1. Which topics and theorems do you think are important out of those we have studied?
Um... I think that everything we've talked about since the last exam will be pretty important... It seems like the base for everything we have done in this class is knowing the definitions of groups and rings and using those definitions to prove different types of results. So, I'd say that knowing those definitions along with the theorems with names will be pretty darn important.
2. What do you need to work on understanding better before the exam?
Cauchy's Theorem. OMG, I have reviewed that thing four or five times since that day in class, and it still gives me nightmares and makes me want to cry.
3. How do you think the things you learned in this course might be useful to you in the future?
The things I have learned from this class, cheesy but true, are the importance of doing readings and homework before class because it really helps with understanding concepts explained in class. As I mentioned before when I did the reading I was able to listen more closely to things that seemed confusing during the reading, rather than trying to learn something during class and having more questions during homework. I also learned about proof techniques. While I still don't feel completely comfortable writing them on my own, I definitely got in the hang of using "If" statements to prove "then" statements and carefully examining definitions to help with the proofs.
Tuesday, April 12, 2011
Sunday, April 10, 2011
8.3
1. What was the most difficult part of the reading for you?
I think the hardest part of the reading for me is similar to the problems I've encountered at the end of the previous two unit's. It seems that the closer we get to tests the more all of the theorems kind of blend together. This is especially true as we are approaching the end of the semester where we have so many 'prime,' 'order,' 'isomorphic,' etc. proofs that things are kind of bleeding together. It certainly didn't help that they didn't really provide proofs for these theorems, but rather just said, "Here are a bunch of theorems you aren't really ready to prove, so we'll do that later." If I wasn't sitting here with my book in front of me, I doubt I'd be able to recount to you what any of the proofs actually said.
2. On a more positive note...
I know I just sounded like a big baby about all of the theorems bleeding together, but on a more positive note it is helpful that we have covered so many theorems that are so similar to each other because it has also made it more 'natural' to prove things, because it always feel like we've proven something similar before. So, hopefully I won't fail the final or the class, because as educational as this semester has been, I'd rather not do it again. :)
I think the hardest part of the reading for me is similar to the problems I've encountered at the end of the previous two unit's. It seems that the closer we get to tests the more all of the theorems kind of blend together. This is especially true as we are approaching the end of the semester where we have so many 'prime,' 'order,' 'isomorphic,' etc. proofs that things are kind of bleeding together. It certainly didn't help that they didn't really provide proofs for these theorems, but rather just said, "Here are a bunch of theorems you aren't really ready to prove, so we'll do that later." If I wasn't sitting here with my book in front of me, I doubt I'd be able to recount to you what any of the proofs actually said.
2. On a more positive note...
I know I just sounded like a big baby about all of the theorems bleeding together, but on a more positive note it is helpful that we have covered so many theorems that are so similar to each other because it has also made it more 'natural' to prove things, because it always feel like we've proven something similar before. So, hopefully I won't fail the final or the class, because as educational as this semester has been, I'd rather not do it again. :)
Tuesday, April 5, 2011
8.1
1. What was the hardest part of the reading for you?
Um... I'm kind of confused by Theorem 8.1, which sucks since it's the first theorem of the chapter. I am just not sure that I know exactly what this theorem is saying or what the proof is showing (other than proving the theorem).
2. May I say something about the final?
I'm not sure how else to say this, but I thought this was a pretty good place... For the love of everything holy, can we please not prove that theorem you proved yesterday in class on the test? Seriously, I was so confused and lost and with all of the new vocabulary and procedures, I wanted to cry and/or die. Or, if you're going to put it on the test, can you please review it or something? I mean, I wrote the whole thing down and as I was re-reading it after class it was more confusing than ever... Please, show some humanity. :)
Um... I'm kind of confused by Theorem 8.1, which sucks since it's the first theorem of the chapter. I am just not sure that I know exactly what this theorem is saying or what the proof is showing (other than proving the theorem).
2. May I say something about the final?
I'm not sure how else to say this, but I thought this was a pretty good place... For the love of everything holy, can we please not prove that theorem you proved yesterday in class on the test? Seriously, I was so confused and lost and with all of the new vocabulary and procedures, I wanted to cry and/or die. Or, if you're going to put it on the test, can you please review it or something? I mean, I wrote the whole thing down and as I was re-reading it after class it was more confusing than ever... Please, show some humanity. :)
Sunday, April 3, 2011
7.10
1. What was the hardest part of the reading for you?
Honestly, and I am embarrassed, I struggled with the whole thing. I'm just not exactly sure what this chapter is saying. It's obvious that it's all aimed at proving theorem 7.52, but I'm not exactly sure what that theorem is saying (because I can read) but I really struggled to follow the proofs of the lemmas and the meaning behind the theorem.
