11.5, due on April 1

The difficulty I had with this section is that there were no examples of application of relatively prime integers.  It was stated in the text that we would see that they are very useful.  However, I only read about theorems and lemmas and proofs thereof.  Perhaps we will see their usefulness later on.  I suppose, actually, that there was one reference to its usefulness.  At the end of the section it stated:

"By Theorem 11.16 then, if we wish to show that 12, say, divides some integer c, we need only show that 3|c and 4|c since 12 = 3 · 4 and 3 and 4 are relatively prime."

It didn't actually show how to do this, but it did mention it.  I typically retain the knowledge better if I see some examples.

Looking back on previously learned mathematics, I don't recall ever hearing about relatively prime numbers.  I am grateful for the opportunity in this class to learn about a lot of new topics, definitions, and mathematical ideas.  When I come across something new in this class, I always think - I should have learned this, or at least heard about this, before during one of my mathematics classes or textbooks.  It is surprising to me that I haven't.  It leads me to believe that there is more to learn than I could even imagine.  After all, many mathematicians let scholastic discovery be their life's pursuit and I'm sure there is no way to ever know it all.

Sections 11.3-11.4, due on March 29

This may seem trivial, but I always find it difficult to follow equations or proofs when there are subscripts.  It really slows me down.  Is there a simple way to understand them?  That is kind of a silly question - it probably is just something that will get easier with more exposure and more familiarity with them.  For example, the proof of Lemma 11.9 introduced r(sub k)q(sub k+1) and then r(sub n-4) = r(sub n-3)q(sub n-2) + r(sub n-2).  In the end, it is not impossible to understand; however, it just takes me a while to get what each variable represents when they have the subunits.  I wish it flowed faster and didn't delay my reading.

The Euclidean Algorithm is pretty cool.  I really liked the way in which the gcd was found by dividing and then dividing again with the remainder into the result just acquired.  Sorry, I'm not using the proper term names, but my point is that I found it very neat how it worked out so smoothly.  Lots of things in mathematics are really fascinating.  There is such an order and it really is beautiful.  Anyway, this algorithm struck me as really neat and clever.  I'm not sure if there are future applications of it or if we build off it to learn other topics, but even if not, it was cool to learn in and of itself.

11.1-11.2, due on March 27

The proof of The Division Algorithm was very difficult for me to follow.  As mentioned in the text, the theorem is quite easy to understand because I've used it so often in my life, but indeed the proof is not obvious.  Maybe it has to do with the fact that it is a proof about something so fundamental, I'm not sure.  It's like trying to describe the taste of salt.  How do you start?  What do you say?  I don't know, maybe it's not exactly like that, but The Division Algorithm, with the quotient and remainder, is something that seems so easy that no proof is required.  But, of course, a proof is required!  It was a hard one for me to understand.

Number theory seems like it will be interesting.  I don't know a lot about it, but the introduction to the chapter seemed to hype it up quite nicely, praising it for its appeal, clarity, simplicity, and elegance.  I look forward to learning about it.  I don't really think it will tie in too strongly to some of my career goals in statistics, but I like the way that Prof Jenkins explained the study of some of the material in this class.  To paraphrase: 'It might not save the world, but it's cool and interesting, so we study it.'  I feel like that is a great way to think about it for me.

Rest of Section 10.5, due on March 25

Once again, I see a proof in which the function is split up.  In the proof of Theorem 10.19, it stated that s(sub n) = 1 if n in S and s(sub n) = 2 if n not in S.  Also, it was stated that t(sub n) = 1 if n in T and t(sub n) = 2 if n not in T.  I don't really understand, still, what this means.  Why are they separated and equal to different numbers in different scenarios.  My shallow understanding is to show that they have unique decimal expansions, but I'm not connecting the dots on how that shows it is unique.  I should have probably asked a question in class a few weeks ago when a proof with a similar format was presented, but I didn't.  Now I see it again and am still confused.

I find it kind of amusing that Cantor is left out of the Schroder-Bernstein Theorem.  Poor Cantor!  It is interesting who gets credit for what with scholarly discoveries.  Will you be remembered forever for what you accomplished in your field or will your work be forgotten?  While I'm fairly certain that I will not leave my mark in mathematics, it is inspiring to see these mathematicians excel in their field and be remembered and studied about by those that followed them.  It inspires me to be the best I can be and to leave an impression on this world - even it is on a smaller scale, like with my children perhaps.  That I can leave this world having benefited it or making it better for those who follow would be something to strive for.

