Math 250 -- Spring 2018
Foundations of Mathematics
About this course
"There are two situations when it's good to know what's under the hood of a car: one is when it breaks, the other is when you need to make a new car. Neither of them should prevent you from getting behind the wheel and learning to drive. And if you know what a car does and when it breaks, you'll be surprised by how much more sense the fancy technology under the hood will make.
That said, a lot of great math has been done by digging around under the hood." -- Dmitry Vaintrob
We will cover the following topics.
Here is an official pdf of the syllabus for this course. (There is no information on this pdf that is not on the webpage.)
- Logic -- statements, negation, converse, etc.
- Sets, relations, functions -- the building blocks of mathematics
- Techniques of proof -- proof by contradiction, induction;
non-constructive proofs, "consider a minimal X" proofs, pigeonhole principle
- Cardinality -- different sizes of infinity
- Numbers -- what they are, transcendental numbers vs. algebraic numbers (i.e., the difference between pi and 2^(1/2))
- Mathematical prose and rigor-- how to write mathematics correctly and in complete sentences.
This class will meet 28 times. I will cover roughly one section of
our text each class. Some sections will be skipped and many will be
covered out of order.
There will be many short in class activities in addition to
The midterm is worth 25 percent and the quiz is worth 10.
The final exam will be comprehensive and will count for 30 percent.
Homework is worth 35 percent. Seriously! If you do not stay on
top of the homework, you will have a bad time in this class.
The quiz and midterm dates below are tenative (and may be adjusted if
the pace of the course is adjusted), but the date of the final exam is set in stone;
make your summer travel plans accordingly. If you have a conflict
with the final exam (e.g., another final) please let me know ASAP.
| Homework || 35% || (Weekly) |
| Midterm || 25% || (Feb 25 (Tentative)) |
| Quizzes || 10% || (Apr. 5 (Tentative)) |
| Final Exam || 30% || (May 3, 8-10:30am, W302)||
Calculators, notes, and textbooks are not allowed in exams or quizzes.
There will be homework assigned every week, usually on Thursday. There
will be many simple problems, checking your understanding of the
definitions, that will be collected and graded for completness but not
correctness. Most weeks there will be a number of proofs assigned. You
are expected to write them up very carefully.
There is a grader for the course. He will very carefully grade 3-5 of these per week and check the rest for completeness. Re-writes will be allowed and in fact expected, and students will be able to recover up to half of the missed points. Homework assignments will typically be worth 20-60 points, depending on the length of the assignment.
The homework assignments are available at this link, and will be updated after each lecture.
The exam problems will be similar to the homework and worksheet
problems (some will be identical, and some will be similar but not identical).
Remember that copying another student's work is a violation of the
Honor Code and will be treated as such. If you must leave class during
an exam for any reason, please leave all of your belongings
handheld supercomputer phone!).
For homework: you are free to consult any sources (animate or
inanimate) while doing your homework (working in groups is
encouraged!), but if you use anything (or anyone) other than your
class notes or the texts listed above, you should say so on your
homework -- please state at the end of every problem any sources used.
On the other hand, you are expected to make an honest attempt to do every problem on your own before consulting other sources. Remember that copying another student's work is a violation of the Honor Code and will be treated as such.
A good rule of thumb to avoid plagarism is the following -- when doing
the final write up of a problem, do not have any text books, web
pages, or classmate's write up in front of you. If you get stuck when
writing up an assignment, go back and look again; just make sure that
you organize the mathematics in your head before writing a proof
rather than copying a solution from some source. This is a
generous homework policy. Please do not abuse it.
There is a helpful, 10 minute video of every subtopic we will cover on Khan Academy. When you are confused about a concept, please watch the corresponding video! Remember, you can pause and rewind a video, but you can't pause and rewind lecture!