MATH 347 D1H: Fundamental Mathematics (Honors Section)
Fall 2012
Professor A.J. Hildebrand
http://www.math.illinois.edu/~hildebr/347/
Final Exam Results and Course Grades
Final exam scores and course grades are now available. The median
score was 147/175, or 84 percent.
You can access your scores as usual
usual under
this link. The grade shown at the end of the online score display
(including plusses and minuses) is your course grade;
it is based strictly on the total number of
accumulated points.
Cutoffs between letter grades
were chosen to minimize hardships and close calls. As a result,
nobody was within a few points from the next higher (or next lower)
grade; in fact, the smallest gap between letter grade ranges (A+, A, A, B+, etc.)
was 17 points.
If you want to see your final exam and the solutions, stop by my office, 241 Illini
Hall, Monday afternoon, or send me an email (ajh@illinois.edu) to set up a time.
Otherwise, focus on your other finals, then relax, enjoy the holidsays, and come
back refreshed and primed for another hard (?) Math class in the spring!
Course Policies, Exams, Grades

Link to Online Scores.
Click on this link and log in with your NetID and password to access
your scores. The display shows all the scores on all assignments and exams
given out so far, and your total accumulated score. If a score is missing
or incorrect, let me know right away.
 Course Information.
BE SURE TO READ THIS FIRST.
First day handout. Contains basic course information such as text and
syllabus, exam and homework information, and grading policies. Also addresses
the honors nature of this course, how this section differs from regular
(nonhonors) Math 347 section, and whether this course is right for you.
Important dates
 Add/drop deadlines: Sept. 10 and Oct. 19. The first date
(Sept. 10) is the last day on which you can add a course; if you want to switch
to a nonhonors section of Math 347, or replace this course by another (easier)
one, you have to do so by that date. The second date (Oct. 19) is the campus
deadline for dropping a full semester undergraduate course. Note that
Engineering students may need to get their Dean's approval if they drop a course
after the 10th day of class.
 Midterm Exam 1: Friday, Sept. 21.
 Midterm Exam 2: Friday, Oct. 19
 Midterm Exam 3: Wednesday, Nov. 14
 Final Exam: Friday, December 14, 8:00 am  11:00 am.
This is the official Final Exam slot for classes meeting MWF 11 am.
Please keep this date in mind when making travel plans.

Sample exams. Links to sample exams from past Math 347 Honors
classes. These should give you a good idea of what to expect.
Announcements
 Illinois Geometry Lab Open House Thursday, Dec. 13, 2 pm  4 pm, 239
Altgeld. The IGL (Illinois Geometry Lab)
is a recently established center that provides opportunities for
undergraduates to participate in research projects.
If you are interested,
come to the IGL Open House, to see this semester's research projects and to
learn about next semester's research projects, or
visit the Illinois Geometry Lab
website.
 Final Exam Information.
The Final will be on the scheduled date/time, Friday, Dec. 14, 8 am  11 am.
Click on the link above for more information and a syllabus.

Getting started with LaTeX.
LaTeX is a technical typesetting system that enables you to produce
professionally looking mathematics. This page shows you how to get started if
you are new to LaTeX, and it has links to tips and resources for those familiar
with LaTeX.
 Midterm Exam 3:
We decided, by class consensus, to have the midterm on Wednesday,
Nov. 14. The exam will be in class, in the regular classroom, 447
Altgeld. The exam will cover the "epsilonics" material (Chapters 13 and 14).
I will post a detailed exam syllabus and study guide by the end of this week.
 Open House schedule: Primary Open House hours are Sundays,
beginning at around 3 pm, in 159 Altgeld, and Tuesdays, beginning at around 5
pm, in 141 Altgeld. Additional hours are after 6 pm Mondays, Wednesdays, and
Thursdays, in 141 Altgeld.
The Open House is an informal office hour and
gettogether for students in my classes, and serves as the primary point of
contact for my students. I will stay as long as necessary  typically at least
an hour, and sometimes much longer.
 Extracurricular opportunities: The following events are unrelated to
this class, but may be of interest to some of the students.
 IGL (Illinois Geometry Lab).
