12.1, Due on 1 December

The most difficult part of the material was proving that a series was divergent.  I understand that there is no limit, but the formatting of the formal proof seems a whole lot more difficult then it needs to be.  Especially deciding which e to choose and so on.

The most interesting part is this remind me a lot of calculus.  I remember in multiple dimensional limits, we learned how to use the definition of a limit to formally show that a limit exists, but it was always easier to show that a limit did not exist by approaching the same point along different lines.

November 25

What I learned in the course:
I learned a lot about sets, and subsets, and cardinality, and functions, and injections, and serjections, and power sets, and primes, and evens, and odds, and divisibility, and integers, and latex, and proofs, and more proofs, and more proofs...

How will they help me in the future:
They will help me to think logically, keep the goal in mind when working to solve something, and sometimes realizing things are much easier if you can prove something else first.

11.5-11.6, Due On Monday November 24

The most difficult material were simply the proofs, especially of the fundamental theorem of arithmetic.  The theorem makes sense intuitively, but using induction is always difficult.

The most interesting part were the tests for divisibility of certain integers.  Those 4 rules seem really useful, especially the second one.  I remember using it as a kid to remember the multiples of 9, 18, 27,36, 45, 54, 63, 72, 81, 90, as summing the digits give you 9.  It is really neat that that extends to 1233, and other super large numbers.

Exam Review

• Which topics and theorems do you think are the most important out of those we have studied?Countability, The SB theorem, and also the Division Algorithm

• What kinds of questions do you expect to see on the exam?                                              Proving theorems, proving whether something is countable, finding bijective functions, finding greatest common denominator.

• 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.                                                                                                                                       Just a bunch of examples of an injective function from an infinite set to a finite set would be great.  That always seems to be hard to do.

11.3-11.4, Due on 19 November

The most difficult part of the material was understanding the proof for the Euclidean Algorithm's lemma.  The Euclidean algorithm proof itself was pretty straight forward though.  I understand how to use the lemma though.  It is not hard to use it.

The most interesting part is how Euclid came up with this so many years ago.  I wonder if he ever had to prove it like we prove it, or if his proof looked like something else.  I looked up more on Euclid and he seems pretty interesting as a Greek living in Alexandria. It says his textbook was used up to the late 19th century, so his reasoning is pretty sound.

11.1-11.2,due 16 November

The most difficult part of the material was the proof for the division algorithm. I did not understand it the first time, but after reading over again it made more sense. Using the algorithm itself was not too difficult to understand.  I think I have been using it since elementary school.

The most interesting part of the material is that we could actually prove something that just seems obvious.  I had never though about trying to prove something so simple, but we need to if we are ever going to use it.

10.5, Part 2 due on 14 november

The most difficult part of the reading was still wrapping my head around the proof.  It is going to take me some time to understand it, but it makes since that something that is bigger than the natural numbers is equivalent to the real numbers, but that is not to say that all sets bigger than the natural numbers is equivalent to the real numbers which they showed by looking at the power set of the real numbers.

I found it interesting that the Axiom of Choice and theorem b were equivalent.  I guess the book does not go into that much detail.  It was also interesting to see a German presenting a "defective" proof, and it took us over 50 years to finally come up with the correct one.

10.5 Part 1, due 11 November

The most difficult part of the reading was the proof for 10.17, and how they got for the method they used.  It seems they used a proof by cases and defined a subset as an example, but because this example turned out to be bijective we know there exists a bijective function from A to B.

The most interesting part was simply the restriction.  If we can't make something one to one, we will simply cut it in half so that it is one to one.  Restrictions especially in calculus can seperate things into different integrals making the math a whole lot easier.

10.4 Due on November 10

The most difficult part of the material was the new terminology for the cardinality of the natural numbers and the real numbers.  I have never seen an aleph null before, and the term continuum is pretty new, or at least now I know how to spell it.  The proof by contradiction was not too difficult, but I need to get used to using them for abstract infinite sets.

The most interesting part is that there is no largest set.  No matter how big a set is, even if it is infinite, there will always be a set with a larger cardinality.  So there is something bigger than our universe, and something even bigger than that and so on and so forth.

10.3, Due on November 6

So the reading was not too difficult as it is simply the opposite of section 10.2.  By far the most difficult part was understanding the proof that an open interval is not a denumerable set.  I understand that it has to do with something being expressed as .40000 and .39999 and that not making it onto, but not sure the stuff in between.

The most interesting thing of course was simply that .40000 and .39999 are the same number.  That kind of blew my mind, but it makes sense.  By that reasoning is .40000000...1 also the same number?

10.2, due on November 4

To be honest the material was really easy, simply because we already went over most of it on Monday.  The most difficult part though is simply thinking that something can be a subset of something else but still retain the same cardinality.  Weird things happen at infinity.

The most interesting part of the reading was of course the square type function used to count the ordered pairs.  I wonder who came up with that and if there is a way to express the function whiteout simply showing a pattern.

10.1, Due on November 3

The most difficult part of the material was looking up Theorem 9.11 and Corollary 9.8 to review those theorems, but after reviewing them the proof for theorem 10.1 made a lot more sense, which it did not at first.

The most interesting part of the material was the theorem that if there exists a bijective function from A to B and A R B, then R is an equivalence relation.  This may seem trivial for finite sets, but when applied to infinite sets, this is really useful.  This also is because infinite sets are said to be able to have the same cardinality.