Course Description
COURSE NUMBER and NAME
MTH 500 Mathematics for Computing Sciences
UNITS
3
LENGTH OF CLASS
8
COURSE DESCRIPTION
This course provides students with an understanding of mathematics, that is the concepts and structures, which underpin general computing and algorithmic design. The content will include investigation of sets, relations, functions, general algorithms, number systems, logic, induction and recursion, proofs, Boolean algebra, languages and automata, circuits, and graphs and trees. Throughout, the course will emphasize computational thinking that leads to the solution of computing problems.
REQUIRED TEXT
Hein, L. J. (2017). Discrete structures, logic, and computability (4th ed.). Jones & Bartlett Learning.
Print ISBN: 9781284070408
eText ISBN: 9781284116328
INSTRUCTIONAL METHOD
Online / On-Campus
Summary of Graded Work and Assessments
Graded work and assessments offer students the opportunity to show the degree of mastery for each CLO. The following table shows how assessments and CLOs align (link).
Assignments Totals Weight CLOs
Engagement and Professionalism (Rubric) including live class activities 200 20%
Week 1 Discussion 25 2.5% 1, 3, 5
Week 1 Assignment 75 7.5% 1, 2
Week 2 Discussion 25 2.5% 1, 4, 5
Week 2 Assignment 75 7.5% 1, 4, 5
Week 3 Discussion 25 2.5% 1, 3, 4, 5
Week 3 Assignment 75 7.5% 2, 3, 4
Week 4 Discussion 25 2.5% 1, 3, 5
Assignments Totals Weight CLOs
Week 4 Assignment 75 7.5% 1, 2
Week 5 Discussion 25 2.5% 2, 4
Week 5 Assignment 75 7.5% 1, 2
Week 6 Discussion 25 2.5% 1, 3, 5
Week 6 Assignment 75 7.5% 1, 3, 5
Week 7 Discussion 25 2.5% 1, 3, 5
Week 7 Assignment 75 7.5% 1, 2
Week 8 Discussion 25 2.5% 1, 3, 5
Week 8 Assignment 75 7.5% 1, 2
Total Points/Percentage 1000 Points 100%
Course Policies
For Westcliff’s course policies, please see the Course Policies document.
Discussion Requirements
For all discussions, the primary response is due by Thursday at 11:59 p.m. Pacific Time. The primary response must be at least 200 words in length and fully address the topic, demonstrating critical thinking and understanding. Each student must then also post a minimum of two responses to other students in the discussion by Sunday night at 11:59 p.m. Pacific Time. Each peer response must be at least 50 words in length and substantively engage with the other student’s original post, continuing the discussion in a professional manner. If at any time information or material is brought in from an outside source or website, it must be properly cited following APA 7th edition guidelines and a full reference must be provided.
Assignment Requirements
Each assignment deliverable is specifically defined in the assignment instructions, such as page length, citations and references, audio or video, presentations, tables, etc. For all written assignments, the required page length does not include the cover or references pages. Refer to the specific requirements as stated in each assignment, and reach out to your instructor for additional information as needed. All graded submissions are due by Sunday at 11:59 p.m. Pacific Time.
All written work must adhere to APA 7th edition academic formatting requirements including core components such as the cover page, page numbers, headings, citations, 1” margins, paragraph indentations, left alignment, double spacing throughout, and the final references using hanging indents.
Participation Requirements
Students are required to attend each live class session either in person or virtually as stipulated in the course policies. Participation in the live class session is determined by actively engaging, answering or asking questions, providing comments, interacting in group activities, etc., as required by the instructor. Students who are unable to attend the live in-class or virtual sessions must follow the VCS submission requirements as stated in the Course Policies document.
Writing Center
The Westcliff University Writing Center is dedicated to providing quality support to students and faculty. From assignment review, to in-class workshops, to dissertation support, to publication help, the Writing Center is committed to empowering individuals to use the written language to articulate and disseminate knowledge.
Course Learning Outcomes (CLOs)
Learning outcomes are statements that describe significant and essential scholarship that students have achieved and can reliably demonstrate at the end of the course. Learning outcomes identify what the learner will know and be able to do by the end of a course – the essential and enduring knowledge, abilities (skills), and attitudes (values, dispositions) that constitute the integrated learning needed for successful completion of this course. The learning outcomes for this course summarize what students can expect to learn, and how this course is tied directly to the educational outcomes ofthe degree.
