Course
Bulletin Board
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Wrapping
Things Up
Final
Exam Solutions
Course
Spreadsheet (Final Letter Grades via WebStac
General information of grading (from the beginning of the semester) iws
at the bottom of this page.
Final Exams and all remaining homework papers are in the Math Office
(Cupples I, room 100). After about a week, I will remove
whatever's left there to my office. I will keep them until a few
weeks into the fall semester, after which all leftovers will be
discarded.
Thanks for all the hard work many of you put into the course. I
enjoyed it.
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Class
attendance is important. Some parts of the
course will
follow the text fairly closely, but on many days there will be things
in the lecture that are not in the text. At certain points in the course
there will
probably be some large chunks of material not in the text at all.
(When this happens, I will give out some additional notes.) I will also do some
topics in an order different than the text.
Week
of
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Reading
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Hand-In Homework: see instructions under "General
Information"
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Recommended Homework
(not to hand in)
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April 21
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Read the
notes from class on
Countable
Sets These notes cover
the material in 5.3 more efficiently and "maturely" than the textbook.
Read the notes on
Transcendental
Numbers
Read Section 5.4
Read the notes on
Comparing
Sizes of Cardinals, Part I
Georg
Cantor Information about the mathematician who discovered
that some infinite sets are countable but others uncountable.
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Your
work
should always contain a justification for your answer.
HW 13: Due in class on Friday, April 25.
Sec 5.3: 2, 12(b,c), 13
Sec 5.4: 2, 5(a,c,d), 8(b),11(d)
Supplementary
Problems for HW 13
Homework
13
Solutions
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Sec. 5.3:
12(a)
Sec 5.4: 5b), 8(a),11b)
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April 14
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Read
Section 5.2, 5.3
The
Rational Numbers
This is a draft of a set of notes that Prof. Shareshian will talk about
Friday when I'm away. The first part of the notes is review and
"consolidation" of material about fields. The remainder, about
constructing the rationals, is very much in the spirit of how we
constructed the integers.
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Your
work
should always contain a justification for your answer.
HW 12: Due Friday, April 18, in class
Sec 5.1: 17a) 20(a,b)
Sec 5.2: 1b), 2(e,f), 6(a,b,c)
Supplementary
Problems for HW 12
Homework
12
Solutions
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Sec
5.2 2d),3
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April
7
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Exam II
Solutions
Complete
the reading of all the materials about sequences. Read Section
5.1.
Sequences
(the complete notes)
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Your
work
should always contain a justification for your answer.
HW 11: Due Friday, April 11, in class
Sec 4.5: 4(b,g)--give a formal proof using the definition
(using "epsilon - N ")
4(e,--you need to prove the result, but you may do so using any of the
theorems proved in text/notes.
6f)
Supplementary
Problems for HW 11
Homework
11
Solutions
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Sec
4.5: 3e) (you
should be able to prove the result using our theorems, not just
"estimate" the result as the text requests), 3l)
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March 31
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Exam II: Monday, April 7, in class
Information
about Exam II
Finish
reading 4.2; read sections 4.3, 4.4, 4.5
Functions,
Part III
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Your
work
should always contain a justification for your answer.
HW 10: Due Friday, April 4 in class
Sec 4.2: 1j), 3f), 6,11, 15(b,e) (see definition for
"decreasing" in problem 14), 19d)
Comment on a
HW problem
Sec 4.3: 1L), 2L), 8(b,d,f), 14(a,c), 17(g,h)
Sec 4.4: 4(b,c), 6f), 8, 11c), 14(c,f),17
Homework
10
Solutions
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Sec 4.2 :
1i), 2i), 14d), 15d), 19a)
Sec 4.3: 1k), 2k), 8a,e)
Sec 4.4: 4a), 6(c,d), 14(a,d), 16
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March 24
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Exam II: Monday, April 7, in class
Finish reading material about constructing the
integers.(Complete set of notes now posted with last week's materials.)
Begin
Chapter 4. Start reading Sections 4.1 and 4.2 as well as the
supplementary material about functions:
Functions,
Part I
Functions,
Part II
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Your
work
should always contain a justification for your answer.
HW 9:
Due Friday,
March 28 in class
Sec 4.1: 1(j); 3(i); 6(a); 8(a,d); 14(b,c,d)
Supplementary
Problems for HW 9
Homework 9
Solutions
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Sec
4.1: 1(a, g), 3(d), 4, 8(b,e); 14(a)
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March 17
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Be sure to
brush up on the last week's reading material before Monday's class.
Read Section 3.2 and any notes distributed in class.
Relations,
Part IV
Relations,
Part V
Constructing
the Integers
Andrew Wiles and
"Fermat's Last (or Great) Theorem" (NOVA)
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Your
work
should always contain a justification for your answer.