2. Figure out what Theorem 7.52 is saying
Okay, so I'm going to break this down and looks stuff up. The theorem states "For each n =/ 4, the alternating group An is a simple group."
Alright, I know the chapter addresses this, but I'm still not sure about n not equaling 4... but let's move on from there since it's just the assumption. Well, let's review "Alternating Groups". Chapter 7.9 says that an alternating group is the set of all even permutations (even permutations can be written as the product of an even number of transpositions [2-cycles]) in Sn. The set of these permutations is written An... And a simple group is a group whose only normal subgroups (or subgroups that have left and right cosets equal to each other) are <e> and itself (G). So, what the theorem is saying, as far as I understand it, is if the set of all permutations (not with 4 elements... or unique mappings containing the numbers 1-4, since A4 really has 24 elements...) that has permutations that are even, or can be written as an even number of 2-cycles, is a subgroup that only has the normal subgroups of <e> and itself (the alternating group of elements of Sn that has all even permutations.)
Honestly, and I am embarrassed, I struggled with the whole thing. I'm just not exactly sure what this chapter is saying. It's obvious that it's all aimed at proving theorem 7.52, but I'm not exactly sure what that theorem is saying (because I can read) but I really struggled to follow the proofs of the lemmas and the meaning behind the theorem.
2. Figure out what Theorem 7.52 is saying
Okay, so I'm going to break this down and looks stuff up. The theorem states "For each n =/ 4, the alternating group An is a simple group."
Alright, I know the chapter addresses this, but I'm still not sure about n not equaling 4... but let's move on from there since it's just the assumption. Well, let's review "Alternating Groups". Chapter 7.9 says that an alternating group is the set of all even permutations (even permutations can be written as the product of an even number of transpositions [2-cycles]) in Sn. The set of these permutations is written An... And a simple group is a group whose only normal subgroups (or subgroups that have left and right cosets equal to each other) are <e> and itself (G). So, what the theorem is saying, as far as I understand it, is if the set of all permutations (not with 4 elements... or unique mappings containing the numbers 1-4, since A4 really has 24 elements...) that has permutations that are even, or can be written as an even number of 2-cycles, is a subgroup that only has the normal subgroups of <e> and itself (the alternating group of elements of Sn that has all even permutations.)
Thursday, March 31, 2011
7.9
1. What was the hardest part of the reading for you?
I was fine with the reading (though this new notation feels confusing and silly to me so far) until the part about transpositions. I DO NOT UNDERSTAND THESE!!!!! I don't understand, with all of the establishing of cycles and stuff, where these transpositions come from or what they even mean. Because of that, I really felt stuck on the alternating groups section.
2. New Notation
I kind of understand the reasoning behind this new notation, and as annoying as it can be to write symmetric groups with the older notation (with the two lines) I would much rather write that because it is something you can track.... Like, I feel like I'm just going to get so lost when it comes to function composition if I'm just looking at a string of numbers (especially if it isn't separated by a comma! Oh Commas, how I love you.)
I was fine with the reading (though this new notation feels confusing and silly to me so far) until the part about transpositions. I DO NOT UNDERSTAND THESE!!!!! I don't understand, with all of the establishing of cycles and stuff, where these transpositions come from or what they even mean. Because of that, I really felt stuck on the alternating groups section.
2. New Notation
I kind of understand the reasoning behind this new notation, and as annoying as it can be to write symmetric groups with the older notation (with the two lines) I would much rather write that because it is something you can track.... Like, I feel like I'm just going to get so lost when it comes to function composition if I'm just looking at a string of numbers (especially if it isn't separated by a comma! Oh Commas, how I love you.)
Tuesday, March 29, 2011
7.8
1. What was the most difficult part of the reading for you?
I think that I understand the idea of simple groups, that doesn't really bother me. However I really struggled to follow the reading after the first paragraph of that section. I mostly think that the problem is that there aren't really very many examples given (other than Zp) but I think I really just need a class with examples to really grasp the concept and the following theorems and stuff.
2. Quotient Groups and Homomorphisms
As I mentioned in the last reading, this was the part where I really really got confused with rings and such. I think it's a mix of having so many theorems in this chapter and the notations make me dizzy and confused. For example, in the third isomorphism theorem for groups when it uses the notation (G/N)/(K/N), like... really? Am I reading this right? Is this "(G mod N) mod (K mod N)"?!? What is that? That is so crazy!