10.5 up to Theorem 10.18, due on March 22

The most difficult part of the material for me was Theorem 10.17.  Well, I should say that the difficult part was in the proof.  The theorem itself appears to be true, upon first reading, and I was surprised to see such an intricate proof for it.  I don't understand why the ran f was brought into the proof.  Also, I find it confusing when all of this extra notation is involved.  Like f^n(x) and B'.  It just kind of hurts my head when I read a proof like that.  I'd have to reread it many, many times before understanding it, and, even then, would not really get what it is saying.  I find it encouraging that we haven't had to prove theorems on exams.  That would be really hard.

While the material is interesting and beneficial in the field of mathematics, sometimes I wonder what the application is for me, personally.  I appreciate that mathematicians study and enjoy this type of material, but, as I stop to think about my life and career goals, I wonder how much of the material - or let's say, the Schroder-Bernstein Theorem, specifically - will I use later in life.  I don't think that I would be cut out to be a math major and study years of mathematics.  I have a lot of respect for the professors and other gifted minds in the math department.  This material is hard to understand for me.  When I feel like the material is going over my head, I think about things like this, to be honest, while I search for motivation to fully understand the material.

10.4, due on March 20

I have had some difficulty with proofs involving showing a function is bijective.  I still need to get to a point where I easily comprehend and can adequately construct similar proofs.  I would expect them to keep appearing in this section, not to mention the final exam.  I fail to understand the logic of the proof of Theorem 10.14.  Aside from the fact that I get tripped up when I see different characters, some with sub-characters, I still don't understand how you set things equal to each other and then that shows an injection.  Then you set other things equal to each other and it is onto.  I really need to focus on the fundamentals of these types of proofs.  At least I understand that it is a bijection if it is injective and onto...at least. 

The Continuum Hypothesis cannot be proven or disproven.  That is interesting.  I would think that there are other hypotheses and conjectures that are impossible to prove or disprove, but I'm not that familiar with mathematics to name one.  In a field that I sometimes see as very black and white, I am always surprised when I see that mathematical ideas cannot be proven or disproven.  I guess there are many mysteries.  I wonder what the opinion is for mathematicians - if it can't be proven, then is it false or, rather, if it can't be disproven, then is it true?

10.3, due on March 18

Since we discussed in class a little bit of 10.3 already, before I read the section, it made a little more sense.  However, the proofs as seen in class and in the text seem very difficult.  In fact, the proof of Theorem 10.8  "The open interval (0, 1) of real numbers is uncountable" was shown in the book and in class, but I still don't really get it.  What does it mean when b (sub) i is separated into a sort of piecewise function of 4 if a (sub) ii equals 5 and 5 if a (sub) ii does not equal 5?  I don't see the connection between doing this and showing that it is not onto.  I recall that it may have something to do with the game.

Speaking of the game, although it wasn't mentioned in the reading, it was discussed in class and has applications, I believe, to the proofs of denumerable and uncountable sets.  I kind of enjoyed the game, although it fast became easy and only Player 2 has a winning strategy.  It reminds me of tic-tac-toe and other such games in that once you know some tactics, you can never lose.  I thought that it might be fun to try the game from the homework out with my son who is in kindergarten.  He might think that it is fun with the X's and O's and it might teach him to think a bit more logically.  No doubt he'll eventually never want to be Player 1.  Anyway, I thought that the game was interesting and maybe I can use it to understand the logic and proofs in this section better.

10.2, due on March 15

This section 10.2 started off fine as I understood the definitions of denumerable, countable, and countably infinite.  Superficially, uncountable means not countable.  So that means not finite or not denumerable.  But denumerable is infinite.  Then it said that an uncountable set is necessarily infinite.  I'm not sure what the difference is between a denumerable infinite set and a necessarily infinite set.  Even after the proofs, which were very intricate and difficult for me to follow, did not clarify this for me.  I must say that this section was the most difficult of all I've read.  The diagrams and tables were more confusing then helpful, unfortunately.

I have always been interested in the concept of infinity, so I'm interested to learn more about this and actually understand it.  I look forward to class tomorrow to gain a better understanding.  To be honest, I was a bit discouraged that I understood so little in this section.  I feel that understanding this will be of importance as we begin the third of three sections in which this class material was divided.  Usually in class, the proofs and examples are more clear, so hopefully that is the case, because the proofs in the book were hard to follow for me. 

