12.3 due on November 30

Perhaps it's something in the previous section that we weren't assigned or not, but what does it mean by the deleted neighborhood of a? It seems like it includes everything except for a, which would kind of make sense because we're discussing limits and quite often limits aren't defined at the point. The toughest part for me about the chapter is choosing the values for epsilon or sigma. How exactly do we know what we want to specify sigma as? It varies from problem to problem and I haven't yet gotten the pattern of it.

The coolest thing about this class for me is actually understanding things that I've grown up learning. While I wouldn't say that this section actually helped me understand limits and calculus, it has definitely helped. Something that was neat was the need to specify a range of acceptable 'closeness'. In programming you often need to put stopping criteria in a loop which is analagous to limits and their 'stopping criteria' of epsilon.

12.1, due on November 28

I'm having a difficult time understanding the use of the ceiling of a number to prove what it converges to. In the proof of result 12.1, we choose a ceiling of 1/e. Since 1/e is always less than one, wouldn't the ceiling then be 1? Then we let n be an integer greater than that, or greater than one. This makes 1/n less than e. I understand all this; the reason why I'm stating it is I suppose to check my understanding. What I don't understand is how that proves that the sequence converges to 0. Or I suppose how that proves that it converges at all. Not understanding this concept makes understanding the rest of this section very tough! Unfortunately I won't be in class tomorrow due to an interview, so I won't be there for the explanation.

Not understanding that concept of the ceiling has really taken all the fun out of this section. Every proof uses that fact to prove some divergence or convergence. So I lack something specific that I enjoyed about the section. More generally it's fascinating to me that we can prove convergence in a different way than how I learned in calculus. In calculus we always ended up taking limits and using L'Hopitals rule or some other rule to show that something either converged or diverged. It's neat to have another way to show that.

Exam 3 Assignment, due on November 26

Which topics and theorems do you think are the most important out of those we have studied?

I feel like showing sets are denumerable or uncountable was something very important we talked about. As well, the Euclidean Algorithm and the Fundamental Theory of Arithmatic were also very important.

What kinds of questions do you expect to see on the exam?

I would expect to see a free response dealing with each of the above stated topics. The true false for this section might be fairly tough. I'm not entirely sure what to expect because a lot of what we did dealt with extensive proofs. Perhaps some questions might deal with divisibility of integers or cardinality of sets.

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.

I'm concerned on the Schroder-Berstein Theorem on knowing how to prove it. If we don't need to do that then great, but the proof hasn't really clicked for me. Also, I get confused when the restrictions are added and the 'new codomain' of a set and how that helps in a proof. If we don't need to prove the Schroder-Berstein Theorem then I won't really worry about it, however if we do then I would love to review that proof.

11.6 and 11.7, due on November 19

The Fundamental Theorem of Arithmetic has a neat result but I found myself lost in the proof. The first question arose on page 257 when it says that "by the induction hypothesis, each of a and b is prime or can be expressed as a product of primes." Is this just a restatement of what we are trying to prove? Because if not that almost seems backwards. Right after this sentence it goes on to say that by the principle of induction, that the theorem is true. Next question deals with Theorem 11.20, dealing with the relation between n^.5 being rational and an integer. While writing this I actually figured it out...I think that the use of several theorems that we have proved in the past was a little confusing.

We talked about this today in class, but the shortcuts for divisibility of several numbers are really neat! We had learned certain parts of these growing up and now it's neat to be able to see why they work. I really enjoyed the rules for 3, 9 and 8. However, 11 seems to me just a little too much work! Although I'm not sure why it works and how useful perfect numbers are, perfect numbers are super cool. Is that just a coincidence for why the numbers are a sum of 1 to the largest prime divisor?

11.5 due on November 19th

This section mainly centers on Euclid's Lemma and its corallaries. The actual proof of the Lemma is neat, however, what is more interesting to me is the usefulness of the lemma in proving other things. In the book it talks about several corallaries and uses the lemma to prove them. Something else that caught my eye was in the proof of theorem 11.16. Two results for 'c' were found and both were substituted in the same equation allowing us to pull out the 'ab' necessary to show that ab|c. I thought that was clever and something I would have missed on my way through the proof for sure.

This section was fairly short and easy for me to understand. At first I had to reason in my mind what it meant for two numbers to have a gcd of 1. Actually, I guess I still don't quite understand that set of numbers. It states that the two numbers are integers, so it seems like to have a gcd of 1 and all the coeffecients to be integers as well, the two numbers (integers also) would need to be zero and one or some combination of negative and postive integers. I would like it if we could go over some examples of integers with gcd of 1 just so I can have a concrete understanding.