HW 8:
Due Friday,
March 21 in class
Sec 3.1: 9(g),
10(e,n,p)
Sec 3.2: 2(e,f); 4(c,e,j), 8(b),16(a)
In 4e), you may assume the relation is an equivalence relation and
just sketch the requested equivalence classes; in both 4c and 4j, show
that the relation is an equivalence relation as part of the problem
Supplementary
Problems for HW8
Homework 8
Solutions |
Sec
3.1: 9(d), 10(g,o)
Sec 3.2: 1(a,j,k), 2(d),4(g)
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March
3
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Finish
reading the notes about Peano Systems and the Whole Numbers
Read Section 3.1 ( the material on representing a relation as a
digraph is worth reading, but optional.)
Read the notes handed out in class:
Ordered
Pairs, Products, and Relations
Relations,
Part II
Relations,
Part III
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Your
work
should always contain a justification for your answer.
HW 7:
Due Friday,
March 7 in class
Sec 3.1: 3(d), 4(b), 5,
8(f), 18(a)
Supplementary
problems for HW 7: open this pdf file
Homework 7
Solutions
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Section
3.1: 1b), 3a), 6c), 7c), 8e)
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February
25 |
Exam 1
Solutions
I let my cat try taking the test: here
he is afterward.
Exam 1
Scores
Continue reading the notes on
Peano Systems and the Whole Numbers
Part IV
Part V
Complete Set
of Notes on Whole Numbers and Pean Systems (Parts I-V, with minor
revisions)
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Your
work
should always contain a justification for your answer.
HW 6: Due Friday,
February 29, in class
Sec 2.4: 11
Supplementary
problems for HW 6: open this pdf file
Homework 6
Solutions
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| February 18 |
Read the notes on Peano Systems and
the Whole Numbers:
Part I
Part II
Part III
distributed in class
(to be continued)
About Giuseppe
Peano |
Your
work
should always contain a justification for your answer.
HW 5: Due Friday,
February 22, in class
Sec
1.7: 10, 12(b)
Note: Typo in #10: problem should
say "If q is a natural number bigger than 1 and q has the property..."
See this
file about the meaning of Problem 10.
Supplementary
problems for HW 5: open this pdf file
Homework 5
Solutions
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February 11
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Return to
Section 1.7 and read from last paragraph at the bottom of p. 61 to the
end of Section 1.7.
Read
Sections 2.4-2.5
and the notes on sums of powers of natural numbers.
In 2.5, focus on the examples; there will be a handout covering
the equivalence of PMI, CMI, WOP that will be, I hope, a little clearer
than the text.
Be sure to look at recommend problem 15(a) in Sec 2.3
Equivalence of
PMI, PCI, WOP (handed out in class)
Division
Algorithm for N
(this proof was rushed at the end of class Monday)
Sums of
Powers of the Natural Numbers
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Your
work
should always contain a justification for your answer.
HW 4:
Due Friday,
February 15, in class
Sec 2.4: 7(b),
8(d, r, t), 9(f), 15(c)
There is a typo in problem 8r: the product should
be for i = 1 to n, not n+1 )
Sec 2.5 6(a,b), 12, 15(a)
Three
Supplementary
problems for the homework set : open this pdf file
Homework 4
Solutions
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Sec
2.4: 4(f), 8(k,l,v),11,15(a)
Sec 2.5: 2,3
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February 4
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Read
Section 2.1-2.3. Also read the Notes About
Sets distibruted in class (and linked, below)
Notes About
Sets (1)
Notes About
Sets (2)
A
History of Zero
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Your
work
should always contain a justification for your answer.
HW 3: Due Friday,
February 8, in class
Sec 1.5: 3(g)
Sec 1.6: 7(j), 8(g)
Sec 1.7: 3(a,b,c), 7(b), 12(d) (you may assume, as the
"proof" states, the fact that "pi" is irrational.)
Sec 2.2: 13(b,c) (the second part of 13c) requires
justification from you)
Sec 2.3: 1(j,k), 6(b)
One
additional problem: open this pdf file
Homework 3
Solutions
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Sec
1.6: 7(h,m), 8(e)
Sec 1.7: 3(d), 12(e) (compare to 12(d) )
Sec 2.1: 8,10,19(f,g)
Sec 2.2: 4(c,d)
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January 28
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Be sure to
read and memorize the
short list of definitions from number theory on p. xii (in the "Preface
to the Student")
Read Section 1.4-1.5-1.6.
In Section 1.7, read up to the Example titled Rolle'sTheorem near
bottom of p. 60. You can skip the rest of 1.7 for the moment.
Extra Notes
for Reading Section 1.6
Proof
that there is a rational between any two reals
One More
Proof that SquareRoot(2) is Irrational
More
Practice with Quantifiers
Informal
Overview of "Theorem" and "Proof"
Some
Writing Tips
Some
Necessary Definitions
History:
Euclild and Axiomatic Method
History:
Q.E.D. as an "end of proof" marker
An
Excellent History of Math Site: Biographies and More
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Your
work
should always contain a justification for your answer.
HW 2: Due
Friday, February 1, in class.
