Course
Bulletin Board and Weekly Schedule
"What is
written without effort is in general read without pleasure."
Samuel Johnson (1709-1784)
"Easy reading is damned hard writing"
Nathaniel Hawthorne
(1804-1864)
To complete the course: as discussed in class, you may either
i) Choose to take the final examination at the
scheduled time (Wednesday, May 6, 1 p.m.-3 p.m). In that case, all the calculations toward your final score will
proceed as advertised at the beginning of the course (see near the
bottom of this web page).
Material covered on the final will start with Chapter 3. (Of
course, you still need to know general stuff from the beginning of the
course: for example, if I ask for a proof by contraposition of
some result in Chapter 4, you need to know what that means.)
You should review all the material assigned in the text or in
supplementary handouts beginning (see schedule below) beginning in the
week of March 2-6. The styles of
question will be similar to previous exams--with maybe a little more
emphasis on questions that are quick to grade (t/f, short answer, give
an example of, ...etc). The reason is that the exam ends at 3pm
and, supposedly, I'm to have grades in the College Office by noon the
next day. I've already told the College Office "this is not
humanly possible" and they've agree to give me a short extension.
But, in any cases, I have to give an exam that I can grade
relatively quickly. I'm not
sure about the length but I don't plan that it will be "double the
length" of the previous exams: but probably a little longer. Definition & Theorem List for the final exam
OR INSTEAD
ii)
Complete an extra homework assignment (problems for the Option HW 12 are linked below). This extra homework would be
due, at the latest, by noon on the day of the final exam , but early turn-ins would be much appreciated.
If you choose this option, this additional
homework score will be put into your homework average (although this
extra homework cannot be one of the homework scores that is discarded). . Here is the link:
OPTIONAL HOMEWORK 12
Your total score for the course will
then be computed by counting four components equally: HW Average,
the Average on HW that I graded, Exam 1 score, and Exam 2 score.
Week
of
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Reading
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Hand-In Homework (see instructions under "General
Inormation") and Solutions
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Recommended Homework
(not to hand in) |
Week of April 20-24 | Read:
Class notes for Introduction to the Reals and the Completeness Property
Equivalent Sets and Arithmetic of Cardinal Numbers This is the complete and final set of notes, covering all the material up through the lecture on Wednesday, April 22.
Notes on Constructing the System Q of Rational Numbers
| OPTIONAL HOMEWORK 12 | | April 13-April 17 | Exam 2 on Friday April 17 in class.
You should read Section 5.1-5.3 up through Theorem 5.17 and the comments about the Infinite Hotel on p. 239-240. We will cover the reamining material in 5.3, but I think the book's treatment is confusing: there will be notes.
Read the notes on Equivalent Sets --notes ONLY through the lecture of Wednesday, April 15.
Notes about the "founder" of the theory if infinite sets:
George Cantor (i) George Cantor (ii)
| Homework 11: Due in class on Wednesday, April 22 (delayed because of Exam 2) I may add to these problems up through Sunday, April 19. This is the last homework unless you choose the "Extra Homework option (see information above with the Exam 2 information.)
Section 5.1 20b), 21c) Section 5.2 1g), 2( c,e,f), 5(b,f) (justify your answers!), 6, 7 Section 5.3: 2,8b), 10a) (there is a bijection f between N and A; use f to describe a bijection between N and A - {x}, 16d)
Supplementary Problems for HW 11, Part i)
Supplementary Problems for HW 11, Part ii)
Homework 11 Solutions
Exam 2 Solutions
| Section 5.2: 1(a,c,e) 2(a,b,d),3, 4a)
| | | OLDER STUFF BELOW
| | April 6-April 10 | Read Sec. 5.1 up through p. 224. If you want to read further, read 5.2
Read the notes on Equivalent Sets (Parts 1-3) | HW 10: Due in class on Friday, April 10.
