Stuff about the Math 131AH midterms
- The first midterm is on Friday, Jan 31, 12:00 pm, at WG Young
2200 (NOT MS 5148!)
- There will be five questions, all worth 10 points; only your best
4 out of 5 will be graded, so read the questions beforehand to decide
which 4 to attempt. They
will all be short proof type questions (similar to the homework
questions, but somewhat shorter), covering Weeks 1-3. (The week 4
material, on limits of sequences of real numbers, will not be covered).
- The exam is open book, open notes; you may bring any course
material or your own written notes. Bringing the solutions to
your own homework may be a particularly good idea. Calculators
are permitted but are very unlikely to be useful. No computers or
other sophisticated electronic equipment is allowed.
- In addition to the open book and open notes, a page of
definitions will be supplied with the midterm. This is so that
you do not have to cycle through your notes just to find a particular
definition.
- I expect your proofs to be fairly detailed - it should be clear
to the grader which of the equations you write down are your
assumptions, which ones are the things you are trying to prove, which
ones are hypotheses, etc. Using English words such as "since",
"assume", "by hypothesis", "by a theorem in the notes", etc. is highly
recommended. On the other hand, you do not need to supply
explanations for each and every step; for instance, the manipulations
of high school algebra can be done without much comment. You do
not need to give precise references (i.e. page numbers or Proposition
numbers) for the results from the notes; it will be acceptable to say
things like "By a theorem in the notes, ..." without specifying exactly
which theorem. (This is, of course, assuming that the theorem you
are quoting IS indeed in the notes).
- Unless otherwise specified in the question, you are allowed to
use any result from the course notes or from the textbook when
answering a question. (There is no danger of circularity, since
the course notes do not rely on using results from the midterm).
- Regarding supplemental material (sets, functions, etc.) - this
material is not the focus of this course, and so we will not be as strict
about enforcing rigor, etc. for set theory as we will be for things like
limits of sequences of numbers. So when arguing using sets it will be
acceptable to use a fairly informal approach (e.g. using Venn diagrams
is acceptable).
Some tips and tricks:
- There is no practice midterm; however I would recommend going
through the homework assignments carefully prior to taking the midterm.
In particular you should try to do as many of the homework
problems as possible by yourself, without external aids (except for
things like the class notes). You are encouraged to bring your
own homework solutions to the midterm. (Bringing someone else's
solutions (eg. the TA's) is much less likely to be helpful, unless you
understand how the solutions work, in which case you probably didn't
need to bring the solutions in the first place).
- Familiarity with the course notes is highly recommended; many of
the proofs in the midterm may be easier if you use some of the
propositions and theorems from the notes. Note that you should
take some care reading the hypotheses of each proposition before using
them; a proposition may apply to integers but not to rational numbers,
or to convergent sequences but not to divergent sequences, or to finite
sets but not to infinite sets, etc. This "fine print" is
sometimes very important, so don't just apply the conclusion of the
proposition blindly! You may find it helpful to create summaries
of the course notes for quicker reference when taking the midterm; also
the very process of creating summaries may help you organize the
material mentally.
- Each midterm question is likely to bear some resemblance to a
homework question, or perhaps a lemma in the notes, which suggests that
the proof method is likely to be similar. For instance, if the
midterm question looks like a homework question, and you used induction
to do the homework question, it is likely that induction will be useful
for the midterm question as well. However, while the methods may be similar, the actual content of the proof may be somewhat
different, so merely copying a proof from the homework or notes word
for word into the midterm is unlikely to impress the graders.
- It may help to write down exactly what the hypotheses are, and
(in a separate location on the paper) write down what conclusions you
want to reach. This may help suggest what propositions or proof
methods may be helpful, and can help convince the grader that you know
what you are doing. Note that one should always take care to
separate the hypotheses from the conclusions; mixing the two up is
almost always a bad idea.
- It may be helpful to use the definitions of terms to break up a
complex statement into more elementary components. For instance,
if one of the hypotheses is that "a_1, a_2, .... is a Cauchy sequence",
it may help to use the definition of a Cauchy sequence to break this
statement up into more elementary statements (e.g. using
epsilon-steadiness). Of course, this can be taken too far... in
most cases you don't want to go all the way back to natural numbers and
++, as most statements become ridiculously lengthy when expanded that
way.
- The most common method to try to prove something is the direct or forward approach: write down your
hypotheses, see what you can derive from them, and eventually work your
way toward the conclusion. Sometimes, however, it is better to
try a backward approach: look
at the conclusion and think
about the possible ways one could get to that conclusion. This
allws you to replace your conclusion with successively easier
conclusions, working your way back towards something which will follow
easily from the hypotheses. A typical use of the backwards
approach is to take the conclusion, and use some definitions to break
up any complex concepts in that conclusion to simpler concepts.
Often this makes the conclusion more elementary, and thus easier
to reach.
- In many cases a statement may look obviously true, and yet you do
not know how to prove it. In this case you might ask yourself "is
there any possibility that the statement is in fact false?".
Often times you will find a good reason for why the statement
cannot be false - and this means that you should probably prove your
statement via contradiction.
- At certain times in a proof, you may want to use some result
which you are sure is true, but is not explicitly stated in the lecture
notes. In that case, you may wish to set that result up as a
Lemma (also called a Claim or an Auxiliary result), use that lemma to
finish the proof, and then go back and prove the lemma. Even if you don't manage
to prove the lemma, you can still get partial credit if you can prove the original
question assuming the lemma is true. One word
of caution, however - sometimes the lemma you wish to use is more complicated
than the original question, so this technique may not necessarily save
time. You should use your own judgement to see if your lemma
really is a more basic and fundamental fact than what you are trying to
prove.
- If you are completely stuck in proving a statement, you can at
least score partial credit by demonstrating to the grader that you at
least comprehend the statement.
Some ways in which you can do this is by drawing a picture, or
giving an example, or by explaining in informal English why you believe
the statement to be true, even if you can't explain it precisely enough
to qualify as a proof. Also, the very process of drawing a
picture or finding an example may lead you to understand better why the statement is true, and
perhaps give you an idea as to how to prove it.
- If you find that a proof is getting very complicated or very frustrating, you should set it aside and
work on a different problem instead. The questions are intended to have relatively short proofs (unlike the homework,
which is intended to be much more time-intensive), if you have
the right approach and know what you are doing; so if you find yourself getting nowhere then you are probably
on the wrong track. In which case it is often profitable to think about another question for a while and
come back to it later.
- In this course it is important to know the precise definitions of terms; if you misquote a definition when using it in a proof, you are likely to not be able to complete the proof correctly. Many of the questions in the Java Quiz will test your knowledge of various definitions; it might be worth giving that quiz a try just to identify your strengths and weaknesses. I especially recommend the Logic, Set theory, Function, and Sequence quizzes (though some of the material in the sequence quizzes will not be covered until Week 4).