George M. Bergman  undergraduate course materials
Index to this page.
(Dates indicate most recent revision.)
Introduction
Nontextspecific handouts
Mathematical induction, 5p., Dec. 2010
(for a calculus class)
The idea of a matrix, 1p., Oct. 2004
(for lower division linear algebra)
A useful principle in solving
differential equations, 6 pp., Nov. 2014
(for lower division differential equations)
Some notes on sets, logic, and mathematical
language, 12 pp., July 2008
(general background for upper division courses)
Proof that the group A_{n} is
simple for all n≥5, 3pp., July 2000
When finite abelian groups are
cyclic, 2pp., July 2000; 1p., June 2009.
Sketch of my favorite proof of the First Sylow
Theorem, 1p., June 2009.
The Greek alphabet, 1p., Aug. 2008
Mathematical symbols in email.
1p. Sept. 2001
On `Incomplete' grades, 1p., Sept. 2004 (Berkeleyspecific)
Textspecific material (e = supplementary exercises,
c = comments and corrections, n = notes on specific topics,
qa = answers to students' questions)
Re J. Stewart, "Calculus, Early Transcendentals",
and its Berkeley Custom version "Single Variable Calculus",
e:26pp., March 2015; qa:298K, Fall 2010+2012+2014.
Re R. Hill, "Elementary Linear Algebra with
Applications", 3rd ed, e:11pp., June 2002
Re Boyce and DiPrima, "Elementary
Differential Equations and Boundary Value Problems",
7th ed., e:7pp., June 2002.
Re K. Rosen, "Discrete Mathematics and Its
Applications", 5th ed., e:26pp., Dec. 2005.
Re Friedberg, Insel and Spence, "Linear Algebra", 4th ed.
e:16pp. + n:2pp + qa:128K, Jan. 2009.
Re Beachy and Blair, "Abstract Algebra", 3rd ed.
e:10pp. + qa:183K (+ qa:107K for 2nd ed.),
May 2007.
Re Joseph J. Rotman, "A First Course in Abstract
Algebra", 2nd ed. e:18pp. + qa:117K, March 2003.
Re Dummit and Foote, "Abstract Algebra", 3rd ed.
e:21pp. + qa:70K, June 2009.
Re I. Stewart, "Galois Theory", 2nd and 3rd eds.
3rd ed.: e:18pp. + c:12pp + qa:129K;
2nd ed.: c:3pp + qa:144K,
June 2005.
Re W. Rudin, "Principles of Mathematical Analysis", 3rd ed.
e:93pp. + c:6pp +qa:300K, Dec. 2006.
Re Stewart and Tall, "Complex Analysis". e:18pp. + qa:208K,
June 2004.
Introduction.
This webpage contains material of various sorts that has come out of
undergraduate courses I have taught; mostly supplementary notes,
supplementary exercises, and collections of answers to students'
questions from upper division courses.
Most of the handouts are PostScript files; a few are in pdf.
(The source files are in locally enhanced troff, so I can't
provide ^{T}E^{X} files, but here is a
link
to software that can be used in viewing PostScript on a Windows
system.)
The collections of answers to students' questions are
in plain text.
I required submission of such questions in my nongiantlecture
courses, so the undergraduate courses for which I have such collections
are the upper division ones, and honors calculus.
Nontextspecific handouts.
I begin with the mathematical handouts, ordered roughly
according to the level of the course in which it was used.

Mathematical induction.
An elementary introduction to mathematical induction and why it works.
Written for an honors calculus section, but usable in the nonhonors
version of that course.
Here is a version without references
to the calculus text, which might be used in other courses
(e.g., in precalculus, or for review in higherlevel courses).
Last revised Oct. 2014, 5 pp.

The idea of a matrix.
Discussion of an mxn matrix as representing a linear
transformation from ntuples of real numbers to mtuples
of real numbers.
Written for our two sophomore calculus courses, Math 53
(multivariable calculus without linear algebra)
and Math 54 (linear algebra and differential equations).
I also give it out as a review sheet in the upperdivision abstract
algebra course, Math 113. 1p., last revised Oct. 2004.

