Since one of the main objectives of this course is for you to learn how
to write mathematical proofs, you will create a portfolio that contains
both examples of perfect proofs and examples of your best work
throughout the semester. For the perfect proofs, I will
assign problems for you to do and grade
(or
regrade) your proofs on an acceptable/not acceptable basis. For
examples of your best work, you should collect proofs from your
homework
every three weeks or so.
- Types of proofs
- Direct proof - Prove that the cross-product of two bipartite
graphs is bipartite by describing the partite sets of the product and
showing that there are no edges connecting any two vertices which are
both members of the same set.
- Proof by contrapositive - Prove that every graph with fewer
edges than vertices has at least one connected component which is a
tree.
- Proof by contradiction - Suppose that G is a graph with n vertices and that for each
vertex, v, deg(v)>=(n-1)/2. Prove that G is
connected. (Hint: How many vertices must be in each
connected component?)
- Weak induction - Suppose that T is an m-ary tree of
height h. Prove that the number of leaves that T can have is at
most m^h.
- Strong induction -
- Perfect Proofs
Initially, I will assign particular problems and ask you to prove them
in a specific way. You may find that you have a different
technique for proving a problem (e.g. you can give a direct proof where
I asked for one by induction). In this case, you should come and
talk with me about whether to submit the proof you have or look for a
different one.
You must hand in each of the proofs assigned above. Beyond
that, you should hand in two other perfect proofs. You need to do
two different types, using the types described above. You can
chose these from practice problems that involve doing proofs,
from
homework problems that you did (provided that you had a score no higher
than 2), from other problems in the text, or from a small list of
problems. If you have questions about whether a problem is ok,
(or
finding a problem) check with me. If you hand in a revised
homework problem, be sure to hand in your original (graded)
version. You should also be sure to clearly identify which type
of
proof you're submitting.
- Homework samples
You need to collect a sample of graded homeworks that show what
kind of progress you've made throughout the semester. I suggest
looking at every other homework assignment, or possibly every third
one.
Some of you made a huge jump in the overall quality of your work
-
you be sure to include the first assignment where that happens and the
one immediately before that. Others have had very creative
solutions; be sure to include those.
- What to hand in
Keep all your proofs in your purple folder. Completed work should
go on the left side; work for me to grade should go on the right side.
You may hand in up to five versions of each proof. If you
decide to rewrite a proof, you should hand in previous versions,
together with any comments.
At the end of the semester, I will ask you to write a
short evaluation of your progress for this semester. In this
evaluation, you should give yourself a grade, using your portfolio
proofs as evidence.
- Grading
Perfect proofs are graded on an acceptable/not acceptable basis.
If a proof is not acceptable, I will write some comments about why, and
I will be quite happy to discuss these with you. I will use
your written evaluation and the percentage of perfect proofs as a basis
for your portfoio grade.