Math 221 Discrete Mathematics UOPX Make sure all of the are fresh work with proper citation APA format, no used work will be accepted thanks so much. To be substantive, in addition to meeting the minimum content expectations, your posts must include proper APA style citations at the point of use of information from a source. A citation is a notation of the form (AuthorLastName, date, page). A page reference is required if the information is a direct quote. Citations are required even if the information is not directly quoted, but rather paraprased or summarized. You are also required to provide references formatted in APA style. Make sure while you write any of those answer please use it proper citations as you can see above red lines from my instructor he is very restrict about citations, I request you to read that and follow his instructions thanks so much. Week-1 Combinatorics Question 1. The job assignment problem The following problem is a little fanciful, and not really a practical application of mathematics, but it is a good way to make your mind think about counting problem. There are actually several different ways to correctly address it. Suppose you have a print shop and you need to print 8 copies of a print job. You have four identical printers (A, B, C, D) to work with. 1. How many different ways cand you parceel out the 8 print jobs? (One way to do this would be to send all 8 jobs to printer A, for example). 2. How many different ways can the job be done if you have to ensure that each printer prints at least one copy? 3. Do not just give a numerical answer, but explain how you got your solution. Question 2. A Decision Problem Probability is used to help decision making in circumstances in which the outcome is not certain. If we can determine which outcomes are most likely to occur, we can choose a course of action based on probability. Suppose you have 3 coins in your pocket: 2 coins are fair and have a head on one side and a tail on the other, 1 coin has a head on both sides of the coin. You reach into your pocket, pull out a coin, and place it on the table. You see a head. You wish to make a decision as to whether the coin a fair coin or not. 1.What is the probability that it is one of the fair coins? 2. Did seeing that there was a head on that coin make a difference? 3. What is the process you used to come up with your answer? 4. Suppose that instead of 2 fair coins and 1 biased coin, the situation was reversed - what would the probability that you had a fair coin given that the coin you saw came up heads? Individual Assignment will be posted once it available from instructor and it's the lab student connect I have to wait for him to get the information so please wait for that we might can work on this assignment together via TeamViewer. Week2 Logic & Set Theory; Boolean Algebra; Relations & Functions Class discussions:- 4 questins Question 1: A Possibily Ambiguous Ad Suppose a job posting states, "Applicants must have one to two years of experience with SQL and certification in Linux or a graduate degree in computer science." Assume that the use of the the word or is the tradtional inclusive or. 1. Having clarified the meaning ofof the word "or: is it still possible to interpret this job description in different ways? Why or why not? 2. Use propositional logic, set theory or Venn diagrams as support for your answer to part 1. 3. If you find that that the ad is ambiguous, reword it in a way that makes it not ambiguous. Question 2: The Confusing Comedian A paradox in mathematics is a statement that cannot be assigned a value of true or false, because if it is false, it must be true and if it is false, it must be true. The following video link shows a statement made by the comedian Groucho Marx on 1950s TV show. https://www.youtube.com/watch?v=kJHUres_2xU&feature=player_detailpage#t=244 In answering this discussion question, don't get tied up in technical details. Don't worry about "Russell's Paradox," because considering that issue will make things seem even more confusing. I may follow up with you later on that topic, but let's look at this question from a basic (non-technical) point of view. You should be able to approach this with your own reasoning. 1. A club will not accept Groucho. Can he join? 2. He is invited to join the Friar's Club. Based on his own words, can he join? (Don't try to psycholanlyze Groucho, take his words at face value) 3. Again, based on his own words, what kind of a club can he join? (Again take his words at face value.) 4. Is the statement "Groucho can belong to a club" an example of a paradox or not? Explain your answer. Question 3: Examples of Relations To help us better understand the properties of relations on a set as discussed in Chapter 9, consider the relation on people of friendship, using your interpretation of the word? We will define a relation on the set H of all humans by the following defintion: (a,b) R if a and b are friends. 1. Is this relation reflexive, symmetric, antisymmetric, or transitive? Is it an equivalence relation? Explain your answers for each of the five properties. 2.Suggest another relation among people that would be an equivalence relation and show that it is one. Question 4: Boolean Algebra Boolean Algebra seems to be a rather confusing system, but it is an example of what is known as an abstract mathematical system. With an abstract mathematical system, one provides a collection of objects and operations to apply to those objects that possess certain propertiess. Usually abstract systems are designed to include the essentials of systems that we are familiar with. Boolean algebra was designed to include the essentials of both propositional logic and set theory. 