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Course Learning Outcomes (CLOs)
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PLOs
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1. Analyze structures relevant to computing and problem solving.
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3
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2. Evaluate algorithms for performance and/or comparative performance.
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6
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3. Design algorithms and procedures relevant to proper functioning of computing systems.
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5
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4. Select heuristics, procedures, and algorithms for specific problem scenarios.
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5
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5. Solve various problems that are amenable to existing procedures.
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3
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Detailed Course Outline
The following outline provides important assignment details for this course, unit by unit.
Students are responsible for all ofthe assignments given. Please refer to the Detailed Description of Each Grading Criteria in the syllabus for specific information about each assignment.
Module 1
Class Preparation Materials:
● Reading:
● Chapter 1- Elementary Notions and Notations (pp. 1-15)
● Section 1.1 (scan the remainder for awareness)
● Chapter 6 - Elementary Logic (pp. 411-473)
Assignments:
● In-Class Activity: Logic Fundamentals
● Discussion: Reasoning and Computation
● Discussion Response to 2 Peers
● Assignment: Logic Problems (6.2)
Week 1 In Class Activity: Learning Logic Fundamentals - CLO 1, 3, 5
Students will explore logic fundamentals, specifically elementary logic. Elementary logic defines logic as the examining of reasoning and the scientific method of judging the truth or falsity of statements. A proposition is a statement that is either true or false. For this in-class activity, students, in groups, will:
● Examine reasoning in everyday language. What elements of language constitute:
o Implication?
o Evidence?
o Proof?
o Counterexamples?
o Induction?
o Deduction?
● Discuss the connections and/or distinctions in reasoning logically between everyday language and elementary logic.
● Discuss how reasoning logic is connected to computation:
o Boolean Logic
o Decision Trees
o Expert Systems
o Graphs
o When can all examples of something be presented as proof of a certain ‘theory’ computationally?
● See whether you can convince yourself, or a friend, that the conditional truth table is correct by making up English sentences in the form “IfA then B.”
● Create 2-3 specific problems whereby logic is used in computation.
o For example, you could use a simple “If” statement if you were searching for an item in a list or if you are determining whether or not a symbol or a number meets a certain criteria.
Week 1 Discussion: Reasoning and Computation - CLO 1, 3, 5 (Rubric)
For this discussion you will examine the ideas regarding reasoning and proofs. In your discussion, post and provide an example of the following:
● Compare reasoning and valid arguments in the three disciplines of law, mathematics, and everyday language. Pay some attention to why two of these areas require more rigorous and verifiable reasoning than the other one.
● Examine and compare legal reasoning versus mathematical reasoning.
● Examine how reasoning is manifested in computing.
o For example, consider logical statements and code. Complex or conditional
statements with multiple criteria on which to base a judgment. Also consider the situation with AI in which we have machines making decisions based on
functions we cannot see and the inability to explain their reasoning.
Week 1 Assignment: Logic Problems- CLO 1, 2 (Rubric)
For this assignment students will solve problems of logic. Specifically, students will:
● Solve the following problems for sections 6.1 & 6.2:
o Problem # 5
o Problem #6
o Problem # 9 a,b,c
o Problem # 10 f,g,h
o Problem # 16a,c,e
● Solve the following problems for section 6.3:
o Problem # 3
o Problem #4
o Problem # 5 a,b,c
o Problem # 6 f,g,h
o Problem # 8
o Problem #9
● Students must show all the steps of their work, providing a specific rationale as to why they solved the problem in the manner they did.
● Students will submit their work, via GAP. You must provide a clear screenshot of all of your work, including your processes and procedures.
Module 2
Class Preparation Materials:
● Reading:
● Chapter 1- Elementary Notions and Notations (pp. 18-41)
● Section 1.2 (scan the remainder for awareness)
● Chapter 2 - Facts and Functions (pp. 81-139)
Assignments:
● In-Class Activity: Sets and Functions
● Discussion: Functions and the Beginning of Processing
● Discussion Response to 2 Peers
● Assignment: Logic Problems (6.3)
Week 2 In Class Activity: Sets and Functions - CLO 1, 2
Students will explore sets and functions, specifically examining function and outcomes and the relationship of functions and decision-making. For this in-class activity, students, in groups, will:
● Discuss the connections between the relationship of function and outcomes.
● Discuss what the idea of function means in terms of everyday decision making.