11.3-11.4, due on November 16

This was a good section, fairly easy to understand. I got a little confused with gcd because we show that the set of all the numbers that are linear combinations of the two numbers in question (i.e. a and b where we're looking for gcd(a,b)) has a least element, even though we're looking for the greatest common denominator. However if we have the greatest common denominator, then the linear combination of the two should be the smallest possible linear combination, or the least element of the set on page 251.

The Euclidean Algorithm is a useful way to find the gcd of really big numbers! The proof was tough to follow, but then the example (example 11.10) laid out actually how to explicitly do it and it made sense. I like the sequence of these sections as well, how we first learned about the division algorithm and then about gcds and now we use all of those in the Euclidean Algorithm.

11.1-11.2, due on November 14

The Division Algorithm is a very basic concept, but the proof is crazy. I especially get tripped up when we introduce q and r into the proof. At first I didn't understand the aspect of it being unique. However, I noticed the restriction 0<= r < a, which makes sure that q and r are unique. In the actual proof, why do we consider the set of integers where b-ax>=0? Also why are the integers that satisfy the qualifications for the set postive? It seems that a negative value for x would make sure that b-ax was always greater than zero. I understand that that wouldn't work with our division algorithm, however the inequality and relation for the set don't seem to take this into account.

As we've had types of these problems before, it's neat that we get to learn more about them now. For instance, I thought the application of the division theroem to divisibility of integers (I'm not sure how to say that; where we let a=2,3,4, ... etc and then we know how to write any integer as a product aq+r. For example 2q+1). We learned about this and used it earlier, taking it as true without really learning the proof behind it. Now we know!

Rest of Section 10.5, due on November 12

On page 239, the actual proof of the Schroder-Bernstein Theorem is tough to follow. It seems like the function g1 doesn't really mean anything, but rather it's just some random function set to equal the function g. This serves to give us a bijective function and an inverse, but how is it relevant? Where does the function come from?

The idea behind the theorem is really neat and it makes sense. It reminds me of calculus and that theorem dealing with limits. I forgot the name, but it could be called the squeeze theorem. We're almost forcing the value of the cardinality due to the limits on either side, which is super common when applied to real numbers and the like, but this is applied to sets and so it's doubly cool.

10.5, due on November 9

Several parts of this section were difficult to understand. The introduction of a restriction went fine, however when applied to the Reals, that threw me. Once I reread it and understood that the restriction is a subset of the initial function, or rather that the restriction is taken on a set that is a subset of the initial domain, I understood how the restriction g1 could be one-to-one. Effectively we are only taking the positives and thus leaving out the negatives that would yield the same values of the range when squared.
Lemma 10.16 is still throwing me and I would appreciate it if we spent some time on it in class. I don't understand why we take values that are in the union of A and B. It seems weird to be taking values from the domain and range of a function.

The definition of a function from the union of two sets to the union of their corresponding sets that make up the range of two individual functions was super cool. I hadn't really thought about that before. It also makes sense that the two sets A and C would need to be disjoint for the function h to be considered a function (pg 237).

How Data Analytics is Transforming our Lives by Jack Thompson

Honestly this address was very hard to follow and wasn't very enjoyable. He relied on a number of videos to demostrate what he was talking about but didn't explain them very well. He didn't spend a lot of time on any one topic either. Overall, it could have been a lot better

Nearing the end, he finally talked about something significant and relevant to us today...Facebook! No but really he discussed a little the significance of social media and the fate of privacy in the future. Who's data will be who's? What constitutes your data? He asserted that in the future we won't be able to protect our data, that there will be so much technology spread around that anything that was ever on the Internet will be there to stay etc. That was a neat part for me! Analyzing data, seems like an interesting direction and line of work, but I definitely know it's not for me.

10.4, due on November 7

Most tough to understand for me was the inital proof of 2^A being equivalent to P(A). Where does this proof come from? Well, I know that we used it for finding cardinality but where does the actual proof come from? How would we think to make a function equal to that piecewise function?

That being said, it was neat that out of nowhere this function comes from the set of subsets and it relates the pairs of elements of P(A) and 2^A. I don't quite understand it, but it works out nicely. Other than that, this section is really short and I don't have a lot else to say!

10.4, due on November 5

Maybe I just haven't spent enough time with decimal expansions yet, but there are two proofs given in the text that I'm not sure where and how they arrive at their contradictions. One is 10.8 and the other is 10.12. The first is the set of real numbers that is uncountable. I get lost somewhere in the defining of another decimal expansion that helps us reach the contradiction. And then with 10.12, it seems like in 10.11 they state the exact opposite.

The idea of decimal expansions seem neat to me. There was one proof on my homework I did one time where the TA said I should have taken the decimal expansion of the number to prove it was irrational/rational and so ever since I've been excited to learn how.