1. What are the differences that you can see between Boolean algebra and the usual algebra that you are used to working with? The major properties of Boolean algebra are described in Table 5 of Chapter 10. You should use this table as a listing of the properties of a Boolean algebra. 2. How is Boolean algebra similar to the operations in logic? Be as specific as you can. Again, refer to Table 5. 3. How is Boolean algebra similar to the operations in set theory? Again, be as specific as you can and refer to Table 5. Individual Assignment will be posted once it available from instructor and it's the lab student connect I have to wait for him to get the information so please wait for that we might can work on this assignment together via TeamViewer. Week3 Algorithmic Concepts Class Discussions: 4 questions Question 1: Big O notation - a specific example Suppose we are given the following program segment: F(x) = 0 Input n For x = 1 to x = n F(x) = x(x+3)+2 Print x Next x 1. Determine the number of steps the program will execute before entering the loop. 2. Determine the number of steps the program will execute each time it passes through the loop. 3. Determine the total number of steps that will be executed by the program if the value of n is 3,4,5, n 4. This program segment is considered to be O(n). Explain why. 5. Suppose that another program segment is O(n). Does this mean that the second segment will always require more steps than this segment? Question 2: Explaining Induction Many people when first ecountering induction find it very confusing, because it appears that one of the steps in proving a statement by induction is to assume that the statement we are trying to prove is true. Suppose you are discussing this concept with a friend, and you friend tell you that induction does not prove anything. How would you explain to this person why it is that induction actually does work. In my follow up responses to you, I may ask you questions that your friend might as in response to your explanation. Try to phrase those responses as though you were responding to somebody who was very confused. Question 3: Recursion and Repetition Here are two functions that both compute the same quantity. One is a recursively defined function and the other is defined by repetition. Function A(n) Sum : = 0 For x := 1 to n do Sum:= Sum + x end do A(n) := sum end Function A(n) Function B(n) If n = 0 then B(n) : = 0 If n > 0 then B(n) := n + B(n-1) end Function B(n) 1. 2. 3. 4. What do these two functions compute? Which function is recursive and which function is repetive? Which one do you find easiest to follow What are the differences in the implementation of each function? Individual Assignment will be posted once it available from instructor and it's the lab student connect I have to wait for him to get the information so please wait for that we might can work on this assignment together via TeamViewer. Week4 Graph Theory and Trees Class Discussions:- 2 questions Question 1: An Application of Graph Theory in Network Design Suppose that we have a communication network with 5 nodes, and each pair of nodes is connected directly. This network can be represented by a graph with 5 vertices {A,B,C,D,E}, and 10 edges {AB,AC,AD,AE,BC,BD,BE,CD,CE, DE}. Assume the the network has been hit with a disaster in such a a way that each link has a 50% probability that the link is intact. Simulate that situation by flipping a coin 10 times, once for each edge. If the coin is heads, the link has remaind intact. If the coin is tails, the link has been lost. 1. Draw the resulting graph 2. How many edges does the resulting graph have? 3. Is the graph that results connected? What does this mean for the ability to continue to communicate through the network? 4. Do you think that if a 6th node were added to the network but we were limited to having only 10 edges if the situation would change? How would you arrange the edges make it as likely as possible that the network would remain connected? This is actually a discussion related to an interesting topic in graph theory known as random graphs. This theory can explore the chances that various network configurations remain intact based on various probabilities of failures of links in the network (we might change the probability of a failure from 50% to some other value). Question 2: Trees and Computations Trees occur in various venues in computer science: decision trees in algorithms, search trees, and so on. One important kind of trees computer languages is the parse tree. It is used to break down statements in a programming langue into a form that can be converted into machine code. The most familiar of these kinds of trees are used to break down Arithmetic statements into trees. For example, the computation 2*3 + 4, can be parsed into the following tree + / \\ * 4 / \\ 2 3 On the other hand the computation 2 * (3 + 4) would have a different parse tree * / \\ 2 + / 3 \\ 4 1. What are the similarities differences between the two trees in terms of depth, roots, and breadth? 2. What do those differences between the trees tell us about the differences between the steps need to do each computation. 3. Provide an arithmetic computation with at least three operations in it, and determine its parse tree. Individual Assignment will be posted once it available from instructor and it's the lab student connect I have to wait for him to get the information so please wait for that we might can work on this assignment together via TeamViewer. Please don't forget to help me on this week Individual assignment Week5 Applications of Discrete Mathematics Class Discussions:- 4 questions Question 1: Solutions to Practice Problem 1 Practice Problem (1) on Counting Techniques There are four practice problems available for you to discuss this week. They focus on content from all four of the previous week of the course. Use these problems to test your readiness for the Final Exam. Towards the end of the the week, I will post solutions to the problems. Your responses in this discussion do count for participation credit Suppose you are organizing a business meeting and are in charge of facilitating the introductions. A. Suppose there are 5 people in the group 1.How would you arrange the group so each person can shake hands with every other person? 