For the final exam, there is so much to choose from as far as topics and theorems go. It is comprehensive, so we could really see anything that we've studied all semester. I think that the most important topics are the following, in no particular order:
1. Sets and subsets
2. Set operations
3. Proofs: direct, contrapositive, and contradiction
4. Equivalence classes
5. Relations: reflexive, symmetric, and transitive
6. Functions: injective, surjective, bijective
7. Sets: denumerable, countable, uncountable
8. Linear combinations and the division algorithm
I feel like the class was well-organized, with principles learned at the end of the semester having been effectively built on a foundation of what was learned earlier. When I look back at the semester, I think that I need to work on many things to better understand the material. I have read all of the readings and am familiar with the material, but I just don't know it enough to perform well on tests. I think that I really need to practice, no matter how time-consuming that may be, many sample problems. Especially the proofs. I need a variety of sample problems and I need to be able to work them out on my own without study aids.
I know there are study guides for the first three exams, but they don't include the last chapter we studied. Perhaps I'd like to see a sample problem of a typical calculus proof that might appear on a final exam, if possible.
In this course, I have learned a lot. The material was interesting, for the most part, and the professor was really great. Rather than list specific mathematical topics that I learned, I can say that, overall, I learned that I am not as strong at mathematics as I thought or I hoped I would be. For some strange reason, the material never made a permanent dwelling place in my mind. It would visit, stay awhile, and then leave. There are many bright mathematical minds and I really look up to the teacher and some of my classmates for how they just "get it" so well. So, I guess what I'm trying to say is that I learned that I didn't easily "get it" in this class and I've come to be okay with it. This will help me as I modify my future academic and career paths.
Sincere thanks for the good class!
Saturday, April 13, 2013
Thursday, April 11, 2013
* Make Up For Missed Reading Assignment * - Talk by Daniel Willingham
I attended the talk "What every student and teacher should know about human memory" by Daniel Willingham at 4:00 PM in 1170 TMCB on Tuesday March 12.
This was a topic that really interested me because - and this might sound silly - I feel like my memory isn't as good as it used to be. I was interested to see if he talked about why that could be. I already feel like an older undergraduate student at 32 years old. Particularly in Math 290 I haven't been able to memorize as well or remember things on tests like I feel like I could as a younger student. I was wondering if that had to do with "old age" (ha ha) or why it might be.
Anyway, the talk was very interesting. I thought that Daniel Willingham was a very good speaker, presenter, and teacher. He had a lot of applicable advice for his target audience. Three of the main points he made were:
1. Memory is the residue of thought.
2. Repetition, but repetition of the deep-thinking.
3. Testing yourself is good.
I've applied the "testing yourself" idea ever since I heard his talk and I think that is effective. It seems to really have prepared me better for my exams. Another suggestion I've tried is explaining the material to someone. I've tried this with my wife and, although she couldn't care less about the subject material, she listens and it's helped me know the material better through explaining to someone else.
Another thing he mentioned is that students stop studying when they get the feeling of knowing, but that this is not always reliable. I've definitely fallen victim to this false sense of security. I think I know something because I am familiar with it and then I go into the test and have forgotten, realizing that I might not have known it that well after all. I am only familiar with the material, not a master of it.
The final thing that I'll mention is that he talked about "overlearning," meaning he suggested still studying even after you know it. I agree that the advice is very good and would indeed help in earning A's on exams; however, I kind of had to laugh because I feel lucky when I'm able to put in just enough time to know the material, not "overlearn" it. I feel like I'm far too busy. So, if life ever slows down for me enough to "overlearn" something, I'll definitely do it - that is, if I can remember.
This was a topic that really interested me because - and this might sound silly - I feel like my memory isn't as good as it used to be. I was interested to see if he talked about why that could be. I already feel like an older undergraduate student at 32 years old. Particularly in Math 290 I haven't been able to memorize as well or remember things on tests like I feel like I could as a younger student. I was wondering if that had to do with "old age" (ha ha) or why it might be.
Anyway, the talk was very interesting. I thought that Daniel Willingham was a very good speaker, presenter, and teacher. He had a lot of applicable advice for his target audience. Three of the main points he made were:
1. Memory is the residue of thought.
2. Repetition, but repetition of the deep-thinking.
3. Testing yourself is good.
I've applied the "testing yourself" idea ever since I heard his talk and I think that is effective. It seems to really have prepared me better for my exams. Another suggestion I've tried is explaining the material to someone. I've tried this with my wife and, although she couldn't care less about the subject material, she listens and it's helped me know the material better through explaining to someone else.
Another thing he mentioned is that students stop studying when they get the feeling of knowing, but that this is not always reliable. I've definitely fallen victim to this false sense of security. I think I know something because I am familiar with it and then I go into the test and have forgotten, realizing that I might not have known it that well after all. I am only familiar with the material, not a master of it.
The final thing that I'll mention is that he talked about "overlearning," meaning he suggested still studying even after you know it. I agree that the advice is very good and would indeed help in earning A's on exams; however, I kind of had to laugh because I feel lucky when I'm able to put in just enough time to know the material, not "overlearn" it. I feel like I'm far too busy. So, if life ever slows down for me enough to "overlearn" something, I'll definitely do it - that is, if I can remember.
12.4 and 12.5, due on April 12
I find that these particular proofs in calculus are difficult to read speedily. I often have to reread the proofs in order to understand them fully. Also, I find myself rereading individual sentences several times just to comprehend what it is saying. A lot of the proofs are equations and inequalities that take time to see how it ties into what is trying to be proven. I'm glad that it is one of the later chapters because, to me, it takes some time to get them. Having been taught all that we've learned in this course, it goes from nearly impossible to understand (like it was for me in Math 112/113) to only time-consuming. The theorems are of particular difficulty, in my estimation. It seems easier and more practical if the proofs involve something in addition to the theorems. Proving the theorems themselves seems so hard - as was the case for all of the theorems presented in the text throughout the year.
I remember seeing the limit laws before - the ones involving their sums and products, for example. That was rather familiar and I readily made the connection. Knowing how to apply the theorems when solving problems in calculus makes it easier to understand the proofs, I think. It is nice when you know that what your learning has an application in another math class. Overall, I think that the text and the course tied into a lot of math I've learned and built upon it or deepened my understanding. The course and text were well-organized in the order in which things were taught - I think it was the ideal order. I learned a lot and it was continually surprising how many things were introduced that I hadn't learned before.
I remember seeing the limit laws before - the ones involving their sums and products, for example. That was rather familiar and I readily made the connection. Knowing how to apply the theorems when solving problems in calculus makes it easier to understand the proofs, I think. It is nice when you know that what your learning has an application in another math class. Overall, I think that the text and the course tied into a lot of math I've learned and built upon it or deepened my understanding. The course and text were well-organized in the order in which things were taught - I think it was the ideal order. I learned a lot and it was continually surprising how many things were introduced that I hadn't learned before.
Tuesday, April 9, 2013
12.3, due on April 10
Once again, I don't know what it is about the greek alphabet, but when I see greek letters in math my mind suffers from some sort of disconnect. Delta was introduced. Then delta schemed with my old nemesis epsilon to try and create superconfusion for me. I only half-joke. This material was familiar in that I've studied calculus and limits; however, I feel like these proofs would make more sense to me if they were explained again in class. Even the graphs and diagrams in the text do not really help me understand the proof better. I think that I just need a teacher to explain them to me in "English" and not "Greekmath." Another thing: in these proofs, how do you choose the values accurately so that the proof comes out nicely? How do you know what to choose?
I did find it interesting, though - the proofs of limits, I mean. The proofs that were proven were fairly easy limits, as far as calculation goes. Still, the proofs were complicated, in my eyes. The proof about the limit of 1/x as x approaches infinity was interesting. I remember studying in calculus about 1/x not converging, but how 1/x^2 does converge. I don't remember seeing a proof, so that was cool. I see that there are so many things in mathematics that I've studied - from the recent study of limits to the past study (way past) of division - and how there are proofs and mathematical theory behind all of it. I always took that for granted. Like the division algorithm, for example, and even now regarding the limit of 1/x. Interesting to read about the proofs.
