Capella MAT2051 Full Course Latest 2020 February

MAT2051 Discrete Mathematics

Unit 1 Discussion

DQ1 Logic and Reasoning

Logic and reasoning play important roles in computer applications, systems management, database management, and error management. In this discussion, you will examine logic and deductive reasoning, symbols and definitions, and an application involving database and error logging.

Deductive Reasoning, Part 1

Given the following true logical statement:

If inputs x1, x2, and x3 are all true, then output y is true.

Write this statement as a logical statement using the appropriate logic symbols for conjunction and implication.

Write the converse and contrapositive of this statement.

Is the converse true? Explain why or why not.

Is the contrapositive true? Explain why or why not.

Deductive Reasoning, Part 2

Assume you are a systems manager and that you know a server will log an error message if a database has a retrieval error. With that, complete the following:

Why is it important for errors to be logged?

What is a database retrieval error, and what might cause this type of error?

If you notice that there is an error logged, can you infer that a retrieval error has occurred without reading the log? Explain why or why not. (Hint: This is a question about reasoning.)

Prior to posting your response, review the Discussion Participation Scoring Guide. This is the recommendation for all discussions in this course.

Response Guidelines

Read your peers’ posts and respond to two. Do you agree with your peer’s assessment of the problem? Explain.

DQ2 Practice Problem Set Review

The problem sets are assigned to help you practice solving problems and prepare for the corresponding quiz and the final exam. As mentioned in the studies, the problem sets are not graded or collected.

Each unit you will have an opportunity to discuss the problem sets with your fellow learners. Use this discussion to ask questions, make comments, ask for help, and assist your peers with the problem sets. Teamwork is encouraged.

Four posts are required in this discussion: two initial and two response. Additional posts are optional and recommended, but not required for your grade:

For the first post, select a problem from this unit’s problem set, write it out fully, solve it fully, and post it.

The second post can be a problem you cannot solve or another fully solved problem from the problem set. If there is a problem in the assigned problem set that you are confused about, write it out fully and show any work you have started. Note that you are stuck and ask for help. If you are able to solve every problem in the set without difficulty, post at least one other fully solved problem from the set as your second post.

The third and fourth posts are response posts. Response guidelines are provided below and in every discussion.

Take advantage of this discussion area to work together as a class on the problem set. Post as many problems as you can and review as many of your peers’ posts as you can. Ask questions. Offer answers. If you are stuck on a problem, post your question. Working as a team will help each person gain a better understanding of the problem set and the concepts covered in this unit.

Response Guidelines

The third and fourth posts, the response posts, are your chance to help your peers. Explore the posts made by your peers and find people who are stuck. Help at least two peers through the problems they are stuck on. If you are unable to locate a peer who is stuck, you may instead choose a peer post, write out your solution to the same problem, and compare solutions and methods. The goal of this area is for the entire class to work together as a group on the entire problem set. You are also expected to complete and understand the problem set on your own in preparation for the quizzes and final exam.

 

MAT2051 Discrete Mathematics

Unit 2 Discussion

DQ1 Relational Databases

Relations and functions play important roles in computer applications, systems management, database management, and error management. In this discussion, you will look at relational databases and their functions and applications.

Databases are common computer and Internet technology applications. Online banking, shopping, iPhones, music, video, personal files, businesses, governments, and academic institutions all use databases to store, process, and retrieve information, including images, video, and music.

Suppose you are a database manager and oversee a large database with the following four tables: EMPLOYEE, DEPARTMENT, SUPPLIER, and BUYER.

Now, turn to page 168 in the text. Under Exercises, you will see these four tables listed as 3.6.4, 3.6.5, 3.6.6, and 3.6.7.

Part 1: Table and Database Operations

Complete the odd problems from 5 through 15 on page 169.

Write out each question fully.

Write a sequence of table operations to answer the query required for each question.

Provide the final answer to the query for each question.

