Math 300 B: Introduction to Mathematical Reasoning
Spring 2017

Instructor: Alexandru Chirvasitu

Class times and location:
Office hours: Monday 12:30 - 2:30, Wednesday 9 - 11 or by appointment
Email: chirva AT uw.edu

As the name of the class suggests, this is supposed to be your first encounter with the formal, proof-based scaffolding of mathematics.

The one take-home point I would like to convey is that the only effective way of developing your mathematical abilities is to do math, in the form of problem-solving. The categorization and taxonomy of proof types and the instructions for how to put these together (to the extent that such instructions even exist) are things you then pick up naturally and organically, in the course of practicing.

You do not go into a problem armed with a cooking recipe of what goes in, since no algorithm will ever be able to lay out a precise path to a solution. Rather, the goal we're trying to achieve here is to get exposed to enough mathematics to build our confidence and working (rather than explicit) knowledge of proof techniques, and to then let your natural problem-solving instincts take over whenever you encounter a mathematics problem.

For all of these reasons I would like to stress that there is no memorization component to any of this, which is why all of our tests are open book (see below, in the website sections describing tests). I want you to get accustomed to doing mathematics as it is usually done ``in the wild'': with full access to whatever reading material you think will be useful, and free to exercise your creativity and ingenuity in building proofs without any extraneous worries of whether or not you ``know the material''.

Textbook

We will be using John P. D'Angelo and Douglas B. West's Mathematical Thinking (2nd edition). The textbook is absolutely necessary, both for the reading and in order to do the assignments.

It's a good textbook because it covers a wide array of topics (too many for us to cover, certainly) and is rich in problems. This makes it a good fit for the above-mentioned approach to learning how to do mathematics, based on problem solving.

We will cover Chapters 1-4 as a kind of base for everything else, plus a few other selected topics (see the reading and homework tables below for the breakdown of the course).

It's essential that you do it as assigned. I won't have time to go over every relevant example in class, and you'll need a good grasp of the material in order to do the homework. In fact, I encourage you to regard the reading as part of your homework.

The phrase 'Chapter x', page numbers and result numbering (such as 'Theorem a.b') always refer to our textbook.

Due date Assignment Remarks
1 Wed Mar 29 Chapter 1
stop at `Functions'
2 Fri Mar 31 Finish Chapter 1
3 Mon Apr 03 Chapter 2
Stop at `Elementary Proof Techniques'
4 Wed Apr 05 Finish Chapter 2
5 Fri Apr 07 Chapter 3
before Proposition 3.16
6 Mon Apr 10 Continue Chapter 3
stop at `Strong Induction'
7 Wed Apr 12 Finish Chapter 3
8 Fri Apr 14 Chapter 4
stop at `Injections and Surjections'
9 Mon Apr 17 Continue Chapter 4
stop at `Cardinality'
10 Wed Apr 19 Finish Chapter 4 Eeverything up to here
Fri Apr 21 No meeting that day
11 Mon Apr 24 Study for the midterm Treat this meeting as a Q&A session
or as in-class office hours.
We can go over problems, I can clarify
things you think need going over again, etc.
We end as soon as you run out of questions, so
make sure you don't.
Wed Apr 26 Midterm
12 Fri Apr 28 Chapter 6
stop at `The Dart Board Problem'
13 Mon May 01 Finish Chapter 6
skip `More on Polynomials'
14 Wed May 03 Chapter 10
`The Pigeonhole Principle'
15 Fri May 05 Finish Chapter 10
16 Mon May 07 Chapter 11
stop at `Isomorphism of Graphs'
17 Wed May 10 Continue Chapter 11
stop at `Connection and Trees'
You'll need to have a look at
the section `Relations' on page 140
for background on equivalence relations.
Fri May 12 No meeting that day
18 Mon May 15 Chapter 11
`Connection and Trees'
We are again not meeting.
19 Wed May 17 Chapter 11
`Bipartite Graphs'
20 Fri May 19 Chapter 13
`The Completeness Axiom'
21 Mon May 22 Chapter 13
`Limits and Monotone Convergence'
Wed May 24
Fri May 26
Mon May 29
No meetings on those days.
I am away on the first two,
while the 29th is Memorial Day.
22 Wed May 31 Finish Chapter 13
Study for final
There will be some time for
a pre-exam Q&A, as before the midterm.
23 Fri June 02 Study for the final some more No meeting;
will have some extra office hours
during exam week.

