Last day due by April 11

A.Some of the most important topics this semester: Logic, Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove problems, Equivalence Relations, Functions, Cardinalities of Sets, Number theory algorithms,

B. One of the things that I would like to review is proofs involving the union of sets.
Also a review of the properties for a question like
Let R be a relation defi ned on the set Z by a b if and only if a2 + b2 is even. Then R is
(remember to choose the most correct answer)... transitive, symmetric, and/or reflexive.

C. This semester I have learned how grateful I am for the mathematics we have today. It is a beautiful thing to be able to solve life's problems and make it a better place by analyzing numbers and making calculations. I have learned that some of the simple rules and theorems we enjoy today are actually very difficult and complicated to prove. Some of the number theory we have covered has been my favorite thing we have studied.

Thank you Professor Jenkins for your help and teaching.

12.4 and 12.5 due by April 8

A. In the previous sections of this chapter I was able to follow the format of the proofs. However finding the real numbers ceilings has been difficult. The following sections require more of this. I would appreciate reviewing the steps to getting these problems started. The side work we do is great and I would like to be able to do it myself.
B. The other day I was able to have a chalk board to myself and it was great working out the problems. I could visually get a grasp on the arithmetic and process that was required to prove these limits.

12.3 due by April 5

A. I liked that there were a lot of examples. It seemed that it came down to rearranging terms and finding the right delta so that we can use some math to come prove by the closeness the limit. It looks like the trick to get things going is this first part of finding the right delta and placing it correctly in the inequality.
B. I like to rearrange terms and see how things simplify. These examples used a lot of this and I am excited to work through these and prove the limits.

12.1 due by April 4

A. It has been a while since I have studied Calculus and i feel that I have forgotten a lot of it. I am excited for this chapter though to help bring it back and be able to use it again.
B.Looking through this section there is a lot of arithmetic that looks very interesting.  A lot of working with the numbers and determining the divergence or convergence.

Test review, due by March 29

A. I think that the Schroder-Bernstein Theorem, division algorithm, and Euclidean algorithm are important topics that we have covered. These are the topics that we have built up to and the smaller theorems have helped us prove and use these.
B. I expect to see definitions, sets where we must prove are denumberable, or uncountable, and apply the theorems and algorithms we've learned.
C.On previous exams I haven't gotten full points on the definitions. This exam that is my goal. I would like to just do a quick run through of the terms we've covered in these chapters.
Thank you.

11.6-11.7 due on March 28

A. The part about perfect numbers didn't make much sense to me. I saw how they took the primes and added them but then wasn't sure where they went from there. It was pretty neat though that there was a connection to sports with the Ruth-Aaron pairs.
B. I find it very interesting the different ways to look at numbers. All of the divisibility tricks are really cool. It is fun to see shortcuts and strategies to determine the divisibility of a number. Its fun to impress friends with as well.

11.5 due by March 25

A. I found in this section and in sections previous, that induction has been used. This is done when there is a a1,a2,...an situation. The wording on these can get a little tricky for me. The proofs in section 11.5 looked a little more simpler but will require practice to get the pattern down.
B. I found it very interesting and fun playing with the numbers. When I have taken the time to dig a little deeper and plug in numbers it is pretty neat. At time it is hard to follow when it is just letters and abstract. But to see it in action is fun.

11.3-11.4 due on March 23

A. This section has a lot more definitions. In the last exams we have had I have struggled with getting the definitions right. I think that for this section I will write down multiple times the definitions in order to get a handle on the specific wording.
B. I thought the use of the Euclidean Algorithm was neat. It was like working through a jawbreaker. At first it appears big but going through layer by layer you get through the flavors and break it down.

11.1-11.2 due on March 20

A. I think the most difficult part will be keeping up in the homework. I haven't been able to devote all the time I would like to this class as I would like to and it becomes difficult when I fall behind. Keeping the proofs straight and doing them correctly for the Division Algorithm it looks will take paying attention in lecture and getting after my homework early.
B. I thought it was interesting that this dates back so far. It has been around almost back until the time Lehi left Jerusalem.

