Answers For Exercises In Elements Of Advanced Mathematics

Course page: ON CAMPUS: http://www.math.fsu.edu/~mesterto/MGF3301.html (this page)
OFF CAMPUS: http://www.math.fsu.edu.proxy.lib.fsu.edu/~mesterto/MGF3301.html (with your FSUID username and password) Professor: Dr Mesterton-Gibbons, or Dr M-G for short Office: 202B Love Office hours: Please click here. Office hours are subject to change during the semester at 24 hours notice, but current times are always posted online. Note that office hours are primarily for personal matters that cannot be addressed in class (as opposed to tutorial help, for which see under How to study below) Phone: (850 64) 42580 Main website: Professor M-G's Home PageEmail: Goal : To introduce the methods of mathematics through a variety of classical and modern topics. Axioms and proofs will be emphasized throughout Class meets: In 106 LOV on Mondays, Wednesdays and Fridays at 10:10—11:00 a.m. Text: Krantz,Elements of Advanced Mathematics, 3rd edition (CRC Press, 2012; ISBN 978-1-4398-9834-5 Hardback, 978-1-4398-9843-7 eBook). We will cover most (but not all) of Chapters 1-4 and 6, and a selection of material from other chapters and elsewhere Credit: 3 semester hours Eligibility: You must have the prerequisites listed below, and you must not have credit for MAD 2104 Discrete Mathematics I, MAA 4224 Introduction to Analysis, MAS 4302 Introduction to Abstract Algebra, MAA 4226 Advanced Calculus I or their equivalents. Prerequisites:

(i)	MAC 2312 Calculus IIand
(ii)	self-motivation and industriousness and
(iii)	the patience to believe in yourself—you may not get it right away, but given (ii), if you are patient, then understanding will come
My philosophy of learning is perhaps best expressed by the green curve in the diagram below:

ALTERNATIVE THEORIES OF LEARNING VERSUS PAIN

For further details, please click here. Communication: I will send email to your FSU email account on a regular basis. It is your responsibility to check it regularly (or arrange to have my messages forwarded, if you prefer to read your email elsewhere) Your name:

I don't know who you are (for the most part), but because everything works so much better when I do, I would like to learn your name as soon as possible. So, please take a sheet of card stock (or even paper), fold it in half, write your name in large letters on one side and stand it up on your desk so that I can see it. (Write what you want me to call you: if you're a William who likes to be called Dubya or a Margaret Jane who likes to be called Dee Dee, write Dubya or Dee Dee, not William or Margaret Jane.) Please bring your nameplate to every class until I have finally learnt your name (which will take significantly longer than it used to take when I started out)

Course format: The course will be based on my lectures, your reading of the text and other materials (supplied by me) and much interactive problem solving—on which we plan to spend most of our time in class. After each period I will set homework for the following period, either at the end of class or soon afterwards by email. This homework will consist of one or more of the following:

Exercises on the current topic or topics
An assigned reading, typically on the next topic(s), hence typically the next section or two of the text
A problem or problems associated with the assigned reading

