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.

8.3-8.4, due on February 21

Difficult: Although I found the idea about equivalence relations easy to understand, I got confused when the concept of an equivalence class was introduced. I don't understand what exactly it means for something to be categorized as an equivalence class and what makes it different from an equivalence relation.

Reflective: I thought that the concept in section 8.3 about equivalence relations was very simple to understand. Since we have already learned about the relation properties reflexive, symmetric, and transitive, I already had all of the main ideas of the concept and just needed to apply them in another way where all three are included in the definition. 

8.1-8.2, due February 17

Difficult: The most difficult section in this reading was 8.2. The beginning of this section was not difficult at all to understand regarding the properties of reflexive relations and symmetric relations. However, I thought that transitive relations were a little confusing.  I'm not really sure what exactly it is referring to and what it means and then as a result helps us

Reflective: My favorite section in this reading was 8.1 about Relations. I liked this section best because it was easy to understand since I already know what it means for something to be a subset of A x B. So it was easy to understand the principle of relations because it is so closely related Cartesian products.

7.1-7.3, due February 15

Difficult: I did not think any of these sections were very difficult when I read them. However, I am afraid that section 7.3 about testing statements will be difficult when actually applying the idea to examples. This is mostly based on my experience with proofs thus far in this class. It seems like the strategies for proving or disproving statements can vary and sometimes I have a problem figuring out which techniques work best in certain situations.

Reflective: My favorite section in this reading assignment was 7.1. I liked how 7.1 talked about conjectures and palindromes. Since both of these terms were very familiar to me before I read this section I found it easier to understand the concepts that were being taught. I also like how the mathematical applications were introduced by real world applications such as example words and sentences that were palindromes. Also, I thought it was cool how a two digit number that isn't a palindrome, when added to its reverse it can create a palindrome. Or if the process is repeated enough then a palindrome will eventually result.

6.2, due February 10

Difficult: The difficult part of this section is the actual application of the principle of mathematical induction. I don't really understand the logic behind it and how it works or perhaps why it works.

Reflective: I understand the idea of the principle of mathematical induction. I understand the background information such as the Well-Ordering Principle. It was helpful to read this section after having had the induction background from the last section.

6.1, due February 6

Difficult: I thought the idea of whether something was well-ordered was the most difficult concept. I specifically do not understand the part in the book where it says that the closed interval [0,1] is not well-ordered because (0,1) is a subset of [0,1].

Reflective: The easiest part of this section was the principle of a least element. This is so similar to just the idea of finding the number that has the lowest value in a list of numbers. Therefore, it was very simple to make the conclusion of it being the lowest element in a set.

The topic that I think is the most important for this class is simply the discussion on statements. I chose this topic because this class is all about analyzing different results in math and figuring out their truth value.

The kinds of questions that I expect to see on the exam would be just like the homework examples. Ones where we will be given a statement which we will need to prove, perhaps with a given type of proof technique. Also I would not be surprised if there was a question that asked for a given proof to be evaluated.

I mostly need to work on figuring out how to decide which approach to take for different kinds of proofs. Not really how to start but how to completely answer the proof. For example, when to use more than one case and when to make the proof draw more than one conclusion. I would like to see a problem worked out like to see some trivial and vacuous proofs worked out.

5.4-5.5, due February 3

Difficult: The more difficult of the two sections in this reading was 5.5. Although it was the harder concept of the two, I feel like I have a pretty good understanding of the idea. Therefore, the only reason why I would classify it as the more difficult concept is because it takes the idea presented in section 5.4 and then builds the next step onto it. Basically, just that instead of stating that there exists something, you're proving that there is not anything that exists that is the opposite of the statement.

Reflective: I thought that section 5.4 was rather easy for this reading. Basically it is simple to understand the idea of an existence proof. It is not proving what example it is that makes the statement true, but just that there is something that exists where the statement is true. Also, existence is a common word in most people's vocabulary, which makes it easier to understand the idea because of previous knowledge.