10.5, due on March 19

A. In Theorem 10.19 they ended using the Schroder-Bernstien Theorem to tie it all together but to get to that point it was hard to follow. It looks like they found a 1-1 function that maps from an interval to the powerset of N then defined a function going the other way so that the powerset of N maps to a set of real numbers within the interval. Then showed that that was 1-1 and applied the theorems to wrap it up.

B. From what I could see, Theorem 10.17 was interesting because is shows that just by restricting the range of the function to exclude the elements which make it not onto, it becomes bijective and numerically equivalent. It just takes the elements on the set within a set that make it work.

10.4 due on March 15

A. One of the things I found challenging with this section is the understanding what belongs to each set. It has taken time and practice to see what elements are in the sets we are trying to prove.
B. I think it is very interesting to have different sizes of infinity. Also the continuum hypothesis is interesting. because it can't be proven or disproven. I like the new symbols we get to use too- "aleph null."

10.3 due on March 13

A. I understand the general process for proving a function is bijective and numerically equivalent. The part I find difficult is understanding from the beginning what the problem is asking. Going from seeing the 2 sets then being able to pull that apart and prove they are 1-1 and onto gets tricky and its a bit hard to keep up in class.
B. I like the challenge of writing in LaTex some of the challenging charts and functions. I have found that online there are some helpful tips and hints to get it to come out right.

March 5

A. I think the work we have done studying functions is the most important. Functions are used everywhere and being able to understand them in a mathematical theory way will help with higher level classes.
B. I  think there will be questions addressing the definitions we have covered as well as results to prove by the different methods we have covered.
C. I would like to review the strategies for proof by induction.

9.6-9.7 due on March 1

A. In example 9.12 I had a hard time following the progression of a proof. I think I don't understand too well how to interpret it from the beginning but after that I could see how they collected the terms and got their result. So my challenge is understanding how to interpret that statement and begin but after that I can work through it.

B. I found it interesting that in determining the number of different permutations you can use factorials. I like to be able to see what the total number of different combinations is and then work to fill them in. Kind of like what we do in class with list the beginning and end and then fill in the middle. It is a good guide.

9.5 due on February 28

A. I think that some of the functions and their properties will be a little hard to remember. It talked about some calculus terms that have kind of gotten blurry. Over all I was able to understand the concept so maybe just the specifics will take practice to remember and work through.
B. I remember doing composite functions in the past. This is a new angle at on them and I liked how the examples and figure 9.2 were simple and I understood what was going on.

9.3-9.4 due on February 26

A. One of the things i found to be challenging on the test was applying the principle we learned in class to prove similar results. I had the general idea but the formalities held me up. I think that the methods of solving these proofs and the format in which they must be presented require time and practice to master. Thanks Professor Jenkins for going over the proof strategies in class.
B. I liked some of the names. They were unique and interesting. The differences between onto, surjective and bijective are important.

9.1-9.2 due on February 23

A. The challenging part in this section was the vocabulary at first. Then as I reread it and followed it with the diagram I was able to understand it better. Functions are where a relation A to B has special properties such that in the ordered pair the first coordinate a is not repeated for the different coordinates of b. Put in a and you get a b. b may repeat but not a. Figure 9.1 helped me see how this works.

B. I liked the formula for determining the number of elements in a set A to B. Plug and chug formulas are fun because in this case it is simple and easy to remember.

8.5-8.6 due on February 21

A. This section also called my attention to the fact that it builds upon the previous. The properties that are in this section are similar to previous ones only applied in a different way. The part that I think looks challenging is coming up with the number tricks that make the proofs simple and work. Also, there were a lot of definitions and theory but  I wasn't able to get a grasp as to the bigger picture and see what this is used for.
B. I did see and liked the part that explains that Division Algorithm because it shows how equivalence relations are used to determine remainders.  This was neat because it was clear and applicable. It made sense of something simple I have seen before.