A useful principle in solving differential
equations.
Points out the following approach to solving firstorder
differential equations: Find a
family of transformations of the plane that must take
solutions of the given differential equation to other solutions,
and make a change of coordinates so that these transformations
become vertical translations.
The differential equation then reduces to an integration.
This is used to motivate the standard methods of solving first order
linear differential equations  first homogeneous,
then nonhomogeneous.
Ends with an optional (for Math (H)1B)
section on equations of the form y' = f(y/x).
Recommended for reading just before textbook's
development of linear differential equations.
Written up Fall 2008 for Math 1B; revised 2010, 2012, 2014
for Math H1B.
Also useful for Math 54. 6 pp.

Some notes on sets, logic, and mathematical
language.
Basic setnotation, logical connectives, quantifiers; how
changing order of quantifiers changes
the meaning of statements; meanings of some phrases
such as "welldefined" and "without loss of generality".
For use as a supplement in any of the basic upperdivision courses.
12pp.; last revised July 2008.
The
version you get by clicking above uses
"blackboard bold" symbols for integers, real numbers, etc..
You can get a similar version with regular
boldface instead (which also notes
the existence of the blackboard bold notation), and four
versions that are tailored for use with specific texts, and that
note points about those authors' notation: One
for Rudin's Principles ...,
one for
Beachy and Blair's Abstract Algebra,
one for
Friedberg, Insel and Spence's Linear Algebra,
and one for
Rotman's First course in Abstract Algebra.
The last two of these use ":" rather than "" for "such that" in
setbrackets.
My source file is set up so that if you want a version with a
particular combination of notations for integers/reals/etc.,
for subsets and supersets (namely, with or without a bar on bottom),
and for "such that", which does not refer specifically to one
of the abovementioned texts, I can fairly easily create one for you.
I also have a short file of
Answers to students' questions
on the above handout, accumulated over several semesters.

Proof that the group A_{n} is
simple for all n≥5.
Written Spring 2000 for Math 113. 3pp.

When is a finite abelian group cyclic?
(written to go with Fraleigh's First Course in Abstract Algebra,
for Math 113, Spring 2000; 2 pp.) and
The criterion for a finite abelian group to be cyclic
(rewritten to go with Dummit and Foote's Abstract Algebra
for Math H113, Spring 2009, 1p.).
To show that a finite subgroup of the multiplicative group of a field
is cyclic, one needs to know that an abelian group which, for
each n, has at most n elements
a satisfying a^{n}=e is cyclic.
Both the above texts prove this using
the structure theorem for finitely generated abelian groups,
a result I don't try to squeeze into Math (H)113; so I have
prepared these handouts proving the cyclicity criterion directly
(and with the hypothesis on solutions to a^{n}=e
weakened to apply only to prime values of n).
The writeups follow the texts' notations;
in particular, in the version for Fraleigh, the image of
a set X under a map f is
denoted f[X].

Sketch of my favorite proof of the First Sylow
Theorem.
This is the "orbit counting" proof, but with a twist: The
fact that a certain binomial coefficient is not divisible by
p, needed for the proof, is proved not by number theory,
but by running the proof backwards in the case of a particular
group of the same order which we know has a Sylow subgroup
(namely, a cyclic group).
The above version was written for Math 113, taught Spring 1997
from Fraleigh, revised in Spring 2009 when I prepared;
this version,
for Math H113, taught from Dummit and Foote. 1p.
Here is some nonmathematical material relevant to
mathematics students:

The Greek alphabet. 1p.

Mathematical symbols in email.
Because of a requirement which I generally make in my upper division
and graduate courses, that students must submit a question about
the reading on each day when there is a reading assignment,
I have a lot of email correspondence with the class.
In this handout I note that mathematicians today generally
express symbols in their email in ^{T}E^{X} (and give
an example), then recommend some conventions for use in my classes,
which borrow some features from ^{T}E^{X}, but don't
look so technical. 1p. Last revised Fall 2001 for Math 113.

On Incomplete grades. Rules and
procedures regarding the grade of "Incomplete" in Berkeley courses.
The above is in html; to get it to print on one page, you need
to fiddle with the "Print preview" option on your browser.
Here is a PostScript
version which does so without extra work.
Textspecific handouts.
These are mostly additional exercises that I have put on
homework sheets when teaching from the text in question.
Some of the exercises are comparable in difficulty to those in the text,
but fill a gap or give some interesting perspective on the material.
Others I have not assigned, but have noted on the homework sheets
that "students interested further interesting and/or in more
challenging problems" might like to look at them; these might
be appropriate for an honors course in the same material.
Here are the collections for lowerdivision courses:

Exercises supplementing those in J.Stewart's "Calculus,
Early Transcendentals",
and its Berkeley Custom version "Single Variable Calculus",
26pp., March 2015
Answers to students' questions in Math H1B,
Fall 2010, Fall 2012, and Fall 2014, taught from the above text.
In the above collection, I have done my best to adjust
the references to the 7th Edition of the text, although the
in 2010 and 2012 the
course was actually taught using earlier editions.