Part 2: Researching the Web for Database Applications

Create two paragraphs that describe two current database application packages. For example, you might choose Oracle and SQL Server.

Read about what these packages offer, how they are alike, how they are different, and what applications they are best suited for.

What language is commonly used to manipulate and communicate with database applications? Offer a few examples of statements from this language.

Review the Discussion Participation Scoring Guide prior to posting.

Response Guidelines

Read your peers’ posts and respond to two. Do you agree with your peer’s assessment of the problem? Explain.

DQ2 Practice Problem Set Review

This discussion allows you to work with your peers to complete and understand the assigned problem set for this unit. Remember, two initial posts and two response posts are required. Further posts are optional and recommended:

For the first post, select a problem from this unit’s problem set, write it out fully, solve it fully, and post it.

The second post can be a problem you cannot solve or another fully solved problem from the problem set. If there is a problem in the problem set that you are confused about, write it out fully and show any work that you have started. Note that you are stuck and ask for help. If you are able to solve every problem in the set without difficulty, post at least one other fully solved problem from the set as your second post.

The third and fourth posts are responses. Response guidelines are provided below and in every discussion.

Take advantage of this discussion area to work together as a class on the problem set. Post as many problems as you can and review as many of your peers’ posts as you can. Ask questions. Offer answers. If you are stuck on a problem, post your question. Working as a team will help each person gain a better understanding of the problem set and the concepts covered in this unit. Although the problem sets are not graded they will help you prepare for the quizzes in this course.

Response Guidelines

The third and fourth posts, the response posts, are your chance to help your peers. Explore the posts made by your peers and find people who are stuck. Help at least two peers through the problems they are stuck on. If you are unable to locate a peer who is stuck, you may instead choose a peer post, write out your solution to the same problem, and compare solutions and methods. The goal of this area is for the entire class to work together as a group on the entire problem set. You are also expected to complete and understand the problem set on your own in preparation for the quizzes and final exam.

 

MAT2051 Discrete Mathematics

Unit 3 Discussion

DQ1 Networking and Probability

Probability and statistics allow us to make decisions when dealing with uncertain variables. For example, assume you are the network administrator of a large firm and have ordered 100 RAID (redundant array of independent disks) devices. You would like to make sure all of the RAIDs work before sending them to different departments. But you do not have time to check all of the devices. If you check some of them and show they work, how confident could you be that all 100 devices work?

Of course, testing all of the RAIDs would be the best way to be confident, but assume you do not have the time. If you only have time to check 5 of the 100, and none of the 5 are defective, how confident would you be that none of the other microprocessors are defective (that is, all 100 devices are fully functional)?

Answer the following questions, which will help you understand the problem. Explain each question and your answer:

How can you use discrete probability to calculate the probability of testing 5 devices such that all 5 end up being nondefective RAIDs, given there are 20 defective RAIDs total of the original 100 RAID devices?

What is the sample space in this problem?

Is this a ratio of combinations or permutations problem? Explain why and show an example. Hint: Read section 6.5 and see Example 6.5.4 in your textbook.

Assume there are exactly 5 defective RAIDs out of the 100 RAID devices. What is the probability of testing 20 RAIDs at random and finding all of them nondefective?

Review the Discussion Participation Scoring Guide prior to posting.

Response Guidelines

Read your peers’ posts and respond to two. Do you agree with your peer’s assessment? Explain.

DQ2 Practice Problem Set Review

This discussion allows you to work with your peers to complete and understand the assigned problem set for this unit. Remember, two initial posts and two response posts are required. Further posts are optional and recommended:

For the first post, select a problem from this unit’s problem set, write it out fully, solve it fully, and post it.

The second post can be a problem you cannot solve or another fully solved problem from the problem set. If there is a problem in the problem set that you are confused about, write it out fully and show any work that you have started. Note that you are stuck and ask for help. If you are able to solve every problem in the set without difficulty, post at least one other fully solved problem from the set as your second post.