The IGL (Illinois Geometry Lab)
is a recently established center that provides opportunities for
undergraduates to participate in research projects. For more
visit the Illinois Geometry Lab
website.
 Math Contest Activities:
The U of I has one of the most extensive math contest programs in the country.
If you are interested in participating in some of these activities, visit the U of I Math
Contests website. To get on the "Putnam mailing list" for announcements
of upcoming contest activities, send your NetID@illinois.edu email to
me at ajh@illinois.edu
Handouts
Below are pdf files of handouts, worksheets, and homework assignments
distributed in class. For additional practice (e.g., to get ready for an
exam) you may want to print off a fresh copy of the appropriate
worksheets and redo the problems on your own.
Solutions to worksheet problems will be posted here within about a
week of handing out the worksheet.
Solutions to homework assignments will be distributed in class at the time
the assignment is returned, but will not be posted here in order to comply
with departmental policies. If you are missing a solution set, you can
pick up an extra copy at my office.
Class Diary
 Class 43, Wednesday, 12/12
 Class: Last day of class. Final Exam Q and A.
 Class 42, Monday, 12/10
 Class:
Final Exam Information.
Some fun questions and facts about primes. How many primes aree are? (Answer 3:
Exactly as many as needed. By the FTA, any more would create redundancies, while any
fewer would leave "holes" among the integers.)
Is there are formula for primes? (Partial answer: NO if "formula" means a
nonconstant polynomial.)
Cool website: Prime
Pages. The "Guinness book" of prime number records. One of the
most amazing mathematical websites, and the place to go for
anything related to primes.
 Class 41, Friday, 12/7
 Class:
Practice problems from the Relations handout.
 Class 40, Wednesday, 12/5
 Class:
Equivalence relations and equivalence classes.
Connections with partitions: From equivalence relations to partitions and
from partitions to equivalence relations.
 Class 39, Monday, 12/3
 Class:
Congruence classes and their properties.
Relations. Formal definition of a relation on a set S..
Reflexive, symmetric, transitive properties.
Equivalence relations. Proof that the congruence relation is an equivalence
relation.

Handout:
Worksheet: Relations

Homework:
HW 12.

Read: Chapter 7, p. 140141.
 Class 38, Friday, 11/30
 Class:
More fun applications of congruences: Divisibility tests. The "Four 4s" problem.
Sums of squares.
 Fermat number links: If you are interested, here are
some links about the "record composite" Fermat number that I mentioned in class,
and related questions:
 Prime factors
of Fermat numbers. An encyclopaedic site with all sorts of
records on Fermat numbers, their primality status, and factors
found.
 Fermat
Number Record. From John Cosgrave, holder of many Fermat number
records, including the largest known composite Fermat number.
Provides some fascinating details on how these used, including
computer screen shots, and a link to the program used (a Windows 95
.exe file!).
 Cracking
Fermat Numbers. An article about Cosgrove's quest for Fermat
records. Nontechnical, and very readable.
 Class 37, Wednesday, 11/28
 Class:
More congruence magic: Primality of numbers of the form a^n + 1 and a^n  1.
Mersenne and Fermat numbers.
 Class 36, Monday, 11/26
 Class:
The Fundamental Theorem of Arithmetic and Euclid's Theorem
on the infinitude of primes.
Congruences: Definition, properties, and first applications of "congruence
magic".
 Handout: Worksheet: Number
Theory II
 Homework: HW 11.
 Class 35, Friday, 11/16
 Class: Started the final part of the course, Number
Theory (Chapters 6/7). Divisibility, greatest common divisor,
primes and composite numbers. Formal definitions.
 Handout: Worksheet: Number
Theory I
 Homework: Honors HW 4.
 Class 34, Wednesday, 11/14. Midterm Exam 3.
 Class 33, Monday, 11/12
 Class 32, Friday, 11/9
 Class:
The BolzanoWeierstrass Theorem. Proofs of the Monotone Convergence Theorem and the
Cauchy Criterion using the BolzanoWeierstrass Theorem.
 Handout:
Epsilonics IV: Proofs of the Key Theorems.
