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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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...Coal is a fossil fuel like oil and gas. Fossil fuels are all formed out of organic matter deposited, decomposed and compressed, storing all the carbon involved under the earth's surface for millions of years. Some of the advantages of coal are - * Easily combustible, and burns at low temperatures, making coal-fired boilers cheaper and simpler than many others * Widely and easily distributed all over the world; * Comparatively inexpensive to buy on the open market due to large reserves and easy accessibility * Good availability for much of the world (i.e. coal is found many more places than other fossil fuels) * Most coal is rather simple to mine, making it by far the least expensive fossil fuel to actually obtain * Coal-powered generation scales well, making it economically possible to build a wide variety of sizes of generation plants. * A fossil-fuelled power station can be built almost anywhere, so long as you can get large quantities of fuel to it. Most coal fired power stations have dedicated rail links to supply the coal. However, the important issue as of now is whether there are more advantages than disadvantages of fossil fuels like coal! Some disadvantages of coal are that - * it is Non-renewable and fast depleting; * Coal has the lowest energy density of any fossil fuel - that is, it produces the least energy per ton of fuel * It also has the lowest energy density per unit volume, meaning that the amount of......

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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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...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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...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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...Exercise 4.1: 4. A wheel of fortune has the integers from 1 to 25 placed on it in a random manner. Show that regardless of how the numbers are positioned on the wheel, there are three adjacent numbers whose sum is at least 39. Work: Let suppose that there are not three adjacent numbers whose sum is at least 39 then for every set of 3 adjacent numbers their sum is less than 39 Since all the numbers are integers for every set of 3 adjacent numbers their sum is less than or equal to 38 Let select the 24 numbers around the “1”, from these 24 numbers we can create 8 sets of 3 consecutive adjacent numbers , then the total sum is less than or equal to 8(38)+1 = 305 So we have that the sum of 1+2+3+……+25 305 (but this is false) because 1+2+….25 = 25(26)/2 = 325 > 305 Then we proved that regardless of how the numbers are positioned on the wheel, there are three adjacent numbers whose sum is at least 39. 7. A lumberjack has 4n + 110 logs in a pile consisting of n layers. Each layer has two more logs than the layer directly above it. If the top layer has six logs, how many layers are there? Work: The 1st layer (the top one) has 6 logs The 2nd layer has 6+1(2) logs The 3rd layer has 6+2(2) The 4th layer has 6+3(2) ……………. The nth layer 6 +(n-1)(2) Total number of layers is: 6n +2(1+2+3+….+n-1) = 6n + 2(n-1)n/2 = 6n+n(n-1) =......

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...1. Write the first four terms of the sequence whose general term is an = 2( 4n - 1) (Points : 3) | 6, 14, 22, 30 -2, 6, 14, 22 3, 7, 11, 15 6, 12, 18, 24 | 2. Write the first four terms of the sequence an = 3 an-1+1 for n ≥2, where a1=5 (Points : 3) | 5, 15, 45, 135 5, 16, 49, 148 5, 16, 46, 136 5, 14, 41, 122 | 3. Write a formula for the general term (the nth term) of the arithmetic sequence 13, 6, -1, -8, . . .. Then find the 20th term. (Points : 3) | an = -7n+20; a20 = -120 an = -6n+20; a20 = -100 an = -7n+20; a20 = -140 an = -6n+20; a20 = -100 | 4. Construct a series using the following notation: (Points : 3) | 6 + 10 + 14 + 18 -3 + 0 + 3 + 6 1 + 5 + 9 + 13 9 + 13 + 17 + 21 | 5. Evaluate the sum: (Points : 3) | 7 16 23 40 | 6. Find the 16th term of the arithmetic sequence 4, 8, 12, .... (Points : 3) | -48 56 60 64 | 7. Identify the expression for the following summation:(Points : 3) | 6 3 k 4k - 3 | 8. A man earned $2500 the first year he worked. If he received a raise of $600 at the end of each year, what was his salary during the 10th year? (Points : 3) | $7900 $7300 $8500 $6700 | 9. Find the common ratio for the geometric sequence.: 8, 4, 2, 1, 1/2......

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...Discrete Math for Computer Science Students Ken Bogart Dept. of Mathematics Dartmouth College Scot Drysdale Dept. of Computer Science Dartmouth College Cliﬀ Stein Dept. of Industrial Engineering and Operations Research Columbia University ii c Kenneth P. Bogart, Scot Drysdale, and Cliﬀ Stein, 2004 Contents 1 Counting 1.1 Basic Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Sum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summing Consecutive Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Product Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two element subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Important Concepts, Formulas, and Theorems . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Counting Lists, Permutations, and Subsets. . . . . . . . . . . . . . . . . . . . . . Using the Sum and Product Principles . . . . . . . . . . . . . . . . . . . . . . . . Lists and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Bijection Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . k-element permutations of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting......

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...Closed path = 1st Vertex = Last Vertex Simple path = different edges (Vertex can be used twice) Closed simple path: different edges + 1st vertex = last vertex (vertices can be used twice) Cycle = closed simple path (1st vertex = last vertex + different edges) + different vertices Distinct vertices ==> different edges Cycle= closed path e1…en of length at least 3 + distant vertices = path e1…en with n >= 3 + 1st vertex = last vertex + distinct vertices A path has all vertices distinct ==> different edges + no cycles G and H are isomorphic if there exists an isomorphism and γ = V(G) ---> V(H) such that if {u,v} edge in G then {γ(u), γ (v)} edge in H each vertex goes to another vertex degrees are the same shapes are mapped too to prove two graphs are isomorphic check the degree lists….if they match find a mapping between the two graphs Euler path = simple path which goes through each edge exactly......