8.3-8.4 due on February 20

A. With equivalence relationships, it builds on the first part of chapter 8 and ties it together by being reflexive, symmetric and transitive. The other night I worked through the homework and at times it was hard to see what meet the definitions of each. I think that I was able to understand it better and I think that with practice I should be able to understand this as well. The challenging part would be in building successfully on each previous section by staying caught up and on top of it.
B. I think it is interesting how these are building upon each other. The proofs seemed to me to have a pattern in which step by step you have to prove that it meets all the criteria and definitions and therefore the whole is true. It's like taking something apart to show how it works.

7.1-7.3, due on February 14

A.I think that going about a problem with the right strategy will be very important. A lot of time could be wasted if not done right and it takes you down a long dead end road. Some of the examples were really simple to see but to explain it gets complicated. It trying to come up with what you can visually see and put it into convincing logical statements.

B. The conjectures that stood for so long before becoming a theorem were interesting. I remember in high school talking about Fermat and Euler. I know that there are a lot of things that I think about and draw my own conclusions about but it sometimes remains at that. For one reason or another those ideas don't turn into reality. In reading about these conjectures I pictured these great mathematicians thinking about great number theory and having big ideas but they couldn't make the next step to becoming theorems for certain hold backs. Then years later some other mathematician picks up the idea and finishes what was previously started. How cool and makes me wonder what other interesting conjectures are still out there waiting to be proven or disproven.

6.3-6.4, due on February 13

A. One of the difficulties that I have been having lately is the amount of time that I take to do the homework. It is taking me on average 3 hours to do each assignment. I feel that I am beginning to understand the concepts but it is coming at a slow pace.  This is one of the things that is difficult and at times frustrating that I have encountered thus far.
B. I found the Proof by Minimum Counterexample to be interesting because it has the ability to prove something within an interval in which you know where it begins and the interval in which it is still true. So if you had an set where you wanted to know what the highest you could get where it is still true you could do it by this method. If you were budgeting and wanted to know the most of something you could buy while still being within budget it could be done in this manner.

6.1, due February 6

A. Mathematical proofs by induction require a lot of algebra. Common errors could be found in simple algebraic errors. I think that this would be a small fundamental thing that could be difficult. Just remembering how to do certain processes and which rules to follow is what I will need to remember.
B. I liked to story about Gauss in school where he added the sum of the first 100 numbers. Formulas are so cool when they simplify life and make things faster. I liked that.

1. I think the methods in which we have done the proofs have been important. There is a mental process that one has to go through to prove a statement, but the presentation is everything for it to have credit. I feel that presenting our work has been one of the most important things we have studied because it provided an essential foundation for this and higher mathematics.
2. In looking at the example of midterm 1, I saw as I expected questions on definitions and a couple of results to prove.
3. I think that I need to be more sure about the definitions. That I understand them correctly and can apply them in proving results. We have taken results and assuming certain conditions have used definitions to break it down and prove it. I am familiar with the definitions but at times they are still a little cloudy, mostly in set theory.  Also a refresher on cross products would be good.

5.4-5.5 due February 2

A. Some times it is hard to do proofs in which multiple cases are needed to be proved. It is difficult seeing at first what are all the cases that need to be proved and not forgetting any.

B.  The example of the students was interesting because there are time you know something exists however, by the means available it is difficult to prove. This example lead into the method of proofs that deals with proving a result based on the existence of the object possessing some specific property. Like in the example, a specific student possessing x number of hairs.

5.2-5.3 due by January 31

A. In considering the example for the proofs by contradiction it appeared to me that the first step is a difficult one to make. First, one has to assume some x in P(x) which makes R false. To get to that point, however it takes practice and familiarity with these problems to be able to identify what would make it false and how to show it. In short, experience seems to be a fundamental key to proving these results. Getting to that point is frustrating at times because of the growing pains and repetition involved but rewarding in the end.
B. This past week I have been wanting to draw up a table or diagram to depict how to proof a result. I like the table in 5.3 because it helped me see visually how the first step can be taken either wrong or right. "A journey of a thousand miles begins with a single step" as they say and it follow that with proofs. The workings of a great proof are made or broke by the efficiency of the first step. I liked that.