 Read: Read p. 277  279 (Proofs of BolzanoWeierstrass Theorem and
Cauchy Criterion).
 Class 31, Wednesday, 11/7
 Class:
The Completeness Axiom and some consequences: The Archimedean Property, the density of
rationals. Proof of the Monotone Convergence Theorem,
 Read: Read the section "Completeness Axiom and Related
Properties" on the Epsilonics III handout from
last time; in particular, familiarize yourself with the proofs of the
Archimedean Property and the density of rationals using the Completeness Axiom.
Also, read the corresponding section in the text (p. 256  258, "Completeness
Axiom").
 Class 30, Monday, 11/5
 Class 29, Friday, 11/2
 Class: Infinite series, continued. Proof of comparison test, absolute
convergence test, and ratio test.
 Handout:
Worksheet: Epsilonics II: Infinite Series.
 Do: Do/rework the problems from the Infinite Series worksheet. Most of
these problems we have been discussed in class, but try to reconstruct the proofs
without referring to your class notes.
None of these proofs are particularly difficulty, and you should try to master all of
them, get an intuitive understanding of why the results are true, and be able to
construct formal proofs.
 Class 28, Wednesday, 10/31
 Class: Infinite series. Definition of convergence and
divergence. Cauchy Criterion for convergence of sequences and its application to
infinite series. The harmonic series and geometric series. Proof of nth term
test.
 Read: From Chapter 14 read the section "Cauchy Sequences"
(p. 276  you can skip the discussion of subsequences and the
BolzanoWeierstrass Theorem on 277/278, since we will cover this later)
and the first part of the section "Infinite Series" (pp. 280  282).
I will spend one more class hour practicing proofs involving infinite series,
such as proofs of comparison test (14.29) and the ratio test (14.31).
 Class 27, Monday, 10/29
 Class: The ArithmeticGeometric Mean (AGM). Proof of
convergence of AGM algorithm.
For more about the AGM and and the computation of Pi, google "Pi and the AGM"
or view the following links:
 Class 26, Friday, 10/26
 Class:
Some cool applications of the Monotone Convergence Theorem: Iterated expressions
such as infinite stacked squareroots (e.g., sqrt[2 sqrt[2 sqrt[2 ...]]]),
infinite stacked fractions (e.g., 1 + 1/(1+ 1/(1 + ... ))),
infinite "towers" (e.g., a^a^a^a ...). Proper interpretation of such infinite
expressions as limits of corresponding truncated expression. Proving
convergence via Monotone Convergence Theorem. Finding the limit once its
existence is known.
 Class 25, Wednesday, 10/24
 Class:
More proof practice with problems from the Epsilonics worksheet. Some tricks of the
trade (epsilon/2, N=max(N1,N2), epsilon=1, finite/infinite split, etc.).
 Homework: Honors HW 3. Due in two weeks (Nov.
7).
Correction: In Problem 3, in the definition of the limit f(x) it should
say (on the left) lim f(x)=L, NOT lim f(x)=f(x_0).
 Do: Work/rework the nonasterisk problems (i.e., Problems 1  10) from the
Epsilonics I worksheet. Several of these we did in class, and the remaining problems
can be done with similar methods. These problems illustrate important techniques and
tricks, and you should get completely comfortable with all of them, develop an
intuition for the proofs, and practice writing up formal proofs.
For the problems we discussed in class, try to reconstruct the argument and write up
formal proofs; for the other problems, try to come up with the proof idea using
similar reasoning (draw pictures!), and then write up formal proofs.
 Class 24, Monday, 10/22
 Class:
Returned midterm exam and solutions.
Started the "epsilonics" part of the course (Chapters 13 and 14 of the text).
Formal (epsilon) definition of the limit of a sequence, and the intuition behind it.
Worked out some proofs involving the epsilon definition of a limit..
 Handouts:
Sequences and Limits: Definitions and Theorems.
Worksheet: Epsilonics I.
 Do/read: Read p. 259263 (Limits and Monotone Convergence) in the
text. (We will cover the first part of Chapter 13 (Completeness Axiom) later.)
Also, work on this week's HW problems.
 Class 23, Friday, 10/19. Midterm Exam 2.
