MAA 3200 Schedule, Fall 2018 |
Last modified on
Overview
We'll begin with some basic logic and practice some simple proofs, focusing on finding a good plan and adopting a good clear style. I urge you to read the textbook by Velleman, How to Prove It, which is ideal for the early stages, especially if you are a beginner. Use your judgement about how much to read, but include most of Chs 3 to 6, at least, perhaps omitting some later sections of Chs 5 and 6. Focus more on building skills than memorizing facts about odd numbers, etc. I hope to finish the Velleman material in about 6 weeks.
After that, our main text will be the one by Kane, which emphasizes "proof templates" for beginners, and provides examples mostly from Analysis (roughly speaking, from Calculus). We will shift to that asap and cover most of Chs 1 to 4, and some of 9. I listed some other optional texts on your syllabus, but do not expect you to read them unless you need more explanation or examples.
After an intro to proofs, we will survey many mainstream advanced mathematics topics, including foundations (these topics may overlap with Discrete Math), topology and analysis. As time permits, we will study cardinality and the construction of our main number systems, including the real numbers, which may require some online reading or handouts. The optional book by Morash is good for this topic.
On this web page, I've grouped the class meetings into seven blocks of two weeks each, with a brief description of the reading, lectures and assigned work for each block. I will complete the latter blocks as we go along. and may have to adjust some dates and content from time to time. I will do so as early as I can, and will announce any major changes in class, but recommend rechecking this page regularly. The HW lists below are intended to be the minimal required to get by. You should do at least 50 per cent more, or until you have mastered the material.
As you can see from the first table below, I expect to cover Ch 1 of our textbook in the first 3 lectures. Note: Homework 1 is due on 9/6, and covers Ch 1 thru 4.3. Exam 1 is is 9/27. I will usually post study advice, here on this page, about a week ahead of each exam, so check back often. Also, see my exam page, with past exams and answer keys to your tests. Please contact me or our TA if you don't understand anything, or if you see a mistake.
Our TA is Giancarlo Sanchez, gsanc018@gmail.com, TR 3 to 5pm in DM 413A, which is Prof Yotov's office. If he is not there, try the common room, DM 409A. You can also use gsanc018@fiu.edu. Meet with Giancarlo as often as you can. I give a little extra credit for regular active attendance.
Weeks 1-2
You should be reading as much of the Velleman text as you can, to get the main ideas, especially the idea of a proof strategy (Ch 3). You can skim over Chs 1-2 if you have seen that before. In general it is unlikely I will test you on specific examples from this book, but you should pick up any vocabulary and theorems (there aren't really very many) mentioned in my lectures or related to the assigned homework.
The exercises for Weeks 1-4 are from Velleman. I intend to use page numbers with Velleman and section numbers with Kane later on to help minimize confusion. If you have time, you can start reading Kane's text thru Sec 2.4 [pages 1-30] or so, which overlaps Velleman a bit, but with a different approach to proofs. Both are useful.
Just to check you understand, here's an example. The fourth HW exercise below is on page 64 of Velleman, problem 7, which begins "Are these . . . ". The asterisk (*7) indicates there is at least a partial answer in the back of the book. You should include some work or reasoning even though the exercise does not say so. I am suggesting that you do this on 8/23, after that lecture, but you do not have to hand it in until 9/6.
| Day | Date | I give you | You give me | Lecture topics | HW |
| 1 | 8/21 | Website | Logic | p24 - 4, 9 p33 - 5 |
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| 2 | 8/23 | [more Logic] | p64 - 7 p72 - 1, 3 |
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| 3 | 8/28 | Sets | p81 - 2, 4, 11, 15 | ||
| 4 | 8/30 | Ch 3 Proofs | p93 - 1, 3, 4, 8 |
Weeks 3-4
Double-check that you are registered for this course.