Section 4.4: 4(b,c) (Note: in case you're looking, answer for 4a) in back of text is wrong: for 4a) , correct answer is the empty set.) 5e), 6(a,d),9, 14f), 17
Open pdf file for: HW 10 Supplementary Problems (final update at 4pm Monday 4/6)
Homework 10 Solutions
| Section 4.4: 4a), 5(b,d), 7(a,c),16 | March 30 -April 3 | Read Sec. 4.3, 4.4 and Supplementary Notes on Functions (Complete set of printed notes)
Skip Sec 4.5 for now
Read Sec. 5.1 up through p. 224. If you want to read further, read 5.2 | HW 9: Due in class on Friday, April 3.
Section 4.1: 3e), 5(e,f), 6c), 8(d,f), 14(b,c,d)
Section 4.2: 2g), 3f), 6,11, 12(b,c), 15(b,e) For 15b): read this information
Section 4.3: 1h), 2(e,l), 4,6, 8(c,f), 9b), 14d), 16(a,b), 17(g,h)
Section 4.4: 2b), 3b)
Homework 9 Solutions
| Section 4.1: 2,9,11
Section 4.2: 1(f,j), 3d), 12a), 15(d,f)
Section 4.3: 1(i,k),3, 9(b,c), 17d)
Section 4.4: 2a),3a) | March 23-27 | Read the Notes:
Constructing the Integers (complete notes )
Read Sec. 4.1, 4.2 Supplementary Notes on Functions (Parts 1 and 2)
| HW 8: Due in class on Friday, March 27. Print the following pdf files.
HW 8, Part i
HW 8, Part ii
Homework 8 Solutions
| | March 16-20 | Read Section 3.3 (I
covered the material in Theorem 3.5 in class. You need to pick up
the rest ofthe material about "partitions" (there's not much) on your
own reading.
Read Supplemental Notes: Equivalence Relations and the Algebraic Systems Zm: Complete Notes (Parts i-ii-iii)
Constructing the Integers, Part i
Of interest:
From the Nova Series on PBC: Andrew Wiles and the Proof of Fermat's Last Theorem
A History of Zero | HW 7 : Due in class on Friday, March 20
Section 3.1: 8(f,h) (write the final answer for each part of 8 in the form { (x,y): y = ... } )
9e), 10L)
Section 3.2: 1i), 2(a,e,f) (in each part of 2, make your example R so that dom(R) = A unless that is impossible), 4(c,f,j) (in each part, take time to convince yourself that the relation given is an equivalence relation; but you don't need to include that in the written solutions) 8, 16a)
Section 3.3 2(a,b,e), 3d), 6a), 7, 8b)
Homework 7 Solutions
| Section 3.2: 1L), 4(a,d)
Section 3.3: 2a, 3(a,b), 6(b,d), 8a | March 9-13 | SPRING BREAK
| | | March 2-6 | Read Section 3.1-3.2 and supplement on Ordered Pairs and Relations | HW 6: Due on Wednesday, March 4:
Supplement Problems for HW6 (pdf file) (Based on the Peano System notes) and Section 3.1: 3d), 4(a,b)
Homework 6 Solutions
| Math 310 Scores, including Exam 1, up through March 5, 2009 | February 23-27 | Peano Systems and the Whole Numbers: the complete set of notes
You
should finish reading these notes this week -- although only the
material up through "Defining an Order in a Peano System" is for Exam
1. | Because of the exam Friday, HW 6 will be due on Wednesday, March 4. Therefore I might still add additional problems -- but none after Fridy noon.
