Math 330   -    Number systems, Section 6   -    Spring 2024  

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Last update: May 10, 2024 - 3:30 PM
NOTE that this document specifically pertains to Section 6 of the course!
Visit this page frequently for important changes and additions!


Course Material for Section 6 of Math 330

   
Math 330 - Lecture Notes, by Michael Fochler. (the "MF doc" or just "MF") -- REQUIRED:
   
  1. 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.
  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:

    2024-05-08. version       This is the last version for the Spring 2024 semester.
    2024-04-16. version Colored boxes are now done in all of ch.13-15.
    2024-04-14. version Colored boxes are now done in all of ch.12.
    2024-03-30. version More colored boxes in ch.11 and 12. Addendum to ch.10.
    2024-03-18. version More colored boxes in ch.9 and 10.
    2024-01-28. version I changed the def of group homomorphisms! New theorem follows it.
    2024-01-17. version Very minor changes and corrections. Exception: Chapter 6 has been reorganized.
    2023-09-23 version This is the first version published for the Spring 2024 semester.

   
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"):
   
  1. 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.
   
Additional course material: The B/K (Bryant/Kirby) course notes.
    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.
   
  1. 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.
  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 safely skip the following:
    • Section 62.11. 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 6 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 6 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 is a good reference.
   
  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.
    Here is the link to a recorded Linear Algebra tutorial , I gave for my Fall 2020 Math 330 students.
    Latex Documentation: 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).
   
  1. (a): The Not So Short Introduction to LaTeX, available at http://cs.brown.edu/about/system/managed/latex/doc/lshort.pdf , It contains more about LaTeX than you possibly need to know.
  2. (b): Notes on getting started using LaTeX by David Biddle, Binghamton University.
   
Miscellaneous additional course material: This list might be expanded in the future.
   
  1. Here is a template for written homework assignments , . It is particularly useful for HW1.