2. How many times will each person shake hands with someone else? 3. How many handshakes will occur? B. Suppose there is an unknown number (n) people in the group 1.How would you arrange the group of n people so each person can shake hands with every other person? 2. How many times will each person shake hands with someone else? 3. How many handshakes will occur? This answer should be expressed as a general counting formula that depends on the value of n. Question 2: Solutions to Practice Problem 2 Practice Problem (2) on Logic There are four practice problems available for you to discuss this week. They focus on content from all four of the previous week of the course. Use these problems to test your readiness for the Final Exam. Towards the end of the week, I will post solutions to the problems. Your responses in this discussion do count for participation credit Use a truth table or Venn diagram to show that the statement p v (q ^ r) is equivalent to (p v q) ^ (p v r) Show that this statement is not equivalent to (p v q) ^ r. Be sure to explain your answer and don't just provide a truth table or Venn diagram. Why does your table or diagram verify the result? Question 3:- Solutions to Practice Problem 3 Practice Problem (3) on Relations There are four practice problems available for you to discuss this week. They focus on content from all four of the previous week of the course. Use these problems to test your readiness for the Final Exam. Towards the end of the week, I will post solutions to the problems. Your responses in this discussion do count for participation credit Define the following relation on the set of positive integers xRy if x - y is an even integer 1. Show that R is an equivalence relation. With an equivalence relation, the set on which the relation is defined is divided into subsets called equivalence classes. These subsets consist of all elements that are equivalent to each other. The equivalence class of 1, denoted by [1] consists of all elements that are equivalent to 1 under the relation. 2. How many distinct equivalence classes are there in this example? Can you describe the sets? Question 4:- Solutions to Practice Problem 4 Practice Problem (4) on Tree Traversal Algorithms There are four practice problems available for you to discuss this week. They focus on content from all four of the previous week osf the course. Use these problems to test your readiness for the Final Exam.. Your responses in this discussion do count for participation credit The following algorithm describes a postorder tree traversal Postorder(tree) If left subtree exists then Postorder(left subtree) If right subtree exists then Postorder(right subtree) Print root end Can you apply that to the following tree + / \\ / \\ * - / \\ / \\ 2 3 * + / \\ / \\ 4 2 1 5 What is the result of the executing the algorithm? In the terminology of Week 3, what kind of algorithm are we considering here? Individual Assignment:- Case Study Application Paper Choose one of the following Case Studies: Food Webs Coding Theory Network Flows Write a 750- to 1,250-word paper in which you complete one of the following options: Option 1: Food Webs Case Study Explain the theory in your own words based on the case study and suggested readings. Include the following in your explanation: Competition Food Webs Boxicity Trophic Status Give an example of how this could be applied in other real-world applications. Format your paper according to APA guidelines. All work must be properly cited and referenced. The correct reference for this source is the following: McGuigan, R. A.. (1991). Food Webs. Retrieved from McGuigan, R. A., MTH221 - Discrete Math for Information Technology website Submit your assignment to the Assignment Files tab. Option 2: Coding Theory Case Study Explain the theory in your own words based on the case study and suggested readings. Include the following in your explanation: Error Detecting Codes Error Correcting Codes Hamming Distance Perfect Codes Generator Matrices Parity Check Matrices Hamming Codes Give an example of how this could be applied in other real-world applications. Format your paper according to APA guidelines. All work must be properly cited and referenced. The correct reference for this source is Rosen, K. H.. (1991). Coding Theory . Retrieved from Rosen, K. H., MTH221 - Discrete Math for Information Technology website. Submit your assignment to the Assignment Files tab. Option 3: Network Flows Case Study Explain the solutions for examples 1, 2 and 3 from the text. Explain the theory developed including capacitated s,t graphs and the lexicographic ordering rule based on the case study and suggested readings. Give an example of how this could be applied in other real-world applications. Format your paper according to APA guidelines. All work must be properly cited and referenced. The correct reference for this source is: Hobbs, A. M. (1991). Network Flows. Retrieved from Rosen, K. H., MTH221 - Discrete Math for Information Technology website. Attached PDF file has all three documents for the week 5 case study application paper Supporting documents... Week -5 Food Webs Week 5 Mt Week -5 Mth theory.pdf h221_r3_coding_theory_case_study.pdf 221_r3_network_flows_case_study.pdf There are 2 individual assignments on week 5, I only have one information so far and that is above listed and waiting for the other one. He is using this book Discrete Mathematics and Its Applications 7e Kenneth H. ROSEN