Math 245 - Writing Assignment
Writing Assignments/Technology Assignments You will complete a total of six writing/technology assignments throughout the semester. You can choose these from the list below. They can all be writing assignments, or they could all be technology assignments or some can be writing and some can be technology. The dues dates are listed in the syllabus.
Writing Assignments Requirements
Each writing assignment is worth 20 points should include the following sections: Background (3 points): This is a discussion of how the non-‐mathematical and mathematical portions of your topic fit together. You might include a historical background of the topic, definitions of terms, the discrete mathematics ideas that are addressed (e.g. induction, logical fallacy, etc.), and some explanation about why these ideas were useful.
Examples (10 points): In most of your writing assignments you are asked to discuss and describe an aspect of discrete mathematics. Give two of three examples or techniques of the topic under discussion. Give general information and also specific examples of the topic.
Bibliography (2 points): List the references you used to complete this report. Just list title and author for any books and articles you used. You should also include a list of people that you consulted or any other form of help that you received. For example, you might obtain some of your information from the internet; in this case, you could include the website. You'll need at least one book or article as a reference, preferably two, and a total of at least two references.
You'll notice that there are still 5 points unaccounted for. The remaining 5 points are for style: clarity, neatness, flow, design, organization and creativity-‐-‐it's important to be able to communicate your ideas.
Note: you don't have to put your report in the precise order given above. You may prefer to use the assigned problems to illustrate how the ideas of the subject fit together with the mathematical ideas that you will be using, in which case Background and Examples would be interwoven. Just make sure that these aspects appear in your report.
Technology Assignments Requirements
Each technology assignment is worth 20 points should include the following sections: Background (5 points): This is a written paragraph of how the non-‐mathematical and mathematical portions of your topic fit together. In other words, you need to talk about what you needed to know about your topic in order to solve the problems and how discrete mathematics fits into the picture. So you might include the definitions of terms, the discrete mathematics ideas you used (e.g. induction, logical fallacy, etc.), and some explanation about why these ideas were useful.
Solution (15 points): In most of your technology assignments you are asked to write a program to solve a problem. Copy and paste your code in a Word document and annotate each section of the code with explanation of what are you doing in each section. Include a screen shot of the output of the program. If you are using EXCEL, include the excel file as a separate attachment.
Writing and Technology assignments will be submitted in Canvas and checked with SafeAssign. Make sure to submit your own work and give written explanation using your own voice. Compile your work in one document and save it in pdf format and submit it by clicking on the assignment title.
Writing Assignments: Combinatorics: 1. Describe some of the earliest uses of the pigeonhole principle by Dirichlet and other mathematicians 2. Describe the different models used to model the distribution of particles in statistical mechanics, including Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac statistics. In each case, describe the counting techniques used in the model. Logic: 3. Discuss logical paradoxes, including the paradox of Epimenides the Cretan, Jourdain’s card paradox, and the barber paradox, and how they are resolved. 4. Describe how fuzzy logic is being applied to practical applications. Proofs and Induction: 5. Look up some of the incorrect proofs of famous open questions and open questions that were solved since 1970 and describe the type of error made in each proof. 6. Describe the origins of mathematical induction. Who were the first people to use it and to which problems did they apply it? Algorithms: 7. Describe six different NP-‐complete problems. 8. Describe the historic trends in how quickly processors can perform operations and use these trends to estimate how quickly processors will be able to perform operations in the next twenty years. Relations and Functions: 9. Discuss the concept of a fuzzy relation. How are fuzzy relations used? 10. Describe how equivalence classes can be used to define the rational numbers as classes of pairs of integers and how the basic arithmetic operations on rational numbers can be defined following this approach. Recursion: 11. Describe a variety of different applications of the Fibonacci numbers to the biological and the physical sciences. 12. When are the numbers of a sequence truly random numbers, and not pseudorandom? What shortcomings have been observed in simulations and experiments in which pseudorandom numbers have been used? What are the properties that pseudorandom numbers can have that random numbers should not have? Number Theory: 13. Describe the history of the Chinese remainder theorem. Describe some of the relevant problems posed in Chinese and Hindu writings and how the Chinese remainder theorem applies to them. 14. Show how a congruence can be used to tell the day of the week for any given date. Graph Theory: 15. Discuss the applications of graph theory to the study of ecosystems, to sociology and to psychology. 16. Explain how graph theory can help uncover networks of criminals or terrorists by studying relevant social and communication networks.
Programming Assignments: Sets: 17. a) Given two finite sets, list all elements in the Cartesian product of these two sets. b) Given a finite set, list all elements of its power set. Combinatorics: 18. Given an equation 𝑥! + 𝑥! + ⋯+ 𝑥! = 𝐶, where C is a constant, and 𝑥!,𝑥!,… ,𝑥! are nonnegative integers, list all the solutions. 19. Input the English alphabet (a string of 26 letters):
a) Generate all the permutations of a set with four elements b) Generate all the combinations of a set with four elements
Logic: 20. Given the truth values of the propositions p and q, find he truth values of the conjunction, disjunction, exclusive or, conditional statement, and biconditional statement of these prepositions. 21. Find as many positive integers as you can that can be written as the sum of cubes of positive integers, in two different ways, sharing this property with the number 1729 Algorithms: 22. Given an ordered list of n distinct integer, determine the position of a specific integer on the list using:
a) A linear search algorithm b) A binary search algorithm c) A tertiary search algorithm
23. Given a list on n integers, use the greedy algorithm to find the change for n cents using quarters, dimes, nickels and pennies. Relations and Functions: 24. Display all the different reflexive, symmetric and transitive relations on a set with six elements. Recursion: 25. Given a nonnegative integer n, find the nth Fibonacci number using recursion. 26. Determine which Fibonacci numbers are divisible by 5, which are divisible by 7, and which are divisible by 11. Prove that your conjectures are correct. Number Theory: 27. Given integers n and b, each greater than 1, find the base b expansion of this integer. 28. Given a positive integer, determine whether it is a prime number or a composite number using trial division. If the number is composite, find the prime factorization of the number. 29. Given two positive integers, find their least common multiple. Graph Theory: 30. Given the vertex pairs associated to the edges of a graph, construct an adjacency matrix for the graph. (Produce a version that works when loops, multiple edges, or directed edges are present.) 31. Given the vertex pairs associated to the edges of a multigraph, determine whether it has an Euler circuit and, if not, whether it has an Euler path. Construct an Euler path or circuit if it exists. 32. Given the list of edges and weights of these edges of a weighted connected simple graph and two vertices in this graph, find the length of a shortest path between them using Dijkstra’s algorithm. Also, find a shortest path.