I did find it interesting, though - the proofs of limits, I mean. The proofs that were proven were fairly easy limits, as far as calculation goes. Still, the proofs were complicated, in my eyes. The proof about the limit of 1/x as x approaches infinity was interesting. I remember studying in calculus about 1/x not converging, but how 1/x^2 does converge. I don't remember seeing a proof, so that was cool. I see that there are so many things in mathematics that I've studied - from the recent study of limits to the past study (way past) of division - and how there are proofs and mathematical theory behind all of it. I always took that for granted. Like the division algorithm, for example, and even now regarding the limit of 1/x. Interesting to read about the proofs.
Saturday, April 6, 2013
12.1, due on April 8
I have read about epsilon twice - once in Math 112 and once in Math 113. Both times, I had difficulty with it. I remember missing questions on tests about epsilon as I didn't really get how to calculate it. I saw it again in this reading and it involves a proof and I instantly became confused. Epsilon has become my least favorite Greek letter. I may just chalk it up as a lost cause, but if we do go over any proofs involving epsilon in class, then maybe it might click for me. Based on the reading, all of the convergence proofs seemed to involve epsilon in order to prove convergence. It would probably be a good idea to get a better handle on it.
In my Math 113 class that I'm taking right now, we just got through the section about sequences. This tied into the reading about sequences and convergence and divergence. I didn't gain a full understanding in my Math 113 study about this topic. There is still time in the semester to gain a better understanding. Perhaps my study in this Math 290 class right now will help me in my other math class. I remember reading in my calculus book about these proofs, but kind of reading through them quickly. Now I have the opportunity to seriously revisit the proofs and I do really hope that it will help me get a richer, deeper understanding of calculus and, of course, help with this class. It may end up being a blessing.
In my Math 113 class that I'm taking right now, we just got through the section about sequences. This tied into the reading about sequences and convergence and divergence. I didn't gain a full understanding in my Math 113 study about this topic. There is still time in the semester to gain a better understanding. Perhaps my study in this Math 290 class right now will help me in my other math class. I remember reading in my calculus book about these proofs, but kind of reading through them quickly. Now I have the opportunity to seriously revisit the proofs and I do really hope that it will help me get a richer, deeper understanding of calculus and, of course, help with this class. It may end up being a blessing.
Thursday, April 4, 2013
Preparation for Exam 3, due on April 5
I'm sure that the cardinality of sets will be very important. The proofs are difficult for me, so I plan to study hard to understand how to prove that a set is denumerable or uncountable. I would expect to see this topic on the exam.
Also, I plan to see the Division algorithm and the Euclidean algorithm. Fortunately, this seemed to make a little more sense than Chapter 10 and the cardinality of sets.
An important theorem is the Schroder-Bernstein Theorem. I would think that we would have to be able to prove that two sets are numerically equivalent. That appeared in the homework and I kind of struggled with it with regards to when to use it and quote it exactly.
It seems that the questions on the test, at least the proofs and free response, are similar in some ways to homework questions, which is nice. I feel that if you can do your homework well, then you should know it well enough to perform on the test without the textbook or notes.
As for the definitions, we went over many, as usual. Denumerable, countable, uncountable - it will be good to know these on the test. I need to memorize the definitions exactly as I wrote them down from class notes. I've found that I lose points on exams when I don't provide the exact definition from text/notes, even though what I've written on the test for a definition is correct. I suppose it wasn't the "correct" correct answer. Definition memorization will be a part of my preparation.
I would please like to see a problem showing how one set has a larger cardinality than another.
Also, I plan to see the Division algorithm and the Euclidean algorithm. Fortunately, this seemed to make a little more sense than Chapter 10 and the cardinality of sets.
An important theorem is the Schroder-Bernstein Theorem. I would think that we would have to be able to prove that two sets are numerically equivalent. That appeared in the homework and I kind of struggled with it with regards to when to use it and quote it exactly.
It seems that the questions on the test, at least the proofs and free response, are similar in some ways to homework questions, which is nice. I feel that if you can do your homework well, then you should know it well enough to perform on the test without the textbook or notes.
As for the definitions, we went over many, as usual. Denumerable, countable, uncountable - it will be good to know these on the test. I need to memorize the definitions exactly as I wrote them down from class notes. I've found that I lose points on exams when I don't provide the exact definition from text/notes, even though what I've written on the test for a definition is correct. I suppose it wasn't the "correct" correct answer. Definition memorization will be a part of my preparation.
I would please like to see a problem showing how one set has a larger cardinality than another.
Saturday, March 30, 2013
11.5, due on April 1
The difficulty I had with this section is that there were no examples of application of relatively prime integers. It was stated in the text that we would see that they are very useful. However, I only read about theorems and lemmas and proofs thereof. Perhaps we will see their usefulness later on. I suppose, actually, that there was one reference to its usefulness. At the end of the section it stated:
"By Theorem 11.16 then, if we wish to show that 12, say, divides some integer c, we need only show that 3|c and 4|c since 12 = 3 · 4 and 3 and 4 are relatively prime."
It didn't actually show how to do this, but it did mention it. I typically retain the knowledge better if I see some examples.
Looking back on previously learned mathematics, I don't recall ever hearing about relatively prime numbers. I am grateful for the opportunity in this class to learn about a lot of new topics, definitions, and mathematical ideas. When I come across something new in this class, I always think - I should have learned this, or at least heard about this, before during one of my mathematics classes or textbooks. It is surprising to me that I haven't. It leads me to believe that there is more to learn than I could even imagine. After all, many mathematicians let scholastic discovery be their life's pursuit and I'm sure there is no way to ever know it all.
"By Theorem 11.16 then, if we wish to show that 12, say, divides some integer c, we need only show that 3|c and 4|c since 12 = 3 · 4 and 3 and 4 are relatively prime."
It didn't actually show how to do this, but it did mention it. I typically retain the knowledge better if I see some examples.
Looking back on previously learned mathematics, I don't recall ever hearing about relatively prime numbers. I am grateful for the opportunity in this class to learn about a lot of new topics, definitions, and mathematical ideas. When I come across something new in this class, I always think - I should have learned this, or at least heard about this, before during one of my mathematics classes or textbooks. It is surprising to me that I haven't. It leads me to believe that there is more to learn than I could even imagine. After all, many mathematicians let scholastic discovery be their life's pursuit and I'm sure there is no way to ever know it all.
Thursday, March 28, 2013
Sections 11.3-11.4, due on March 29
This may seem trivial, but I always find it difficult to follow equations or proofs when there are subscripts. It really slows me down. Is there a simple way to understand them? That is kind of a silly question - it probably is just something that will get easier with more exposure and more familiarity with them. For example, the proof of Lemma 11.9 introduced r(sub k)q(sub k+1) and then r(sub n-4) = r(sub n-3)q(sub n-2) + r(sub n-2). In the end, it is not impossible to understand; however, it just takes me a while to get what each variable represents when they have the subunits. I wish it flowed faster and didn't delay my reading.
The Euclidean Algorithm is pretty cool. I really liked the way in which the gcd was found by dividing and then dividing again with the remainder into the result just acquired. Sorry, I'm not using the proper term names, but my point is that I found it very neat how it worked out so smoothly. Lots of things in mathematics are really fascinating. There is such an order and it really is beautiful. Anyway, this algorithm struck me as really neat and clever. I'm not sure if there are future applications of it or if we build off it to learn other topics, but even if not, it was cool to learn in and of itself.
The Euclidean Algorithm is pretty cool. I really liked the way in which the gcd was found by dividing and then dividing again with the remainder into the result just acquired. Sorry, I'm not using the proper term names, but my point is that I found it very neat how it worked out so smoothly. Lots of things in mathematics are really fascinating. There is such an order and it really is beautiful. Anyway, this algorithm struck me as really neat and clever. I'm not sure if there are future applications of it or if we build off it to learn other topics, but even if not, it was cool to learn in and of itself.
Tuesday, March 26, 2013
11.1-11.2, due on March 27
The proof of The Division Algorithm was very difficult for me to follow. As mentioned in the text, the theorem is quite easy to understand because I've used it so often in my life, but indeed the proof is not obvious. Maybe it has to do with the fact that it is a proof about something so fundamental, I'm not sure. It's like trying to describe the taste of salt. How do you start? What do you say? I don't know, maybe it's not exactly like that, but The Division Algorithm, with the quotient and remainder, is something that seems so easy that no proof is required. But, of course, a proof is required! It was a hard one for me to understand.