The third and fourth posts are responses. Response guidelines are provided below and in every discussion.

Take advantage of this discussion area to work together as a class on the problem set. Post as many problems as you can and review as many of your peers’ posts as you can. Ask questions. Offer answers. If you are stuck on a problem, post your question. Working as a team will help each person gain a better understanding of the problem set and the concepts covered in this unit. Although the problem sets are not graded they will help you prepare for the quizzes in this course.

Response Guidelines

The third and fourth posts, the response posts, are your chance to help your peers. Explore the posts made by your peers and find people who are stuck. Help at least two peers through the problems they are stuck on. If you are unable to locate a peer who is stuck, you may instead choose a peer post, write out your solution to the same problem, and compare solutions and methods. The goal of this area is for the entire class to work together as a group on the entire problem set. You are also expected to complete and understand the problem set on your own in preparation for the quizzes and final exam.

 

MAT2051 Discrete Mathematics

Unit 4 Discussion

DQ1 Machines, Intelligence, and Neural Networks

In this discussion, you will research the history of artificial intelligence (AI). Write two to three paragraphs on the history of AI and how the ideas of logic, resolution, and unification were implemented within machines so they could demonstrate “intelligence.” In your post, answer at least two of the following questions:

What is the resolution and unification algorithm, and what is an algorithm?

What is the Turing test, and who is Alan Turing?

What is a neural network?

Can machines really demonstrate intelligence?

In order to answer these questions, you will have to search the Internet. Suggested search terms include history of artificial intelligence, Turing test, and so on.

Review the Discussion Participation Scoring Guide prior to posting.

Response Guidelines

Read your peers’ posts and respond to two. Assess your peer’s post. Do you agree or disagree? Explain.

DQ2 Practice Problem Set Review

This discussion allows you to work with your peers to complete and understand the assigned problem set for this unit. Remember, two initial posts and two response posts are required. Further posts are optional and recommended:

For the first post, select a problem from this unit’s problem set, write it out fully, solve it fully, and post it.

The second post can be a problem you cannot solve or another fully solved problem from the problem set. If there is a problem in the problem set that you are confused about, write it out fully and show any work that you have started. Note that you are stuck and ask for help. If you are able to solve every problem in the set without difficulty, post at least one other fully solved problem from the set as your second post.

The third and fourth posts are responses. Response guidelines are provided below and in every discussion.

Take advantage of this discussion area to work together as a class on the problem set. Post as many problems as you can and review as many of your peers’ posts as you can. Ask questions. Offer answers. If you are stuck on a problem, post your question. Working as a team will help each person gain a better understanding of the problem set and the concepts covered in this unit. Although the problem sets are not graded they will help you prepare for the quizzes in this course.

Response GuidelinesThe third and fourth posts, the response posts, are your chance to help your peers. Explore the posts made by your peers and find people who are stuck. Help at least two peers through the problems they are stuck on. If you are unable to locate a peer who is stuck, you may instead choose a peer post, write out your solution to the same problem, and compare solutions and methods. The goal of this area is for the entire class to work together as a group on the entire problem set. You are also expected to complete and understand the problem set on your own in preparation for the quizzes and final exam.

 

MAT2051 Discrete Mathematics

Unit 5 Discussion

DQ1 Data Mining, Time Complexity, and Algorithms

In Appendix C, you read about different programming control structures used to write pseudocode and actual computer algorithms, such as if statements, while and for loops, and function calls. For this discussion, assume you work for a data mining company and your job is to write a program to find information on various Web sites pertaining to sales of the Lenovo X200. After your algorithm finds this data, more complex analysis will be done to extract more meaningful information from the data.

Your algorithm is going to scan different sites and search for the character string “Lenovo X200.” Assume you decide to use an algorithm similar to Text Search (see algorithm 4.2.1 on page 178 of your text for an explanation of what this is). If the algorithm finds a site that contains the string (that is, Lenovo X200), assume that it then stores all data or all the text on that particular site into a storage area.

