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.

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.

5.2-5.3, due on February 1

Difficult: The most difficult section in this reading is section 5.2. However, I would not say that 5.2 is actually a difficult idea to understand. For one, the concept of a contradiction should not be new. Therefore, it is easy to apply previous knowledge and experience regarding what constitutes a contradiction to this mathematical application involving proofs. That being said, I would have liked the example to have been more consistent when it came to the variables that were used. At first the variable used are P and Q, but then the book jumps to R and C which caused me to be confused for a bit.

Reflective: I particularly like section 5.3. I thought that it was very helpful to me regarding how to begin proofs. Sometimes, the most difficult part of a proof is simply deciding how to approach it. I also liked the review of all the different kinds of proofs such as direct, contrapositive, and contradiction. I honestly wish that this section had come earlier in the course.

4.5-4.6,5.1 due on January 30

Difficult: The most difficult section from this reading assignment is section 4.6 about proofs involving Cartesian products of sets. I'm mostly just not sure exactly what the book is trying to say. I think that it is talking about that if A x B then it will equal every integer in set A and every integer in set B. But I'm just not a hundred percent sure.

Reflective: My favorite section in this reading was 4.5, which might come as no surprise. I liked this section because it was familiar since it reviewed Commutative Laws, Associative Laws, and Distributive Laws. All of which I remembered learning about in previous readings. Although, I did not remember De Morgan's Laws, it appealed to my love of history and reminded me of J.P. Morgan, so it was still familiar and less intimidating. Basically, this section was just a review. So I was simply building onto a foundation that I had already learned previously.

4.3-4.4, due January 27

Difficult: I thought that both sections for this reading were not too complicated. However, the one that was more difficult was probably 4.4. I think the main reason why this one was more difficult is because I have less experience with sets and their intersections and unions. I really think that the Venn diagrams help me visualize the different ideas that are being presented. I think what is most difficult is actually knowing when to apply each different method and then simply how to start or approach each proof.

Reflective: I particularly like 4.3 for this assignment. I thought that 4.3 was essentially a review which was nice. I like how the book actually takes the time to make sure that old principles are refreshed and that they are actually understood and a base for any concepts that may relate to them. By having a review it made new ideas that were introduced much easier to understand because I already had a foundation. However I am slightly unsure of what exactly a triangle inequality is.

How long have you spent on homework assignments? Did lecture and the reading prepare you for them?
On average I have spent about 2-3 hours on homework assignments. I think that both the lecture and the reading were helpful. The reading was helpful because, although I may not have understood everything, or even really anything, after reading the chapter, the ideas were still introduced. That way, during the lecture I was not starting from complete scratch. However, I think that the lectures are the most helpful for teaching me the concepts. I understand much better with that type of teaching rather than just reading from the textbook.

What has contributed most to your learning in this class thus far?
I would say that the lectures have probably contributed most to my learning so far. Also, the problems from homework of course have been helpful in practicing the different concepts on my own.

What do you think would help you learn more effectively or make this class better for you?One thing that would be very helpful to me would be to work through a problem on my own for a bit in class and then come back together as a class and work through the problem. That way I would be able to try and see if I knew how to do it and then got to check it and see where I was right and where I may have gone wrong. That way I would have branched out on my own a little before I was completely on my own.

4.1-4.2, due January 25

Difficult: The difficult part of this section's reading was 4.2.  For the beginning of the reading for 4.2 I was on track since it was mostly just review of past sections. However, I started to get a little lost when the book moved to introducing the two terms congruent and modulo. I do not understand what the definitions of these two terms are. I assume that congruent has the same meaning now as it has in previous math concepts, but I was not able to verify that for sure. On the other hand, I have no idea what modulo is. As I continued to study this section on congruence of integers, I started to notice a pattern for modulo. But I do not trust it without a secure grasp on the term's definition.

Reflective: I liked section 4.1. I thought that it was written in a way that it made sense to me. This section was probably also easier to understand because the terms that were introduced here were already familiar words so I just had to understand their application for this situation. The idea of this section relates directly to algebra which is actually one of my favorite areas in math. Therefore, it was easy to draw parallels between the new concepts being introduced with knowledge from previous algebra classes.

3.4-3.5, due January 23

Difficult: Section 3.5 was the more difficult section of the two for me. It was hard for me to understand exactly what was going on with having the result coming first. It seemed like the book was trying to make it be different, but in a way it seems that for most proofs you start with the end result and then go about trying to work your way to prove that result.  So I didn't really understand what they meant by that.