Number theory seems like it will be interesting. I don't know a lot about it, but the introduction to the chapter seemed to hype it up quite nicely, praising it for its appeal, clarity, simplicity, and elegance. I look forward to learning about it. I don't really think it will tie in too strongly to some of my career goals in statistics, but I like the way that Prof Jenkins explained the study of some of the material in this class. To paraphrase: 'It might not save the world, but it's cool and interesting, so we study it.' I feel like that is a great way to think about it for me.
Number theory seems like it will be interesting. I don't know a lot about it, but the introduction to the chapter seemed to hype it up quite nicely, praising it for its appeal, clarity, simplicity, and elegance. I look forward to learning about it. I don't really think it will tie in too strongly to some of my career goals in statistics, but I like the way that Prof Jenkins explained the study of some of the material in this class. To paraphrase: 'It might not save the world, but it's cool and interesting, so we study it.' I feel like that is a great way to think about it for me.
Saturday, March 23, 2013
Rest of Section 10.5, due on March 25
Once again, I see a proof in which the function is split up. In the proof of Theorem 10.19, it stated that s(sub n) = 1 if n in S and s(sub n) = 2 if n not in S. Also, it was stated that t(sub n) = 1 if n in T and t(sub n) = 2 if n not in T. I don't really understand, still, what this means. Why are they separated and equal to different numbers in different scenarios. My shallow understanding is to show that they have unique decimal expansions, but I'm not connecting the dots on how that shows it is unique. I should have probably asked a question in class a few weeks ago when a proof with a similar format was presented, but I didn't. Now I see it again and am still confused.
I find it kind of amusing that Cantor is left out of the Schroder-Bernstein Theorem. Poor Cantor! It is interesting who gets credit for what with scholarly discoveries. Will you be remembered forever for what you accomplished in your field or will your work be forgotten? While I'm fairly certain that I will not leave my mark in mathematics, it is inspiring to see these mathematicians excel in their field and be remembered and studied about by those that followed them. It inspires me to be the best I can be and to leave an impression on this world - even it is on a smaller scale, like with my children perhaps. That I can leave this world having benefited it or making it better for those who follow would be something to strive for.
I find it kind of amusing that Cantor is left out of the Schroder-Bernstein Theorem. Poor Cantor! It is interesting who gets credit for what with scholarly discoveries. Will you be remembered forever for what you accomplished in your field or will your work be forgotten? While I'm fairly certain that I will not leave my mark in mathematics, it is inspiring to see these mathematicians excel in their field and be remembered and studied about by those that followed them. It inspires me to be the best I can be and to leave an impression on this world - even it is on a smaller scale, like with my children perhaps. That I can leave this world having benefited it or making it better for those who follow would be something to strive for.
Thursday, March 21, 2013
10.5 up to Theorem 10.18, due on March 22
The most difficult part of the material for me was Theorem 10.17. Well, I should say that the difficult part was in the proof. The theorem itself appears to be true, upon first reading, and I was surprised to see such an intricate proof for it. I don't understand why the ran f was brought into the proof. Also, I find it confusing when all of this extra notation is involved. Like f^n(x) and B'. It just kind of hurts my head when I read a proof like that. I'd have to reread it many, many times before understanding it, and, even then, would not really get what it is saying. I find it encouraging that we haven't had to prove theorems on exams. That would be really hard.
While the material is interesting and beneficial in the field of mathematics, sometimes I wonder what the application is for me, personally. I appreciate that mathematicians study and enjoy this type of material, but, as I stop to think about my life and career goals, I wonder how much of the material - or let's say, the Schroder-Bernstein Theorem, specifically - will I use later in life. I don't think that I would be cut out to be a math major and study years of mathematics. I have a lot of respect for the professors and other gifted minds in the math department. This material is hard to understand for me. When I feel like the material is going over my head, I think about things like this, to be honest, while I search for motivation to fully understand the material.
While the material is interesting and beneficial in the field of mathematics, sometimes I wonder what the application is for me, personally. I appreciate that mathematicians study and enjoy this type of material, but, as I stop to think about my life and career goals, I wonder how much of the material - or let's say, the Schroder-Bernstein Theorem, specifically - will I use later in life. I don't think that I would be cut out to be a math major and study years of mathematics. I have a lot of respect for the professors and other gifted minds in the math department. This material is hard to understand for me. When I feel like the material is going over my head, I think about things like this, to be honest, while I search for motivation to fully understand the material.
Tuesday, March 19, 2013
10.4, due on March 20
I have had some difficulty with proofs involving showing a function is bijective. I still need to get to a point where I easily comprehend and can adequately construct similar proofs. I would expect them to keep appearing in this section, not to mention the final exam. I fail to understand the logic of the proof of Theorem 10.14. Aside from the fact that I get tripped up when I see different characters, some with sub-characters, I still don't understand how you set things equal to each other and then that shows an injection. Then you set other things equal to each other and it is onto. I really need to focus on the fundamentals of these types of proofs. At least I understand that it is a bijection if it is injective and onto...at least.
The Continuum Hypothesis cannot be proven or disproven. That is interesting. I would think that there are other hypotheses and conjectures that are impossible to prove or disprove, but I'm not that familiar with mathematics to name one. In a field that I sometimes see as very black and white, I am always surprised when I see that mathematical ideas cannot be proven or disproven. I guess there are many mysteries. I wonder what the opinion is for mathematicians - if it can't be proven, then is it false or, rather, if it can't be disproven, then is it true?
The Continuum Hypothesis cannot be proven or disproven. That is interesting. I would think that there are other hypotheses and conjectures that are impossible to prove or disprove, but I'm not that familiar with mathematics to name one. In a field that I sometimes see as very black and white, I am always surprised when I see that mathematical ideas cannot be proven or disproven. I guess there are many mysteries. I wonder what the opinion is for mathematicians - if it can't be proven, then is it false or, rather, if it can't be disproven, then is it true?
Saturday, March 16, 2013
10.3, due on March 18
Since we discussed in class a little bit of 10.3 already, before I read the section, it made a little more sense. However, the proofs as seen in class and in the text seem very difficult. In fact, the proof of Theorem 10.8 "The open interval (0, 1) of real numbers is uncountable" was shown in the book and in class, but I still don't really get it. What does it mean when b (sub) i is separated into a sort of piecewise function of 4 if a (sub) ii equals 5 and 5 if a (sub) ii does not equal 5? I don't see the connection between doing this and showing that it is not onto. I recall that it may have something to do with the game.
Speaking of the game, although it wasn't mentioned in the reading, it was discussed in class and has applications, I believe, to the proofs of denumerable and uncountable sets. I kind of enjoyed the game, although it fast became easy and only Player 2 has a winning strategy. It reminds me of tic-tac-toe and other such games in that once you know some tactics, you can never lose. I thought that it might be fun to try the game from the homework out with my son who is in kindergarten. He might think that it is fun with the X's and O's and it might teach him to think a bit more logically. No doubt he'll eventually never want to be Player 1. Anyway, I thought that the game was interesting and maybe I can use it to understand the logic and proofs in this section better.
Speaking of the game, although it wasn't mentioned in the reading, it was discussed in class and has applications, I believe, to the proofs of denumerable and uncountable sets. I kind of enjoyed the game, although it fast became easy and only Player 2 has a winning strategy. It reminds me of tic-tac-toe and other such games in that once you know some tactics, you can never lose. I thought that it might be fun to try the game from the homework out with my son who is in kindergarten. He might think that it is fun with the X's and O's and it might teach him to think a bit more logically. No doubt he'll eventually never want to be Player 1. Anyway, I thought that the game was interesting and maybe I can use it to understand the logic and proofs in this section better.
Thursday, March 14, 2013
10.2, due on March 15
This section 10.2 started off fine as I understood the definitions of denumerable, countable, and countably infinite. Superficially, uncountable means not countable. So that means not finite or not denumerable. But denumerable is infinite. Then it said that an uncountable set is necessarily infinite. I'm not sure what the difference is between a denumerable infinite set and a necessarily infinite set. Even after the proofs, which were very intricate and difficult for me to follow, did not clarify this for me. I must say that this section was the most difficult of all I've read. The diagrams and tables were more confusing then helpful, unfortunately.
I have always been interested in the concept of infinity, so I'm interested to learn more about this and actually understand it. I look forward to class tomorrow to gain a better understanding. To be honest, I was a bit discouraged that I understood so little in this section. I feel that understanding this will be of importance as we begin the third of three sections in which this class material was divided. Usually in class, the proofs and examples are more clear, so hopefully that is the case, because the proofs in the book were hard to follow for me.