To understand this problem fully, answer the following questions:

What is data mining?

What is a character string?

What is the worst case run time of this algorithm in terms of p, m, t, n (that is, what is O)?

How long do you think it will take this algorithm to run? Note the time complexity as O (run time in terms of n).

Assume that each Web site, on average, has character strings of length 10,000 and that the length of the character string “Lenovo X200” is 11. How many computations will the algorithm need to make per site?

Why is speed and the analysis of algorithm speed so important?

Review the Discussion Participation Scoring Guide prior to posting.

Response Guidelines

Read your peers’ posts and respond to two. Did you arrive at the same time complexity calculation? Explain why or why not.

DQ2 Practice Problem Set Review

This discussion allows you to work with your peers to complete and understand the assigned problem set for this unit. Remember, two initial posts and two response posts are required. Further posts are optional and recommended:

For the first post, select a problem from this unit’s problem set, write it out fully, solve it fully, and post it.

The second post can be a problem you cannot solve or another fully solved problem from the problem set. If there is a problem in the problem set that you are confused about, write it out fully and show any work that you have started. Note that you are stuck and ask for help. If you are able to solve every problem in the set without difficulty, post at least one other fully solved problem from the set as your second post.

The third and fourth posts are responses. Response guidelines are provided below and in every discussion.

Take advantage of this discussion area to work together as a class on the problem set. Post as many problems as you can and review as many of your peers’ posts as you can. Ask questions. Offer answers. If you are stuck on a problem, post your question. Working as a team will help each person gain a better understanding of the problem set and the concepts covered in this unit. Although the problem sets are not graded they will help you prepare for the quizzes in this course.

Response Guidelines

The third and fourth posts, the response posts, are your chance to help your peers. Explore the posts made by your peers and find people who are stuck. Help at least two peers through the problems they are stuck on. If you are unable to locate a peer who is stuck, you may instead choose a peer post, write out your solution to the same problem, and compare solutions and methods. The goal of this area is for the entire class to work together as a group on the entire problem set. You are also expected to complete and understand the problem set on your own in preparation for the quizzes and final exam.

 

MAT2051 Discrete Mathematics

Unit 6 Discussion

DQ1 Cryptography and Security

Cryptography is an exciting IT field that focuses on secure communications, typically on a network. Many encryption methods use number theory to encode and decode messages. This number theory typically draws upon ideas such as prime factorization, greatest common divisors (GCD), modulus, and so on.

For this discussion, read Chapter 5, section 5.4, which starts on page 250 of your text, which discusses number theory and applications to cryptography. Post an explanation of how the RSA public-key cryptosystem works that includes answers to the following questions:

What is a key?

What are public and private keys, and how are they used in the RSA public-key cryptosystem?

How are messages encoded and decoded using mod and GCD?

If you are using p = 17, q = 23, and n = 31, what would the encryption of 101 be, using public key z = pq?

Where is security used, and why is it important to everyone?

Review the Discussion Participation Scoring Guide prior to posting. Make sure you include a minimum of three posts and that your first, or main, post responds to all parts of the question posed.

Response Guidelines

Read your peers’ posts and respond to two. Did you reach the same encryption as your peer? Explain why or why not. Show your work.

 

DQ2 Practice Problem Set Review

This discussion allows you to work with your peers to complete and understand the assigned problem set for this unit. Remember, two initial posts and two response posts are required. Further posts are optional and recommended:

For the first post, select a problem from this unit’s problem set, write it out fully, solve it fully, and post it.

The second post can be a problem you cannot solve or another fully solved problem from the problem set. If there is a problem in the problem set that you are confused about, write it out fully and show any work that you have started. Note that you are stuck and ask for help. If you are able to solve every problem in the set without difficulty, post at least one other fully solved problem from the set as your second post.

The third and fourth posts are responses. Response guidelines are provided below and in every discussion.