Reflective: I like the section about proof by cases and subcases found in 3.4. What I like about this is the idea of breaking things down into smaller pieces. This reminds me of art paintings that are composed of just a bunch of dots. Where when you look up close, all you see are all the different colors of dots, but then when you step back you begin to see the whole picture. This seems to be the same way of these types of proofs where there are a bunch of little pieces, or dots, that come together to form the whole picture.

3.1-3.3, due on January 20

Difficult: I did not really understand what the difference between a trivial proof and a vacuous proof is.  I think I picked up that for a vacuous proof every x that is an element of s for P(x) implies Q(x) but where every element is false for P(x).  But I didn't really get how to define a trivial proof.  So I would like it to be a little more clear as to how to differentiate between the two.

Reflective: My favorite section for this reading was 3.2 on Direct Proofs. I thought that it was helpful how the text introduced a small part of a direct proof, followed by another, and so on.  By breaking up the different aspects for the conditions of the proof I was able to follow the thinking and the explanation easier than in other past times.  I also liked how their addressed the section about properties of integers as something that was "familiar" and "elementary".  It seems sort of ridiculous but just by the text alluding to the simplicity, made the whole thing seem easier to understand.

Mathematical Writing, due January 18

Difficult: This section of reading was rather simple to understand, especially since the basics all relate to writing instructions which have been given in English class since elementary school.  However, perhaps the only semi-confusing portion of reading in this section was the part about the difference between when to use "than" or "which".  It sounds like for the majority of the time, than is usually preferable. Although I liked the example separate from the usual mathematics samples about the favorite math textbook, I would have liked more examples that did directly relate to mathematics and how each of these words would be applied to writing math.

Reflective: The comparisons in this chapter to everything that I have learned in English class for writing the common essay are extensive in this section of reading.  The first part I made this connection was in the discussion about having an outline before beginning to write down anything. How many times have I heard a teacher suggest creating an outline to organize my thought before I began my paper? An even more simple connection was that of simply capitalizing the beginning of a sentence. Then, more importantly, word choice. Whether it's deciding between "than" or "which," this chapter emphasizes the importance of choosing the right words and constructing each sentence's structure correctly so that it reads well and makes sense.

2.9-2.11, due on January 13

Difficult: The most difficult part of this section of reading was 2.10 which is on Quantified Statements.  I don't really understand what it does beyond making an open sentence into a statement. I do not understand what the different symbols for a universal quantifier and an existential quantifier. I understood the part of the section that talked about whether or not a quantified statement was true or false. But I am not sure I could apply it since I don't think I understand what it means to be honest. I also found that I have been getting a little confused about adding R into a truth table. I am not always sure if I am doing it right or not.

Reflective: Section 2.9 about the fundamental properties of logical equivalence mentions commutative laws, associative laws, and distributive laws.  When I read these three different types of laws, they immediately sounded familiar and connected me back to other principles that I remember learning.  Unfortunately, I cannot seem to pin point what exactly they remind me of.

2.5-2.8, due on January 11

Difficult: Although I would not really say the concept is difficult, the hardest idea for me to completely understand in this section of reading is the biconditional that is found in 2.6.

Reflective: The principle of a contradiction found in 2.7 is just like a normal contradiction that would be found in English.

2.1-2.4, due on January 9

Difficult: Probably the most complicated concept to wrap my head around in this section of reading is the distinctions between disjunction and conjunction statements.  Although I understand it just fine, there was just some parts that I had to reread and make sure I understood the different symbols and the conditions.

Reflective: What I like about this section of reading is the fact that it doesn't have to do with theorems or principles necessarily, but yet these statements are so important to the rest of mathematics. Without the use of statements it would be difficult to teach the different concepts and principles that make up math.

1.1-1.6, due on January 6

1. Difficult: the most difficult concept in this chapter was subsets.  I keep thinking I have it down, but then I just get confused.
2. Reflective: In a way sets kinda reminds me of sigma notation that I learned in math 112. It is the same idea of using a list to show a group of objects like a pattern. This connection helps me increase my understanding of both of these principles.

Introduction, due on January 6

• I am a junior and am majoring in Secondary History Education
• I have previously taken math 112 and am currently enrolled in math 113
• I am taking this course for my minor which is math.  I am minoring in math so that I can also qualify to teach the course in secondary schools if a history position is not available.
• My math teacher who was most effective was my calculus teacher in high school. He was kinda crazy and would shoot kids with a water gun which made class entertaining.  But he was effective as a math teacher because he would go over a concept and then assign a lot of examples for that one principle as homework which really ingrained the methods and techniques into my brain.
• I cannot straighten my arm completely. It is always crooked.
• Your office hours work for me.