Exercises supplementing those in Richard O. Hill's
"Elementary Linear Algebra with Applications",
3rd Edition.
Ends with 2pp. of errata to the text. 11pp., June 2002

Exercises supplementing those in William Boyce and
Richard DiPrima's
"Elementary Differential Equations and Boundary Value Problems",
7th Edition. 7pp., June 2002.

Exercises supplementing those in Kenneth H. Rosen's
"Discrete Mathematics and Its Applications",
5th Edition. 26pp., Dec. 2005.
(These need to be updated to go with the 6th edition  at very least,
renumbered to match his new arrangement of the material.)
The next few items concern texts for upperdivision courses,
and along with supplementary exercises, they include files
labeled "Answers to students questions".
In these courses, I require every student
to submit a question on each day's reading.
I incorporate the answers to some of these into my lectures;
others I answer by email.
I maintain cumulative webpages of answers
sent by email on points that seemed important enough that other
students in the class might want to refer to them.
(See the handout Mathematical symbols in email above for
the conventions I follow regarding such symbols in these answers.)
Some of the answers in these files have been edited retrospectively
to improve clarity etc.
Unfortunately, answers to the questions asked by the greatest numbers of
students may not get into these files, since they are answered in
class.
Files of the same sort for my graduate courses (only one so far)
can be found at the end of my page of
graduate course materials

Exercises supplementing those in Friedberg, Insel and
Spence's "Linear
Algebra", 4th Edition. 16pp., January 2009.
 Exercises supplementing those in Beachy and Blair's
"Abstract Algebra", 3rd Edition. 10pp., May 2007.

Exercises supplementing those in Joseph J. Rotman's "A First
Course in Abstract
Algebra", 2nd Edition. 18 pp., March 2003.

Exercises supplementing those in Dummit and Foote's "Abstract
Algebra", 3rd Edition. 21 pp., June 2009.

Exercises supplementing those in Ian Stewart's "Galois
Theory", 3rd Edition. 18pp., June 2005.
(Almost all of these can also be used with the 2nd edition,
though the order of material is very different, so the exercises
would be associated with very different chapters.
I recommend teaching from the 2nd edition rather than
the 3rd if you can get copies.)
Corrections and Clarifications to Ian Stewart's "Galois
Theory", 2nd Edition. 3pp., 14 June, 2001.
Answers to students' questions
in Math 114, Spring 2001, taught
from Ian Stewart's Galois Theory, 2nd ed..
Corrections and Clarifications to Ian Stewart's "Galois
Theory", 3rd Edition. 12pp., June, 2005,
updated Jan.'11.
Answers to students' questions
in Math 114, Spring 2005
taught from Ian Stewart's Galois Theory, 3rd ed..

Supplements to the Exercises in Chapters 17 of Walter Rudin's
"Principles of Mathematical Analysis", Third Edition.
This is the most ambitious of these packets.
In addition to my own exercises, it contains information on each of
Rudin's exercises in those chapters, including an estimate
of its difficulty, notes on its dependence on other exercises if any,
and in some cases further comments or hints.
89pp., last major revision December 2006.
(The above link is to a PostScript file;
here's the same in pdf.)
Notes on Rudin's "Principles of Mathematical Analysis",
and same in pdf.
Two pages of notes to the instructor on points in the text
that I feel needed clarification, followed by 3½ pages of
errata and addenda to the current version, suitable for distribution
to one's class, and ending with half a page of
errata to pre1994 (approx.) printings.
Last major revision December, 2006.
Answers to students' questions
in Math 104 (Fall 2003 and Spring 2006)
and in Math H104 (Fall 2006) taught
from Rudin's Principles of Mathematical Analysis, 3rd ed..

Exercises supplementing those in Ian Stewart and David Tall's
"Complex Analysis". 18pp., June 2004.