Math 330   -    Number systems, Section 2   -    Spring 2017  

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Last update: April 30, 2017 - 11:10 AM
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"):
    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"):
    Math 330 - Additional Material by Michael Fochler.
   
  1. The last third of the course is almost exclusively based on ch.8-12 and, if time allows, part of ch.13 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. 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 is in progress 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:

    2017-06-08 version       Intermediate version for the new semester. b. Many changes from the previous version!.
    2017-04-29 version       a. Fixed many errors, mostly in ch.10 and 11, b. Check the addenda to ch.8, 9, 10, 11 for additions and modifications.
    2017-04-17 version       a. Fixed many errors, mostly in ch.10, b. Check the addenda to ch.8, 9, 10 for additions and modifications.
    2017-04-13 version       a. Addenda to ch.8, b. additions to ch.9.2.2, c. Addenda to ch.10
    2017-04-02 version       Additions to addenda and exercises for ch.6 and ch.7.
    2017-03-26 version       More minor mods to ch.8.1.
    2017-03-24 version       Small mods to ch.8.
    2017-03-18 version       Added ch.8.4: Addenda to Ch.8 (Real functions): Contains examples of function sequences.
    2017-03-08 version       Ch.16.7.1 (Base-Ten Representation of Integers) now contains a proof of B/G prop.7.9 (uniqueness of base-ten representation ...).
    2017-03-07 version       Further additions to ch.16 (Addenda to B/G). They are local to ch.16.6.1, ch.16.6.2 and ch.17.
    2017-02-28 version       Additions to ch.16 (Addenda to B/G): Subchapters were added for B/G ch.6 and ch.7. More additions to ch.16 will be forthcoming.
    A new chapter 7.1.1 (Cardinality as a Partial Ordering) was added.
    2017-02-26 version       Additions to the end of ch.6: computational rules for indicator functions.
    2017-02-13 version       Stopgap version: Fixed up the inclusion lemma (ch.5) and its proof. A new version will be forthcoming soon.
    2017-01-30 version       More additions to ch.16. Some corrections.
    2017-01-26 version       Fixed some errors. Now terminating definitions, remarks, examples, ... with a hollow square.
    Updated ch.16 (Appendix: Addenda to Beck/Geoghegan’s "The Art of Proof"): Renamed ch.16.1 to ch.16.2 and added new chapters 16.1 and 16.4.
    2017-01-16 version       This is the first version published for the Spring 2017 semester.
    An update will be published within 10 days, so hold off on printing!

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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 2.11. 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.

  6. Modular Arithmetic by Miguel A. Lerma. This document is not part of the B/K course notes. It 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.