I have always been interested in the concept of infinity, so I'm interested to learn more about this and actually understand it. I look forward to class tomorrow to gain a better understanding. To be honest, I was a bit discouraged that I understood so little in this section. I feel that understanding this will be of importance as we begin the third of three sections in which this class material was divided. Usually in class, the proofs and examples are more clear, so hopefully that is the case, because the proofs in the book were hard to follow for me.
Tuesday, March 12, 2013
10.1, due on March 13
I found this statement to be very interesting: "Certainly, then, understanding infinite sets was not
an easy task, even among well-known mathematicians of the past." I almost find this humorous. If some of the most brilliant mathematical minds had difficulty understanding infinite sets, how in the world do I have a chance at all understanding them? I laughed a little thinking about that. Specifically, I didn't really understand "There can be a one-to-one correspondence between a set and a proper subset of the set" and how it relates to infinite sets. It troubled Galileo, so I guess my lack of understanding should not trouble me too much.
Numerically equivalent sets, from what I read, seem to be tied to concepts learned in Chapters 8 and 9, so I am comfortable with this. The description involving two sets A and B having the same cardinality if they're empty or if there is a bijective function from A to B. Learning about bijective functions already really makes this easier to grasp. Once again, I feel that the text and the course is well-organized and the principles learned build off one another nicely.
an easy task, even among well-known mathematicians of the past." I almost find this humorous. If some of the most brilliant mathematical minds had difficulty understanding infinite sets, how in the world do I have a chance at all understanding them? I laughed a little thinking about that. Specifically, I didn't really understand "There can be a one-to-one correspondence between a set and a proper subset of the set" and how it relates to infinite sets. It troubled Galileo, so I guess my lack of understanding should not trouble me too much.
Numerically equivalent sets, from what I read, seem to be tied to concepts learned in Chapters 8 and 9, so I am comfortable with this. The description involving two sets A and B having the same cardinality if they're empty or if there is a bijective function from A to B. Learning about bijective functions already really makes this easier to grasp. Once again, I feel that the text and the course is well-organized and the principles learned build off one another nicely.
Thursday, March 7, 2013
Preparation for Exam 2, due on March 8
The topics and that I think may be most important out of those we've studied would include induction, relations, equivalence classes, and functions. Specifically, regarding relations, I know that it is important to know if they are reflexive, symmetric, and transitive. Concerning functions, it is important to know if they are injective, surjective, and bijective. There are a couple of noteworthy theorems regarding this in my notes. Also, I think that the theorems about congruence modulo n and equivalence relations are very important.
I am expecting a similar format for this exam as the one used in Exam 1. True/False questions, Multiple Choice, and Free Response, which include proofs. Speaking of proofs, I feel like I need a much better understanding of how to construct proofs. Is there a certain formula or are there guidelines I can follow when doing an induction proof or when I am trying to prove a function is injective, surjective, or bijective. I think that I get stuck with how to start and how to finish, not to mention what to put in between. Really, I sometimes freeze up when thinking about how to write proofs because I don't know how to proceed. I am more used to solving problems - first, take the derivative, then plug it into the formula, then simplify, then solve the integral, etc. I'm not sure if there are clear-cut guidelines for proof construction, but that would be helpful to me.
A problem I would need to see worked out would be, for example, is "Prove that n! > 2^n for every integer n >3." I missed this one on the homework.
I am expecting a similar format for this exam as the one used in Exam 1. True/False questions, Multiple Choice, and Free Response, which include proofs. Speaking of proofs, I feel like I need a much better understanding of how to construct proofs. Is there a certain formula or are there guidelines I can follow when doing an induction proof or when I am trying to prove a function is injective, surjective, or bijective. I think that I get stuck with how to start and how to finish, not to mention what to put in between. Really, I sometimes freeze up when thinking about how to write proofs because I don't know how to proceed. I am more used to solving problems - first, take the derivative, then plug it into the formula, then simplify, then solve the integral, etc. I'm not sure if there are clear-cut guidelines for proof construction, but that would be helpful to me.
A problem I would need to see worked out would be, for example, is "Prove that n! > 2^n for every integer n >3." I missed this one on the homework.
Tuesday, March 5, 2013
9.6-9.7, due on March 6
Theorem 9.11 quickly made sense.
Let f : A → B be a function. Then the inverse relation f−1 is a function from B to A if
and only if f is bijective. Furthermore, if f is bijective, then f−1 is also bijective.
The proof got a little jumbled for me. In order to show that f-1 is a function, I suppose it is as easy as showing that f is one-to-one and onto. I am confused as to why the proof continued with the converse. For me, it has been helpful to follow a proof pattern when writing my own proofs. If I have a guide of what needs to be shown, it is manageable. I don't quite get the pattern of this proof and would need a clarification as to why it was set up the way it was.
I remember learning about permutations in a statistics class I had last semester. Pretty interesting. That part of the material made sense to me, although I remember getting a little tripped up with permutations as they became more complicated. It is interesting to note that permutations are bijective functions and have inverses. I didn'onsider that really when I learned about them last semester, but it definitely makes sense. I look forward to Chapter 13 when they are discussed in more detail.
Let f : A → B be a function. Then the inverse relation f−1 is a function from B to A if
and only if f is bijective. Furthermore, if f is bijective, then f−1 is also bijective.
The proof got a little jumbled for me. In order to show that f-1 is a function, I suppose it is as easy as showing that f is one-to-one and onto. I am confused as to why the proof continued with the converse. For me, it has been helpful to follow a proof pattern when writing my own proofs. If I have a guide of what needs to be shown, it is manageable. I don't quite get the pattern of this proof and would need a clarification as to why it was set up the way it was.
I remember learning about permutations in a statistics class I had last semester. Pretty interesting. That part of the material made sense to me, although I remember getting a little tripped up with permutations as they became more complicated. It is interesting to note that permutations are bijective functions and have inverses. I didn'onsider that really when I learned about them last semester, but it definitely makes sense. I look forward to Chapter 13 when they are discussed in more detail.
Saturday, March 2, 2013
9.5, due on March 4
I really didn't get the last part of the material: Theorem 9.7, Corollary 9.8, and Theorem 9.9. I should say that the theorems made sense and were acceptable, but the proofs were a little different. For example, in Theorem 9.7, it was proven ultimately by implying that g of f was injective and surjective. I didn't get how that implication could be made. At the end, Example 9.10's solution stated that the ran f was {0,1,4,9}. While I understand that it is necessary to show that ran f is a subset of B', I failed to see how the ran f was defined how it was.
Once again, this material connects with something I've learned in calculus. There were a few ties to derivatives, including the product rule and the chain rule. While I appreciate the tie-ins, I get a little frustrated with it being easier for me to understand the material when it talks about calculus than in other situations. What I mean by this is that I wish it would click in my head easier when reading about non-calculus material in this text. I do think, though, that the text does a good job of bringing in calculus references and other things we've surely learned in previous classes. This forms a nice transition and I generally understand the material better upon first reading.
Once again, this material connects with something I've learned in calculus. There were a few ties to derivatives, including the product rule and the chain rule. While I appreciate the tie-ins, I get a little frustrated with it being easier for me to understand the material when it talks about calculus than in other situations. What I mean by this is that I wish it would click in my head easier when reading about non-calculus material in this text. I do think, though, that the text does a good job of bringing in calculus references and other things we've surely learned in previous classes. This forms a nice transition and I generally understand the material better upon first reading.
Saturday, February 23, 2013
8.5-8.6, due on February 25
I'm having difficulty when there are discussions about equivalence classes. I think I need to spend more time or attention to them. I did not really understand the addition and multiplication table for Z sub 6. It seems to have something to do with remainders when you add or multiply. That is all I can gather for now and I'm thinking about what the point must be. Then, it went on to talk about the sums or products of two equivalence classes as being well-defined. This also wasn't entirely clear, although the proof of Theorem 8.9 generally made sense from a technical standpoint
I found the following interesting: "If you have more mathematics in your future, it is likely that you will see the equivalence relation we are about to describe again – indeed often." Honestly, before this class I hadn't even heard about Congruence Modulo n. It sounds like it is a noteworthy topic in mathematics and I've appreciated learning about it and all of its properties; however, I am surprised that I hadn't learned really anything about it before now, despite taking math every year in all my years of schooling. Kind of funny, really.