Supplementary material

On occasion, I'll post extra notes, comments, etc. in this space.
• Some notes meant to complement the lecture on Monday, May 01 2017.

Homework

It's due Wednesdays at the beginning of class, before the lecture. No late homework for any reason, but I will drop the lowest score.

I am denoting the problems by x.y, as the textbook does (to mean problem y from Chapter x). Whenever you solve a problem, you can cite any preceding problem or theorem in your solution.

Because of time constraints your TA will grade a couple of problems (tops) for correctness and the rest for completeness. I won't be telling you in advance which problems are graded for correctness though..

Due date Assignment Remarks
1 Wed Apr 05 1.15, 1.20, 1.24, 1.25, 1.38, 1.39, 1.50
2.9, 2.21, 2.32, 2.44, 2.49, 2.52

There is a typo in problem 1.50:
Replace all ⋃ with ⋂ and solve
that version of the problem.
See next table row for a follow-up.

Fri Apr 07 Patch problem 1.50:
Prove part (a) as stated originally;
Is it true that you always have equality in part (a)?
Optional; you do not need to turn it in, but if
you do and it's correct it'll be worth 1% extra credit.
2 Wed Apr 12 3.1, 3.2, 3.16, 3.18, 3.28
3.35, 3.41, 3.50, 3.61, 3.65
3 Wed Apr 19 4.5, 4.10, 4.20, 4.26, 4.27
4.33, 4.37, 4.42, 4.43, 4.47
Your homework problems up to here
are midterm prep.
Mon Apr 24 Spillover problem
from Apr 14 lecture
Optional again; you do not need to turn it in, but
worth 1% extra credit.
No hw on Apr 26
because of the midterm
4 Wed May 03 6.18, 6.24, 6.28, 6.29
6.37, 6.41, 6.47
10.3, 10.5, 10.9, 10.18
Fri May 05 Another extra problem that came up
during the May 01 lecture
Optional and worth 2% extra credit
5 Wed May 10 10.28, 10.32, 10.35
11.1, 11.3, 11.5, 11.10
11.12, 11.13, 11.17
6 Wed May 17 11.7, 11.8, 11.21, 11.28
11.29, 11.35, 11.36, 11.37
For 11.37 you'll need the definition
of the complete bipartite graph
Kn,n from page 225.
Fri May 19 A problem inspired by a question asked
in class on Friday, May 05
Optional; 1% extra credit
7 Wed May 24 13.3, 13.4, 13.8
13.12, 13.25, 13.28
Do not turn in!
This is un ungraded assignment, due to my being away.
It is meant as a suggestion for practice problems.
8 Wed May 31 13.6, 13.7, 13.14, 13.21
13.34, 13.37, 13.38, 13.39

Midterm

It's happening in class (usual time and place):
• Wednesday Apr 26, 11:30 - 12:20, room 226 Sieg Hall
The test is open book. This means that apart from not communicating with anyone (in class or electronically) during the test, anything goes: physical textbook, computers, the internet, class notes, old homework you've brought with you, etc. etc.

Final

The date and time are set by the department and I cannot change it. Early / late exams are not an option, so please plan accordingly.
• Wednesday, June 07 2017, 2:30 - 4:20 PM, room 226 Sieg Hall
Both the midterm and the final are open book exams. You get to bring your book, class notes, past homework, whatever (just do not collaborate with anyone during the tests).

• Homework : 20%
• Midterm : 35%
• Final : 45%

As mentioned above, we'll drop the lowest homework score.