At the start of each class, I will always assume that you have both read (though not necessarily fully understood) any assigned reading and at least seriously attempted (though not necessarily completed) any associated problem(s). I will also assume that you are going to complete Item 1 in a timely fashion, though not necessarily in full by the following period.
Typically, we will begin class by reviewing up to three different solution attempts that you have written on the board. A lecture will often follow. Early in the semester, that lecture will tend to be very brief. Its purpose will be to highlight the key points of an assigned reading, largely to jog your memory of reading it earlier, perhaps also partly to add some fresh perspective—but in any event based on the assumption that you have actually read it. As the semester progresses, however, lectures are likely to lengthen, because I will introduce material not covered by the text. Either way, I will work from notes that get posted here as soon after class as I can manage.
We will then proceed to work on problems, beginning with Item 3 if there exists an Item 3. Note the use of we—interactive problem solving implies working together, and in this class your active participation is vital. To encourage you in that regard, 15% of your final grade will be based on the extent to which you have participated in class.
In particular, on any given class day, up to three of you can earn 3% of your grade by writing a sufficient attempt at a homework problem on the board immediately before class. You must clear your choice of problem beforehand with me. I will then notify the class that the problem is taken, and up to two more of you can pick a different problem. The rest of you will have to wait for a later class. There are 37 non-test periods after the first, yielding capacity for 111 sufficient attempts; the enrollment is 22 and 5 x 22 = 110 < 111, so all of you in principle can earn all your participation points this way. But you can also earn them in class in other ways—by asking very good questions, by exemplary contributions to interactive problem solving, etc., etc. (where "etc., etc." just means whatever other ways I might have overlooked).
Questions may be asked at any time—and should be, if there's anything you don't understand. (Perhaps you have a question about the assigned reading that you anticipate being answered by one or more of the problems we work together; in which case, it may well be socially minded not to ask your question at the outset. However, if it turns out that you anticipated incorrectly, then be sure to ask your question before the class is over!) Test format: You must write your answers in blue or black ink. If you make a mistake, then just cross it out and make a correction, which is far more efficient temporally than erasing pencilling—moreover, it can even earn you otherwise unavailable partial credit (if it turns out that you were right before you changed your mind). Begin each question (but not subsequent parts of the same question) on a fresh sheet of paper, use one side of the paper only, and have your solutions stapled together in order at the end of the examination (without the question sheet, which is yours to keep). Please do not use dog ears (I will bring a stapler to the classroom). Needless to say, your name must appear legibly on Page 1 of your solutions (as opposed to on the question sheet, which will not help, because you are keeping it) Grades: Will be based on participation in class (15%), four tests (15% apiece) and a cumulative final exam (25%), for all of which you must use blue or black ink.
Precise cut-off points for A, B and C will be determined by the distribution of grades at the end of the semester, but are likely be in the vicinity of 90%, 80% and 70%, respectively. In borderline cases, a smaller number of perfect answers will carry more weight than a proportionate number of imperfect answers (e.g., a ten and a zero will trump two fives); later scores will carry more weight than earlier scores; and—above all—an exemplary record of class participation will carry far more weight than a mediocre one. Plus or minus grades may be assigned in a manner consistent with standard University practice.
A score for a question worth 10 points should be interpreted as follows:

10	Practically perfect
9	Still very good, but lacking—or wrong about—a significant detail
8	Still good, but lacking—or wrong about—significant details
7	Minimally satisfactory. You have—just—managed to demonstrate that you basically understand and are at least capable of getting all details correct (although it clearly did not happen this time)
6	A grade that will not be given
5	Half right in some appropriate sense (e.g., there were two parts, each worth 5 points, and your first part was practically perfect)
1-4	Not even half right and showing little understanding, but some degree of positive effort
0	Zero effort, or submitted in pencil