I found the following interesting: "If you have more mathematics in your future, it is likely that you will see the equivalence relation we are about to describe again – indeed often." Honestly, before this class I hadn't even heard about Congruence Modulo n. It sounds like it is a noteworthy topic in mathematics and I've appreciated learning about it and all of its properties; however, I am surprised that I hadn't learned really anything about it before now, despite taking math every year in all my years of schooling. Kind of funny, really.
Thursday, February 21, 2013
8.3-8.4, due on February 22
The textbook has chosen to represent relations with the capital letter R. This would make sense because the word starts with an R. However, it is still confusing to me. I see the capital R and it is between other letter variables and it is just a little confusing. I appreciated in Wednesday's lecture that Prof. Jenkins used the "star" symbol instead of the R. This seems to make more sense and I actually make the quick substitution in my head when doing the reading. This helps a little bit. The subject matter in pretty complicated for me, so having this R issue just adds to the problem. Kind of minor, but I wish the text would have been more intuitive like the lecture.
I never really noticed it before, but I now see the fundamental importance of relations in mathematics. Sure, since first learning arithmetic, we solve problems that are equations. I've taken these relations-problems for granted, but it is interesting that we are focusing on the relations now in this class. Equivalence relations seem to make sense, for the most part, especially when relating it to other math I've done involving equalities. That familiarity helped with the comprehension of the reading. It was good to grasp that because the rest of the material was a little hard to follow, as I mentioned in the first part of the post. I'm looking forward to tomorrow's lecture to learn the material better - the lectures always help.
I never really noticed it before, but I now see the fundamental importance of relations in mathematics. Sure, since first learning arithmetic, we solve problems that are equations. I've taken these relations-problems for granted, but it is interesting that we are focusing on the relations now in this class. Equivalence relations seem to make sense, for the most part, especially when relating it to other math I've done involving equalities. That familiarity helped with the comprehension of the reading. It was good to grasp that because the rest of the material was a little hard to follow, as I mentioned in the first part of the post. I'm looking forward to tomorrow's lecture to learn the material better - the lectures always help.
Monday, February 18, 2013
6.3-6.4 due on February 19
The reading was pretty challenging. I especially find it difficult to understand how or where the author comes up with the steps of the proof. To me it is not something I see myself doing. Perhaps, as more familiarity is acquired with The Principle of Mathematical Induction and The Strong Principle of Mathematical Induction, it will be easier to follow and then create similar proofs. For now, I am frustrated by the thought of it. It seems that there is a plan at work when new variables are introduced, but it kind of confuses me. We start with n, then we introduce m, then there is always k. Now we see i and let's do a proof involving a and b. In order to prove, we need to also introduce x and y. I think that last proof in the reading did all of this and it hurt my brain.
This second principle seems to tie in nicely with The Principle of Mathematical induction. We are learning integration techniques in my Math 113 class right now. For certain integrals, there is an easier way, and as it was taught to me, an only way to integrate certain integrals. I think of this when I learn new proof methods. I need to be smart enough to know which proof method will be appropriate for the given problem or statement. I look forward to the challenge, knowing that there are solving similarities in this class and other math classes I've had.
This second principle seems to tie in nicely with The Principle of Mathematical induction. We are learning integration techniques in my Math 113 class right now. For certain integrals, there is an easier way, and as it was taught to me, an only way to integrate certain integrals. I think of this when I learn new proof methods. I need to be smart enough to know which proof method will be appropriate for the given problem or statement. I look forward to the challenge, knowing that there are solving similarities in this class and other math classes I've had.
Thursday, February 14, 2013
6.2 due on February 15
The proofs for mathematical induction are a bit difficult to follow. Perhaps I need more practice with them or more examples. I'd expect that the homework assignments to help with the understanding. It always does. I am starting to understand the principle and the logic behind it all. As it was with past proofs, I find it challenging to know what to do. Why would you manipulate the equation algebraically like the author of the book did? How would you know to do that in order to get the desired result? I found it particularly tricky and I had to slow down to see the logic behind the manipulation of the exponenents in many of the proofs. I hope that in time I'll have the intuition to do the same.
We studied the cardinality of the power set in the past and it even showed up on the first exam. It was good to see the proof of it and that it was an induction proof. I hadn't considered that it would be proved by induction, but it makes a lot of sense, when you think about it, that that is how you would prove it. I like to see how things we learn about are being tied to things we learned about earlier in the course. I can see that things are only going to get more challenging, but tying the material to what we've learned before will help out a lot.
We studied the cardinality of the power set in the past and it even showed up on the first exam. It was good to see the proof of it and that it was an induction proof. I hadn't considered that it would be proved by induction, but it makes a lot of sense, when you think about it, that that is how you would prove it. I like to see how things we learn about are being tied to things we learned about earlier in the course. I can see that things are only going to get more challenging, but tying the material to what we've learned before will help out a lot.
Tuesday, February 12, 2013
6.1, due on February 13
The Principle of Mathematical Induction was interesting. The section went into a lot of detail and there was quite a bit of algebra. It was challenging following all the algebra as it became a bit of a blur after a while. I read about how it is critical to perform the algebra correctly and I can see that. I kind of didn't get the point of induction proofs, but the section gave some examples and, though I'm not certain, I suppose that you would only use induction proofs in certain situations. I expect to get more clarity on this principle in the coming sections.
I noticed the induction proofs for n(n+1)/2 and n(n+1)(2n+1)/6. I recognized these from my Math112 class last semester. I found it interesting to see the induction proofs. I don't remember going over them or seeing them in the Math112 class. My professor did, though, tell us the story about Gauss and his adding-integer task. I find it amazing that he was that brilliant. I think that brilliance and genius become evident at a young age and that story was a fun one to share with my family here at home.
I noticed the induction proofs for n(n+1)/2 and n(n+1)(2n+1)/6. I recognized these from my Math112 class last semester. I found it interesting to see the induction proofs. I don't remember going over them or seeing them in the Math112 class. My professor did, though, tell us the story about Gauss and his adding-integer task. I find it amazing that he was that brilliant. I think that brilliance and genius become evident at a young age and that story was a fun one to share with my family here at home.
Saturday, February 9, 2013
Preparation for Exam 1, due on February 11
As I study for the exam, I think that the study guide and practice exams provide nice direction on where to focus my study time. I'm grateful that they are available as resources. As I review, it looks like definitions will be very important. This is not surprising as we went over many definitions in the text and in lecture. I know they are at the foundation of the course and, while I know most off-hand, I'll need to get those down. I think that proofs involving even and odd integers will be important. There were many on the homework assignments and it looks like they appear on the practice exam. Speaking of the practice exam, I see a combination of True/False, Multiple Choice, and Free Response. This seems fair and I'd expect a similar format for the actual exam.
I need to work on my understanding of sets, subsets, and empty sets. I didn't really grasp the topic the first time through, so now is the time to master it. It seems very basic and I think that I just need it to click in my mind. Also, I would like to get more confidence in knowing which method of proof I should choose. Is there a good way or technique in knowing? Or does it just come with experience? The homework guided me well based off of what section the homework problem was in, but the exam will surely be different.
There was a problem on the second practice exam that I would like to see worked out.
Prove or Disprove: Let A be a set. If A U B not= emptyset for all sets B, then A not= emptyset.
I need to work on my understanding of sets, subsets, and empty sets. I didn't really grasp the topic the first time through, so now is the time to master it. It seems very basic and I think that I just need it to click in my mind. Also, I would like to get more confidence in knowing which method of proof I should choose. Is there a good way or technique in knowing? Or does it just come with experience? The homework guided me well based off of what section the homework problem was in, but the exam will surely be different.
There was a problem on the second practice exam that I would like to see worked out.
Prove or Disprove: Let A be a set. If A U B not= emptyset for all sets B, then A not= emptyset.
Thursday, February 7, 2013
7.1-7.3, due on February 8
There is still something in the textbook that gives me some trouble. Negations and empty sets. I find it difficult to recall what the negations are when they involve "For every" and "There exists" statements, especially when you add in the mathematical language with the upside down A and the backwards E symbols, if you will. It takes some extra time for me. Also, the empty set. We haven't had it in the reading or homework lately, so I kind of dismissed it or tucked it away in a corner of my brain. I am pretty unfamiliar with the empty set and why they are used since they are empty. Anyway, this question and others like it that I have will probably seem silly to me once I get a better handle on the empty sets. I think I'm going to seek assistance or do more study on them.
