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Course Material for Section 5 of Math 330
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Math 330 - Lecture Notes, by Michael Fochler.
(the "MF doc" or just "MF") -- REQUIRED:
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Most of ch.2-3, 5-13, and, if time allows, the beginnings of ch.14
will be taught, some of it concurrently with the corresponding material
from the B/G text.
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Note that your instructor is the author of this document.
For that reason it is much more likely to contain errors than the ones
you buy at the store or view on the internet as those
have probably been vetted by many viewers
before having been made accessible.
Caveat emptor!
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There are reading instructions just after the table of contents.
Be sure to look at them first as they tell you what parts
of the material are optional, which ones you should understand,
and which ones you must study intensively.
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This document is work in progress and will be modified as
the course unfolds but, once reading is assigned from
this document,
I will make an effort not to alter the numbering of the definitions,
theorems, ... by doing the following:
New material (as opposed to error corrections) which might influence
the numbering of those earlier chapters) will be placed
into an appendix of the main chapter to which it belongs.
Doing this will not change the numbering of the material
outside those appendices.
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Older editions of the document will eventually be deleted.
You can find them posted in reverse chronological order in this table:
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The Art of Proof: Basic Training For Deeper Mathematics,
by M. Beck and R. Geoghegan (Springer, 2010).
(the "B/G text" or just "B/G"):
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The MF doc refers to the B/G text for several proofs and also cross--references
propositions and theorems whenever possible.
Note though that the B/G material is in many cases not presented in the generality
to be found in the MF doc.
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Additional course material: The B/K (Bryant/Kirby) course notes.
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The B/G text and the MF doc provide material
about sets, functions and logic but they do not have sufficiently many
good examples. I have found course notes from
Florida State University, written by John Bryant and Penelope Kirby.
The link to both the entire PDF and separate chunks is
http://www.math.fsu.edu/~pkirby/mad2104/CourseNotes.htm
which help to close this gap.
These notes will be referred to as the B/K notes.
The material was pointed out to me by Prof. Marcin Mazur.
The following comments pertain to these course notes.
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Chapters 2 and 3 are very well written notes
on the subject of logic and its use for writing formal proofs.
Reading some of this material, in particular looking at its many
examples, will help you to understand ch.3 in
the B/G text on logic better.
MF ch.4 on logic (almost none of which will not be discussed in lecture)
was written with the same goal in mind
but it also is lacking enough examples.
I give no homework assignments on logic as this is not done
in the B/G text either (there are only projects),
but understanding the basics of logical
reasoning and its terminology is might help you to
understand the proofs given in the other course material and
creating your own.
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Sets part 1:
This is ch.1, section 1 of B/K (Introduction to Sets),
a very basic introduction to sets which
many of you should be able to skim through in a hurry,
but you should skip nothing and be sure you understand
all examples.
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Sets part 2:
This is ch.4, section 1 of B/K (Set Operations).
Note that this article needs a higher level of sophistication
but you should have enough of an intuitive knowledge of sets
to understand the material rather quickly.
Be sure you learn the notation. Some of it deviates
from the notation used in B/G and/or in MF.
You can safely skip the following:
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Section 52.11. Set Identities:
Everything starting with ``Proof 2'' until the end on p.105
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All of section 1.15. Computer Representation of a Set.
Recommendation:
If you are a computer scientist I recommend you take
a look at this stuff simply because it will probably
interest you.
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Functions part 1:
This is ch.1, Section 5 of B/K ( Introduction to Functions).
It is a very brief document but you will need more time
per page to understand its contents. You can skip
chapter 2.4. Floor and Ceiling Functions.
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Functions part 2:
This is ch.4, Section 5 of B/K (Properties of Functions).
It focuses on injective, surjective and bijective
(invertible) functions.
Pick up your copy of Stewart's Calculus and review the chapter on inverse functions.
You will see material on injective functions (Stewart calls them one-to-one)
and on inverse functions. This will help you understand the document.
Skip all proofs as the important ones are given in B/G. but be sure to understand
the definitions and examples and draw pictures with functions that you understand
well to get a feeling for why the theorems are true.
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Additional course material: Linear Algebra
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If you did not take a linear algebra class then you will have
to educate yourself about a few basics.
Here is a good reference.
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The
lecture notes from Paul Dawkins on linear algebra,
available at
https://www.cs.cornell.edu/courses/cs485/2006sp/LinAlg_Complete.pdf
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have the advantage that they cost nothing.
You should look at the following:
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Vector Spaces, p.182, def. 1,
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Subspaces p.193: def.1, thm 1,
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Span, p.202: def 1, def 2, thm 1,
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Linear independence, p.210: def.1,
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Basis and dimension, p.220: def.1, thm 2, def 2, thm 3.
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Here is the link to a recorded
Linear Algebra tutorial
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I gave for my Fall 2020 Math 330 students.
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Additional course material: Latex Documentation
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Here are some links to useful Latex references in case you would
like to learn how to typeset mathematical work (NOT REQUIRED IN THIS COURSE).
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The Not So Short Introduction to LaTeX,
available at
http://cs.brown.edu/about/system/managed/latex/doc/lshort.pdf
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It contains more about LaTeX than you possibly need to know.
Notes on getting started using LaTeX
by David Biddle, Binghamton University.
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