Sec 1.4: 5(b),6(e),7(k),9(a),11(b)
Sec 1.5: 3(d),5,7(b),9,10
Sec 1.6: 5(g) (f you think it's true, then using basic
calculus for the proof is OK), 7(i)
Homework 2
Solutions
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Sec
1.4: 5(e), 6(d),7(e),9(e), 11(c)
Sec 1.5: 3(h),6(d),11,12(a)
Sec 1.6: 5(b,h),7(g)
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January
14-25
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Before
class on Wednesday, 1/23, you should have read all of Sections
1.1-1.3. If you like, peek ahead at 1.4.
Useful Handouts from Class:
Inserting
Parentheses
Useful
Equivalences
Practice
with Quantifiers
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Your
work
should always contain a justification for your answer. For
example, in #4, do not just answer "equivalent" or "not
equivalent".
HW 1: Due Friday, January 25
in class.
Sec 1.1: 4(j), 5(b), 8(a),10(m)
Sec. 1.2: 11,13(f), 14
Sec 1.3: 5(k),7(j,k),8(b): in 8(b), simply give an
example of an open sentence A(x) and a universe U where the converse of
the statement given is false.
Homework 1
Solutions
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Sec.
1.1: 1,2(c,d,l); compare c,d), 5(a,d), 10(b,f,h,j,l)
Sec 1.2: 4(a,c,e), 5(a,c,d,k), 6(e),13(e,g)
Sec 1.3: 1(a,f,m),5(b,e,k),6(d,g),8(c),10(a,b)
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General
Information About the Course
In-Semester Tests There will be two exams (= E1,
E2) in
class during the semester. The dates of these exams will be
announced at least 1 week in advance.
Final Examination The final examination
(= F) will be on Wednesday, May 7, from 10:30-12:30. The
final will count either as one exam or as
two exams (if it helps you).
Homework In most weeks a set
of homework exercises, to be handed in, will be posted on this web
page. These assignments will
normally be due in class on Fridays. Late homeworks will
not be accepted unless
there is a legitimate special reason
such as illness. When a
list of
assigned problems is posted here, it may not
be
complete. However, the list of problems due on Friday
will always be complete by noon on the preceding Tuesday. Be sure
to check after noon Tuesday that you have the complete set of
problems.
Writing Homework
Solutions The hand-in homeworks should be written up on 8.5 x 11 paper
with "clean edges" (no ragged edges torn out from a spiral
notebook).
Since one goal of the course is good writing (in particular,
of proofs), the solution of each problem should be written up quite
clearly and legibly, and should follow the "mathematical writing"
guidelines that will be handed out in class. (In particular,
check yourself
by reading aloud the words and symbols you wrote down, exactly
as written. The result should be smooth sounding English.)
Your work should always
contain a
justification for your answer. For example, in Sec 1.1, #4, do
not just answer
"equivalent" or "not equivalent".
Homework Collaboration
Talking with other people about
problems is an excellent way to
learn, but each student must write up his or her own
set of solutions. Therefore, solutions from different students
should not look much alike. After all, two people will say things
in
their own way, make up their own notation as needed, etc. A good
way to avoid "copying" even inadvertently from another student is to
talk about problems together without taking any notes away from
the conversation. This lets you share understanding and
ideas,
but forces you to reconstruct your own understanding on paper.
Grading of homeworks
1) For
about half the homework sets, I will
select one problem to grade myself, and keep a separate record of the
scores. At the end of the semester, I will discard the
lowest one
of these scores. The average of the remaining scores (= H1) will
count as one exam in determining your final course grade.
2) A graduate student (Peter
Townsend) will grade the
other hand-in homework problems. At the end of the semester, the
lowest two of these homework scores will be discarded.
The
average of the remaining scores (= H2) will count as one exam in
determining your final course grade and assign you a score (=
H2)
for them.
Additional Homework
There will usually be some additional
problems recommended for you to do but not hand in. It's important to
work on a lot of these. The number of hand-in problems is kept
reasonably small for the sake of the grader.
Total Score and Grades for the Course
All scores for E1, E2, F, H1 and H2 will be converted to
%'s. Your total score (T) for the course will be the larger
of the two numbers
(E1+E1+F+H1+H2)/5
(E1+E1+2*F+H1+H2)/6
I will base
grades on the number T.
I
will
not make up a grading scale until the end of the course, but
the following "floor" for grades is guaranteed. It is possible
that some of the grade cutoffs
will be a bit lower, but there is no guarantee of that.
90-100
A
(possibly +/-)
80-89
B
(possibly +/-)
65-79
C
(possibly +/-)
50-64
D
<
50 F
If you are taking the course with a
pass/fail grading option, then you will need to earn a C- or
better to receive the grade of "pass."
Course Evaluations Online Course
Evaluations will
be available toward the end of the semester. I urge all of you to
participate. Thoughtful and accurate feedback is valuable to both
the instructor and to your fellow students.
Academic
Integrity This link gives the general policies of the
University on academic integrity. Please also see the
comments,
above, about homework collaboration.
Anonymous
Feedback to Professor Freiwald. Of course, I'd really
prefer open
feedback and
discussion about the course at any time. However,
this link is provided as a way for students to offer suggestions and
comments anonymously. I'll keep this link here as long as it's
constructively
used. (I can't
respond, of course, to your
anonymous e-mail.)
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