities given for the second ball. For example, if the first ball is red, then four out of the remaining seven balls are red, so the probability that the second ball is red is 4/7 (and the probability that it is yellow is 3/7). On the other hand, if the first ball is yellow, then five out of the remaining...
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...Lecture 1. Logic. Propositions. The rules of logic specify the precise meaning of mathematical statements. For instance, the rules give us the meaning of such statements as, “There exists an integer that is greater than 100 that is a power of 2”, and “For every integer n the sum of the positive integers not exceeding n is ”. Logic is the basis of all mathematical reasoning, and it has practical applications to the design of computing machines, to artificial intelligence, to computer programming, to programming languages, and to other areas of computer science. A proposition is a statement that is either true or false, but not both. Letters are used to denote propositions, just as letters are used to denote variables. The conventional letters used for this purpose are p, q, r, s, … The truth value of a proposition is true, denoted by T, if it is a true proposition and false, denoted by F, if it is a false proposition. We now turn our attention to methods for producing new propositions from those that we already have. Many mathematical statements are constructed by combining one or more propositions. New propositions, called compound propositions, are formed from existing propositions using logical operators. Let p be a proposition. The statement “It is not the case that p” is another proposition, called the negation of p. The negation of p is denoted by p. The proposition p is read “not p”. A truth table displays the relationships between the truth values of propositions. Table...
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...Phase 5 Individual Project 03/23/2014 Math 203 Colorado Technical University (Online) Part I: 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} Answers: * AUB- {All BLACK and RED numbers} * A∩D- {All numbers that are both RED and EVEN} * B∩C- {NO numbers intersect between these two sets} * CUE- {All ODD numbers and 00, 0} * B∩F- {2,4,6,10,11} * E∩F- {1,3,5,7,9,11} Part II: 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 even nor odd.) Also include a primary key. What is the value of n in this n-ary relation? The primary key is the trial attempts, the reason for this is because only one attempt can be linked to that trial attempt, therefore making it unique. The value of n is four. Trial Attempt | Number | Color | Odd or Even | 1 | 1 | Red | Odd | 2 | 29 | Black | Odd | 3 | 12 | Red | Even | 4 | 19 | Red | Odd | 5 | 9 | Red | Odd | 6 | 33 | Black | Odd | 7 | 28 |...
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...CARLETON UNIVERSITY SCHOOL OF COMPUTER SCIENCE WINTER 2013 COMP. 3803 - DISCRETE STRUCTURES: II ASSIGNMENT I DUE: FRIDAY JAN 25, 2013 __________________________________________________________________________________ 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. • • • • 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. 1. 2. 3. 4. 5. 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...
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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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