I think that I understand the idea of simple groups, that doesn't really bother me. However I really struggled to follow the reading after the first paragraph of that section. I mostly think that the problem is that there aren't really very many examples given (other than Zp) but I think I really just need a class with examples to really grasp the concept and the following theorems and stuff.
2. Quotient Groups and Homomorphisms
As I mentioned in the last reading, this was the part where I really really got confused with rings and such. I think it's a mix of having so many theorems in this chapter and the notations make me dizzy and confused. For example, in the third isomorphism theorem for groups when it uses the notation (G/N)/(K/N), like... really? Am I reading this right? Is this "(G mod N) mod (K mod N)"?!? What is that? That is so crazy!
Sunday, March 27, 2011
7.7
1. What was the hardest part of the reading for you?
Oh man, I hate to be a Negative Nancy here, but this is where I started to get lost and feel frustrated when talking about rings and quotient rings and yadda yadda. I feel nervous about going into this chapter and what it'll do for the rest of the semester. The part that was difficult for me was the part that talked about the structure of the groups. For example, Theorem 3.37 kind of lost me, along with 7.38. I don't like that they changed notation of the center of the group during the proof -- who just does that?!? and I really just didn't follow that proof very well.
2. Why the confusion?
This was the question I asked myself at the very beginning of this section. The first thing this chapter says is, "Let N be a normal subgroup of a group G. Then G/N denotes the set of all right cosets of N in G." Okay, my question is this: Why state that N is a NORMAL subgroup then make a notation for the right cosets? Why can't G/N just be notation for the normal cosets? Is there going to be a different notation for left cosets? I just don't really understand this part of it. I know that if N is normal to G then the right and left cosets of N will be the same, but why make the distinction here when talking about a normal set? That is what I would like to know.
Oh man, I hate to be a Negative Nancy here, but this is where I started to get lost and feel frustrated when talking about rings and quotient rings and yadda yadda. I feel nervous about going into this chapter and what it'll do for the rest of the semester. The part that was difficult for me was the part that talked about the structure of the groups. For example, Theorem 3.37 kind of lost me, along with 7.38. I don't like that they changed notation of the center of the group during the proof -- who just does that?!? and I really just didn't follow that proof very well.
2. Why the confusion?
This was the question I asked myself at the very beginning of this section. The first thing this chapter says is, "Let N be a normal subgroup of a group G. Then G/N denotes the set of all right cosets of N in G." Okay, my question is this: Why state that N is a NORMAL subgroup then make a notation for the right cosets? Why can't G/N just be notation for the normal cosets? Is there going to be a different notation for left cosets? I just don't really understand this part of it. I know that if N is normal to G then the right and left cosets of N will be the same, but why make the distinction here when talking about a normal set? That is what I would like to know.
Thursday, March 24, 2011
7.6 Part 2
1. What was the hardest part of the reading for you?
I know I'm naughty because I haven't blogged or done my homework in a couple of days. Call it my spring break. That being said, tonight's reading was pretty easy for me, mostly because we talked about the whole chapter in class last time... but the thing that I found the most 'interesting' or something that I think is kind of tricky is the stuff in bold letters on the top of page 212: "The condition aN=Na does NOT imply that na=an for every n in N."
2. Theorem 7.34
While proving this in class yesterday, the question was asked, "Why do we need all 5 of these pieces of the theorem? Aren't 2 and 3 kind of redundant?" I thought the answer to that question was kind of interesting, and that was "Though 4 and 5 are stronger statements than 2 and 3, it's easier to prove a weaker statement, and from the weaker statement it becomes easier to prove the stronger statement, and that was evident from the proof of 7.34.
I know I'm naughty because I haven't blogged or done my homework in a couple of days. Call it my spring break. That being said, tonight's reading was pretty easy for me, mostly because we talked about the whole chapter in class last time... but the thing that I found the most 'interesting' or something that I think is kind of tricky is the stuff in bold letters on the top of page 212: "The condition aN=Na does NOT imply that na=an for every n in N."
2. Theorem 7.34
While proving this in class yesterday, the question was asked, "Why do we need all 5 of these pieces of the theorem? Aren't 2 and 3 kind of redundant?" I thought the answer to that question was kind of interesting, and that was "Though 4 and 5 are stronger statements than 2 and 3, it's easier to prove a weaker statement, and from the weaker statement it becomes easier to prove the stronger statement, and that was evident from the proof of 7.34.