4.5,4.6,5.1, due January 29

A. In reading these sections, I think the hardest thing will be remembering the set operations and correctly applying them in proofs. The proofs we have been doing have a lot of steps in order to sufficiently define and clarify. I like the examples we are using and look forward the TA sessions. The proofs of sets operations are hard to follow. I drew a Venn diagram and that helped a little to see what was happening.
B. In 5.1 it was interesting how one disproves a statement. It looks to me that my careful examination and playing with numbers, you can conclude which elements make the statements false. An interesting example is when mathematicians tried to come up with a formula for deriving prime numbers. They had a good one until they found a count-example and latter offered a $1 million prize to anyone who could prove or disprove it.
 ( http://akoaotearoa.ac.nz/ako-hub/good-practice-publication-grants-e-book/resources/pages/using-counter-examples-enhance-learn )

4.3-4.4, due January 26

A. When I was studying chapter 1 and working through the problems, I kind of struggled wrapping my head around the concept of sets. I started to catch on but then we moved on to logic and now we are tying it back in. I think that it will be difficult at first getting going on these proofs.
B. I really like the proof strategies for the examples in the book. They explain the thinking behind the work and I would like to go back and carefully read through these strategies to try and align my thinking with theirs.
C. I usually spend an average of 1-2 hours on a given homework assignment. At times I get frustrated and have to break it up because I cant get my head around certain problems.  Using LaTeX, I find enjoyable but now it is taking a bit more time. I follow my lecture notes for proof strategy examples.
D. I think that working through problems helps a lot. I am excited to have a TA to help with the explanation. I think it will be very helpful to have someone to coach us on to success in the homework and in our mathematical logic lives.
E. Honestly I think that working in groups would help a lot. Having a study session with a TA would do wonders at least for me.

4.1-4.2, due January 24

A. There have been a lot of new definitions and symbols in these last couple of sections. I am really working on keeping up and keeping things straight but it is a little bit tough remembering what things mean and what is the proper way to go about a certain problem.  I enjoyed and found helpful the review we had at the beginning of class. Also is there anything happening in the math lab for our class? It would be great to work in groups to discuss strategies and get better acquainted with these style of proofs.

B. I thought the proofs involving congruence of integers were particularly interesting. It takes the divisibility of integers to the next step. By looking at the result and evaluating what it is saying and thinking about how to do it, they can be taken apart and proved. I like the proof strategy in result 4.12.

3.4-3.5, due on January 22

A. I think because this builds upon the previous things we have learned, the difficult part with be continued familiarity with these proofs. Practice makes perfect and the more we work them out the better we will see and develop on our own mathematical proving strategies and techniques.  In the text, it stated that part of learning mathematics is making mistakes. We all do but what makes the difference is learning from and not repeating them. There were examples of finding mistakes that others have made and that might be difficult if is a similar mistake to the ones I might be making. It would be important and helpful  to be able to identify and correct those types of mistakes.
B.  These problems and exercises that we have done are building blocks in a foundation for more complex mathematics. It was very interesting to me to see how the previous subjects we have studied have been applied and built upon. We have studied sets, basics proof strategy, and logic, and now we are putting them together to prove more complicating exercises that require this foundation.

3.1-3.3, due on January 19

A. I think that it will be challenging is correctly documenting the proof strategy and/or analysis. This is done to show and explain how certain conclusions were reached in the proof. At times, things work out in our heads but getting that on paper can be something entirely different. I think this will be very important to do correctly and recognize that it will take practice and patience.
B. I thought the section talking about Q.E.D. was neat because it brought me back to the 500 hall in Douglas High School sitting in front of the a little old lady who skydives. I remember math class with Mrs. Barnes and first learning about proofs and how to finish them off with Q.E.D.

