Math 131AH Homework Assignments
General remarks:
- You are permitted to use all resources available to you
(textbook, notes, other students, TA, office hours, virtual office
hours, etc.) to do these homework assignments. However, you will get the most
benefit from the homework if you first attempt to do all the questions
yourself, without outside assistance. Only after you have tried a
question can one truly appreciate the solution. If you find
yourself relying more and more on external help in order to complete
each week's homework, then this is a danger
sign and may indicate that you will have significant trouble with
the midterms and final. Copying your homework from someone else
without first trying it yourself may help you score better on the 15% of
the grade that comes from the homework, but may at the same time cheat
you out of the 85% of the grade that comes from the exams.
(Trying to make up for this later by reading other people's
solutions to the homework may not necessarily be effective. By far
the best way to learn is to do the homework yourself, without
assistance). The homework assignments are definitely long, tough,
and time-consuming - this is an Honors
course, after all - but they will
help you understand the course material far more thoroughly, and the
exams will be much easier to handle if you took the time to do the
homework by yourself. (You will also get a lot more out of the
discussion section if you have already tried all the homework problems
beforehand).
- This course emphasizes rigor, and in particular seeks to avoid
circular reasoning. However, this does not mean that you have to
necessarily prove and justify every single statement in your proofs;
this would of course get very cumbersome, especially in the later
assignments. Generally, material from previous weeks notes can be
used without having to reprove them again. The basic rule of thumb
is that if you know where a result comes from and have some idea how to
prove it, and you are pretty sure that it would not be circular to use
it, then it is OK to use it in your homework. If however you are
quoting a result that you don't know how to prove, then proceed
carefully; it may be that that result can only be proven using more
advanced material than what the homework is covering, which introduces
the risk of circularity.
- One should be careful in checking all the various pre-conditions
and hypotheses of a proposition, when applying that proposition to your
homework and exams. For instance, if a Proposition is only proved when
n is an integer, you cannot apply it to numbers n which are rational or
real. Or a result may only apply to bounded sequences but not
unbounded sequences, or to convergent sequences but not unconvergent
sequences. These distinctions can become quite important, especially
since we are emphasizing rigor.
Assignment 1 (Due Jan 17)
- Available in PDF format here.
Don't forget to read the back page of this assignment!
- Solutions: page 1 page
2 page 3 page 4 page 5
- Some common errors: propositions which applied to natural numbers
or integers do not necessarily apply to rational numbers or real
numbers. Also, while induction is extremely useful for proving
statements about natural numbers, it does not work when trying to prove
statements about integers, rationals, or reals (unless one somehow
rewrites that proposition in terms of natural numbers instead).
Assignment 2 (Due Jan 24)
- Available in PDF format here.
- Errata: There are some errata to Q2: see the virtual
office hours or the bonus point page. Also, despite what it said in
(an earlier version of) the week 2 notes, Proposition 19 is not
officially part of Assignment 2. Also, in Q5, "1-close" should read
"1-steady" in the hint. Finally, in the hint for Q8, "the sequence is
epsilon-close" should read "the sequence is epsilon-close to the zero
sequence 0,0,0,0,...".
- Solutions: page 1 page
2 page 3 page 4 page 5 page 6 page 7
Assignment 3 (Due Jan 30)
Assignment 4 (Due Feb 7)
Assignment 5 (Due Feb 14)
- Available in PDF format here.
- Solutions available here
- Errata: In Q5(b) (i.e. Proposition 10(b) of the notes),
the right-hand side should be f(x_0) rather than 0. For some hints
on Q1(a), see the Virtual Office Hours. For Q3(b), M should be assumed to be positive. For Q2(e), "x^q < x^r" should read "x^q > x^r".
Assignment 6 (Due Feb 21)
Assignment 7 (Due Feb 28)
Assignment 8 (Due Mar 7)
Assignment 9 (Due Mar 14)