Supplement Problems for HW6 (pdf file) (Based on the Peano System notes)
and
Section 3.1: 3d), 4(a,b)
The next HW (#7) won't be due until March 20 (the Friday after Spring Break) | EXAM 1 IN CLASS FRIDAY, FEBRUARY 27 Information about Exam 1
Exam 1 Solutions
Comments about a few problems on Exam 1 | February 16-20 | Read skipped material in Section 1.7, from bottom of p. 61 through p. 64
We then abandon the textbook for several lectures: read the handout material on
Peano Systems and the Whole Numbers: the complete set of notes
By Friday, you should have read at least up through the material on addition in Part ii). | HW 5: Due Friday, February 20
Section 1.7: 8c), 11(f,h), 12b) (in f, "Euclid's Lemma" is a typo for "Euclid's Algorithm"
Supplementary Problems to hand in with HW 5
Homework 5 Solutions | Section 1.7: 9b), 11a), 12a) | February 9-13 | Comments on Writing a Proof -- Homework Example
Read Sections 2.4-2.5 & the notes on Mathematical Induction
Mathematical Induction
Sums of Powers of Natural Numbers
Some Induction Examples from Class
Read skipped material in Section 1.7, from bottom of p. 61 through p. 64
The Division Algorithm for N and for Z (This goes beyond what's in pp. 61-64 of the text.) | HW 4: Due Friday, February 13
Section 2.3 : 1j), 8d) (prove it's true or give a counterxample if it's false), 12(a,b), 18a)
Section 2.4 : 8(d,m), 9f),11,15c)
Section 2.5: 6b), 12(a,b) ( use PCI for part b) , 15a) Supplementary Problems to hand in with HW 4
Homework 4 Solutions | Section 2.3: 1(g,i), 8b)
Section 2.4 : 8(k,s), 9c), 15a)
Section 2.5: 5a), 15b) | February 2-6 | Read Sections 2.1-2.3 Start reading Section 2.4. There will also be some supplementary notes covering this material.
It is important to learn to state definitions correctly. There will be questions on each exam like "Define ... "
A History of Zero
Supplementary Notes: Basic Set Theory, I
Class handout: unions & intersections | HW 3: Due Friday, February 6
Section 1.6 : 3, 5(c,g,h), 7(i,j), 8(e,g) (Note:
in 7j, modify the problem slightly to read: "there exist integers L and
G where L < G and such that ..." Otherwise you could get off
easy by sayi)ng: let L=1, G=0. Then for every x, L
< x < G would be false and the conditional would be true.)
Section 1.7: 2c), 6a)
Supplementary Problems to hand in with HW 3
Section 2.1: 5d), 6d), 8(b,c), 9(h,j,l),19g) (for any answer in 8 or 9 which you think is false, give an explanation)
Section 2.2: 3(m,n), 4(c,e), 11e),13(b,c), 17f)
Supplementary Problems, Part II, to hand in with HW 3
Homework 3 Solutions
| Section 1.6 : 1c), 2(b,c), 5(i), 7(g), 8d)
Section 1.7: 7a)
Section 2.1: 4(a-k), 9(a-k)
Section 2.2: 4f) 10c)13(a,d), 17d)
| January 26-30 | Read Section 1.5 - 1.6 In
Sec. 1.7, everybody should read up to the example called Rolle's
Theorem on p. 60. There is really no "new" material here, just more
examples. Reading further in 1.7 is optional. We will
return to the material on pp. 62-64 later.
Fourteen Different Proofs that SQRT(2) is Irrational
Some information about Ramanujan
Handouts from Wednesday Lecture: Proofs involving Quantifiers | HW 2: Due Friday, January 30
Section 1.4: 5b), 6e) (You can assume the standard algebra for manipulating inequalities, for example: if x < y, then - x > - y ; try to use as few cases as possible, or avoid them completely.) 7(i,j), 9a), 11b)
Section 1.5: 3g), 5, 7b)
Supplementary Problems for HW 2 to hand in: print this pdf file; you can write the solutions to these problems on the printed sheets.
Additional supplementary Problems for HW 2 to hand in
Homework 2 Solutions
| Section 1.4: 5d), 6d), 7(b,f)
Section 1.5: 3e)., 6(b,d) |
January 12-23
|
Reading from the textbook:
Preface to the Student (pp. x-xii) . Be sure you know the basic definitions on p. xii.
Sections 1.1 - 1.5
Some extra comments about propositional connectives
Connectives and Parentheses
Some Comments About Writing Solutions
Practice with Quantifiers
Handouts from class about proofs
Euclid & the Axiomatic Method
Wikipedia: Euclid and the Axiomatic Method
Russell-Whitehead & Principia Mathematica
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Your
work
should always contain enough justification for your answer that the reader can undersand how you got your answer. For
example, in #4, do not just answer "equivalent" or "not
equivalent".