 Class 22, Wednesday, 10/17
 Class 21, Monday, 10/15
 Class 20, Friday, 10/12
 Class: Cardinality and countability, continued. Cantor's
diagonalization method, revisited. Uncountability of the real numbers, R, and
some consequences. Fun application/paradox: The Hotel Infinity Problem.
 Handout:
Worksheet: Cardinality II: The Hotel Infinity Problem.
 Links for the Hotel Infinity Problem: Here is a much
embellished short
story version of the "Hotel Infinity" problem. For a much drier
analysis of this problem, read the
the Wikipedia article on the subject.
 Homework: HW 7. Due Wednesday, 10/17.
 Do/read:
Finish any remaining problems on the two cardinality worksheets handed out so far.
These are problems you should get fully comfortable with; try to develop a "feel"
for what is and what is not countable. Most of the problems on the first
worksheet have come up in class, and we have done all except the last of the
Hotel Infinity problems. The remaining problems can be done by similar
techniques.
 Class 19, Wednesday, 10/10
 Class: Cardinality and countability, continued. Cardinality of
power sets and sets of functions. Encoding of subsets via "membership
functions". Proof of the uncountability of P(N) using diagonalization method.
 Do/read:
Work through remaining problems on the cardinality worksheet.
For many of the given sets you can prove countability or uncountability by
finding a natural bijection to a known countable or uncountable set (or a
subset/superset of a countable/uncountable set); for others you can prove
countability by building up the set from known countable (or finite) sets
using unions and cartesian products.
Also, go over the diagonalization method used in class to prove uncountability
of P(N) and of binary sequences, try to fully understand why it works, and try
to adapt the method to prove the uncountability of the real numbers.
 Class 18, Monday, 10/8
 Class: Cardinality, countable and uncountable sets. Examples of countable
sets. Proof of the countability of the rational numbers.
 Handout:
Worksheet: Cardinality I: Countable and Uncountable Sets.
 Do/read:
Start working through the problems on the cardinality worksheet; in each case ether
try to find a bijection from the given set to a known countable or uncountable set,
or try to "build up" the given set from known countable sets and use results on
countability of unions and cartesian products.
Read pp. 87  90 (the first part of the section on cardinality) in the text.
 Class 17, Friday, 10/5
 Class: More about functions. Injective, surjective, bijective functions.
Sample proof involving injectivity.
Some interesting/nonstandard examples of functions. Sequences as functions. Operators
(e.g. derivative operator) as functions.

Homework: HW 6. Due Friday, 10/12.
 Read:
pp. 80  87 in Chapter 4 (injections, surjections, bijections, composition and
inverses). Also, if you have not done so read pp. 10  13 in Chapter 1
(domain, target, image, and graph).
 Class 16, Wednesday, 10/3
 Class: Finished up the induction chapter with a discussion of
the WellOrdering Principle and a derivation of the Induction Principle
from the WellOrdering Principle.
Started Chapter 4 on functions. Formal definition and intuition behind the
concept of a function. Domain, target, image of a function. Graph of a
function.

Homework: Honors HW 2. Due Monday,
October 15.
 Do/read:
If you have not done so, read up in the text on the "WellOrdering
Principle", and its application to the irrationality
of the squareroot of 2 (p. 64/65). Also, read the section on functions in
Chapter 1 (pp. 10  13), with introduces basic concepts like domain, target,
and graph.
 Class 15, Monday, 10/1
 Class: Some nonformula induction examples: Number of subsets of
an nelement set; number of regions in the plane created by n lines. The
piethrowing problem.
 Handout:
Worksheet: Induction, IV: Miscellaneous.
Solutions to Induction III Worksheet.
 Do/read:
Read up on the "WellOrdering Principle" and its application to the
irrationality of the squareroot of 2 (p. 64/65). If you are
interested, try to come up with an induction proof for the piethrowing
problem, and for the Tower of Hanoi problem (see 12.1 in the text for a
description).
 Class 14, Friday, 9/28
 Class 13, Wednesday, 9/26
 Class: More applications of strong induction. Even/odd Fibonacci
numbers. Representation of integers as products of primes, and the
Fundamental Theorem of Arithmetic. Representation of integers as sums of
Fibonacci numbers.
