Thoughts, but mostly a question.

While studying for the test today, Ryan and I had a question.

How is it possible that (N) X (N) is countable, but 2^N is uncountable...? Because, isn't 2^N just (N) X {0,1}? That is all.

Studying for Exam 3 due March 31

Well, honestly I haven't spent a lot of time studying for the test yet. I will, though. I've been doing a lot of studying as we go though, because this unit has been quite hard for me (as you know). Based on what we've been doing in class and on homework, I would say that the most important things from Chapter 11 would be finding the gcd and the smallest prime factor. This is easy enough to be tested, but hard enough that we would have to think in order to do it. These seem like valid multiple choice questions. For Chapter 10, I would expect much simpler questions than the ones that we did on the homework. I might be wrong here, but some of those homework questions were ridiculously hard and long. So...simplistically, I think we could show what sets are denumerable and which are uncountable. We probably are going to be accountable for the definitions as always.

Hopefully I will do alright on the exam this time around. Monday-Wednesday are by far the worst days for me.  >.<

Also, I had tickets to Calculus the Musical and totally mixed up the times and missed it. My weekend was ruined by this. I hope you got to attend and enjoy it for me.

11.4-11.5 due March 28

My thoughts on the "relatively prime." My first observation was that we are trying to get the linear combination to add to one. This was unique to me because the definition of a prime is that its only factor is itself and one. So in this case of relatively prime integers, we are trying to get their greatest common factor to be one...even though they might be composite numbers. So I guess from what I understand, "relatively prime" essentially means that the linear combination is prime. I love how the text states, "This is extremely useful." That is a biased statement, and I'd like to see us tell the English majors how "useful" that statement is.

Next thought...I at first was pretty fascinated by the idea that every integer exceeding 1 has a prime factor. Well. That was interesting until I remembered that 1 is a prime number...and everything as 1 as a factor. So, does this corollary even tell us anything? These are the things I wonder about.

And these divisibility rules are pretty fantastic. Why did no one ever tell me that if I summed the digits in a number and it was divisible by three that the number itself was divisible by three? I feel like this information would have caused me much less stress in math over the years. It also would have saved paper because I wouldn't have done so much long division.

The proofs in this unit are really hard to figure out how to do. I wish I could get along with prime numbers. Can we go back to doing even and odd proofs? I liked those. That is all.

11.3-11.4 due March 26

So I want to say first off that this post is going to be essentially worthless...but not because I didn't try to do the reading. I just didn't understand the reading from last time, which made the reading for today pretty close to impossible to understand. I tried to do the homework, and it was equally confusing. I am so nervous about this unit. The last few homework assignments have been terrible and I haven't hardly figured a single problem out without the help of the back of the book, lecture, or another student. So that's where I'm at. Hopefully my work schedule will cooperate and we can have some bonding time at office hours soon.

Linear combination proofs look like the proofs we were doing earlier in the semester with "Every number greater than 3 can be written as a combination of 2x+y=n. I would be lying if I didn't tell you that I just barely remembered what a linear combination is...and DUH that's what we were doing earlier.

The well-ordering principle has got to be the most confusing thing I've learned so far in this class. I don't understand its usefulness at all...and yet the book keeps using it...I feel like there's going to be a test question about it and I'm going to totally blank.

Have I mentioned to you that I dream in proofs? Yes. And the other day, the temple ceremony was coming into my head in the form of mathematical proofs. Something about this class has me pretty excited, because it has made my brain function differently.

10.5 - The End due March 21

So...mind blown...

I am overwhelmingly fascinated by the fact that we can define the cardinality of an uncountable set. And that we can do so simply! I already knew that we could call it "c," but the defining of 2^A being 2^aleph-null...it just seems so simple. Well, I suppose that nasty looking aleph null symbol doesn't make it look too simple, but it is!

The main problem I have with these proofs is that they seem to all be proving the theorem. (They are.) I just don't understand how I'm going to figure out how to do these problems without the infinite hints of the example. For instance. In the proof of Theorem 10.19, I don't think I would've even known how to define the sets without the help of the example. To put it simply, I am must more scared of this test than I was of the previous two tests.

What exactly is the benefit of knowing that the cardinality is less than another cardinality? I see great benefit in knowing that if (in the language of LaTex) A\leq B and B \leq A, then A=B. But what if the only thing you can do is prove A \leq B? Then we can assume that A is not equal to B, but that doesn't tell us that it's denumerable, does it? Questions, questions...

10.5 up to Theorem 10.18 due March 19

So what we are trying to do here is take a function that isn't bijective, and make it bijective by changing the domain. If we have a function that isn't one to one, we can change the domain to make it one to one, and therefore bijective. That sounds like it could be very useful. It's almost as if you could change an uncountable set to be a denumerable set. Dreamin' big here.

As for the thing that confused me and made my brain spin around, The proof of theorem 10.17 has me pretty lost. In my logical attempt to understand the theorem itself, I have determined that the identity function works. That's all good. So we're just proving that it works for proper subsets. Therefore, the set B' must be a proper subset of B? Ok. I continued reading through it, and after a study of the proof, I think I actually get it now. There were just a few too many variables for me to understand it the first time through. But it makes sense.

10.4 due March 17th

IT'S MY BIRTHDAY! YES!

I read today about infinity. And double infinity. It was kind of like reading about Chuck Norris.

I think I have always figured that there are different sizes of infinities...now I just have proof of it. It makes sense. I guess the main thought I have is that my favorite Chuck Norris joke is,

"Chuck Norris counted to infinity. Twice."