I enjoyed the part about conjecture. At the end of the section, it talked about Goldbach's Conjecture: Every even integer exceeding 2 is the sum of two primes. This conjecture truly hasn't been resolved? Even with the assistance with the computers we have today? I find that hard to believe and wish that the author had gone into a bit more detail as to why this hasn't been resolved. Anyway, I think it is neat that there is still mathematical study going on with new discoveries. I would be interested to know what the leading mathematicians do in their pursuit of knowledge and mathematical study. I would imagine it would be pretty sophisticated, although, if Goldbach's Conjecture is still unresolved, maybe there are basic pursuits of study, too.
I enjoyed the part about conjecture. At the end of the section, it talked about Goldbach's Conjecture: Every even integer exceeding 2 is the sum of two primes. This conjecture truly hasn't been resolved? Even with the assistance with the computers we have today? I find that hard to believe and wish that the author had gone into a bit more detail as to why this hasn't been resolved. Anyway, I think it is neat that there is still mathematical study going on with new discoveries. I would be interested to know what the leading mathematicians do in their pursuit of knowledge and mathematical study. I would imagine it would be pretty sophisticated, although, if Goldbach's Conjecture is still unresolved, maybe there are basic pursuits of study, too.
Tuesday, February 5, 2013
5.4-5.5, due on February 6
I had difficulty understanding the existence proof regarding irrational numbers and the one involving the square root of 2 to the power of square root of two. I agreed that the proof seemed unsatisfactory since two specific irrational numbers a and b were not identified to confirm a^b was rational. It would have been far easier to understand if specific numbers were given to produce the desired result. To prove it in theory was hard to comprehend, but a crafty way, I suppose, in proving the result without giving two actual numbers.
I noticed that I had recently studied about the Intermediate Value Theorem of Calculus in Math 112. That tied in nicely to something I just studied. It made sense and in the Calculus homework we had to prove that an equation set to equal zero had a solution within a certain interval. I remember that I threw together some sort of rough proof and called it good, but now I believe I have the capacity to produce a more sophisticated and thorough proof. It would have been nice for that assignment. I think it was fine, though, because, from what I could tell, not many of the Math 112 students in my class had skill mastery in proofs. This Math 290 class is beneficial.
I noticed that I had recently studied about the Intermediate Value Theorem of Calculus in Math 112. That tied in nicely to something I just studied. It made sense and in the Calculus homework we had to prove that an equation set to equal zero had a solution within a certain interval. I remember that I threw together some sort of rough proof and called it good, but now I believe I have the capacity to produce a more sophisticated and thorough proof. It would have been nice for that assignment. I think it was fine, though, because, from what I could tell, not many of the Math 112 students in my class had skill mastery in proofs. This Math 290 class is beneficial.
Saturday, February 2, 2013
5.2-5.3, due on February 4
One thing that I have difficulty with is how to know how to manipulate the equations or inequalities within the proofs. For example, look at Result 5.18:
Result 5.18 Let x be a nonzero real number. If x + (1/x) < 2, then x < 0.
Direct Proof Assume that x + (1/x) < 2. Since x does not equal 0, we know that x^2 > 0. Multiplying both sides of the inequality x + (1/x) < 2 by x^2, we obtain x^2(x + (1/x)) < 2x^2. Simplifying this inequality, we have x^3 + x − 2x^2 < 0; so x(x^2− 2x + 1 ) = x(x − 1)^2 < 0.
The proof concludes shortly thereafter. My problem is not understanding that the proof is valid, but, rather, it is how would I know to multiply both sides of the inequality by x^2? Is this simply a trick-turned-technique? Should I memorize this method? Maybe multiplying by x would make sense to me, considering you have a (1/x), but the multiplying by x^2 just stood out to me as unique. "That's great and all that it worked out," I thought to myself. "But how would I have come up with that idea to multiply by x^2 on my own?"
The "story" problem with the three prisoners was brilliant. I really liked it. My 6-year old son entered the room right when I was re-reading and pondering the problem and asked, "What are you studying?" I told him there was a story problem about prisoners and he wanted to hear it. After hearing the story, my son answered the question about the prisoner correctly identifying the color of his dot: "Maybe he just saw the other two guys with red dots and thought, 'I probably have a red dot, too.'" Sounded like a good explanation to me! It was satisfying to me that, only once I was explaining it out to my son, the answer to the question correctly came to my mind. I had figured it out - and only had done so once I started to put into words what I was thinking. This illustrated something to me that I didn't realize: talking a problem out can be a powerful way to help you arrive at a desired answer or result.
Result 5.18 Let x be a nonzero real number. If x + (1/x) < 2, then x < 0.
Direct Proof Assume that x + (1/x) < 2. Since x does not equal 0, we know that x^2 > 0. Multiplying both sides of the inequality x + (1/x) < 2 by x^2, we obtain x^2(x + (1/x)) < 2x^2. Simplifying this inequality, we have x^3 + x − 2x^2 < 0; so x(x^2− 2x + 1 ) = x(x − 1)^2 < 0.
The proof concludes shortly thereafter. My problem is not understanding that the proof is valid, but, rather, it is how would I know to multiply both sides of the inequality by x^2? Is this simply a trick-turned-technique? Should I memorize this method? Maybe multiplying by x would make sense to me, considering you have a (1/x), but the multiplying by x^2 just stood out to me as unique. "That's great and all that it worked out," I thought to myself. "But how would I have come up with that idea to multiply by x^2 on my own?"
The "story" problem with the three prisoners was brilliant. I really liked it. My 6-year old son entered the room right when I was re-reading and pondering the problem and asked, "What are you studying?" I told him there was a story problem about prisoners and he wanted to hear it. After hearing the story, my son answered the question about the prisoner correctly identifying the color of his dot: "Maybe he just saw the other two guys with red dots and thought, 'I probably have a red dot, too.'" Sounded like a good explanation to me! It was satisfying to me that, only once I was explaining it out to my son, the answer to the question correctly came to my mind. I had figured it out - and only had done so once I started to put into words what I was thinking. This illustrated something to me that I didn't realize: talking a problem out can be a powerful way to help you arrive at a desired answer or result.
Thursday, January 31, 2013
4.5-4.6 and 5.1, due on February 1
The proof of Theorem 4.21 looks just like a big mess on the page at first glance. Then it still looks that way at second glance. It only looks slightly better after reading. I was getting lost in the proofs and wanted just to say, "Ok, I'll take your word for it." The theorem itself is fine as the commutative, associative, and distributive laws are similar to the same laws we learned as children. De Morgan's Laws are easily accepted and understood. Maybe that is the problem - they are accepted and understood, so I haven't seen them proved before. I think that all of the symbols made the proofs look jumbled and crowded. It was difficult for me to read them straight through.
The section 5.1 on counterexamples seems a bit more like what I've been used to in previous math courses. It's more like solving equations and getting "answers." You look at the statements and you've got to find real numbers or integers which disprove the statements. This was definitely more in my comfort zone as I've found myself in past proofs trying to plug in integers, for example, and calculate the equations to confirm their validity.
The section 5.1 on counterexamples seems a bit more like what I've been used to in previous math courses. It's more like solving equations and getting "answers." You look at the statements and you've got to find real numbers or integers which disprove the statements. This was definitely more in my comfort zone as I've found myself in past proofs trying to plug in integers, for example, and calculate the equations to confirm their validity.
Tuesday, January 29, 2013
4.3-4.4, due on January 30
Why is "absolute value" so hard to understand? Frustratingly, it is something that doesn't come naturally to me in any of the mathematics classes I've ever had. The proof involving the absolute value was difficult for me to understand. I'm not sure if I just get intimidated when I see those absolute value signs | |, but I seem to struggle. Perhaps it was even harder because there were 3 cases and a couple of subcases. We haven't yet read much about those things and it looks like we will be getting into some more complicated or deeper proofs. Should be a challenge.
I find the "Proof Analysis" parts helpful because it offers an explanation as to why the author included certain parts in the proof and gives insight into the logic behind the proof. There was one for Result 4.18 that added much needed clarification. Then the analysis for the proof of Result 4.20 taught me a lot about if/when you need to tell the reader which technique you are using. Without the proof analysis, sometimes, I find it hard to understand why something was done or written the way it was in the book, so they are very helpful.
