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.

11.3-11.4, due March 23

Difficult: The most difficult part in this reading in my opinion was the Euclidean Algorithm in section 11.4. i am not really sure I understand anything about this Algorithm. I get that it is a continuation of the idea behind the division Algorithm but I'm not sure if I really understand that completely yet either... i would like to see lots of examples and applications of both algorithms.

Reflective: I liked section 11.3. This section talked about the common divisor and the greatest common divisor. I liked this part because both of these concepts are simply to understand and I have used before many times. Therefore it made it easier to know what the book was talking about in the rest of the section.

11.1-11.2, due March 21

Difficult: The most difficult part of this chapter is the division algorithm theorem presented in section 11.2. This theorem on the surface seemed as if it should be fairly simple to understand. However, after I looked at it and studied, I came to the conclusion that i did not really understand the logic and how the conclusion of the theorem was made. I kind of have a little bit of an idea as to how we get the first part that b=aq+r but I am not sure I understand why the condition that 0 is less than or equal to r which is less than a needs to be included.

Reflective: My favorite part of this section can be found in 11.1 about the primes and composite numbers. I actually don't remember if I have ever hear integers that are not primes be referred to as composite numbers, but in a way, the name makes sense. What I really liked though was Theorem 11.2. i liked this theorem because it reminded me of the conditions for whether or not relations can be categorized as being reflexive, symmetric, and transitive and therefore be equivalence relations.

10.5, due March 19

Difficult: In this section, I really did not understand the whole introduction that was talking about the restriction f1 of f to D. I am so confused that I cannot even seem to decipher what it is that I am confused about. I just don't understand what it is saying at all.

Reflective: In this section, i thought that the Schroder-Bernstein Theorem was relatively easy to understand. Because in order for two sets to say that A is less than or equal to B and that B is also less than or equal to A, then it is obvious that we can conclude that A and B must be equal to each other.

10.4, due March 16

Difficult: The most difficult concept for me to understand in this section of reading is Theorem 10.14. Pretty much I don't really get any part of this theorem. Part of the problem I think is that I am kind of having a difficult time figuring out the principles of something being uncountable. Also I think I still feel sort of confused on what exactly the set 2^A consists of.

Reflective: I liked Theorem 10.15 in this section. It just seemed to actually make sense whereas some of the other theorems I don't understand the logic or anything. But it makes sense to me that the cardinality of a set A would be less then the cardinality of the power set of that set A.

10.3b, due March 14

Difficult: The whole idea that sets (0,1) and R could be numerically equivalent seemed really weird to me at first. I have a hard time with the abstract sense to this proof. However, when I would think about them each as two separate sets, (0,1) and R, and focus on them individually then I understood the logic a little bit better.

Reflective: What I liked about this proof was how it used concepts from calculus to help prove it. I really like calculus and more mechanical forms of math that focus more on arithmetic type stuff. I have an easier time understanding those ideas because they seem less abstract to me. This sense of familiarity helped me understand the idea behind this proof.

10.3a, due March 12

Difficult: I don't understand when something is uncountable. I think I get when things are denumberable since I can find a bijective function with the natural numbers, but I have a hard time understanding when you can tell that something is uncountable without just thinking about it. Most of the theorems for this section i didn't understand at first, and sometimes even with the proof i didn't understand it, but then I would sit and think about it in relation to a denumberable set and then I could figure it out. Is there a different way to think about it though?

Reflective: I liked the part that talked about rational and irrational numbers being written as decimals. I think that this will help me with differentiating between rational and irrational numbers in the future because it is an additional way of thinking about it that I never really considered before. This also, in theory, helps with understanding the whole idea of something being uncountable.

10.1-10.2, due on March 9

Difficult: The most difficult concept for me to understand in this reading is the portion about when sets are denumberable in section 10.2. I think the main thing that I am unsure of regarding denumberable sets is the idea of sets that are infinite. Also, I do not really understand what it means for a set to be denumberable, as in how that affects the set.

Reflective: The easiest section of this reading was 10.1. I thought that this section was easy to understand because it is based on what we already know and understand regarding cardinality. It just shows a new way to figure out if two sets have the same cardinality. Then it introduces the term numerically equivalent sets and defines it as two sets that have the same cardinality. All of it is basic and since I already have an understanding of what cardinality is, it was simple to build on that knowledge.

9.6-9.7, due March 2

Difficult: The most difficult part from this reading assignment had to do with permutations in section 9.7. I guess I feel like I am just missing something for this principle. I understand that a permutation is a bijective function so it has to be one-to-one as well as onto. However, I don't understand what the difference between a bijective function and a permutation is. If there is no difference, then why do we need to learn both?

Reflective: For this section I liked reading about inverse relations in section 9.6. I liked inverse relations because it was easy to understand. Its terminology relates to its definition and what it does. This made it make sense very quickly. This then made understanding the idea of an inverse function relatively easy since I already had the foundation for it. However, I would like to work through examples of inverse functions and apply both inverse functions and relations to help my understanding of how to use them.

9.5, due February 29

Difficult: I thought that the most difficult part in this section was the part about associative functions. I think I understand the basics of the idea but I just thought it was difficult to get a clear understanding of the information simply by reading it.

Reflective: I liked reading the part about composition. I liked this section because I already understand the foundation of the principle after having taken calculus. Plus the idea is fairly simple and makes sense to me without having to go into too much detail.

9.3-9.4, February 27

Difficult: The most difficult part of this reading would probably the part about identity functions in section 9.4. At first this concept seemed rather simply, but the more that I read about them, the more that I got confused. I understood the part in the section about bijective functions for the most part, however I am still a little unsure about all of the details on a surjective element. What are the similarities and differences between one-to-one and onto functions?

Reflective: I thought that the most simple concept to understand for this reading was one-to-one functions. I liked this part in the reading because I already have a foundation for this concept. After taking calculus and studying what entails something to be referred to as being one-to-one, it was rather simple to apply new terminology to the old idea. Although I think I understand the basics for onto functions, I'm not very confident with them.

9.1-9.2, due February 24

Difficult: The difficult section for this reading was 9.2. I just don't think i really understood most of it. I didn't understand what it meant by B^(A) and why that notation was chosen or what it really meant.

Reflexive: I enjoyed section 9.1 for this reading which was on functions, domains, codomains, ranges, images, and mapping. I felt like all of these terms were based of fundamental concepts that I already understand. However I don't think that I understood why for f: A > B that A is the domain and B is the codomain.

9.1-9.2, due February 24

Difficult: The difficult section for this reading was 9.2. I just don't think i really understood most of it. I didn't understand what it meant by B^(A) and why that notation was chosen or what it really meant.

Reflexive: I enjoyed section 9.1 for this reading which was on functions, domains, codomains, ranges, images, and mapping. I felt like all of these terms were based of fundamental concepts that I already understand. However I don't think that I understood why for f: A > B that A is the domain and B is the codomain.

8.5-8.6, due on February 22

Difficult: For this reading I found section 8.6 more difficult to understand. This is probably partly because I still feel a little unsure of things when it comes to equivalence classes. This probably stems to the confusion regarding how equivalence classes relate to residue classes. I think that once I get more comfortable with equivalence classes though then this section will make more sense because I will have more of an understanding of the foundation for it.

Reflective: I thought that section 8.5 in this reading was very easy to understand. This is due to the fact that I have already learned the ideas and concepts behind the information that is given in section 8.5 regarding congruence when dividing and modulos. Then the section applies all of this information to relations, without really adding any new information.