Syllabus           Math 210 Discrete Structures Fall 2025

Margaret Menzin       Office: Suite E-425-D Office Phone: X2704               
                                    Email: menzin@simmons.edu   Home Phone: 781-862-5107
My  zoom office hours is https://simmons.zoom.us/j/4672991761

                                    The class on Monday Sept. 22, 2025 will be rescheduled.

 Office Hours: MWF 7:15-8:00 a.m and MWF 12:00- 2:00 and
                                                              other days and times by appointment –
I am almost always around/on email,
but give me a heads-up the day before.

Note:  The Computer Data and Mathematical Sciences
(CDMS) Department "eats" at the Fens on many
Wednesday 12:00-1:00.
Whenever feasible we will be outside in front of the Fens.

Common Syllabus Statements about Accessibility, Academic integrity and Sexual Harassment Policies will be found here .

The Specifics of This Course

Textbook: Susanna S. Epp – Discrete Mathematics and Its Applications –
4th Edition
You do not need the solutions manual or student study guide, as I post
solutions.
I urge you to have a physical copy of this book! I own both physical and
electronic copies, and the eBook is very hard to use.

The moodle site for the course is at https://moodle.simmons.edu/course/view.php?id=49929

The VoiceThread invitation is at https://voicethread.com/login/join/group/23610488/unpromoted/e09565be3

What this course is about

This course has three major goals: the first is to master certain content, and the second is to expand students' ability to prove theorems and reason abstractly, logically, and numerically. (This second goal is sometimes described as 'become more mathematically sophisticated.'). The third goal is to examine how mathematics and computer science interpret and use many abstract structures which are foundational to both disciplines, and how those differing interpretations enrich each other.

These goals support each other: one learns to reason in the context of a particular topic, and the topics (content) are such as to expose the methods of reasoning. Further, being able to examine a topic from different perspectives helps us understand it.

As we master new content we will reflect upon the methods we are using, and be explicit about the patterns in our methods of proof and applications. For example, when we study mathematical induction we will identify and apply three different patterns. Knowing how these patterns work will, then, allow us to tackle new problems by asking which pattern we should try first. We will tie this into how recursion is used in computing (in recurrence relationships and in recursive proofs) and how induction is used to prove the correctness and complexity of computer algorithms.

Another tool for 'getting our heads' around a concept is to be able to write about it. So in this course there will be a significant amount of writing about the topics we are studying, and especially about the differing – mathematical and computational – perspectives. Being able to write about a technical subject is also a valuable skill – one used in writing math proofs, writing documentation for a program, or making a presentation.

This is a large class and we will try various settings for our writing – at home, in class in pairs or small groups, in class by yourelf. For work done at home and in class, you may not use AI tools like ChatGPT unless the assignment explicitly says it is allowable.

We will need to translate back and forth between 'real world' (English) statements and their logical/programming representations. Phrases like 'there exists' and 'for all' are important in both mathematics and computer science, but often unstated/implicit in our ordinary conversation.

We will strive to make our explanations and proofs as clear and orderly as we can. Of course, we will use grammatically correct English.

Because we will be doing so much writing I have asked to have a Writing Assistant for the course (through the Writing Center.)

Student Learning Outcomes

Students will be able to construct short, clear proofs about functions, sets, and basic number systems, and formulas using direct and indirect methods of proof, and mathematical induction.

Students will be able to distinguish between examples and proofs, and will use examples appropriately. Students will be able to explain the roles of examples and counter examples in testing mathematical statements and program correctness and in assessing outcomes from generative AI.

Students will translate statements in English, both with and without quantifiers,
into logical format, use logical methods to deduce conclusions as needed, and
translate the conclusions back into English..

Students will be able to compare and contrast the perspectives of mathematics
and computer science on the topics studied in the course.

