12.4, due April 9

1. (difficult) Following the proofs of theorems 12.23, 12.24, 12.25, 12.26, and 12.27 was pretty difficult. After last class period, I now understand the purpose of the deltas, but now I am struggling particularly with the manipulation of delta1, delta2 and delta together to prove the theorems. However, I appreciate the tools that the theorems provide.
2. (reflective) Theorems 12.28-12.31 are applications of the above listed theorems. I can't see the broad application because they seem very specializes or specific to certain problem types rather than problems in general.

12.3, due April 7

1. (difficult) I didn't understand the selection of delta in the last section, and delta becomes a much more used term in this section and in proving the limits of functions. I am starting to see the patterns that develop in choosing delta, but no real explanation has been given here either.
2. (reflective) Deleted neighborhoods is an interesting concept, and I think the name for it is what causes it to seem difficult at first, but is entirely easy to understand upon examination. The images they use to describe it remind me of function mapping from earlier sections.

12.2, due April 4

1. (difficult) While trying to understand the proofs for convergence of an infinite series, I am having a hard time grasping the concept of epsilon ad it's necessity in the proof. I am also very confused by the notation in the book. It is possible that it is a misprint, but on page 275 directly above the second Lemma 12.8 It has an almost bracket looking symbol around 1/epsilon. I don't have a clue what that means.
2. (reflective) I am realizing as I am trying to understand these proofs how integral an understanding of epsilon is, and I don't know what it means, nor can I glean any sort of understanding from the text. All I know about it is that it is crucial to these proofs.

12.1, due April 2

1. (difficult) The proofs for divergence or convergence are difficult to follow, with many cases and new assumptions.
2. (reflective) Most of the work on these proofs is done in the proof strategy -  the proofs are relatively short. It seems like if you have a good strategy, you won't need to have a long proof.

Questions, due March 31

1. Schroder-Bernstein theorem, continuum hypothesis, division algorithm, euclidean algorithm
2. Gcd questions, comparing cardinality, Denumerability, and infinite sets.
3. Questions like 10.19.

11.5-11.6, due March 28

1. (difficult) It is hard for me to think abstractly about the relatively prime numbers that are introduced here I needed to come up with real examples on my own in order to make sense of the proofs.
2. (reflective) I used to always wonder why the greatest mathematicians loved prime numbers so much, and with the work we keep doing, I can see how these prime numbers are really the building blocks of factorization and many other theorems we are studying.

11.1-11.2, due March 24

1. (difficult) The only difficult part I had was in following the proof of the division Algorithm. I first struggled when the book introduced the value t=r-a, but after some careful looking and reading I understood that. The one spot that I still do not understand is the introduction of, characteristics of , and manipulation of the variables r' and q'.
2. (reflective) I thought it was nice to finally come back and expand on the understandings that we have gained about the integers being able to be described as 2x, or 2x+1, or further as one of the following: 3x, 3x+1, 3x+2. It only makes sense that we come to understand all numbers as being written as aq+r where r<q.

10.4, due March 17

1. (difficult) The gravity of the continuum hypothesis has not hit me yet, and it seems like a statement that says "well, duh."
2. (reflective) It is really interesting that the cardinality of a denumerable set is defined as a letter of the Hebrew alphabet. With mathematics begin mainly written in Greek symbols, I wonder how aleph has survived as being a standard symbol. I'm sure at some point the mathematicians would run out of Greek letters and symbols to use, but Hebrew is a surprising alternative.

10.2, due March 12

1. (difficult) A lot of this section was difficult to understand, beginning with the ideas of Countably infinite, uncountable, and denumerable. The book never explicitly says that these are the same thing, but from what I can tell from the reading, they are. Also, the proof of the theorem "Every infinite subset of a denumerable set is denumerable" is two pages long and very hard to follow.
2. (reflective) I am not quite sure of what use it is to know that a set is denumerable or uncountable. No purpose was given in the text, and I have a hard time finding one on my own. Since I am very tied to practical applications, this may be a difficult concept for me to grasp.

9.6_ 9.7, due March 5th

1. (difficult) Nothing was particularly difficult. I think permutations could get confusing quickly when the sets are much larger.
2. (reflective) I think the idea of an inverse relation is a natural following from learning of functions and relations. It simply follows that there should exist some inverse of them. Not surprisingly, the rules for these inverses also follow the same as the functional inverses that were taught in early math.

9.5, due March 3

1. (difficult) I thought it was frustrating that in this section there was no explicit definition of what it meant to be injective or surjective. I can kind of understand from the examples, but can't be totally sure because of the lack of definition.
2. (reflective) I think it is really cool that we can see the truths and rules of being derivatives and composite functions be proven by the relations and sets that we have used to prove so many things. It just goes to show you how all of these mathematics laws are truths as much as they are anything else.