I often give a little extra credit for catching website errors, or for making some similar contribution to our world. Just email me, or stop by DM 419B
On 8/27/18, I revised the exercise lists below. I inserted an extra lecture on Ch 3 on 9/4, with some exercises to go with it. Every list after that got pushed ahead a day. Notice that HW 1 now ends at page 153 (not page 186 as stated on an earlier version of this web page). I plan to stick with the Velleman book a bit longer, mainly to avoid confusion, but would suggest also reading the Kane text through Ch 2 by the 5th week. Those 40 pages in Kane roughly match the 250 pages in Velleman, but they omit Relations.
| Day | Date | I give you | You give me | Lecture topics | HW |
| 5 | 9/4 | Ch 3 Proofs | p106 - 3, 5, 12 p122 - 4, 5, 19 p144 - 8, 13 p153 - 1, 6 |
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| 6 | 9/6 | HW 1 [thru p153] | Relations | p170 - 6, 12 p178 - 2, 9 p186 - 12, 17 |
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| 7 | 9/11 | Relations |
p199 - 1, 4 p222 - 2, 3, 4 |
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| 8 | 9/13 | Functions | p233 - 1, 4, 9 |
Weeks 5-6
Weeks 5 to 8 - Induction, Cardinality, N, Z, Q and some R, and ordered fields, perhaps some basic Algebra. We will use Velleman's book through cardinality. The optional book by Morash (see the syllabus) is best for the number systems. But I will try to cover the main points in class and/or provide sufficient notes online. Then we will switch to Kane's book, learning limit proofs and some basic Analysis.
Exam I will include everything up to Induction, but not Cardinality. It may include examples similar to HW 1 and 2, and to the various proofs in Ch 3. Starting in Ch 4, the actual topics start becoming more important. So you should know the definition of total order, equivalence realtion, 1-1, bijection, etc. Prepare to answer standard kinds of questions on these topics (in addition to writing basic proofs). You can find some old MAA 3200 exams on my exam page. Also, see my Help Pages, if interested, but these may be outdated - different textbooks, etc.
On 9/21, I added a few exercises on induction. If you have trouble with pg.296-4, it is a standard type covered in most Discrete Math books. One student has suggested a link to Prof Gorman's notes on Discrete Math [scroll down to Discrete Math / Course Pack].
Remember that Giancarlo has changed his TA office hours. They are Wed 9AM to Noon + Thurs 3PM to 5PM (no Tuesday hours).
| Day | Date | I give you | You give me | Lecture topics | HW |
| 9 | 9/18 | Functions | p243 - 2, 5, 8 | ||
| 10 | 9/20 | HW 2, thru Functions | Induction | p265 - 2, 9, 14 p277 - 12 p286 - 8,14 p296 - 4, 9 |
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| 11 | 9/25 | Cardinality | p312 - 3, 4, 9, 11 p320 - 1, 3, 4 |
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| 12 | 9/27 | Exam 1, topics thru 9/20 | NA | NA |
Weeks 7-8
Approx 10/2, read this about constructing the number systems N, Z and Q. Then do these problems for HW3 [these two web pages were updated on the morning of 10/2. If you see the old 2013 versions, refresh your browser.]. The construction of the reals, R, is similar but harder. We will come back to that as time permits after learning about Cauchy sequences.
The exercises below, starting 10/9 are in Kane's book. I suggest reading Chs 1-2 of Kane, mainly for his ideas on proof-writing, and some review of topics we have already covered. Read more carefully the properties of R in Ch.2.5 (and then Ch3, etc).