Take advantage of this discussion area to work together as a class on the problem set. Post as many problems as you can and review as many of your peers’ posts as you can. Ask questions. Offer answers. If you are stuck on a problem, post your question. Working as a team will help each person gain a better understanding of the problem set and the concepts covered in this unit. Although the problem sets are not graded they will help you prepare for the quizzes in this course.

Response Guidelines

The third and fourth posts, the response posts, are your chance to help your peers. Explore the posts made by your peers and find people who are stuck. Help at least two peers through the problems they are stuck on. If you are unable to locate a peer who is stuck, you may instead choose a peer post, write out your solution to the same problem, and compare solutions and methods. The goal of this area is for the entire class to work together as a group on the entire problem set. You are also expected to complete and understand the problem set on your own in preparation for the quizzes and final exam.

 

MAT2051 Discrete Mathematics

Unit 7 Discussion

DQ1 Computer Networks, Distributed Systems, and Mobile Agents

Computer networks typically consist of nodes and edges—they are graphs. The nodes can typically be computers, routers, servers, and so on. The edges indicate a communication path between nodes. Assume you are a network administrator for a distributed system and that you wish to perform maintenance of all the servers in your system. You have decided to do this by using a mobile agent. To get started, look up and answer the questions in Part 1. After that, complete Part 2:

Part 1

What is a distributed system?

What is a mobile agent?

Since this agent is autonomous and moves from computer to computer, it would be smart to equip it with shortest path algorithms, so that it can go from server to server, reaching all the servers only once, in the shortest amount of time. This is the traveling sales person (TSP) problem. Note that computer networks can be modeled using graphs and that shortest paths solutions, Hamilton cycles, TSP, and so on can be found using graph-based algorithms.

Part 2

How would you model a computer network using a graph? What are potential nodes, edges, and weights?

Which of these traversal algorithms should the agent be equipped with? Which will help the agent traverse the network in an efficient manner? Explain why.

What is the TSP problem, and how does it relate to this question?

Review the Discussion Participation Scoring Guide prior to posting.

Response Guidelines

Read your peers’ posts and respond to two. Compare your response to your peer’s response. Did you come up with the same results? Explain why or why not.

DQ2 Practice Problem Set Review

This discussion allows you to work with your peers to complete and understand the assigned problem set for this unit. Remember, two initial posts and two response posts are required. Further posts are optional and recommended:

For the first post, select a problem from this unit’s problem set, write it out fully, solve it fully, and post it.

The second post can be a problem you cannot solve or another fully solved problem from the problem set. If there is a problem in the problem set that you are confused about, write it out fully and show any work that you have started. Note that you are stuck and ask for help. If you are able to solve every problem in the set without difficulty, post at least one other fully solved problem from the set as your second post.

The third and fourth posts are responses. Response guidelines are provided below and in every discussion.

Take advantage of this discussion area to work together as a class on the problem set. Post as many problems as you can and review as many of your peers’ posts as you can. Ask questions. Offer answers. If you are stuck on a problem, post your question. Working as a team will help each person gain a better understanding of the problem set and the concepts covered in this unit. Although the problem sets are not graded they will help you prepare for the quizzes in this course.

Response Guidelines

The third and fourth posts, the response posts, are your chance to help your peers. Explore the posts made by your peers and find people who are stuck. Help at least two peers through the problems they are stuck on. If you are unable to locate a peer who is stuck, you may instead choose a peer post, write out your solution to the same problem, and compare solutions and methods. The goal of this area is for the entire class to work together as a group on the entire problem set. You are also expected to complete and understand the problem set on your own in preparation for the quizzes and final exam.