9.1-9.2, due February 26

1. (difficult) One difficult thing about this section was that with all the talk of the function mapping the coordinate points in the relation, there was not a single graph given to illustrate the point. Therefore, I must assume that the actual graph matters very little to the purpose of the section. It would have been helpful in illustrating I think.
2. (reflective) It's pretty interesting that the number of sets from A to B is |B^A| = |B|^|A|. It's pretty cool that there is usually a way to calculate these things that would be insane to try and count from just writing down all the possibilities.

8.6, due February 23

1. (difficult) The explanation of the equivalence classes and their operations was super confusing at first, have [2]*[3]=[0] for Z_6, but it definitely makes sense now. The book does a good of job of explaining, especially with its likening the operations to a clock that resets at 12 'o-clock to essentially be a zero hour.
2. (reflective) The idea of being well-defined seems to be of little/no use, or have little application in real life. I would really like to have more application for these things in real life scenarios.

8.1 - 8.2, due February 17

1. (difficult) The most difficult part of the reading to understand is the use of the words Relation, Domain, and Range that departs from any other mathematical and english usage that I have ever used or been familiar with. It is actually very frustrating to try to understand what is being described when the words mean things that my mind does not immediately understand.
2. (reflective) I can't see any application yet for a relation to take on the properties of being reflexive, symmetric or transitive. Is there going to be a way to shortcut to figure that out? Because checking each value to be sure its inverse is also in the set seems to be a waste of time.

5.4-5.5, February 4

1.(difficult) Everything was pretty straight-forward. However, the proof the book gives of Result 5.24 is barely follow able. I could not see where the basis for the proof came from. It was confusing to say the least.
2.(reflective) There is one sentence that would have made all of the difference in my initial interactions with proofs: "In a proof, we are not required to explain where we got the idea for the proof". I often felt like I had to explain where my knowledge of the beginning of the proof come from. This is KEY information!

5.2-5.3, due February 2

1. (difficult) The most difficult concept I ran across in this reading is the in depth look at proof by contradiction. On the surface, and in the problems that they gave as examples, it looked fairly easy. But, after the story problem example, I was thoroughly confused. It seems in the story problem as if the proof by contradiction was shaky, and not totally convincing as a proof. I began to wonder if this is often the case with Proof by contradiction?
2. (reflective) I've  been looking for a chart like the one that is given in 5.3 to help a person to decide what course to take with their proof. I was actually contemplating making a chart myself, but this will be of use instead.

4.5 - 4.6 & 5.1, due January 30

1. (difficult) The difficult part is wrapping my head around the wording of the proof symbols for sets. It will take a little while to note what is actually being stated by set notation. Also, mixing cartesian products with set algebra is confusing.
2. (reflective) We are finally granted a way to disprove statements with the information in 5.1. It was difficult to just push that option in the back of my mind as we were constantly proving the statements given. Counterexample seems to be the only way something can be disproved, but I have been wrong before about what is available. I might get surprised.

4.3-4.4, due January 29

1. (difficult)  I don't understand result 4.16 and the subsequent proof for it. I read over it a couple of times, and I don't see how the proof arrived at. It seems to take the exact same expression and reworks it to prove itself. Something to the effect of: Prove 5x=y. Well, 5x-y=0, so 5x=y. It just doesn't seem very logical to me.
2. (reflective) The set algebra is tricky. I was honestly hoping that we'd left behind the set notations and intersections, differences, subsets, unions, compliments, etc... But of course we will be using them probably on and off through the course.

• How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
• I have spent roughly one hour on each assignment and one hour on readings and bloggings per assignment. However, the LaTeX assignments usually takes me longer than just writing it. The reading and lectures adequately prepared me for the assignments.
• What has contributed most to your learning in this class thus far?
• reading and writing blog posts about the material
• What do you think would help you learn more effectively or make the class better for you? (This can be feedback for me, or goals for yourself.)
• I often do the homework assignments during class lectures, so I would like to have them done either before or afterward and take that time to listen and ask my questions.

4.1 - 4.2, due January 27

1. (difficult) Modulo is a difficult concept to grasp, especially when the following are equivalent: 

a_= b (mod n)  and n|(a-b). It seems like such a strange way to express an equivalent expression. There must be some way that it is more convenient to use the "mod" in it.

2. (reflective) These mod statements mixed with proofs by cases can be very complex things, in fact the most complex we've studied so far. It's difficult to wrap your mind around at first, but it becomes easier after time is spent trying to really understand what the book is saying about the various statements. It's just going to be another tool in our belt to use while proving results.

3.3 - 3.4, due January 23

1. (difficult) I had a difficult time understanding the proof by contrapositive because it seemed like it was simply the opposite of trivial proof. In other words, the statement is always false when Q is false. I know that is not true, so seeing the truth table to prove P(x)->Q(x)---(~Q(x))->(~P(x)) was helpful.
2. (reflective) It is interesting that WOLOG only applied to the case where x and y were of opposite parity. It totally makes sense that the proofs are the same for simple proof cases where the x and y are of the same parity, etc... It will be interesting to see how complicated cases can get and still be able to use the WOLOG in the proof.