If you are interested in becoming a McNair Fellow, click here, and / or see me. In my opinion, this is a good opportunity for almost any math /sci /eng major.
| Day | Date | I give you | You give me | Lecture topics | HW |
| 13 | 10/2 | N, Z, Q | see above | ||
| 14 | 10/4 | N, Z, Q | |||
| 15 | 10/9 | HW 3 thru NZQ | R, Algebra | Kane pg 40 - 1, 4, 6 | |
| 16 | 10/11 | Limits | p49 - 1, 2, 3, 4 |
Weeks 9-10
Weeks 9 to 12 topics will include more proofs about functions, sequences, limits and continuity. Also some famous theorems of Calculus such as the EVT and IVT, with proofs. And basic topology, compact sets, the BW and HB theorems. In general, write your HW proofs in the same manner as Kane (emphasizing the definition of limit rather than shortcuts).
Exam II is expected to cover Cardinality through Limits (ending on page 67 or 68 of Kane). This includes Kane Ch.2.5 and the online pages on number systems. I am also posting a review page for the exam. Upon request, I am also posting some answers to the nzq homework (problems 4,5,6).
Note that Giancarlo has changed his office hours to W and F, 10am to noon.
Also, note that HW4 should have included p59, though there seems to be some problem with the previous posting or the web migration. We will sort this out asap, but in any case prepare for these exrecises before Exam II.
If you are interested in a summer research program with the DHS (with a stipend), I can forward an email to you.
| Day | Date | I give you | You give me | Lecture topics | HW |
| 17 | 10/16 | Limits |
p54 - 1, 4, 8 p56 - 5 p59 - 1, 5, 6 |
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| 18 | 10/18 | HW 4, thru p59 | Limits | p67 - 1defg, 2a, 3, 6 p74 - 5, 8 |
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| 19 | 10/23 | Limits | p79 - 1, 2, 3, 4 p81 - 1, 4 |
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| 20 | 10/25 | Exam II, thru p67 |
I think the DR date is Monday 10/29. If you are in the C to D range and need your Exam 2 score by Monday, please email me by appprox 2pm, including ID.
Weeks 11-12
| Day | Date | I give you | You give me | Lecture topics | HW |
| 21 | 10/30 | Limits |
p89 - 4, 8 p93 - 1 |
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| 22 | 11/1 | Continuity |
p97 - 1, 2abc, 3 p105 - 1 |
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| 23 | 11/6 | HW 5 thru pg 105 | Continuity |
p109 - 3,4,7 p117 - 1,2,4,7 |
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| 24 | 11/8 | Continuity | p120 - 1,3,5 p123 - 1,3 |
Any exercises beyond HW 6 may be considered practice problems for the final exam. I will not collect them, but will assume you have done them. Questions on them are welcome, of course. I will probably not accept late work after 11/27. Any doctor's notes, regrading requests, etc, are due then. Note that 11/22 is a holiday.
Practice problems for 11/20 were given in class and not also appear below. We'll finish by outlining a construction of the real numbers, including a proof of the Completeness Axiom. Read these online notes and try the exercises in them. If possible, let me know by 11/27 of any typos, or questions. The book by Morash contains a full treatment of the subject, and is still on reserve in the library.
I plan to post more notes about the final exam soon, including an updated list of textbook proofs. A good preparation would be to finish the exercises beyond HW 6 and the related reading. You can use old finals on my exam page for practice, though as usual the topics covered in class will vary a bit from year to year.
Weeks 13-14
| Day | Date | I give you | You give me | Lecture topics | HW |
| 25 | 11/13 |
Continuity, Topology |
p130 - 3,4,5 p274 - 1,2,3,6,8 |
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| 26 | 11/15 | Topology | p278 - 3, 4, 5, 8 p282 - 1, 4 p285 - 1, 5 |
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| 27 | 11/20 | HW 6 thru p278 | Metrics | p305 - 2, 7 p308 - 5 p311 - 2 p314 - 1 |
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| 28 | 11/27 | All is due | The reals | See Real notes | |
| 29 | 11/29 | The reals |
The final will be Dec 4, 12pm to 2pm in GC283A. It covers everything, but at least half will cover topics after Exam II. Here is a review page for the final. It includes a proof list (mainly the pre-image thm, IVT and Cauchy converges thm).