Washington University in St. Louis
Math 310
Foundations for Higher Mathematics
Spring 2009

Instructor:          Professor Ron Freiwald 
Office:               Cupples I, Room 203A, 935-6737
Office Hours:     
M 3:30-4:30, W 2:30-3:30, Th 2-3, and by appointment
 
Time:        M-W-F 1:00-2:00 p.m.
Location:   Duncker Hall, Room 101 
Text:         A Transition to Higher Mathematics by Douglas Smith, Maurice Eggen & Richard St. Andre (Sixth Edition)
               
Errors, and other comments about the material in the textbook
                Note: these were sent to the authors & publisher.  They are written in a way that addresses that audience, and has some
               "style" comments that might not interest students in the class. 


Class attendance is importantSome 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 written notes.)  I will also do certain topics in an order different than the text.

Because there are a lot of handouts, you should decide on a way to keep them organized.  Probably you should date copies when you receive them and keep them in order in a folder (or punch holes in them and keep them in a three-ring binder).

General information about the course (homework pollicies, exams, grading, ...) is located on thie web page below the following table of
weekly assignments.

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)



All Scores through April 25

Final Exam Solutions

Final Course Scores

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
Reading
Hand-In Homework (see instructions under "General Inormation") and Solutions

Recommended Homework
(not to hand in)
Week of April 20-24Read:

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 17Exam 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 10Read 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 3Read 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 23-27Read 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 16-20Read 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 2-6Read Section 3.1-3.2 and supplement on Ordered Pairs and RelationsHW 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 23-27Peano 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 16-20Read 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 9-13Comments 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 2-6Read 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

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

         
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)


    


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.