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.
Tuesday, February 28, 2012
Sunday, February 26, 2012
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.
B. I liked some of the names. They were unique and interesting. The differences between onto, surjective and bijective are important.
Thursday, February 23, 2012
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.
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.
Tuesday, February 21, 2012
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.
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.
Monday, February 20, 2012
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.
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.
Tuesday, February 14, 2012
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.
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.
Saturday, February 11, 2012
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.
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.
Monday, February 6, 2012
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.
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.
Thursday, February 2, 2012
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.
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.
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