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...Robert Gibbins MATH203-1401A-05 Application of Discrete Mathematics Phase 1 Individual Project January 12, 2014 Part I Demonstrate DeMorgan’s Laws using a Venn diagram: The portions below define 2 sets, and a universal set that exists within the 2 sets.. They state the union and intersection of the 2 sets and the complements of each set. The Venn Diagram gives a visual to give a demonstration of DeMorgan’s law of sets. Union The Union is the combined elements of both sets A and B: A U B = {Addison, Mabel, Leslie, Matt, Rick, Joel} or U = {Addison, Mabel, Leslie, Matt, Rick, Joel} Set A is expressed below in the shaded area of the Venn Diagram: A = {Addison, Leslie, Rick} Set B is expressed below n the shaded area of the Venn Diagram: B = {Mabel, Matt, Joel} Intersection The intersection are the elements both A and B have in common, in this case both softball and volleyball players. The intersection is expressed below and is represented in the shaded section of the Venn Diagram. A ∩ B = {Matt, Rick} Matt Rick Matt Rick Complements The Complement of a set contains everything in the universe minus what is in the set itself. The complement of Set A and Set B are expressed: A’ = {Mabel, Matt, Joel} B’ = {Addison, Leslie, Rick} De Morgan’s Law 1. The first portion of DeMorgan’s law says the complement of the union of sets A and B are the intersection of the...

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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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...Phase 2 IP Matt Peterson MATH203-1403A-03 Part 1 Consider the following 2 sets of data that list football teams and quarterbacks: D = {Jets, Giants, Cowboys, 49’ers, Patriots, Rams, Chiefs} Q = {Tom Brady, Joe Namath, Troy Aikman, Joe Montana, Eli Manning} 1. Using D as the domain and Q as the range, show the relation between the 2 sets, with the correspondences based on which players are (or were) a member of which team(s). (You can usehttp://www.pro-football-reference.com to find out this information). Show the relation in the following forms: * Set of ordered pairs (Jets, Namath) (Giants, Manning) (Cowboys, Aikman) (49ers, Montana) (Chiefs, Montana) (Patriots, Brady) 2. The relation is a function, no one element of the domain matches no more than one element of the range. * Directional graph Jets Namath Giants Manning Cowboys Aiken 49er’s Montana Chiefs Montana Patriot’s Brady 3. Now, use set Q as the domain, and set D as the range. Show the relation in the following forms: * Set of ordered pairs (Namath, Jets), (Manning, Giants), (Aiken, Cowboys), (Montana, 49er’s), (Montana, Chiefs), (Brady, Patriots) 4. The relation is not a function, one element of the domain matches with two elements of the range. ( Montana, Chief’s & 49er’s) * Directional graph Namath Jet’s Manning Giant’s Aiken Cowboy’s Montana 49er’s Montana Chief’s Brady Patriot’s Part 2 Mathematical......

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...need 50 rows of seats. (a) Write the sequence for the number of seats for the first 5 rows C(1) = 16 + 4 (1) = 16 + 4 = 20 seats in the first row C(2) = 16 + 4 (2) = 16 + 8 = 24 seats in the first row C(3) = 16 + 4 (3) = 16 + 12 = 28 seats in the first row C(4) = 16 + 4 (4) = 16 + 16 = 32 seats in the first row C(5) = 16 + 4 (5) = 16 + 20 = 36 seats in the first row The sequence for the number of seats in the first 5 rows would be the following; 20, 24, 28, 32, 36 (b) How many seats will be in the last row? There are 50 rows, so n = 50 C(50) = 16 + 4 (50) = 16 + 200 = 216 seats in the last row The last row will have 216 seats. (c) What will be the total number of seats in the theater? From our MATH 203 Live Chat Session #4 on Wednesday evening 26 November 2014 we learned that the Sum of terms in a Sequence has the following formula; Sn = [ n (a1 + an ) ] / 2 So with that said n = 50 for the number of rows in the theater. S50 = [ 50 (a1 + an ) ] / 2 So in this case a1 is the first row which equals 20 seats in that row and an is the last row (50th row) which equals 216 seats in that row from (a) and (b) above.. S50 = [ 50 (20 + 216 ) ] / 2 Sum formula from results above S50 = [ 50 (236 ) ] / 2 20 plus 216 equals 236 S50 = [ 11800 ] / 2 50 times 236 equals 11,800 S50 = 5900 11,800 divided by 2 equals 5,900 The total number of seats in the theater will be 5900 seats....

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...statements like “if 2+2=5 then the moon is made of green cheese” are technically true since whenever the antecedent (the part to the left of () is false, the entire implication is true irrespective of the truth value of the consequent (the part to the right of (). This feature of ( turns out to be very important, as we shall see later. Some definitions The CONVERSE of p(q is q(p The CONTRAPOSITIVE of p(q is (q ( (p The INVERSE of p(q is (p ( (q There are 16 (24) logically possible truth tables for two propositions, that is, two possible values in the four cells of the last column. We’ll look at one more although others may be of interest and occur elsewhere when logic is discussed. The expression “if and only if” occurs a lot in math and science, though not so much in daily life where we often say just “if” to mean “if and only if.” (When you say “If it rains I’ll cut classes” you usually mean “If it rains I’ll cut classes and if it doesn’t rain I won’t cut classes,” which amounts to “I’ll cut classes if and only if it rains,” but noone talks that way.) In any event we are interested in the logic of “if and only if” which is denoted by (. So consider: “I will leave tomorrow if and only if I get my car fixed.” Or: p ( q, where p symbolizes “I will leave tomorrow” and q symbolizes “I get my car fixed.” This compound proposition is true whenever both sides have the same truth value: If I leave tomorrow you would assume I got my car fixed and if you know I got......

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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......

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