Also note that a grade of I will not be given to avoid a grade of F or to give additional study time. Failure to process a course drop will result in a course grade of F Solutions to homework problems: Will eventually be posted here. I caution you, however: never read a posted solution until first of all you have seriously attempted the problem yourself. If you have at least made a serious attempt, then even if you were unable to complete the problem yourself, you will benefit from reading the posted solution; if not, then not—rather, you will merely form a false impression of how well you understand ... as indicated by the above learning-versus-pain diagram. To help you in this regard, solutions will be posted only after a significant delay Test solutions: Will be posted here Attendance policy: You are expected to attend class regularly, and bear the full responsibility for learning anything covered during any class that you miss. On the other hand, it would be extremely anti-social to attend class if you either have, or are coming down with, a contagious disease. So please keep me apprised (by email) of any illness or other emergency, so that I can make any necessary adjustments (and please make friends within the class as soon as possible if you haven't done so already, so that there is someone you can call upon to borrow notes if the need should arise) Exam policy: No makeup exams. An absence may be excused given sufficient evidence of extenuating circumstances (in which case, extra weight will be attached to the other exams or—if necessary, and only if necessary—an oral exam may be substituted). But you must either have discussed the matter with me (well) in advance; or, e.g., in the case of illness, have brought me a note from a physician explicitly stating that you were too ill to attend class on the day in question. An unexcused absence will result in a grade of zero Etiquette: You are firmly bound by Florida State University's Academic Honor Code. Briefly, you have the responsibility to uphold the highest standards of academic integrity in your own work, to refuse to tolerate violations of academic integrity in the University community, and to foster a high sense of integrity and social responsibility on the part of the University community. Even more briefly, you must neither cheat nor enable others to cheat. The penalties for violations can be severe. Please carefully read the section in the FSU Student Handbook on the Honor Code and official procedures for dealing with students who violate it. If you are in any doubt at all as to what constitutes acceptable behavior in this regard, you should ask me for clarification.
You are also bound by the ordinary rules and customs of polite behavior that prevail in a civilized society. I assume that you know these rules and customs, and I expect you to comply with them. (In particular, you are not allowed to use a cell phone or otherwise have private conversations with others during class; and you are not allowed to create disturbances through arriving late or leaving early.) Probable test dates: Monday, February 01
Monday, February 22
Monday, March 28
Monday, April 11 Final: Thursday, April 28, 10:00 a.m.—12:00 noon in 106 LOV How to study: It is important that you keep up with the course from the very beginning, always attempting as many as possible (preferably all) of the homework exercises. If you get stuck, then send me a question by email. As soon as I possibly can, which might be as soon as within half an hour, but might also be a day or two later (I have a life, too, you know), I will reply—not to you, but rather to the class alias (after carefully concealing your identity, just in case you are inexplicably bashful about being perceived as smart enough to ask a question). My reply will aim at nudging you over the immediate hurdle so that you can then take the next step by yourself (as opposed to supplying a complete solution, again for reasons encapsulated by the above learning-versus-pain diagram).
Note, however, the following. First, you must identify yourself (i.e., you remain anonymous to the other students in the class, but not to me) in the body of your message (because typically your username does not identify you to me): I don't reply to anonymous email. Second, you should be as specific as possible in describing your difficulty: the more precisely you identify how you got stuck, the more helpful my reply is likely to be. Disabilities: If you have a disability requiring academic accommodations, then not only should you register with the Student Disability Resource Center (SDRC), but also you should bring me written confirmation from SDRC during the first week of class. This and other class materials are available in alternative format upon request.

University Attendance Policy:
Excused absences include documented illness, deaths in the family and other documented crises, call to active military duty or jury duty, religious holy days, and official University activities. These absences will be accommodated in a way that does not arbitrarily penalize students who have a valid excuse. Consideration will also be given to students whose dependent children experience serious illness.

Academic Honor Policy:
The Florida State University Academic Honor Policy outlines the University's expectations for the integrity of students' academic work, the procedures for resolving alleged violations of those expectations, and the rights and responsibilities of students and faculty members throughout the process. Students are responsible for reading the Academic Honor Policy and for living up to their pledge to ". . . be honest and truthful and . . . [to] strive for personal and institutional integrity at Florida State University.'' (Florida State University Academic Honor Policy, found at http://dof.fsu.edu/honorpolicy.htm.)

Americans With Disabilities Act:
Students with disabilities needing academic accommodation should:
(1) register with and provide documentation to the Student Disability Resource Center; and
(2) bring a letter to the instructor indicating the need for accommodation and what type. This should be done during the first week of class.

This syllabus and other class materials are available in alternative format upon request.

For more information about services available to FSU students with disabilities, contact the:

Student Disability Resource Center
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108 Student Services Building
Florida State University
Tallahassee, FL 32306-4167
(850) 644-9566 (voice)
(850) 644-8504 (TDD)
sdrc@admin.fsu.edu
http://www.disabilitycenter.fsu.edu/

On-campus tutoring and writing assistance is available for many courses at Florida State University. For more information, visit the Academic Center for Excellence (ACE) Tutoring Services' comprehensive list of on-campus tutoring options—see http://ace.fsu.edu/tutoring or contact tutor@fsu.edu. High-quality tutoring is available by appointment and on a walk-in basis. These services are offered by tutors trained to encourage the highest level of individual academic success while upholding personal academic integrity.

"Except for changes that substantially affect implementation of the evaluation (grading) statement, this syllabus is a guide for the course and is subject to change with advance notice.''

