Math 5615H, Fall '05

Last changed: November 12, 2005.

Old news:


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NEWS Oct 31: Assignments have been updated: Assignment 8,
Special Problem 5.

NEWS Oct 24: Assignments have been updated: Assignment 7.

NEWS Oct 17: Assignments have been updated: Assignment 6,
Special Problem 4.

NEWS Oct 16: A solution of Special Problem 3 is on the Web.
It contains a solution less than one page long, tho it depends on some
background material (included). The long version is included too, and the
"draft" has been removed.

NEWS Oct 14: A draft of a solution of Special Problem 3
is on the Web. It is very long because it contains background material. A
much shorter version is planned.

NEWS Oct 12b: Assignments have been updated: Assignment 5,
#10 in Chapter 1 should be #11; it's been corrected. Anent Special
Problem 2, due Friday, if I drew a five-pointed star on your paper, your
argument can be made to work without using Special Problem 3. In any
case, it's OK to apply Special Problem 3 to Special Problem 2!

NEWS Oct 6: Assignments have been updated: Assignment 4,
Special Problem 2 rescheduled, Special Problem 3 is still due Oct 10.

NEWS Oct 4:
Questions about Test 1
1. What will Test 1 cover?
A: The parts of Chapter 1 emphasized in class (not the Appendix),
especially the Axioms for the Real Numbers, axiomatic proofs, Arcimedes.
Inductive sets, complex numbers, Cantor's diagonal proof, and so on.
In Chapter 2, thru 2.24, with emphasis on the metric space stuff.

2. Do we need to write the exact definition as in Rudin or described
in the class, or would you be o.k. with other definitions as long as they are
correct and precise?
A: As long as your definition, when written "in logic," is the same
as Rudin's or the one given in class. For example, a logically equivalent
definition that is not the same is a Theorem and not the definition, e.g.
"A set is closed iffi its complement is open" is a Theorem, NOT the definition.

3. While writing proofs in the test, can we just state another theorem(s)
that is(are) proved/covered (this means that it is proved in Rudin) in the
class, unless we are specifically asked to prove that(those) supporting
theorem(s) as well?
A: Questions like this one cannot be answered by anything other
than "It all depends." I repeat what was said in class: you may use other
Theorems proved in class (not theorems in Rudin that have not been
proved in class) if you identify them properly and show the checking of
their hypotheses. However, if the problem asks for a "direct" or "self-
contained" proof, no other theorems may be applied.

NEWS Oct 3: *1* Test 1 is Wednesday Oct. 5. Nothing but
writing instruments (for writing by hand!!) and erasers may be used
while taking this exam. This includes telephones and calculators.
Watches may be used, but only watches that tell time only. I'll indicate,
on the board, the time remaining during the test.
*2* The papergrader's name is not French. The papergrader asks that
you return your Assignment 2 papers to me on Wednesday or to his
mailbox so that he can correctly record your corrected scores. He will
also make his "late homework" policy more explicit.
*3* Beginning about 2pm tomorrow, I'll take questions about Test 1
by email. The asker of the question will not receive an answer by email.
Instead, the question (edited, perhaps added to) and its answer will be
be placed on this Web page. Questions may be sent before 2pm Tuesday.
Questions that arrive after 10pm will not be answered, and no answers will
be posted after 11pm: you need to be well-rested!

NEWS Oct 1: Special Problem 3 is now due October 10, instead of
next Wednesday, October 5, the date of the first Test. The date change has
not yet been made on the Assignment page.

NEWS Sept 29: Special problem 2 is due tomorrow. TEST 1 will
be Wednesday October 5. Some things would be good to memorize, if
possible: Definitions, Statements of Theorems. Go back over your home-
work paper and Special problem 1 too. Also, I have mentioned in class
that it's likely you will have aa Axiomatic proof to do, so be sure you
learn the Real-number Axioms too.

NEWS Sept 26: Assignments have been updated: Assignment 3
has been devised. The papergrader's late Homework Policy: a 2-point per
day late penalty until he's done grading, your score is zero after that!
Note on Special problem 2: You must use the definition of "infinite"
given in class. If you wish to use some Theorem you must include a proof of the
Theorem you want to use, done using the in-class definition of "infinite set!"
The definition was: A set E is infinite if it is not finite; a set is finite
if it is empty, or if there exists a positive natural number n and a function
f: E -> F_n such that f is both one-to-one and onto. Thus a set E
is infinite if (and only if, because this is a definition) for every positive natural number n
and every f: E -> F_n, f is not one-to-one or not onto.

NEWS Sept 22: The date mixup in Assignments reappeared
and has been corrected once more!

NEWS Sept 21: Special Problems 2 and 3 have been posted.

NEWS Sept 19: Reminder: Assignment 1 is due today! A note on
Complex Numbers has been added to the "optional" notes list below. It
has extra details.

NEWS Sept 16: We have a papergrader. Thus Assignment 1 is
due Monday September 19 after all. The dates on the Assignment sheet
have been corrected.

NEWS Sept 15: The Note on Inductive Sets has been changed.

NEWS Sept 12:
A Question about Special Problem 1: May we use a contradiction
argument?
Answer: You may, and it's a very good idea -- maybe essential!
The note on Inductive Sets has been revised and reposted.

NEWS Sept 8: New Notes, on Inductive Sets, have been posted.

NEWS Sept 6: We will start with Chapter 1, but we will modify it.
Modifications will be minor before Theorem 1.19. The really BIG modification
will be that we will assume what
Theorem 1.19 asserts. This is
done so that we can avoid going thru the Appendix to Chapter 1, which relies
on the assumption that the properties of the integers and the rational are
true. We won't assume that -- except for the sake of examples. We will "build"
the natural numbers and then the rational numbers as subsets of the real numbers.

NEWS Aug 8:  Here's a link to the Fall '03 Math 5615H Web site.
The Syllabus and Assignment links below are, for now, from Fall '03.
Later NEWS items will reflect updates.