10.1, due on March 13

I found this statement to be very interesting: "Certainly, then, understanding infinite sets was not
an easy task, even among well-known mathematicians of the past."  I almost find this humorous.  If some of the most brilliant mathematical minds had difficulty understanding infinite sets, how in the world do I have a chance at all understanding them?  I laughed a little thinking about that.  Specifically, I didn't really understand "There can be a one-to-one correspondence between a set and a proper subset of the set" and how it relates to infinite sets.  It troubled Galileo, so I guess my lack of understanding should not trouble me too much.

Numerically equivalent sets, from what I read, seem to be tied to concepts learned in Chapters 8 and 9, so I am comfortable with this.  The description involving two sets A and B having the same cardinality if they're empty or if there is a bijective function from A to B.  Learning about bijective functions already really makes this easier to grasp.  Once again, I feel that the text and the course is well-organized and the principles learned build off one another nicely.

Preparation for Exam 2, due on March 8

The topics and that I think may be most important out of those we've studied would include induction, relations, equivalence classes, and functions.  Specifically, regarding relations, I know that it is important to know if they are reflexive, symmetric, and transitive.  Concerning functions, it is important to know if they are injective, surjective, and bijective.  There are a couple of noteworthy theorems regarding this in my notes.  Also, I think that the theorems about congruence modulo n and equivalence relations are very important.

I am expecting a similar format for this exam as the one used in Exam 1.  True/False questions, Multiple Choice, and Free Response, which include proofs.  Speaking of proofs, I feel like I need a much better understanding of how to construct proofs.  Is there a certain formula or are there guidelines I can follow when doing an induction proof or when I am trying to prove a function is injective, surjective, or bijective.  I think that I get stuck with how to start and how to finish, not to mention what to put in between.  Really, I sometimes freeze up when thinking about how to write proofs because I don't know how to proceed.  I am more used to solving problems - first, take the derivative, then plug it into the formula, then simplify, then solve the integral, etc.  I'm not sure if there are clear-cut guidelines for proof construction, but that would be helpful to me.

A problem I would need to see worked out would be, for example, is "Prove that n! > 2^n for every integer n >3."  I missed this one on the homework.

9.6-9.7, due on March 6

Theorem 9.11 quickly made sense.

Let f : A → B be a function. Then the inverse relation f−1 is a function from B to A if
and only if f is bijective. Furthermore, if f is bijective, then f−1 is also bijective.

The proof got a little jumbled for me.  In order to show that f-1 is a function, I suppose it is as easy as showing that f is one-to-one and onto.  I am confused as to why the proof continued with the converse.  For me, it has been helpful to follow a proof pattern when writing my own proofs.  If I have a guide of what needs to be shown, it is manageable.  I don't quite get the pattern of this proof and would need a clarification as to why it was set up the way it was.

I remember learning about permutations in a statistics class I had last semester.  Pretty interesting.  That part of the material made sense to me, although I remember getting a little tripped up with permutations as they became more complicated.  It is interesting to note that permutations are bijective functions and have inverses.  I didn'onsider that really when I learned about them last semester, but it definitely makes sense.  I look forward to Chapter 13 when they are discussed in more detail.

9.5, due on March 4

I really didn't get the last part of the material: Theorem 9.7, Corollary 9.8, and Theorem 9.9.  I should say that the theorems made sense and were acceptable, but the proofs were a little different.  For example, in Theorem 9.7, it was proven ultimately by implying that g of f was injective and surjective.  I didn't get how that implication could be made.  At the end, Example 9.10's solution stated that the ran f was {0,1,4,9}.  While I understand that it is necessary to show that ran f is a subset of B', I failed to see how the ran f was defined how it was.

Once again, this material connects with something I've learned in calculus.  There were a few ties to derivatives, including the product rule and the chain rule.  While I appreciate the tie-ins, I get a little frustrated with it being easier for me to understand the material when it talks about calculus than in other situations.  What I mean by this is that I wish it would click in my head easier when reading about non-calculus material in this text.  I do think, though, that the text does a good job of bringing in calculus references and other things we've surely learned in previous classes.  This forms a nice transition and I generally understand the material better upon first reading.