I've been spending a few hours on each homework assignment. The lectures and reading have prepared me well in completing the assignments. I think that having something due everyday - be it a blog entry or an assignment - has contributed most to my learning by creating a schedule in which I can steadily and effectively learn the material. I appreciate that the professor and TA are available for assistance - this always helps.
I find the "Proof Analysis" parts helpful because it offers an explanation as to why the author included certain parts in the proof and gives insight into the logic behind the proof. There was one for Result 4.18 that added much needed clarification. Then the analysis for the proof of Result 4.20 taught me a lot about if/when you need to tell the reader which technique you are using. Without the proof analysis, sometimes, I find it hard to understand why something was done or written the way it was in the book, so they are very helpful.
I've been spending a few hours on each homework assignment. The lectures and reading have prepared me well in completing the assignments. I think that having something due everyday - be it a blog entry or an assignment - has contributed most to my learning by creating a schedule in which I can steadily and effectively learn the material. I appreciate that the professor and TA are available for assistance - this always helps.
Saturday, January 26, 2013
4.1-4.2, due on January 28
Sometimes I read a proof in the text and I can follow it; that is to say, it all makes logical sense and looks true. However, I sometimes stuggle to understand how it all proved the result. Of particular difficulty was the last proof in 4.1.
Result to Prove For every integer n ≥ 7, there exist positive integers a and b such that n = 2a + 3b.
The two cases proved:
Since q ≥ 4, it follows that q − 3 ∈ N.
Since q ≥ 3, it follows that q − 1 ∈ N.
I still do not understand how this proves the result. So, what does the proof that q − 3 and q − 1 are natural numbers have to do with the result? As the text stated, this is a result of a somewhat different nature. I'm definitely missing something here. I'll try to keep looking at it to make sense of it.
I appreciate that the text and the professor of the course have prepared the student well to understand the material. As I was reading section 4.2, I thought to myself, "What if I picked up this book at the beginning of the semester and started reading from this section. I would really have no idea what is going on." It's true, the language is coming easier to me and I feel like everything has built up nicely to where we are today. The basics on notation and logic that we learned play a critical role in the comprehension of the material in the current sections and those yet to come, I'd assume.
Result to Prove For every integer n ≥ 7, there exist positive integers a and b such that n = 2a + 3b.
The two cases proved:
Since q ≥ 4, it follows that q − 3 ∈ N.
Since q ≥ 3, it follows that q − 1 ∈ N.
I still do not understand how this proves the result. So, what does the proof that q − 3 and q − 1 are natural numbers have to do with the result? As the text stated, this is a result of a somewhat different nature. I'm definitely missing something here. I'll try to keep looking at it to make sense of it.
I appreciate that the text and the professor of the course have prepared the student well to understand the material. As I was reading section 4.2, I thought to myself, "What if I picked up this book at the beginning of the semester and started reading from this section. I would really have no idea what is going on." It's true, the language is coming easier to me and I feel like everything has built up nicely to where we are today. The basics on notation and logic that we learned play a critical role in the comprehension of the material in the current sections and those yet to come, I'd assume.
Thursday, January 24, 2013
3.4-3.5, due on January 25
After reading the previous sections of Chapter 3 and doing the homework, the reading of these two sections was easier to comprehend. However, there is one thing that I can see will be difficult going forward: which proof method to use. In this reading we learned of another method - proof by cases. There are benefits and advantages to all the methods learned thusfar. I follow the examples of proofs in the text and agree, of course, that the correct method has been chosen. However, I have a fear that I will not be able to quickly choose the correct method when showing my own proofs, resulting in lost time, mounting frustration, or inability to construct a correct proof.
This fear leads to the part of the text that I appreciated most:
When learning any mathematical subject, it is not the least bit unusual to make mistakes along the way. In fact, part of learning mathematics is to learn from your mistakes and those of others.
This is encouraging to me. I've already made many mistakes in this course. I fully expect to make more mistakes along the way. The thing on which I will focus is learning from my mistakes. I'm certain that I will stumble along the way, but I am still excited about learning the material in this course and, in some strange way, looking forward to making mistakes because I know that I will learn from them.
This fear leads to the part of the text that I appreciated most:
When learning any mathematical subject, it is not the least bit unusual to make mistakes along the way. In fact, part of learning mathematics is to learn from your mistakes and those of others.
This is encouraging to me. I've already made many mistakes in this course. I fully expect to make more mistakes along the way. The thing on which I will focus is learning from my mistakes. I'm certain that I will stumble along the way, but I am still excited about learning the material in this course and, in some strange way, looking forward to making mistakes because I know that I will learn from them.
Tuesday, January 22, 2013
3.1-3.3, due on January 23
I found it challenging to follow some of the proofs. Perhaps it is to be expected to have to reread them in order to make complete sense of them. Once I was able to pause and really think about them, they all made sense - just not at first for all of them. I know that this is essential to the course as we begin to construct our own proofs, so I put a high priority on understanding the fundamentals during the beginning stages of our study of proofs. Following the proofs and comprehending them can be quite rewarding.
I really liked the last part of the last line of Section 3.3:
Also, you might think we used a trick to give the second proof of Result 3.14, but, as we will see, if the same “trick” can be used often, then it becomes a technique.
To me, this was very fun to read. Playing sports all my life, I learned many techniques. In soccer, for example, there were dribbling techniques, passing techniques, shooting, defending, and goalkeeping techniques. I liked how techniques were described as "tricks that can be used often." Looking back, essentially that is what they were. Used often enough, they become techniques - techniques a quality player uses whenever he steps out onto the field. I can see that mathematicians can use many techniques to show their quality in the mathematical field. Scoring a goal can be beautiful just like a theorem can be beautiful.
I really liked the last part of the last line of Section 3.3:
Also, you might think we used a trick to give the second proof of Result 3.14, but, as we will see, if the same “trick” can be used often, then it becomes a technique.
To me, this was very fun to read. Playing sports all my life, I learned many techniques. In soccer, for example, there were dribbling techniques, passing techniques, shooting, defending, and goalkeeping techniques. I liked how techniques were described as "tricks that can be used often." Looking back, essentially that is what they were. Used often enough, they become techniques - techniques a quality player uses whenever he steps out onto the field. I can see that mathematicians can use many techniques to show their quality in the mathematical field. Scoring a goal can be beautiful just like a theorem can be beautiful.
Thursday, January 17, 2013
Chapter 0 - pages 5 to 12, due on January 18
The most difficult part was reading about the difference between "that" and "which." Both example sentences sounded correct, so I was wondering what the point was.
The solution to the equation is the number less than 5 that is positive.
The solution to the equation is the number less than 5 which is positive.
Upon re-reading, I see that there is an important difference between "that" and "which," depending on the question you are trying to answer or the point you are trying to make. I can see how these two words are confused with each other and are often not interchangable. I had to stop and think hard about restrictive and nonrestrictive clauses and how they affect the meaning of the sentences.
I feel that this section was very beneficial. I appreciated that creativity and an attitude of being yourself is encouraged in mathematical writing. Often I find myself holding onto the belief that there is a right answer and a wrong answer in mathematics, that there is a right way and a wrong way - very black and white. I found it interesting that, while there are guidelines and tips one should follow, originality can and should exist in mathematical writing, just as it does in other forms of writing. All of the advice given in the chapter will be of great benefit, especially the suggestions and guidelines. There should be order and accepted principles to follow in mathematical writing, especially when it assists the reader in understanding what you are trying to show, prove, or teach. I'm sure I will return to this chapter to use it as a reference guide in the future.
The solution to the equation is the number less than 5 that is positive.
The solution to the equation is the number less than 5 which is positive.
Upon re-reading, I see that there is an important difference between "that" and "which," depending on the question you are trying to answer or the point you are trying to make. I can see how these two words are confused with each other and are often not interchangable. I had to stop and think hard about restrictive and nonrestrictive clauses and how they affect the meaning of the sentences.
I feel that this section was very beneficial. I appreciated that creativity and an attitude of being yourself is encouraged in mathematical writing. Often I find myself holding onto the belief that there is a right answer and a wrong answer in mathematics, that there is a right way and a wrong way - very black and white. I found it interesting that, while there are guidelines and tips one should follow, originality can and should exist in mathematical writing, just as it does in other forms of writing. All of the advice given in the chapter will be of great benefit, especially the suggestions and guidelines. There should be order and accepted principles to follow in mathematical writing, especially when it assists the reader in understanding what you are trying to show, prove, or teach. I'm sure I will return to this chapter to use it as a reference guide in the future.