HW 1: Due Friday, January 23
in class
Section
1.1: 2i, 4n, 5(b,d), 8a ("prove" by making a truth table),
10(f,m) (the denials should be English sentences)
Section 1.2: 8(b,c,g,h), 11,13(b,i,j) (apply the conventions on grouping on p. 8 and p. 17 wherever there's any ambiguity in the given form), 14
Section 1.3: 1 (c,d) (translate using a mix of logical symbols and words, such as the text uses in Exercise 6) 2(c,d), 5(d,e,k), 6c, 7(i,j,k) (In
5,6,7: if a statement is false, give a brief explanation why:
here, the reason doesn't need to be an "airtight" argument, just
something fairly convincing.
In 7, remember that quantifiers are interpreted left-to-right).
Homework 1 Solutions
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Section 1.1: 2c,d (compare the results), 4i, 8c, 10(i,j)
Section 1.2: 1(b,c,e), 4(b,f,j), 5(f,g), 8(i,j,k), 13(g,h)
Section 1.3: 1(e,f), (2(e,f), 6(f,g), 7(f,g)
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General
Information About the Course
Homework
Hand-in Homework Exercises to be handed in will be posted on this web
page (for now, at the bottom). Homework assignments will usually be due in class on Fridays.
When a
list of
assigned homework problems first appears here, it may not
yet 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 12 p.m. Tuesday to be sure you have the complete set of
problems.
Late homeworks will
not be accepted unless
there is a legitimate special reason
such as illness.
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).
If you write in pencil, please be sure the solutions are dark enough to read easily.
One goal of the course is good writing (in particular,
of mathematical proofs), tso he solution of each problem should be written up in clear language (and legibly). . A
solution should always
contain a
justification for your answer. It should be at a level and contain
enough ndetails to make your solution clear and convincing to other
students in the class. For example, in Sec 1.1, #4, do
not just answer
"equivalent" or "not equivalent."Follow the "mathematical writing"
guidelines that will be posted here and handed out in class. First do your work on
scratch paper and then write up what you're going to hand in;
proof-read and rewrite if necessary. In particular,
check yourself
by reading your solution: word-for-word, exactly
as written. The result should be smooth sounding English.All
writing is supposed to communicate something to the reader. Good
writing strives to make what you want to say as clear as possible to
the reader without unnecessary work on the reader's part: the
reader should not have to mentally reorganize your work, fill in gaps in logic, try to
guess what you meant, etc. Some good advice:
"What is
written without effort is in general read without pleasure."
Samuel Johnson (1709-1784)
Do
you really want a confused, irritated and unhappy grader evaluating
your work? Give the grader some pleasure in reading what you turn
in! 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 increase your learning -- and to avoid "copying" even inadvertently from another student -- is to
talk about problems together without taking any notes away from
the conversation. That way you can share your understanding and ideas, but you are forced to reconstruct your own understanding on paper.
Grading of homeworks
1) For
about half the homework sets, I will
select one problem which I will grade. I will keep a separate record of these
scores. At the end of the semester, I will convert these scores to % and discard the
lowest one. The average of the remaining scores (= H1) will
count as one exam in determining your final course grade.
2) Two student graders who took this course from me last spring will grade the
other homework problems. At the end of the semester, these scores will be converted to % and
lowest two be discarded.
The
average of the remaining scores (= H2) will count as one exam in
determining your final course grade.
Additional Recommended Homework
There will usually be some additional
problems recommended for you to do but not hand in. It's important to
work on as many of these as you can. The number of hand-in problems is kept
reasonably small for the sake of the graders.
Exams
In-Semester Tests There will be two exams (= E1,
E2) given in
class during the semester. The dates of these exams will be announced at least 1 week in advance. Exam 1 will probably be in late February and Exam 2 in late March,
Final Examination The final examination
(= F) will be on Wednesday, May 6, 1 p.m.-3 p.m. The
final will count either as one exam or as
two exams, whichever gives you the higher total score. The location of the final will be announced later.
Total Score and Grades for the Course
The 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." Anyone taking
the course on an "Audit" basis needs to talk with me immediately to
discuss the conditions for a "successful audit."
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. Of course, I'm unable to
respond to anonymous e-mails.
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