Introduction to Advanced Mathematics, MGF 3301, Spring 2020.

Location: 106 LOV

Time: TuTh 9:30 - 10:45 am.

Instructor: Dr. Mark van Hoeij

Text: Irving Kaplansky, "Set Theory and Metric Spaces", AMS Chelsea Publishing, 1977. You can buy this book online new for $34 and used for less than $30.

Midterm Tests: Schedule: Test 1: Feb 6, Test 2: MOVED TO MARCH 10, Test 3: April 16 (was April 14). Final exam: Monday April 27, 7:30-9:30 am.

Exam policy: There will be no makeup tests or quizzes. A missed test can only be excused with a valid reason before the day of the test. When a missed test is excused, the grade of the test will be the same as the grade on your final test. A missed test will not be excused on or after the day of the test.

Course objectives: Mathematics before this class primarily involves computations (solve these equations, integrate that function, etc.). In contrast, mathematics after this class primarily involves proofs and concepts. This course is designed to help with the transition from computation-oriented mathematics to proof-based mathematics.

In the undergraduate curriculum, proof-based math is often called advanced math. After this course, proof-based math will simply be called math because nearly all math after this course is proof-based.

To explain the difference with an analogy, in a lower level astronomy book you might learn that the Andromeda Galaxy is 2.5 million light years away. But in advanced courses it is not enough to simply know such facts, you also have to know how people can verify such things. Likewise in math you may have learned things like: there are infinitely many prime numbers, the square root of two is not a rational number, the fundamental theorem of calculus, etc. In advanced math it is not enough to simply know these facts, we have to understand why these things are true. For that, we need to understand proofs in mathematics.

In typical math research, the main goal is to prove new results. There is a consensus among mathematicians about when a proof is valid and when it is not. It takes time to get used to this, you'll have to learn the rules for proofs. The first rule for a valid proof (a proof that other mathematicians will accept) is to obey the rules of logic. So we will begin this class with the study of propositional logic via truth tables. Next we will study topics in math that are particularly suitable for learning about proofs, what makes proofs valid, and how to write valid proofs.

Second chances: It takes time to learn how to write valid proofs, that's why there will be second chances. When the class does not do well on a question in a test, you may be able to retroactively improve your test score by answering a similar question in a quiz. Likewise there will also be second chances for certain questions from homework/quizzes. Don't skip class, quizzes will often be unannounced. If you miss an unannounced quiz, this will be excused if you e-mailed me before class that you will be absent.

Homework: It can be helpful to study with other students but do not copy their homework, for several reasons: (1) The goal in this course is to learn to write proofs. That takes time and will only be accomplished if you write your own proofs. (2) It is not useful to receive feedback on copied work, this wastes time for the grader. (3) Errors in your homework will not affect your grade, the homework grade is based on how many you turn in and not on the number of errors.

Grading: There will be three tests during the semester and one final test, each worth 25%. The average of these grades is converted to a letter grade as follows:

Students who turn in little/no homework and quizzes: 70 = C-, 72 = C, 78 = C+, 80 = B-, 82 = B, 88 = B+, 90 = A-, 92 = A.

Students who consistently turn in homework+quizzes: 60 = C-, 64 = C, 71 = C+, 74 = B-, 77 = B, 84 = B+, 87 = A-, 90 = A.

(in between those if you turn in some homework/quizzes). The reason for this difference is because this class is about learning how to write proofs, and it takes practice (homework/quizzes) to learn that.

Honor code: copy of the University Academic Honor Code can be found in the current Student Handbook. You are bound by this in all of your academic work. It is based on the premise that each student has the responsibility to 1) uphold the highest standards of academic integrity in the student's own work, 2) refuse to tolerate violations of academic integrity in the University community, and 3) foster a sense of integrity and social responsibility on the part of the University community.

ada statement: Students with disabilities needing academic accommodations should: 1) register with and provide documentation to the Student Disability Resource Center (SDRC); 2) bring a letter to the instructor from SDRC indicating you need academic accommodations. This should be done within the first week of class. This and other class materials are available in alternative format upon request.