Math 225 Foundations of Advanced Mathematics section 003 Spring, 2021

Math 225   Foundations of Advanced Mathematics    section 003      Spring, 2021
TuTh, 11:45 am – 1:00 pm;   Online only

Instructor: Michael Shearer     Office: SAS 3228     Phone: 515-3298    Email:

Office Hours: Tu Th  2.00-3.00 p.m. online by zoom. Or by appointment (email me!)

Book: We will use the book by Dr. Jo-Ann Cohen. It is online as pdf files (linked at webpage)

Prerequisite: MA 241.   

Homework, tests and final:
1. Weekly Homework Assignments (10% of final grade)   due Thursdays.
2. Three in-class exams (60%)
:  Dates to be decided
3. Final Exam: (30%):   To be decided

Basic Grading Scale: 90%-100%: A, 80%-89%: B, etc.   (+ and – will also be used.)

Syllabus: complete syllabus and course materials can be found at the class  webpage

Approximate schedule:

Chapter 0. Preliminaries: 1-2 days

Chapter 1. Language, Logic, Sets, and Methods of Proofs: 3-5 weeks

Chapter 2. The Principle of Mathematical Induction: 1-2 weeks

Chapter 3. Functions: 2-3 weeks

Chapter 4. Equivalence Relations: 2 weeks

Chapter 5. Properties of the Integers: 1-2 weeks

Chapter 6. The Size of Sets: 1-2 weeks

Summary: First few weeks: Language and notation of Mathematics; How to write proofs for number properties; Writing proofs for sets.

Then: Proofs using induction (about 1 week);  Properties of functions and mappings between sets (few more weeks); Relations and equivalence relations (couple of weeks); Using these topics to describe how to measure the size of sets (finite, countable, uncountable).

Course Objectives: The primary goal of Math 225 is to prepare you for upper level courses in mathematics, especially courses that rely on understanding and writing mathematics. The course includes an introduction to the format of definition, theorem and proof, the design and writing of logical arguments and formal proofs, and techniques of problem solving in higher level mathematics. During the semester, fundamental concepts in several areas of mathematics are introduced, partly to show the structure of mathematical writing in different contexts, but also to lay the foundations for higher level courses such as abstract algebra and analysis.

Additional Policies:
1. I encourage you to discuss homework with other students, or with me during office hours. You should be aware that the homework is intended for you to learn from the course; working on homework at least in part on your own will help you master the material, keep up with the course and prepare for tests. Homework will be graded on a scale of 1-10. Points will be awarded for amount of homework attempted; selected problems will be graded in detail. Please hand in assignments carefully written; illegible or sloppy homework will not be graded.
2. I expect you to read sections of the book around the time of lectures and homework will be mostly set from those sections. The book has additional examples and discussion that you will find helpful. Some test questions may resemble examples from the book.
3. You are expected to attend all classes on time. Classroom discussion and questions in class help clarify issues in this course, so please feel free to participate by asking questions.
If you are unavoidably absent from a test, a score for that test will be assessed at the end of the semester, based on your performance in homework, the other tests, and on the final, with an emphasis on the material of the missed test.  Attendance regulations can be found at

Academic Integrity Statement: Students are expected to follow university policies on academic integrity and the Honor Pledge,  which may be viewed by following the links at:

Further Statements:
The following statement is required by the University: Students may be required to disclose personally identifiable information to other students in the course, via electronic tools like email or web-postings, where relevant to the course. Examples include online discussions of class topics, and posting of student coursework. All students are expected to respect the privacy of each other by not sharing or using such information outside the course.

Accommodations for Disabilities: Reasonable accommodations will be made for students with verifiable disabilities. In order to take advantage of available accommodations, students must register with the Disability Services Office at Suite 2221, Student Health Center, Campus Box 7509, 919-515-7653. For more information on NC State’s policy on working with students with disabilities, please see the Academic Accommodations for Students with Disabilities Regulation (REG02.20.01) (

Non-Discrimination Policy: NC State University provides equality of opportunity in education and employment for all students and employees. Accordingly, NC State affirms its commitment to maintain a work environment for all employees and an academic environment for all students that is free from all forms of discrimination. Discrimination based on race, color, religion, creed, sex, national origin, age, disability, veteran status, or sexual orientation is a violation of state and federal law and/or NC State University policy and will not be tolerated. Harassment of any person (either in the form of quid pro quo or creation of a hostile environment) based on race, color, religion, creed, sex, national origin, age, disability, veteran status, or sexual orientation also is a violation of state and federal law and/or NC State University policy and will not be tolerated. Retaliation against any person who complains about discrimination is also prohibited. NC State’s policies and regulations covering discrimination, harassment, and retaliation may be accessed at Any person who feels that he or she has been the subject of prohibited discrimination, harassment, or retaliation should contact the Office for Equal Opportunity (OEO) at 919-515-3148.

List of Policies: Students are responsible for reviewing the NC State University PRRs (policies, rules and regulations) that pertain to their course rights and responsibilities:

•Equal Opportunity and Non-Discrimination Policy Statement additional references at

•Code of Student Conduct

•Grades and Grade Point Average

•Credit-Only Courses



•Attendance Policy