Lecture Notes and Problems

Basic logic: the propositional calculus (§§1.1-1.6, Wednesday, January 06)
Basic logic: the predicate calculus (§1.7, Friday, January 08)
Some methods of proof (§§2.2-2.3, 2.5.1-2.5.2, Monday, January 11)
Proof by induction (§2.4, Wednesday, January 13)
Set theory (§§3.1-3.5, Wednesday, January 20 and Friday, January 22)
On counterexamples (Friday, January 22)
Relations, especially equivalence relations (§4.1, Monday, January 25)
Order relations (§4.2, Wednesday, February 3)
Functions and compositions of functions (§§4.3-4.4, Friday, February 5 and Monday, February 8)
More on relations, especially inverse functions (§4.4, Wednesday, February 10)
Set cardinality (§4.5, Friday, February 12)
More on set cardinality (§4.5, Monday, February 15)
Well ordering and further set cardinality (§4.2, §4.5, Wednesday, February 24)
More on proof by induction (§2.4, Monday, February 29)
The integers (§6.2, Wednesday, March 02)
ℤ and other algebraic structures (§6.2, Friday, March 04)
ℤ and other rings (§§6.2-6.3, Friday, March 04 and Monday, March 14)
More on rings, especially ℤ_n (§§6.2-6.3, Monday, March 14)
Further properties of rings (§6.3, Friday, March 18)
Yet more on rings and fields, including ℚ (§6.3, Monday, March 21 and Wednesday, March 23)
Ordered fields (§6.3, Wednesday, March 30 and Friday, April 01)
Constructing ℝ from ℚ (§6.4, Friday, April 01 and Monday, April 04)
ℝ: a complete ordered field (§6.4, Wednesday, April 13 to Wednesday, April 20)
Yet more on set cardinality (§4.5)
The Schröder-Bernstein theorem (§4.5)
Even more on set cardinality (§4.5)

Solutions or Hints for Selected Problems

Problem 1.1
Problem 2.1
Solution 3.1 Problem 3.8(b) Problem 3.09 Problem 3.10 Problem 3.11 Problem 3.12
Problem 4.1 Problem 4.3 Problem 4.4 Problem 4.10
Problem 5.2 Problem 5.3
Problem 6.1
Problem 7.2 Problem 7.4 Problem 7.5 Problem 7.6
Problem 8.2 Problem 8.4 Problem 8.5 Problem 8.6
Problem 9.2 Problem 9.3 Problem 9.7 Problem 9.8 Problem 9.9 Problem 9.10 Problem 9.11 Problem 9.12
Problem 10.1 Problem 10.2 Problem 10.3 Problem 10.4 Problem 10.5 Problem 10.6 Problem 10.7 Problem 10.8 Problem 10.9 Problem 10.10 Problem 10.11 Problem 10.12
Problem 11.1 Problem 11.2 Problem 11.3 Problem 11.4 Problem 11.5
Problem 12.1 Problem 12.2 Problem 12.3
Problem 13.1 Problem 13.2 Problem 13.3 Problem 13.4 Problem 13.5 Problem 13.6 Problem 13.7
Problem 14.2 Problem 14.3 Problem 14.4 Problem 14.5 Problem 14.6 Problem 14.7 Problem 14.8 Problem 14.9
Problem 15.1 Problem 15.2
Problem 17.2 Problem 17.3
Problem 18.2 Problem 18.3 Problem 18.4 Problem 18.5 Problem 18.6 Problem 18.7 Problem 18.8 Problem 18.9 Problem 18.10 Problem 18.11 Problem 18.12 Problem 18.13
Problem 19.1 Problem 19.2 Problem 19.3
Problem 20.1 Problem 20.2 Problem 20.3
Problem 22.1 Problem 22.2 Problem 22.3 Problem 22.4
Problem 23.1 Problem 23.2 Problem 23.3 Problem 23.4 Problem 23.5 Problem 23.6 Problem 23.7
Problem 24.1 Problem 24.2 Problem 24.3 Problem 24.4 Problem 24.5
Problem 26.1 Problem 26.2 Problem 26.3