Math 330   -    Number systems, Section 2   -    Fall 2018  

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Last update: Novenber 23, 2018 - 6:00 PM
NOTE that this document specifically pertains to section 2 of the course!
Visit this page frequently for important changes and additions!


Course Material for Section 2 of Math 330

Course Material:
Textbook (the "B/G text" or just "B/G") -- REQUIRED:
    The Art of Proof: Basic Training For Deeper Mathematics, by M. Beck and R. Geoghegan (Springer, 2010).
   
  1. The first two thirds of the course will follow to a large degree the textbook but some items will be presented in a different order.
Instructor's lecture notes (the "MF doc" or just "MF") -- REQUIRED:
    Math 330 - Additional Material by Michael Fochler.
   
  1. The last third of the course is almost exclusively based on ch.9-13 and, if time allows, part of ch.14 of those notes. The earlier chapters serve to give additional background material for the basics: sets, functions and logic. Only some of this will be actually taught in class and then often concurrently with material from the B/G text. Rather, you will be given reading assignments (as will also be the case for the later chapters).
  2. 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!
  3. 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.
  4. 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.
  5. Older editions of the document will eventually be deleted. You can find them posted in reverse chronological order in this table:

    2018-11-23 version Last version for Fall 2018
    2018-10-25 version Some corrections to ch.10 (Real Numbers). The addenda to this chapter have been greatly expanded with material from B/G ch.11 and 12.
    2018-10-20 version Many corrections to ch.10 (Real Numbers) Check also the addenda to ch.10 for additional material!
    2018-10-01 version Greatly expanded hints for exercise 6.8 (hwk 8, #1), so check it out immediately!
    2018-09-27 version New chapter 20: Sample problems for Exams.
    2018-09-24 version Emergency fix to prop.6.34 (Euclid’s Lemma for Two Factors)
    2018-09-20 version Substantial additions to ch.6: new chapter on binomial coefficients and addenda. Many errors were fixed.
    WITHDRAWN and replaced with ver. 2018-09-24
    2018-09-05 version       Substantial changes to ch.6: The Integers.
    2018-08-20 version       Mostly fixed some typos, but also added more material, in particular to ch.6 (The Integers).
    Another version will be forthcoming before the beginning of next week.
    2018-08-17 version       This version is much more stable but some newly added parts need proof reading.
    Do not print anything beyond ch.3.4: Order Relations in Integral Domains.
    2018-08-16 version WITHDRAWN
    2018-08-02 version       This is a preliminary version published for the Fall 2018 semester.
    An update will be published before the start of the semester, so hold off on printing!

    ----------
Additional course material: The B/K (Bryant/Kirby) course notes.
    The B/G text together with the MF doc provide sufficient some exposure to sets, functions and logic but they are lacking good examples. for this 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. We refer to these notes as the B/K notes The material was pointed out to me by Prof. Marcin Mazur. The following items all are part of these course notes.
   
  1. Chapters 2 and 3 of the B/K notes are very well written notes on the subject of logic and using its tools to write 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.3 on logic 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 invaluable in helping you to make it through the Math 330 course.
  2. 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.
  3. 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 skip the following:
    • Section 22.11. Set Identities: Everything starting with ``Proof 2'' until the end on p.105
    • 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.
  4. Functions part 1: This is ch.1, section 2 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.
  5. Functions part 2: This is ch.4, section 2 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.
Additional course material: Linear Algebra
    If you did not take a linear algebra class then you will have to educate yourself about a few basics. Here are two good references.
   
  1. The lecture notes from Paul Dawkins on linear algebra, available at https://www.cs.cornell.edu/courses/cs485/2006sp/LinAlg_Complete.pdf , have the advantage that they cost nothing. You should look at the following:
    • Vector Spaces, p.182, def. 1,
    • Subspaces p.193: def.1, thm 1,
    • Span, p.202: def 1, def 2, thm 1,
    • Linear independence, p.210: def.1,
    • Basis and dimension, p.220: def.1, thm 2, def 2, thm 3.
  2. The lecture notes for Math 304 (Linear Algebra by Brin/Marchesi might be reusable when you get around to take that course but there is some danger that a new edition will have been published by then. See the chapter "Vectors and Vector spaces" of the MF doc for references to the topics listed above.