How to Succeed in This Course

· Flipped

This course is essentially flipped. Saying the course is flipped means that you will read the book/watch the video before you come to class, that there is essentially no 'lecture' during class, and that class time is devoted to the kinds of activities which used to be homework – solving problems, constructing examples, writing explanations, and proving theorems.

I think that that means the course is more fun. (Also, the material is a lot of fun.) Instead of listening to long lectures in our face-to-face meetings, you will be working together in small groups, talking about the material and working on problems and proofs.

Flipping the course also means that it is critical that you stay current in the material and come to each class session having done the homework.

· Expectations
I expect you to be present, punctual, alert and prepared for every session.
Being prepared means that you have done the work for the meeting( see below), that you are ready to ask questions about anything which is confusing, and that you are ready to attack thought-provoking questions with your classmates.

A lot of the learning in this course takes place in class. If you are not present,
puntual, prepared and alert then you miss work which can not be made up!

The instructor reserves the right to deduct up to 15% of the grade for students who do not meet the expectations of present, punctual, alert and prepared in every session.

A maximum of 4 unexcused absences is allowed and I do take attendance at all our meetings. Anything beyond that – aside from religious holidays and doctors’ notes - is an automatic Fail.

I consider the policy of 4 absences out of 39 meetings to be very generous. I am making the policy generous because I do not want to be in the position of judging what is an excused absence and what isn’t. That said, please be very judicious in your use of absences --- at the start of the semester you have no way of telling if you will feel lousy in the last week.

Anything beyond a very occasional minor lateness will be considered as an absence. If you find that your bus/the T is often late, then you need to leave your apartment earlier.

· The Order of the Work

Flipping a course means that before coming to class you will watch the VoiceThread, answering any questions in it, and then do the reading and problems which are assigned in the VoiceThread.

I have provided a spreadsheet with links to the VoiceThread for each class, and the assignments in the textbook and outside reading are detailed in the VoiceThread.

.Again, the order of work is to first go the VoiceThread, being sure to respond to all the questions, and then go to the textbook/outside reading.

The VoiceThreads are fairly short, and, depending on the topic, they may provide an introduction to the material, or provide a different way (from that in your book) to think about the material. The same is true for the outside reading.

When you go to the problems in our textbook you will find that, for the most part, I have chosen rather straightforward problems.

When we come together for our face-to-face sessions, you will work in small groups on more difficult problems . I will move from group to group, coaching you and helping you to think about the material.

· Getting the most out of our book

We have an excellent book, and, in the past students have been very pleased with it. On the other hand, there is a technique to reading a math or other techical book.

Reading such material is sometimes slow going, and a section will usually require two or three readings. 

It is a good idea to read the section first for the general flow of ideas, skipping over anything that doesn't seem obvious. 

On the second and third readings, now that you know where the book is heading, you should read the book closely, making note of anything that does not make sense, or any calculation you can't follow. 

As you re-read, you may also choose to make a note to yourself to ask about something in class.  For example, 'p. 37, line 4 doesn´t make sense' is a fine thing to ask about in the beginning of class (and a lot more helpful than I'm lost'.)  I also strongly encourage you to email me the evening before class with the same kind of reference to unclear items. Each VoiceThread also provides a place to ask about anything which is unclear and I look at them about 9:30 p.m. the evening before each class.

Lastly, it will help you master the material if, after you read a section, you try to summarize the main ideas in your own words and to ask yourself how this section connects to previous material.

The cognitive science about learning has shown that the learning ‘sticks’ best if you review material in a spaced out way, mix together different kinds of problems, and connect new material to previously learned material. So you can expect the work in this course to try to follow those practises.

Sadly, you also remember things better when they are hard to learn – for example, you need to reach back to material you studied a while ago. When that happens to you, you can comfort yourself by knowing that it means you are doing the hard job of cementing the knowledge in your long-term memory (which is where you want it to be.)

