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Course Material for Section 3 of Math 330
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Course Material:
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Textbook:
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The Art of Proof: Basic Training For Deeper Mathematics,
by M. Beck and R. Geoghegan (Springer, 2010).
The course will follow to a large degree the textbook
but some items will be presented in a different order.
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Additional course material:
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The Beck/Geoghean text provides some exposure to sets and
functions but I shall go beyond that. Here are some documents
for to give you material in addition to that presented in the
textbook. I have taken them from a 330 course that was taught
earlier by Prof. Marcin Mazur.
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Differences in Notation
See this document for some remarks, including potential difference in notation
between the course, the B/G text and the articles listed above.
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Logic part 1:
(Added to this list on Sept. 11, 2015).
Reading this document might help you to understand ch.3 in
the B/G textbook on logic better. I give no homework assignments
on logic as this is not done in the text either
(there are only projects) But understanding the basics of logical
reasoning and its terminology is invaluable in helping you to
make it through the Math 330 course.
Here are some guidelines to help you focus on the essential:
a. Stop reading once you reach section 1.12.:
"NAND and NOR Operators".
b. Note important differences in terminology:
We: "statement", they: "proposition".
We: "double-implication" or "equivalence", they: "biconditional".
c. Important topic not discussed in B/G:
the truth tables that come with the logical operations.
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Sets part 1:
This document is 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 document covers to a large degree the same material as
Sets part 1.
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 will deviate from the notation
used in the text and/or in lecture.
You can skip the following:
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Ch. 1.11. Set Identities:
Everything starting with ``Proof 2'' until the end on p.105
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Ch. 1.13. Infinite Unions and Intersections:
on p. 107:
the use of a ``general index set'' J in the union of
sets Ai where i is an element of J
and exercises 1.14.1., 1.14.2.
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All of the remainder of the document
starting with the ``Discussion'' section on p.107.
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Functions part 1:
This document is very brief and it will probably take you more time
per page to study it. You can skip
chapter 2.4. Floor and Ceiling Functions. Note that I have an important
comment on the use of f - 1(y) for both the inverse image (a set!)
of y and the image of y under the inverse function f - 1 of f
here
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Functions part 2:
This document 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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Lecture Notes:
Math 330 - Additional Material
I, Michael Fochler, am the author of this document.
For that reason it is much more likely to contain errors than the ones
listed above as those have probably been vetted by many viewers
before having been made accessible on the internet.
Caveat emptor!
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 (most of it) and which ones you
must study.
This document is work in progress and will be modified as
the course is in progress.
I shall make an effort not to alter the numbering of the definitions,
theorems, ... by putting new material (as opposed to error corrections)
into appendices for each main chapter. I shall also keep older
editions of the document posted in this table:
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Modular Arithmetic
by Miguel A. Lerma.
This document contains background material on arithmetic
modulo n. I do not plan to teach from it or use anything
in there not covered in the book for quizzes and/or exams.
This material is strictly for your convenience as it might
help you to better understand the material from
B/G ch.6.3 and 6.4.
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