Now I see that that is not nearly as impossible or impressive as it sounded in my youth. You see, if he was counting to infinity on a denumerable set, then it's not impossible. All he needed to do was include all the natural numbers, and he'd be fine. To make this more impressive, I've decided to rephrase the joke to,

"Chuck Norris counted a set greater than the uncountable set of the real numbers. Twice."

There we go. Now that is impressive. I am glad to see that there's no limit to infinity. I would be concerned if there was. The Cosmos (TV show) had an animation of an ancient philosopher's idea of the universe as Cupid standing on the edge of infinity and shooting off an arrow. There are two choices for what happens to the arrow, 1) it continues on forever, or 2) it hits a barrier and doesn't continue, from which he can now stand and shoot another arrow. In either case, it goes to show you that he wasn't actually standing on the edge of infinity, nor will he ever be. It's boundless.

And that's my thoughts for today. I need to go to bed.

10.3 due March 14 (HAPPY PI DAY EVERYONE! This class doesn't do anything with pi, but still.)

1. So I'm not sure what my problem is regarding this proof (0,1) being uncountable. I just don't understand how b1 is not in the set (0,1)...and if it's in the set (0,1), shouldn't it be mapped to? Maybe I just don't understand the way that we defined the function. From the way I am looking at it, there are only 9 elements in the function...why? I don't understand. I wish I could come to office hours tomorrow to discuss this, but I cannot. Perhaps I will come on Monday.

2. On the bright side, the proofs after that all made sense, because they are largely the same proofs as section 10.2, only some are contradiction-style. I can handle that. Once I wrapped my head around infinite sets (as much as you CAN wrap your head around them) all the proofs about them became essentially logical. If you have an infinite set, then it will always be infinite, even if you subtract elements out of it.

What has been killing me is remembering that there are theorems for these proofs, so I don't have to write everything out! I did a problem for this last assignment and it took quite a lot of space. When I checked my answer in the back, there was a theorem. Oh well, at least I know how to prove it. Hopefully.

10.2 due March 12

I figured I would do the blog entry early this week since the homework for Wednesday is from 10.2. Aaaaand I forgot to do the blog for today for section 10.1.

1. By far the hardest thing about these denumerable sets is that you have to try to comprehend the idea of two non-equal sets that are related to each other in some way to have the same cardinality. The example in class today with the hotel rooms was helpful though. It's taking me back to my linear algebra days when I had to attempt to comprehend the n-th dimension. It's just counter intuitive to think that you can "count" an infinite set.

2. I am just loving that everything we've learned is packed together in this one section. We've got subsets, cardinality, types of proofs, induction, least elements, well-ordered sets, the identity matrix...it just never ends! I've never been one to understand application, but I always enjoy seeing it, and since this class is so fascinating to me, I will probably put in the time to understand it. My newest idea for comprehending it is imagining a graph that goes from [0,infinity) that has a range from (-infinity, infinity). This is entirely possible, and would have to be a function, and it would have to be one-to-one and onto.

Studying for the second midterm

• Which topics and theorems do you think are the most important out of those we have studied?
• Well I would have to say that the most important thing is proofs about functions. I say this because I learned a ton when I was learning about them about what functions really are, which changed my entire perspective on math. I also had the hardest time understanding it, so it's most likely the most important thing.
• What kinds of questions do you expect to see on the exam?
• I would say problems that look like the homework. The homework for the most part is very appropriate. The difficulty is exactly as it should be, and it covers the expanse of the material. Historically, the homework I've done for math classes is far harder than anything we ever need to understand at that level, and so it's frustrating and we don't profit from doing the homework at all. But this homework prepared me very well for the last exam, and I would expect the same for this exam.
• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.
• Currently I am focusing on SPMI proofs, but I also need some perfecting in the f-circle-g proofs. It's better than it was when I came to office hours, but I haven't reworked any problems from that section yet. That's my plan for when I finish this blog post. On that note...over and out.

9.6-9.7 due March 5

1. The inverse function. It makes sense that it needs to be both injective and surjective. A function has to have every value of the set A as the first coordinate, so naturally if not every value of the set B is in the range of the function, then when you take the inverse and switch the coordinates, then it wouldn't be a function anymore. So it has to be surjective. And it has to be one-to-one because all the points have to be unique and distinct, whether it's (a,b) or (b,a). The book shows a few examples that are easy to find by algebraic manipulation...so what do we do if it's not possible to find the inverse by manipulation? Is there a way? I don't know if we're responsible to know how because the book didn't teach us how, but it would still be interesting to know.

2. So a permutation...I feel like I've heard about these before. I forgot what they are. So a permutation is just a function of A-->A that is one to one? So basically a subset. That sounds good, but the use of it is not totally known to me. I seem to have this problem all the time. I know how to calculate something, but don't know why I would want to do it. Other than the format that we can write it in, I don't quite know what the difference between permutations and everything else we've been looking at is.

9.5 due March 3

This is a shout out to that awesome moment when I do the reading and then forget to blog about it before falling asleep. I was dreaming about proofs and definitions of functions...does that count for anything?

Calculus and math 290 meet...KAPOW!!! That was exciting. I think I forgot how to do integrals and derivatives though, so hopefully it doesn't come back full force. I did get excited seeing the derivative formulas that I had forgotten though. So it's really true that it was a study of composition functions that lead to the formulas for derivatives? I am still failing to see quite how that works...(are we supposed to understand it? Or was it just some "food for thought"...) I also believe that the dot has a few too many functions these days and I just know that I'm going to forget one in the testing center and accidentally multiply a composite function.

Lastly, what exactly is a "corollary"?  I know what a theorem is, and I figured that Corollary 9.8 was actually just a theorem, but now I'm not so sure. I suppose I will find out in lecture tomorrow. \qed