0, due January 17

A. I had previously read this part in previewing the book and a couple of things caught my attention. The way we present our information and proofs are critical in communicating correctly and effectively right solution. If stated improperly it could be hard to interpret or most likely incorrect. The language of correct notation for answering problems is difficult. In my first homework assignment I struggled with this because I was very unsure of how to do it. Now, I feel I have more practice and I am trying harder but still I would like to greatly improve and get it right.
B. As stated in a previous post, the new symbols caught my attention. I thought in was interesting the how and why of using the symbols. Also, I thought it was challenging but interesting the wording and mathematical definitions of words in writing (example: or, that vs. which, since/then, etc.) and I look forward to being able to correctly use it. 

2.9-2.11, due on January 12

A. I got my first homework back and one of the things that is challenging to me is the notation. These sections have a lot of theorems and ways to make statements in which the correct notation is critical. I found this to be a bit difficult to follow and to effectively carry out in a problem. The examples in the quantitative statements seem to have a procedure that ensures correctness and shows clarity and I want to be sure to understand and able to do it.
B. When first flipping through this book for the first time, the upside down and backwards letters definitely jumped out at me. It made me chuckle because I had no idea what they meant and what the symbols were. Now, reading through the sections I came upon them again and this time was able to get an idea of how I could use the neat backwards and upside down letters to make mathematical statements through quantification.

2.5-2.8, due January 10

A. In reviewing the exercises for the sections I read, there was a lot of work that would require careful organization and mastery of the principle and rules of logic. I feel this could be a challenge in doing the work correctly if I don't understand fully how use the rules of logic. I think the biggest hold up is making the connection between the use of sets and the statements were are studying and getting it right. Sometimes, when I do the set problems I have a hard time understanding where to start and what I am doing. Now, taking it a step further will require more practice and dedication.
B. In the previous homework and in the reading I loved the use of the truth tables. They helped me so much and I understood them. They make the organization and visualization a lot more clear. It seems that that is key to working through the problems and getting the desired result.

2.1-2.4, due January 9

A. The part that I feel will be most difficult will be getting a good handle on the vocabulary and the proper use of it. As I read, I was impressed by the importance of clearly presenting information so that it makes sense to not only you but to others. That is a point that is very important and I feel that it will take some getting used to and a lot of practice to reach that proficiency and accuracy but it is needed and expected.
B. The part that interested me were the truth tables. I liked the visual representation of what was being stated and the options for the statements. This technique is a helpful tool in organizing thoughts and verifying the statements. For example, in the section on implications it was discussed what it is and how to write it. It was seen clearly as a table was drawn and the story backed it up. Very applicable too. How often are worried about our grades and now we can draw a cool table to show our options. Art is a great stress reliever right?

1.1-1.6, due on January 5

A.    As I read through the first chapter, there was a lot of new terminology that I was unfamiliar with. Also remembering some of the terms that I have not used for over 2 years was a bit challenging. The book moved really fast through these concepts and new terms which was a bit tough to keep straight. However, with practice and reading some parts out loud I will get a handle on it.

B.     To me, I found interesting the use of Venn diagrams. At times when some concepts were a bit hard to follow, the use of the Venn diagrams help a lot. Visually, sorting out the information in a way that is easier to read and to see the operations being made was neat.

Introduction, due on January 5

My name is Derrik Jenkins and I am a mechanical engineering major.
I studied for 1 year, then completed a two year mission, and I have just returned to begin my sophmore year at BYU.
I took AP calculus in high school and Math 113 at BYU my freshman year.
I am very interested in attaining a minor in math to go with a degree in mechanical engineering. I enjoy math and would like to get a firm grasp on mathematical concepts that will build a solid foundation for further engineering and mathematical application to what I study and work with.
I greatly enjoyed the examples that were used by previous professors. They explained the concept, then showed how to work out the problems. Problems similar to those we would face later on in the homework or in test situations. Also the past final exams were available to review and study.
I feel that the lectures were at times vague and the why of certain application of principles weren't explained very well. The concepts we covered were shown but not explained very well. However, the TA's did a fantastic job reviewing the material and helping with homework questions.
Well, not my last name. So, I love to be in the outdoors- running, biking, climbing, hiking.. also, I can tear a phone book in half.
I have class until 3pm, so any time after 3 or before 10am would work for me.