Final Blog post, due on December 5

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

Over the whole semester there has been many that we have studied. The most important I think is the methods of proving things: induction, contrapostive, direct, etc. Equivalence relations also fall under the most important, along with functions. Of the more recent things, the Schroder-Bernstein Theorem is probably the most important.

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.

The most recent things are toughest for me (epsilon delta proofs and infinite series in particular). If I had a choice I would like to work out 12.6 or any of the extra problems at the end of the chapter (12.31-12.34). I recognize that some of these were homework problems but I want to make sure that I understand them. We could also just work out similar ones!

What have you learned in this course? How might these things be useful to you in the future?

I've learned a lot about proving things. All of the theorems and topics that I mentioned in the first part are all things that I've learned and have retained. Also I've learned some neat characteristics of numbers, both even and odd, and many, many other proofs and facts!

These sort of things honestly won't help me too much in my career. However, the general mindset of seeking for things that would make a proof not true very much apply. I can use it in engineering analysis of a given situation or problem. Also logic has already helped a ton in the analysis of circuits.

12.4-12.5, due on December 3

Honestly these sections are still like Hebrew to me. I don't have a real concrete understanding of the general format to prove limits; I scooted by just barely in the last homework. Choosing the epsilon/sigma always throws me. I'm sure I'll get it, but as of right now it's my biggest obstacle to mastering these sections. One thing that doesn't make sense to me is the necessity of condition 2 for continuity when we have condition 3. Is there ever a time when condition 3 (on page 289) is satisfied when condition 2 isn't?

I enjoyed the proofs that were showing the addition of two things (whether they be polynomial or just general functions) have the same limit as that of the individual limits added together. The same also goes for the product of two objects. The use of inequality amazes me. It seriously is like the best thing in a proof. You can typically make something smaller (or larger) that makes your proof not only possible but ten times as simple.

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.