 

MAT2051 Discrete Mathematics

Unit 8 Discussion

DQ1Networking, Graphs, Adjacency Matrices, and Server Application

Assume you are a network administrator of a large business firm that wishes to construct an intranet. Your firm has a number of computers, servers, and services such as printers, all connected to five hubs (hubs 1, 2, 3, 4, and 5), which must now be connected to each other. Assume each hub is separated by distances shown in the following adjacency matrix:

Adjacency Matrix. See D-Link for full description. [D]

Each hub must be connected to at least one other hub, therefore, you only need to make four connections to allow communication between all computers. You can further minimize the cost of the cables and time by only connecting the closest hubs.

Consider and answer the following questions to evaluate this problem:

What is an intranet?

What is an adjacency matrix, and how do they relate to graphs?

What is the optimal connection of the five hubs that will minimize the length of the cable required to connect them? Note: The length of the cable will be the distance between the hubs as shown in the adjacency matrix.

How did you arrive at this answer? What algorithm should be used to solve this problem?

Hint: This is a graphical minimization problem. Make sure you comment on the following: What are the vertices? What are the weights on the edges? What are we trying to minimize?

Review the Discussion Participation Scoring Guide prior to posting.

Response Guidelines

Read your peers’ posts and respond to two. Compare your response to your peer’s. Explain the similarities and differences between your posts.

DQ2 Practice Problem Set Review

This discussion allows you to work with your peers to complete and understand the assigned problem set for this unit. Remember, two initial posts and two response posts are required. Further posts are optional and recommended:

For the first post, select a problem from this unit’s problem set, write it out fully, solve it fully, and post it.

The second post can be a problem you cannot solve or another fully solved problem from the problem set. If there is a problem in the problem set that you are confused about, write it out fully and show any work that you have started. Note that you are stuck and ask for help. If you are able to solve every problem in the set without difficulty, post at least one other fully solved problem from the set as your second post.

The third and fourth posts are responses. Response guidelines are provided below and in every discussion.

Take advantage of this discussion area to work together as a class on the problem set. Post as many problems as you can and review as many of your peers’ posts as you can. Ask questions. Offer answers. If you are stuck on a problem, post your question. Working as a team will help each person gain a better understanding of the problem set and the concepts covered in this unit. Although the problem sets are not graded they will help you prepare for the quizzes in this course.

Response Guidelines

The third and fourth posts, the response posts, are your chance to help your peers. Explore the posts made by your peers and find people who are stuck. Help at least two peers through the problems they are stuck on. If you are unable to locate a peer who is stuck, you may instead choose a peer post, write out your solution to the same problem, and compare solutions and methods. The goal of this area is for the entire class to work together as a group on the entire problem set. You are also expected to complete and understand the problem set on your own in preparation for the quizzes and final exam.

 

MAT2051 Discrete Mathematics

Unit 9 Discussion

DQ1 Data Storage, Retrieval, Sorting Algorithms, Time Complexity

In computer systems, the storage, retrieval, and communication of data occurs frequently and therefore must be optimized and fast. In some instances, it is best to store a list of numbers in some order (for example, in an increasing or decreasing order), to make the retrieval of some of the data very fast (for example, the maximum value).

In section 4.2 of your textbook, you read about the insertion sort algorithm (see Algorithm 4.2.3, page 180). This algorithm will sort a list of n numbers in non-decreasing order. However, in section 9.7, pages 477–483 of the text, you read about a faster sorting algorithm.

In order to fully understand this question, answer the following questions:

The name of the faster sorting algorithm is the tournament sort. How is this faster sorting algorithm achieved?

Given a list of n numbers, what is the worst case times for the insertion sort algorithm and the tournament sort algorithm?

In the insertion sort and the tournament sort algorithms, what discrete structures are used to store the data? Explain.

How does the use of the tree structure provide the framework for a tournament sort?

What is a divide-and-conquer algorithm? Is this a divide-and-conquer algorithm?

You may find the link in the resources section useful for this discussion. Review the Discussion Participation Scoring Guide prior to posting.

Response Guidelines

Read your peers’ posts and respond to two. Compare your post with your peer’s. Explain the similarities and differences.

DQ2 Practice Problem Set Review