 Do:
Finish this week's homework (HW 4) if you have not done so.
Then work through the Fibonacci problems (1(a)(d) and the binary and factorial
representation problems (2(c)(d)) on the Induction II worksheet.
For the representation problems use the "greedy" strategy in the induction
step to pick the largest "building block" that fits in. Try to work out the
details on why this algorithm guarantees distinctness of the "building
blocks" involved in the binary representation and the
Fibonacci representation. (For the factorial representation, distinctness is
not required, but there is a limit on the number of times a given factorial
i! can be used, namely at most i times. The greedy strategy, together with
the hint in the problem, can be used to ensure this; work out the details.)
 Class 12, Monday, 9/24
 Class: Strong induction. The Strong Induction Principle, and the
intuition behind it. A sample strong induction proof.
 Handout:
Worksheet: Induction, II: Recurrences and Representation Problems.
Solutions to Induction I Worksheet.
 Do:
First get to work on this week's homework (HW 4) if you have not done so.
This assignment covers the simplest types of induction problems, of the type
we discussed last week, and you should be comfortable with doing those
problems. Then move on to the Fibonacci type problems (Problems 1(a)(d)) on
the new worksheet. Some of these require only standard induction, but for the
last one (1(d)) strong induction is necessary.
 Class 11, Friday, 9/21
 Class: Midterm Exam 1.
 Homework: HW 4. Due Friday next week.
 Do: With the exam out of the way, focus on the Honors homework if you
have not finished it yet, and on the induction proofs on next week's homework (HW 4).
 Class 10, Wednesday, 9/19
 Class: Induction proofs, continued. Focus on logical structure of an
induction proof, and proper writeup. Examples from worksheet of induction
proofs of inequalities (Type II) and statements involving n variables (Type
III).
 Do/read:
Try to work through the remaining problems of the Induction I worksheet by
Monday, and read Chapter 3 through p. 58 in the text. I will have a new
worksheet on Monday, with problems on Fibonacci numbers and representation problems
(e.g., binary representation) that require Strong Induction.
 Class 9, Monday, 9/17
 Class: Started Chapter 3.
The Induction Principle, and why it "works". Intuition: Row
of dominos tipping over, and infinite chain of implications.
A sample induction proof.
 Handout:
Worksheet: Induction, I: Basic Examples.
 Do/read:
Work through the first two types of induction proofs (sum/product formulas and
inequalities) on the worksheet, using the sample induction proof as model for
your writeup. Those types of proofs should become completely routine; pay
attention to the writeup. We will spend another class hour on proofs from
this worksheet, focusing on proofs of type III.
Also, read through the first few pages of Chapter 3. (You can
skip, for now, over the trickier applications (such as those in the Problems
stated at the beginning of the chapter.)
 Class 8, Friday, 9/14
 Class: Finished the discussion of logical statements with a
careful analysis of the statement in Problem 5(b) of the Logic Worksheet.
Interlude on sum/product notation. (This will be needed for our next topic,
induction proofs.)
 Handout:
Worksheet: Sum/product notation.
 Homework: HW 3. Due Wednesday next
week.
 Do: Work the remaining problems on the sum/product notation
worksheet. Also, get started on HW 3, a relatively short assignment with
problems on the sum/product notation and some additional problems on logical
statements.
 Class 7, Wednesday, 9/12
 Class: More problems from the Logic Worksheet.
The importance of the order of quantifiers. Analyzing complex statements such
as those in Problem 5 of the Logic Worksheet.
 Handout: Exam 1 Information, Syllabus, and
Study Guide.
 Do:
Finish working through the Logic Worksheet problems. In particular, you should
by now be comfortable with problems of the type 1  4, and with the definitions
(bounded, increasing, nonincreasing, etc.) in Problem 4.
Try to analyze, and get an intuition for, the epsilondelta statements in
Problem 5, using the strategy illustrated in class: Think of the statement as
a nested object, consisting of several layers, "peel off" quantifiers from left
to right, think of a "for all" variable as one that is given to you, and an
"exists" variable as one you can pick, and keep track of which variables are
fixed at each stage.