● Provide an example of a decision you made in terms of input and function.
o For example, you bought a car. You consider price, mileage and other inputs to make this decision. In terms of function, the criteria you have about a car in order for it to be purchasable.
● Identify and compare the sources of the inputs for functions across mathematics and computing/algorithms (a.k.a., code).
o Is it possible for an input or an output to be something other than a number or a phrase/string?
o How can sets be structured?
o How does a database table serve as a set?
o When would the complement of a set be used for input for a function?
● In mathematics, we often start with a given or known function and use it to calculate
values for some purpose. However, in practice, we often use data to compute functions. Why is this necessary?
Week 2 Discussion: Functions and the Beginnings of Processing -CLO 1, 4, 5 (Rubric)
● For this discussion you will examine the ideas regarding how sets (collections of ‘things’) underlie functions and how functions serve as a means of processing inputs into outputs. In your discussion, post and provide an example of the following:
● Examine how functions are expressed in everyday language.
o Be sure to examine the relationship of implied functions to outcomes.
● ● Discuss what the idea of function means in terms of everyday decision-making.
o Provide an example of a decision you made in terms of input and output.
▪ For example, when you purchase a car you consider price, mileage and
other inputs to make this decision. The output is the purchase decision.
● How are sets and functions manifested and used in computation?
o What things can be input? What things can be output? How are sets and set-like structures used in computation?
Week 2 Assignment: Logic Problems- CLO 1, 4, 5 (Rubric)
For this assignment students will solve problems of logic. Specifically, students will:
● Solve the following problems for Section 1.2: #2, 6, 8, 12, 15, 16, 21, 27 (a, d, g),
● Solve the following problems for Section 2.1: #3, 8, 10, 12, 13, 29
● Solve the following problems for Section 2.2: #2, 5, 6, 9 (a, d, g),
● Solve the following problems for Section 2.3: #4, 8, 12
● Students must show all the steps of their work, providing a specific rationale as to why they solved the problem in the manner they did.
● Students will submit their work, via GAP. You must provide a clear screenshot of all of your work, including your processes and procedures.
Module 3
Class Preparation Materials:
● Reading:
● Chapter 1 - Elementary Notions and Notations
▪ Section 1.3 (scan the remainder for awareness)
● Chapter 9 - Algebraic Structures and Techniques (pp. 617-691)
● Section 9.1
● Section 9.2
● Section 9.3
Assignments:
● In-Class Activity: Algebraic Structures and Techniques
● Discussion: Boolean Algebra
● Discussion Response to 2 Peers
● Assignment: Algebraic Problems (9.1) (9.2) (9.3)
Week 3 In Class Activity: Algebraic Structures and Techniques - CLO 1, 3, 4, 5
Students will explore algebraic structures and techniques, specifically defining algebra and descriptive problems and concrete versus abstract, Boolean Algebra and other functions and concepts. For this in-class activity, students, in groups, will:
● What is algebra? To answer this question, consider the ‘algebra of sets. ’
● What is calculus? To answer this question, consider something called the ‘situation calculus. ’
● What entities form. the (analogs of) sets and functions in an algebra and a calculus?
● What is the importance of Boolean algebra and logic to computation and computers?
Week 3 Discussion: Boolean Algebra -CLO 1, 3, 4, 5 (Rubric)
For this discussion you will examine the ideas regarding Boolean Algebra. In your discussion, post and provide an example of the following:
● What are alternative definitions of a Boolean Algebra?
● Can a Boolean Algebra be defined as the set ofall functions from the domain of binary n-tuples into the set {0,1}?
● Can a Boolean Algebra be defined recursively, beginning with certain primitive elements?
● The author exhibits the power set of a set S (the set of subsets of S) as a Boolean Algebra. How can one gain insight into the relation between Boolean functions by observing the lattice consisting of the power set of S? (please note p. 634 of the text).
Week 3 Assignment: Algebraic Problems- CLO 2, 3, 4 (Rubric)
For this assignment students will solve problems using algebraic structures and techniques. Specifically, students will:
● Solve the following problems for Section 9.1
o Problem #1
o Problem #2 a, b
o Problem #10
● Solve the following problems for section 9.2:
o Problem # 1
o Problem # 2 a,b
o Problem #3
o Problem #4a
o Problem #5 a,c,d,e
o Problem #7 d,e,f
o Problem #12
● Students must show all the steps of their work, providing a specific rationale as to why they solved the problem in the manner they did.
