due April 11

I think that the most important topics and theorems that we have studied are the different kinds of proofs that we can use (direct, contrapositive, contradiction), the Principle of Mathematical Induction (SPMI as well), Equivalence relations/classes, bijective functions, and the Schroder-Bernstein Theorem.

I think that I need to understand chapters ten and eleven more before the final exam. I don't think I understand very well the different ideas and logic behind the uncountable proofs among most of the other materials in chapter eleven. I thought that I did well on the midterm which covered these chapters as I was going through the problems, I felt like I understood them. However, my score was the lowest out of all three midterms. So I would like to spend most of my time reviewing these principles and ideas.

The problem that I would like to see worked out would be one like 21 on the last exam where we needed to prove that the square of every odd integer is of the form 8k+1, for some integer k.

For this course, I have made some connections with previous theorems and logic that I have been taught in other math classes but never went into the details that made up the proof. However, now I think as I continue to take math classes I will be able to see connections with future theorems that I learn and be able to hopefully understand them better than I would have without having taken this class. Therefore, I hopefully will be able to apply them more effectively to problems I am given.

12.4-12.5, due April 9

Difficult: The more difficult part of this reading was section 12.4. I am having difficulty with this section with understanding the different proofs behind the theorems. I can understand a little bit more how to apply these theorems for different types of math problems, however, I have some trouble with the breakdown behind the logic of how these theorems have been proved and how they have been constructed.

Reflective: My favorite part of this reading was section 12.5 about continuity. I liked this best because I remember learning this term and its definition in other math classes such as calculus. I think that it is fairly easy to understand, especially when put in the simple terms as "can you draw the graph without lifting up your pencil?".

12.3, due April 6

Difficult: I did not understand the part about having a deleted neighborhood. I did not understand what this referred to and what exactly it meant. I am assuming that it has to do with finding limits, but other than that I do not know anything about them.

Reflective: The part of this section of reading that I thought was the easiest was the discussion on Limits. I thought that this was relatively easy to understand after having taken calculus and other math classes where I have worked with limits before.

12.1, due April 4

Difficult: I did not really find any specific parts of this reading all that difficult to understand since I have taken calculus before. If I did have to pick though I would probably go with the terms converge and diverge. I remember learning about these and it being very simple, however, I would like to go over them again to be sure that I completely understand them.

Reflective: My favorite part of this section was the discussion of the concept of ceilings. There is nothing really all that special about them per se, however I just remember really liking them when I did calculus. I would however like to know if there is any particular reason that you would want to use a ceiling for a result, or if it is just a preference thing.

due March 30

The most important theorems in my opinion are the theorems from chapter ten, especially theorem 10.17.

The kinds of questions that I expect to see on the exam would be some multiple choice that have to do with mostly definitions and then some proofs for telling if something is denumerable or not and a proof about the cardinality of sets.

What I need to understand better would be concepts such as the schroder-bernstein theorem, uncountable sets, and the division algorithm. I would like to see examples worked through for all three of these but particularly the schroder-bernstein. Perhaps see a proof using schroder-bernstein that shows that |(0,1)|=|[0,1]|

11.6-11.7, due March 28

Difficult: I did not really find any of the topics in either of these sections very difficult. Although I feel like I understand the Fundamental Theorem of Arithmetic, I feel like it is probably the most difficult concept from this reading. I get what the Fundamental Theorem of Arithmetic means and the logic makes sense to me. However, I think I would like to see some more examples for how to use the Fundamental Theorem of Arithmetic and the various ways to apply it.

Reflective: My favorite part of this reading was the part about perfect integers. It is extremely easy to understand the idea behind this concept. Also, I just think it is really cool how an integer can have the sum of its proper divisors be equal to the number itself. I am curious as to how frequent this occurrence is or if it is more of a rare incident.

11.5, due March 26

Difficult: The most difficult portion of this section actually kind of surprised me. I do not understand Theorem 11.12, like at all. I get that that theorem seems to lead to the term relatively prime, but I do not understand how the theorem is useful or how to apply it to prove anything. Also, the whole part about its converse also being true did not make any sense to me.

Reflective: I liked reading about Euclid's Lemma in this section. I am not entirely confident that I completely understand this concept, but I feel like I at least understand a base. But the main reason why I liked this lemma is because I thought it was really cool how you cannot say that just because a|bc that a|b and/or that a|c. But if a|bc and gcd(a,b)=1 then you can conclude that a|c.