 Class 6, Monday, 9/10
 Class: Logical statements, continued. Rules for negation. Implied
quantifiers. Practice with problems from Logic
Worksheet.
 Homework: Honors HW 1, due Monday,
9/24..
 Do: Continue working problems from the Logic Worksheet (and problems
of the same type form HW 2). The first four problems on the Worksheet,
and most of the logic problems on HW 2, should be routine by now and you should
have little difficulty with these. Try to work through these problems by the
next class on Wednesday. I plan to devote part of Wednesday's class to
going over any issues that may come up in these problems before moving on to
some of the more difficult logic exercises such as interpreting the
epsilondelta statements in Problem 5.
 Class 5, Friday, 9/7
 Class: Finished the "Even/odd proofs" worksheet with some examples of
proofs by contradiction; in particular, the irrationality of the square root of 2.
Started a new topic, Logical statements, that will take up the next couple of
hours. Discussed the proper definition of a logical implication, the converse,
and contrapositive, and the consequences of false assumptions (namely,
anything!).
 Handout: Logical statements.
 Handout: Worksheet: Logical Statements.
 Homework. HW 2. Due Friday next week.
 Do: Read the section on logical statements (pp. 2734) in the text.
Also, familiarize yourself with the various English equivalents of logical
notations from the Logic handout, and try to work through the first few
problems on the Logic Worksheet. These problems are mostly routine exercises
and drills in translating an English phrase into a logical statement written in
symbolic form (using the correspondences given on the Logic handout), or
exercises in negating logical expressions (using the rules for negation given
at the end of the Logic handout).
 Class 4, Wednesday, 9/5
 Class: Proof practice with "even/odd" proofs.
 Handout: Proof techniques.
 Handout: Worksheet: "Even/odd" proofs.
 Do: Read the section on proof techniques (pp. 3439) and the
section ``How to approach problems'' (pp. 3944) in Chapter 2.
Also, try to work through the other problems on the "Even/odd proofs"
worksheet by Friday. .
 Class 3, Friday, 8/31
 Class: Proof practice: Proofs of settheoretic relations.
 Handout: Settheoretic proofs.
 Do: Work on next week's homework assignment (HW
1). For the settheoretic proofs in this assignment proceed as in the
examples in today's handout as models.
 Class 2, Wednesday, 8/29
 Class: Proofs: Good, bad, and ugly.
An illustration of common errors in proofs using the twovariable AGM
inequality: No explanations/justifications, no connecting words, and, most
importantly, a proof going in the wrong direction.
Also, presented an example of a false statement resulting from a "proof" going
in the wrong direction.
 Handouts:
"Proofs" of x=x+2. How NOT to do proofs.
"Proofs" of the AGM inequality:
Good, bad, and ugly.
 Homework. HW 1. The first "real" homework
assignment, due Friday next week.
 Read/do: If you have not done so, be sure to study/review the
settheoretic notations and concepts on the handout and on pp.
6  9 in the text, and do the problems on the set theory worksheet (which are
due this Friday.)
Note:
As mentioned on the worksheet, the text "Discrete Mathematics" by Rosen (on
course reserve in the Math Library and the Engineering Library) is an excellent
source for additional examples and practice problems on this material.
If you are a bit shaky on settheoretic concepts like power sets, cartesian
products, etc., I highly recommend you check out the Rosen
text.
 Class 1, Monday, 8/27.
 Class: Diagnostic quiz
and Course Overview.

Handout:
Settheoretic notations and terminology.

Worksheet:
Settheoretic notations and terminology.
 Read: For the next class, study/review the settheoretic
notations and concepts on the handout and on pp. 6  9 in the text,
and do the problems on the settheory worksheet. This material is quite easy,
and you'll probably know much of it already from other classes.
In the next class, we will be using these concepts to practice some
proofwriting.
 Homework for Friday: Do the problems on the settheory worksheet.
This should not take much time. (You can do the work on the sheet itself.)
Last modified: Sat 15 Dec 2012 02:01:32 PM CST
A.J. Hildebrand