● Students will submit their work, via GAP. You must provide a clear screenshot of all of your work, including your processes and procedures.
Module 4
Class Preparation Materials:
● Reading:
● Chapter 1 - Elementary Notions and Notations
▪ Section 1.4 Graphs and Trees
● Chapter 10- Graph Theory (pp. 691-739)
▪ Sections 10.1 - 10.5
Assignments:
● In-Class Activity: Graph Theory
● Discussion: Graph Theory
● Discussion Response to 2 Peers
● Assignment: Graph Theory (10.1 - 10.5)
Week 4 In Class Activity: Graph Theory - CLO 1, 3, 5
Students will explore fundamental concepts in graph theory, including graph types, connectedness, and Eulerian paths. They will work together to make connections while solving graph-related problems and practical applications. For this in-class activity, students, in groups, will:
Select and solve a problem from one of the problem sets:
● Eulerian Path and Circuit:
o Problem: Given a graph, determine if an Eulerian path or Eulerian circuit exists. If it does, find the path or circuit.
o Instructions: Discuss the conditions for the existence of an Eulerian path or circuit. Diagram the graph and walk through finding the path (if possible).
● Graph Traversal (Breadth-First and Depth-First Search):
o Problem: Given a graph, perform. a breadth-first search (BFS) and a depth-first search (DFS) starting from a specific vertex.
o Instructions: Show how BFS and DFS work step-by-step, highlighting the order in which vertices are visited. Discuss the differences between the two traversal methods.
● Graph Coloring:
o Problem: Assign colors to the vertices of a graph such that no two adjacent vertices have the same color. What is the minimum number of colors needed (chromatic number)?
o Instructions: Diagram the graph and experiment with different coloring methods. Try to minimize the number of colors used and justify your choices.
● Shortest Path (Dijkstra's Algorithm):
o Problem: Given a weighted graph, use Dijkstra’s algorithm to find the shortest path between two specified vertices.
o Instructions: Walk through the steps of Dijkstra’s algorithm, illustrating how it finds the shortest path. Present the final path and its total weight.
● Bipartite Graph:
o Problem: Given a graph, determine whether it is bipartite.
o Instructions: Explain how to determine if a graph is bipartite and provide a diagram of the graph with the two sets of vertices identified.
● Connectedness:
o Problem: Determine whether a given graph is connected. If it is not connected, find its connected components.
o Instructions: Diagram the graph, and explain the process of determining
connectedness. If the graph is disconnected, highlight the distinct connected components.
● Each group will discuss the graph they worked on, explaining their problem-solving process and solution and discussing any challenges they faced and how they overcame them.
● Groups will discuss the following:
o What are some real-world problems that can be modeled using graph theory?
How might concepts like graph traversal, shortest paths, or graph coloring apply to real-world networks?
Week 4 Discussion: Graph Theory- CLO 1, 3, 5 (Rubric)
For this discussion you will examine the ideas regarding graph theory. In your discussion, post and provide an example of the following:
● Graphs can model a wide range of real-world systems, from social networks to
transportation routes. How can concepts like graph connectivity, shortest paths, or graph coloring be applied to solve practical problems in areas such as computer science, urban planning, or biology? Can you think of specific examples where these graph theory
concepts have been or could be applied?
● In what ways do graph theory and network analysis overlap, and how do they contribute to advancements in fields like data science and artificial intelligence?
Week 4 Assignment: Graph Theory - CLO 1, 2 (Rubric)
For this assignment students will solve problems of logic. Specifically, students will:
● Solve the following problems for section 10.1:
o Problem # 1a,b,c
o Problem # 2a,b
o Problem # 3a,b
o Problem #6a,b,c
o Problem # 7a,b
● Solve the following problems for section 10.2:
o Problem # 2
o Problem # 3
o Problem # 8a,b
o Problem #10a,b,c
● Solve the following problems for section 10.3
o Problem #1
o Problem #5
o Problem #7
● Solve the following problems for section 10.4:
o Problem # 1
o Problem # 2
o Problem # 3
o Problem # 6
● Solve the following problems for section 10.5
o Problem #3
● Students must show all the steps of their work, providing a specific rationale as to why they solved the problem in the manner they did.
● Students will submit their work, via GAP. You must provide a clear screenshot of all of your work, including your processes and procedures.