Homework - please try hard to keep up - class will get confusing quickly if you don't! Late homework will generally not be accepted without a really good excuse (please let me know if you're having any medical or personal issues that would get in the way of your being able to keep up). Note that your lowest homework score will be dropped at the end of the semester.
Homework solutions:
Induction Proof Example
Assignment 2
Assignment 3
Assignment 4
Assignment 5
Assignment 6
Assignment 7
Assignment 8
Assignment 9
Assignment 10
Reading that has been assigned so far:
Notes - Chapter 1, Chapter 2 and Chapter 3 and Appendix B
Wolf - 1, 2, 3, 4.5, 5 and 8.1 and 8.2
Assignment 10: assigned on Wednesday, August 3rd,
due at any point on Friday, August 5th in the Math 101 folder outside room
108.
Reading: please reread Chapter 3 in the Notes,
and read Chapter 4, sections 1 through 4 in Wolf
Problems:
1) Write out a solution for maximizing the survival rate of the people
standing in line with purple or silver hats on their heads (i.e. the answer
to the last puzzle from Tuesday's class). Recall that everyone stands
in line (there were a dozen students in class that day) and each person
is given a hat that's either silver or purple. You can see all the
hat colors in front of you. You can't see your own, or those of the
people behind you. The "exectioner" walks up the line starting at
the last person (the one that can see everyone else's hat). This
exectioner asks simply for your hat color - if you get it right then you
get to eat your hat, if not, woe betide you.
Incidentally, this problem was referred to in Jim Tanton's newsletter from the Saint Mark's Math Institute - the website address I have for it doesn't seem to be operating right now though! The types of codes that I mentioned in class for another of the hat puzzles are called Hamming codes.
2) Provide one more counterexample to show that an infinite union of closed sets is not always closed. To do this you'll need to specify what the topological space is that you're working with, and what the infinite collection of closed sets is. We have a limited collection of topological spaces that we've seen so far - you might want to review the couple of examples from chapter 1 (in one of the first homework assignments).
3) Please do problems 1, 3 and 4 in section 3.3 in the Notes
to hand in. Note that for question 4, you just need to do the first
half of the problem (skip the part that says "Prove that this metric...")
Also the definition of the metric should include a line saying that d(n,
m) = 0 if n = m and another line saying that d(n, m) = 1/m if n = infinity.
Assignment 9: assigned on Thursday, July 28st, due
at the beginning of class on Tuesday, August 2nd
Assignment 9
Assignment 8: assigned on Tuesday, July 26st, due
at the beginning of class on Thursday, July 28th
Assignment 8
Assignment 7: assigned on Thursday, July 21st, due
at the beginning of class on Tuesday, July 26th
Reading: Please read through sections 2.1, 2.2
and 2.3 in the Notes and read Appendix A as well (this is a repeat
of material in Wolf,
but it should be very interesting to compare the two approaches,
as well as to read through Danny's definition of a Proof on page 29).
and in Wolf please read sections 5.2 and 5.3 (some of
this should look very familiar by now!)
Problems: In the Notes please do problems 2, 3
and 4 at the end of section 1.4
and in Wolf, in section 5.2 please
do problems 2, 3, 6(a and b), 11 and 12 and in section 5.3 please do problems
2, 4(a and b) and 7
Group Project: (due in Emily's weekly section on
Monday, August 1st) meet with 2 or 3 other students in the class, each
group should just turn in one write-up with everyone in the group's name
on it (and so each person will receive the same grade for the group).
Do the Sudoku puzzle in the Boston Globe several times (it's in the new
Sidekick section, or you can try it online at www.sudoku.com). Now,
as a group come up with at least four interesting questions based on this
new logic puzzle. I don't want to list any such questions right now
as examples in order to leave you with as many to find as possible.
Some questions might lead to combinatoric (or counting) issues such as
"how many puzzles are there (that meet a particular constraint)..." or
"what's the most (or least) number of entries so that (the puzzle meets
a particular constraint)..." Finally, present a solution to at least
one (nontrivial!) question that your group has raised. If you answer
more than one question then you can get bonus points for this group project.
If you get stuck with the 9 by 9 puzzle as it's presented, you might want
to consider puzzle variations as well (and this line of inquiry could lead
to some interesting questions too), such as a simpler 4 by 4 puzzle with
four internal blocks of 2 by 2, etc. What about a three dimensional puzzle?
Mathematical theorems arise in the oddest places some times, and who knows
- you might run into a new result while you work on this group project!
Assignment 6: assigned on Tuesday, July 19th, due
at the beginning of class on Thursday, July 21st
Please read chapter 3 in Wolf and do the following (this will be known
as the "A to C" homework for obvious reasons!) from section 3.2 do
problems 3(a to c), 6(a to c) and 7(a to c) and in section 3.3 please do
problems 5(a to c), 7(a to c) and 10(a to c).
Assignment 5: assigned on Tuesday, July 12th, due
at the at the any point on FRIDAY, July 15th (any time on Friday will do).
Please leave your assignment in the Math 101 folder
outside room 108.
Follow up with more logic and questions on the closure operator. Please read chapter 2 section 3 in Wolf, and the rest of chapter 1 in the Notes.
