MAT 511 - Fundamental Concepts of Mathematics

Course Description

This course aims to make the student fluent in the parlance of theorems and proofs, to prepare one for the critical thinking necessary for courses ahead.

Our course will cover five main topics: Logic and Proofs; Elements of Set Theory; Relations between sets; Cardinality; Elementary Number Systems. Below is a brief description of each topic.

In the first section, we will introduce the student to mathematics as a formal language, discussing its grammar (what elements go into a mathematical sentence), syntax (how to structure such sentences), and semantics (what is the meaning of a sentence). Then we shall move on to discuss aspects of this language's deductive apparatus, examine many common inferences used in modern mathematical reasoning, and work on many exercises on proof writing and analysis.

In the second section, we will discuss general notions of set theory, such as membership, containment and set operations. We will then move on to work on a specific example, the Natural numbers, as construced by Peano, and discuss Mathematical Induction in its many forms, as well as basic principles of counting.

The third topic is concerned with relations between sets. We will study the general notion of a relation, constructions of new relations from old, and properties of such relations. After this introduction, we will study special relations: equivalences, orderings, functions.

In the fourth section, we will introduce the notion of equivalent sets, characterize infinite sets, and discuss the many layers of infinity, as well as the Continuum Hypothesis.

Finally, we will study elementary number systems. We will discuss the Integer Numbers and Modular Arithmetic, the Rational Numbers, and finish with a construction of the Real numbers.

Time and location

We meet weekly on Tuesdays and Thursdays, between 6:00 pm to 9:25 pm, in the Earth and Space Sciences (ESS) building, room 177.

Important: Classes will follow a Thursday schedule on Friday, June 28. Plan accordingly.

Textbook

A Transition to Advanced Mathematics, by Douglas Smith, Maurice Eggen and Richard St. Andre (8th edition).

Homework

Your assignments are an important part of the course (and your grades). There will be four problem sets, whose due dates can be found in the calendar above. The assignemnts will be available on this course webpage (on the resources tab)
    Problems will range from simple manipulations of the concepts developed in class on that week to more involved applications of these concepts.
    Not all problems will be graded, but you should attempt to solve them all anyway, as they will serve as your foundation for the problems you will see on your exams. Graded problems will be discussed in class. Solutions to selected non-graded problems will be posted on this webpage weekly. 
    If you would like to know how to solve a problem which was not solved in class, whose solution is not posted here, or if you'd simply like to go over parts of the solution you don't understand, I'd be happy to discuss it with you during office hours, or via e-mail.
    Late homework policy: homework will be considered late if submitted after 7:00pm, on the due date. Such assignments will not be granted full credit, with their worth decreasing by 10% with each day of tardiness.

Exams

The exams will take place in ESS 177 on the following dates and times:

• Midterm: 06/11, between 7:30pm and 9:25pm.
• Final exam: 07/02, between 6:00pm and 9:25pm.

Grading policy

    Your numerical grade will be composed in the following way:

• Homework assignments: 15% each.
• Midterm: 20%.
• Final: 20%.

Information for students with disabilities

If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact Disability Support Services at (631) 632-6748 or http://studentaffairs.stonybrook.edu/dss/.
    They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information go to the following website: http://www.sunysb.edu/ehs/fire/disabilities.shtml