We are all friends in this class, and none of you is yet an expert on the material. I really encourage you to email me, ask questions in the VoiceThreads, and ask questions in class; it is rarely the case that only one student finds something mysterious.  If I do not ask for questions at the beginning of class, and you have some, please ask your questions.

Outline of Course Material

Unit 1 - Logic, and Basic Proofs (Chapters 1 to 4 of text)

In the first part of the course we examine Aristotelian or truth table logic and its application to the design of logic circuits for computers. Importantly, we learn to be precise about our language, and see how that helps us to state our ideas clearly and unambiguously. This also leads us to understand different ways we can prove theorems, and to practice constructing proofs (something we will do a lot of).

Truth table logic is equivalent to certain simple circuits - series and parallel circuits to implement 'and' and 'or' gates in computers.  We will examine these correspondences and see the design of circuits for functions such as addition in a computer.

Quantified logic introduces the phrases 'for all' and 'there exists'. You will translate English, database,and mathematical statements into this format, and learn how to negate quantified statements.  In this context we will discuss the difference between an example and a proof.  We will apply our knowledge to writing focused queries for Internet searches and for databases.

We also look more deeply at set theory and some of the foundational issues in theoretical computer science. This examination leads to the consideration of what it means for a system to be complete, consistent and decidable, as propositional logic is. Quantified logic does not have the neat characteristics (complete, decidable) of truth table logic. We revisit these notions and learn about Russell's paradox, Godel's Incompleteness Theorem (a fundamental result in the foundations of mathematics) and about the Halting Problem and about the famous question 'P=NP?' (the fundamental outstanding problem in foundations of computer science).

Unit 2 - Mathematical Induction, Recursion, and More Sophisticated Proofs (Chapters 5 to 8, part of 11 of text)

In this unit we learn how to use one of the most important proof techniques of mathematics- namely, mathematical induction. Mathematical induction also has an important relationship to recursion, a basic programming technique, which we will examine briefly. Further, many proofs about the complexity of a computer algorithm depend on the use of mathematical induction.

As we learn about induction, we will also improve our theorem proving skills by working with fundamental number systems such as the Z and Z_n and the prime numbers. Z_n is important in computing, both in cryptography and in error-correcting codes.

Finally, a return to the subject of functions (including one to one, onto, and inverse functions) prepares us for the topic of cardinality (size) of sets. Mathematicians, in addition to worrying about what it means to prove something and how to prove things, also spend a lot of time thinking about measuring things.  We will discuss what it means for two sets to have the same number of elements ( or cardinality). For example, we will ask if all infinite sets are the same size. 

In addition to measuring sets, we will also revisit our work on describing how hard a problem (such as factor an integer into primes) is, or the complexity of the algorithm to solve it. That is, we return to the earlier topic of complexity. This has applications to cryptography, since you would like breaking a secret code to be as hard as possible. We will see how hard problems are valuable to people trying to protect computer systems from hackers.

Unit 3 - – Graphs and Trees (Chapter 10)

The basic notions of graph theory are introduced - vertices, edges, degree of a vertex, connected components, directed and undirected graphs, and acyclic graphs or trees. 

A variety of graph theory problems and their solutions will be explored -Eulerian and Hamiltonian graphs, graph isomorphism, graph coloring, minimal spanning trees, etc. - as well as the representation of graphs in computers and their applications to real world problems.

·  Grading
There are 3 written exams, each of which is worth 25% .
There are 6 'graded written assignments', which are graded on a
check/check-plus basis and which together are worth 10%
There is a final written assignment which is worth 15%.

Any assignment is due on the stated date, although you have a grace period until the next class. There are no extensions beyond that unless you have previously been given permission by me or for dire emergencies (death, serious illness, etc.).
I remind you that the instructor reserves the right to deduct up to 15% of the grade for students who do not meet the expectations of present, punctual, alert and prepared in every session. Please re-read the material on absences in the section on Expectations (p.3 of this document.)

This course is a lot of fun and it will be a very sociable experience. So let’s get started!