Problems - section 2.1 (page 24) in Wolf - do problems 1(f and
g), 3(a and b), 6(a and b).
in section 2.2 (on page 33) do problems 2(c, d and e), 3(a and b)
in section 2.3 (on page 42) do problems 2(b and d) and 9
In the Notes, at the end of section 1.3 (on page 15) do problems
1 and 2
at the end of section 1.4 (on page 17) do problem 1
Assignment 4: assigned on Friday, July 8th, due
at the beginning of class on Tuesday, July 12th
For homework you'll be asked to read up on some basic logic - we'll
go over more of this in the weekly section and during upcoming classes,
so don't worry if it's not clear at first - this is just to get you started.
Do try to reread those parts of the text that are confusing to you at first
- remember that reading a math textbook often requires reading through
several times - you learn a bit more each time you go through it.
Also, think through the examples in the text carefully to check your understanding
as you go along.
So, please read in Wolf, Chapter 2 sections 1 and 2 (on Logic) and Chapter 5, section 1 (on basic Set theory - you've already seen a lot of this in the Notes Appendix B reading)
For problems, in Wolf please do problems:
1(just parts a through d), and 2(all) and 4(parts a through e) on page
24 and 25, and
problems 1(a, b and c), 2(a, b and c), 3(all), 5(just a and b), 6(c,
d and e) and 9(just part c) on page 139 - 140.
If you haven't worked with truth tables before, just do your best for
now for the questions from section 2.1 on page 24. I'll ask Emily
to go over some basics for these in section on Monday, which meets from
11 to 12 (classroom to be announced via email later - and a note will be
left on the door to our regular classroom as well).
Assignment 3: assigned on Wednesday, July 6th, due
at the any point on Thursday, July 7th.
Please just leave your assignment in the folder
marked Math 101 located on the bulletin board outside room 108.
You can drop it off at any point on Thursday (I'll
ask Emily to pick up the homework assignments on Friday morning)
Please download the first set of Notes by Danny Goroff (hereafter "Notes" will mean these downloaded notes, and "Wolf" will refer to the textbook that you've already started reading). Please read Chapter 0, Chapter 1 section 1 and Appendix B (note that we'll be reviewing set theory in class over the next several days so don't worry if it doesn't all make sense yet!)
Problems:
(1) (a group problem) Consider integers composed only of the digit
1 (such as the numbers 11 and 1,111,111). Please prove that there
exists an integer M composed only of the digit 1 such that M is divisible
by the number 2001. Hints - this involves the pigeonhole principle
along with the fact that 2001 and powers of 10 have no common factors.
You should also observe what the difference of two numbers composed of
1's looks like. Please find someone else in the class to work with
before you turn in this problem (at the very least you should find someone
to compare your solution with so that you both agree that it's a solution!)
I'll email out everyone's email addresses in a short while if you'd like
to get in touch with someone else in the class through email. You
can work in a group with three people in it if you'd like (but not four
or more, as then it becomes too large and it's easy for someone to feel
left out of the process).
(2) (not a group problem, but you can check your answer with other students
if you'd like) Answer the following two questions that are based on the
reading of Section 1 in Chapter 1 of the notes (just do your best at this
point - these questions are here to check that you're doing the reading)
(i) Suppose set A is the set of
numbers {x | 0 < x < 1} where x is a real number
(so A is the line segment that doesn't include the endpoints 0 and 1).
What do you think K(A) might equal (where you'll need to find out
what the notation K(A) means from the reading)? Is there more
than one possibility?
(ii) Suppose B is the subset of
numbers in set A that are rational numbers (e.g. numbers like 2/3rds, 1/4,
etc.). Now what do you think K(B) equals, and is there more
than one possibility?
Assignment 2: assigned on Thursday, June 30th, due
at the beginning of class on Wednesday, July 6th.
(If you've joined the class late, please be sure to take care of assignment
1 as well)
Please track down a copy of the Proof, Logic and Conjecture
textbook by Wolf (I've left a copy on reserve in Cabot Science Library
in the Science Center). Please read the Note to the Student starting
on page xii, as well as all of Chapter 1, section 5 in Chapter 4, and sections
1 and 2 in Chapter 8 (these might be somewhat challenging to read through
at first, so don't worry if you don't quite follow everything in these
sections yet!)
Please do the following problems: (write them up as best you can to
hand in on Wednesday)
Remember that Emily said that the first two assignments will be graded
on a simple credit/no credit basis, so don't panic if you can't do everything
perfectly at this point! She will provide you with lots of comments
on your work so that you'll be in good shape when she starts grading the
next problem set with point values on it.
Section 1.2 problems 1, 2, 4(just parts a and b), 5, 6 and 10
Section 4.5 problems 4, 8, 9 and 14
Assignment 1:
Please email me a short message introducing yourself (please just send
it to engelward@math.harvard.edu). Please include a little bit about
your math background and why you're interested in taking the class.
Also, please let me know if I can include your email contact information
on a list for the rest of the class to use (and what email address you'd
like to use for that).
Problems: please write these up to hand in
1) Find the error in the "proof" from the end of the class (write down
what the error is and why it causes problems!)
Let a = b
so ab = b2
then ab - a2 = b2 - a2
so a(b - a) = (b + a) (b - a)
so a = b + a
so a = a + a = 2a
so 1 = 2!
2) Prove that -1 x M = -M for any integer M
(hint: examine (-1 + 1) x M and rewrite this expression
in two different ways)