...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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...* 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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...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 energy...
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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 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 sequences...
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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) = (n+5)n ...
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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 owner...
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...Most people think that computer programming just recently had been created but far from that, it started for over a century. Starting from Charles Babbage’s steam driven machine named the Analytical Engine back in 1834. This idea caught the attention of a mathematician Ada Lovelace, she wrote a program to make the Analytical Engine calculate and print a sequence of numbers known as Bernoulli numbers. Because of her work with the Analytical Engine, she is considered as the first ever computer programmer. The first true computer appeared in 1943 when the U.S. Army created a computer called ENIAC that was able to calculate artillery trajectories. To give it instructions, you had to physically flip its different switches and rearrange its cables. However, physically rearranging cables and switches to reprogram a computer proved to be very tasking and clumsy. So instead of physically rearranging the computer’s wiring, computer scientists decided it would be easier to give the computer different instructions. In the old days, computers were taking up entire rooms and cost millions of dollars. Today, computers have shrunk that they are essentially nothing more than a piece of granular bar which are called the central processing unit (CPU), or processor. However in order to tell the processor what to do, you have to give it instructions written in a language that it can understand. That is where programming languages comes in. There many different computer languages and each one of them...
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...L1.1 Lecture Notes: Logic Justification: Precise and structured reasoning is needed in all sciences including computer science. Logic is the basis of all reasoning. Computer programs are similar to logical proofs. Just as positive whole numbers are the fundamental units for arithmetic, propsitions are the fundamental units of logic. Proposition: A statement that is either true or false. E.g. Today is Monday Today is Tuesday The square root of 4 is 2 The square root of 4 is 1 2 is even, and the square of two is even, and 3 is odd and the square of 3 is odd. The Panthers can clinch a playoff berth with a win, plus a loss by the Rams, a loss or tie by the Saints and Bears, a win by the Seahawks and a tie between the Redskins and Cowboys. (Copied verbatim from the sports page 12/26/2004.) Propositions may be true or false and no preference is given one way or the other. This is sometimes difficult to grasp as we have a “natural” preference for true statements. But “snow is chartreuse” and “snow is white” are both propositions of equal standing though one is true and the other false. Non-propositions: What is today? Is today Monday? Questions are not propositions. You can’t judge whether the question itself is true or false, even though the answer to the question may be true or false. Show me some ID! Similarly, imperative statements lack a truth value. 2x=4 x=y Statements with undetermined variables do not have truth...
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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 once Euler...
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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 subsets...
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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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...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 (Points...
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...Relations, Functions, Sequences, and Graphs Part I: Suppose you are developing a statistical database in which information about professional football teams and records are stored. Consider the following 2 sets of data that list football teams and quarterbacks: D = {Jets, Giants, Cowboys, 49ers, 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 use http://www.pro-football-reference.com to find out this information). Show the relation in the following forms: Set of ordered pairs {(Jets, Joe Namath), (Giants, Eli Manning), (Cowboys, Troy Aikman), (49ers, Joe Montana), (Patriots, Tom Brady), (Rams, Joe Namath), (Chiefs, Joe Montana)} Jets Jets Directional graph Tom Brady Tom Brady Giants Giants Joe Namath Joe Namath Cowboys Cowboys Troy Aikman Troy Aikman 49ers 49ers Joe Montana Joe Montana Patriots Patriots Rams Rams Eli Manning Eli Manning Chiefs Chiefs 2. Is the relation a function? Explain. Yes, this relation is a function since for every element on the domain side there is one and only one element on the range side. 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 {(Joe Namath, Jets), (Eli Manning, Giants), (Troy Aikman, Cowboys)...
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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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