Tuesday, January 15, 2013
2.9-2.11, due on January 16
I found the Example 2.24 in section 2.10 to be difficult to understand upon first reading.
"Suppose that we are considering the set A = {1 , 2, 3} and its power set P( A), the set of all subsets of A. Then the quantified statement for every set B ∈ P( A), A− B = ∅. (2.6) is false since for the subset B = A = {1 , 2, 3} , we have A− B = ∅."
I guess I am still struggling with the principles of sets, subsets, power sets, and empty sets. I had to reread many times the following, B ∈ P( A), A− B = ∅ is false, before I grasped what it was trying to say about the relationship between B and the power set P( A). I understand now, but it was difficult for a little while.
The most interesting part to me was the discussion of how mathematicians rarely use the definition in an "if and only if" statement. To me it seemed a valid characterization of an equilateral triangle. "A triangle T is equilateral if and only if T has three equal sides." However, I learned that this is not a characterization, but rather a definition. An accurate characterization of equilateral triangles was presented, "A triangle T is equilateral if and only if T has three equal angles." I quickly understood the difference and found it to be an interesting point as I look forward to other examples of this in future chapters.
"Suppose that we are considering the set A = {1 , 2, 3} and its power set P( A), the set of all subsets of A. Then the quantified statement for every set B ∈ P( A), A− B = ∅. (2.6) is false since for the subset B = A = {1 , 2, 3} , we have A− B = ∅."
I guess I am still struggling with the principles of sets, subsets, power sets, and empty sets. I had to reread many times the following, B ∈ P( A), A− B = ∅ is false, before I grasped what it was trying to say about the relationship between B and the power set P( A). I understand now, but it was difficult for a little while.
The most interesting part to me was the discussion of how mathematicians rarely use the definition in an "if and only if" statement. To me it seemed a valid characterization of an equilateral triangle. "A triangle T is equilateral if and only if T has three equal sides." However, I learned that this is not a characterization, but rather a definition. An accurate characterization of equilateral triangles was presented, "A triangle T is equilateral if and only if T has three equal angles." I quickly understood the difference and found it to be an interesting point as I look forward to other examples of this in future chapters.
Saturday, January 12, 2013
2.5-2.8, due on January 14
One difficult part of the reading is quickly remembering what the symbols and notations mean. I really have to stop and think about what the statements P=>Q and ~P^Q are saying in English. As time passes, I'm sure my mind will process the new language more quickly. It is very difficult to know the truth or falseness of statements without writing up a truth table. I can see that truth tables are very useful, especially since the following situation is a little confusing:
If P is false then Q is true. This is still a true statement. The realization is that I'll have to trust the truth table. I'll also try to remember the effective example from class. "If you get your pajamas on, then you'll get ice cream." Even if the kids get ice cream without putting on pajamas, the statements wasn't a lie to the kids.
I am really looking forward to mathematical application of this study in logic. An example in the text began with the phrase: "Let R be a mathematical statement that we would like to show is true." It gave me some insight and foreshadowing perhaps on what lies ahead in this course. I anticipate seeing how we apply logic in mathematical proofs. I remember taking a math class in high school in which the teacher asked us to do many proofs. It has been a long time since high school so my memory is a little hazy, but it seems that we used logically equivalent statements in the construction of proofs. I look forward to applying what we have read in these sections later in the course.
If P is false then Q is true. This is still a true statement. The realization is that I'll have to trust the truth table. I'll also try to remember the effective example from class. "If you get your pajamas on, then you'll get ice cream." Even if the kids get ice cream without putting on pajamas, the statements wasn't a lie to the kids.
I am really looking forward to mathematical application of this study in logic. An example in the text began with the phrase: "Let R be a mathematical statement that we would like to show is true." It gave me some insight and foreshadowing perhaps on what lies ahead in this course. I anticipate seeing how we apply logic in mathematical proofs. I remember taking a math class in high school in which the teacher asked us to do many proofs. It has been a long time since high school so my memory is a little hazy, but it seems that we used logically equivalent statements in the construction of proofs. I look forward to applying what we have read in these sections later in the course.
Thursday, January 10, 2013
2.1-2.4, due on January 11
Overall, the reading in this section of the book was straightforward and easier to comprehend than the material in Chapter 1. One challenge is all of the definitions presented to the reader. Now we learn of disjunction and conjunction, negation, open sentences and domains, just to name a few. Also, with all of the new symbols, I find it difficult to keep the notation straight without reviewing and realizing, "Oh, yes, that is what that means in English." The challenge presents and exciting opportunity to learn all this class will offer.
The reading was interesting. I like logic and believe I am a logical thinker. Of particular interest was Example 2.6 in the section about The Implication. I found it to be an effective real-world example of an if-then statement. I could relate to the example because I have been, possibly like many other students at one time or another, in a similar situation. The author in the book presents the example: "If you earn an A on the final exam, then you will receive an A for your final grade." It really helped me understand the four different possibilities in the truth table: 1. P and Q are both true. 2. P is true and Q is false. 3. P is false and Q is true. 4. P and Q are both false.
The reading was interesting. I like logic and believe I am a logical thinker. Of particular interest was Example 2.6 in the section about The Implication. I found it to be an effective real-world example of an if-then statement. I could relate to the example because I have been, possibly like many other students at one time or another, in a similar situation. The author in the book presents the example: "If you earn an A on the final exam, then you will receive an A for your final grade." It really helped me understand the four different possibilities in the truth table: 1. P and Q are both true. 2. P is true and Q is false. 3. P is false and Q is true. 4. P and Q are both false.
Tuesday, January 8, 2013
1.1-1.6, due on January 9
The most difficult part of the material for me is empty sets. I find it challenging to understand them conceptually. I feel that it will take some added focus and exercise in comprehending them and their importance. Also the notation is new to me and will take some getting used to; although, I'm sure it will come naturally as I become more engaged with the material in this course.
I find that it very interesting that this is mathematics. I know that it truly is mathematics, but it reminds me that many people I know believe that mathematics is arithmetic. The material is analytical and logical. I really appreciate that aspect of the mathematics. I believe this will be a very useful skill to sharpen as I progress in my academic career and also in my career as an actuary, which is exciting to me.
I find that it very interesting that this is mathematics. I know that it truly is mathematics, but it reminds me that many people I know believe that mathematics is arithmetic. The material is analytical and logical. I really appreciate that aspect of the mathematics. I believe this will be a very useful skill to sharpen as I progress in my academic career and also in my career as an actuary, which is exciting to me.
Introduction, due on January 9
I am a junior and my major is actuarial science.
I have taken Math 112 and Math 113.
I am taking this class to gain a deeper understanding of mathematics with hopes that it will prepare me better for future mathematics classes and all types of learning in my life. This class can potentially contribute towards a mathematics minor.
The least effective math teacher I've had was very hands-off, requiring little from the students. I felt that many of my classmates and I would have learned more if we were required to give more effort and challenged to fully comprehend the material. The most effective math teacher I've had really cared about the students, I believe, and encouraged learning the material while not focusing so much on studying just to pass a test. This helped strengthen a desire within me to embrace learning and master the subject matter.
I am a returning student after being away from BYU for 8 years. My freshman year at BYU was in 1998. I decided I wanted a career change and to finish my undergraduate degree, so my wife and our 3 children moved from Denver, CO to Provo in August 2012 so that I could finish my degree in pursuit of a more satisfying career.
I have no conflict with your scheduled office hours. The indicated times work for me.
I have taken Math 112 and Math 113.
I am taking this class to gain a deeper understanding of mathematics with hopes that it will prepare me better for future mathematics classes and all types of learning in my life. This class can potentially contribute towards a mathematics minor.
The least effective math teacher I've had was very hands-off, requiring little from the students. I felt that many of my classmates and I would have learned more if we were required to give more effort and challenged to fully comprehend the material. The most effective math teacher I've had really cared about the students, I believe, and encouraged learning the material while not focusing so much on studying just to pass a test. This helped strengthen a desire within me to embrace learning and master the subject matter.
I am a returning student after being away from BYU for 8 years. My freshman year at BYU was in 1998. I decided I wanted a career change and to finish my undergraduate degree, so my wife and our 3 children moved from Denver, CO to Provo in August 2012 so that I could finish my degree in pursuit of a more satisfying career.
I have no conflict with your scheduled office hours. The indicated times work for me.
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