Thursday, March 17, 2011
7.5 Part 2
1. What was the hardest part of the reading for you?
I am just feeling a tad confused with the way that all of the theorems are coming together, mostly because I don't remember a lot of them and I'm struggling to keep them straight. It's evident that everything is coming to a head in this section. As I read through the proofs of these theorems, I know I could (and should) spend more time reviewing and looking at the different theorems that are used here, but I didn't... so that is a good part of my confusion.
2. I love sudoku...
The one thing that I did enjoy about these proofs was how similar they were to sudoku, like we have discussed. I like that you start with knowing that something is cyclic and looking at it's order and then filling in the table. The only problem is, it can be tricky to keep these theorems straight! Like I said already...
I am just feeling a tad confused with the way that all of the theorems are coming together, mostly because I don't remember a lot of them and I'm struggling to keep them straight. It's evident that everything is coming to a head in this section. As I read through the proofs of these theorems, I know I could (and should) spend more time reviewing and looking at the different theorems that are used here, but I didn't... so that is a good part of my confusion.
2. I love sudoku...
The one thing that I did enjoy about these proofs was how similar they were to sudoku, like we have discussed. I like that you start with knowing that something is cyclic and looking at it's order and then filling in the table. The only problem is, it can be tricky to keep these theorems straight! Like I said already...
Tuesday, March 15, 2011
7.5
1. What was the hardest part of the reading for you?
Well, now I'm kind of embarrassed and confused. When looking at D4 (like in the example on page 200) under function composition, I realized that I am probably crazy, but h*d would have you look at h on the left side of a chart and over to d on the top of the chart. But, according to this, you look at d along the side then to h on the top. Maybe I've been writing my tables wrong, but that is nuts and confusing. The other thing that confused me is how h was congruent to d mod that subset of rotations. What?!? That made no sense to me. It made sense, I suppose, that congruence classes here are ab^-1 in K... but... what? I also hated that they left me confused then changed concepts.
2. Lagrange's Theorem
Um.... what is this malarky? It's more confusing notation and it's a bunch of confusing stuff, and I feel sad and confused. Pretty much the only thing that I really took out of that section is with these distinct right cosets, where every element of a group belongs to one and only one coset, then of course by adding the number of elements of the right cosets will be the same as the number of elements of a group... But that was pretty much the only thing I took out of that section.
Well, now I'm kind of embarrassed and confused. When looking at D4 (like in the example on page 200) under function composition, I realized that I am probably crazy, but h*d would have you look at h on the left side of a chart and over to d on the top of the chart. But, according to this, you look at d along the side then to h on the top. Maybe I've been writing my tables wrong, but that is nuts and confusing. The other thing that confused me is how h was congruent to d mod that subset of rotations. What?!? That made no sense to me. It made sense, I suppose, that congruence classes here are ab^-1 in K... but... what? I also hated that they left me confused then changed concepts.
2. Lagrange's Theorem
Um.... what is this malarky? It's more confusing notation and it's a bunch of confusing stuff, and I feel sad and confused. Pretty much the only thing that I really took out of that section is with these distinct right cosets, where every element of a group belongs to one and only one coset, then of course by adding the number of elements of the right cosets will be the same as the number of elements of a group... But that was pretty much the only thing I took out of that section.
Sunday, March 13, 2011
7.4
1. What was the most difficult part of the reading for you?
The thing that I'm struggling with is, if there are two groups who work under two different operations, I feel like I'm going to struggle how to use which operation. For example, I struggled to follow the second example on page 192 where were were looking at the additive group of real numbers and it's mapping to the multiplicative group of positive real numbers. This seems confusing and no fun to me.
2. Theorems
It's probably because I haven't done the homework for 7.3 yet, but the different theorems discussed and proven in 7.4 kind of through me for a loop. Again, it's probably because I haven't done the exercises for the things discussed in the theorem (like cyclic groups and such) but these theorems kind of make my head feel mushy.
The thing that I'm struggling with is, if there are two groups who work under two different operations, I feel like I'm going to struggle how to use which operation. For example, I struggled to follow the second example on page 192 where were were looking at the additive group of real numbers and it's mapping to the multiplicative group of positive real numbers. This seems confusing and no fun to me.
2. Theorems
It's probably because I haven't done the homework for 7.3 yet, but the different theorems discussed and proven in 7.4 kind of through me for a loop. Again, it's probably because I haven't done the exercises for the things discussed in the theorem (like cyclic groups and such) but these theorems kind of make my head feel mushy.
Thursday, March 10, 2011
7.3
1. What was the most difficult part of the reading for you?
I think the hardest part of the reading for me today was understanding "Generators of a Group." I'm not convinced that I'm convinced all the way yet about this concept. Well, I see how, for example {7,11} generate the group U15, like in the example.... but.... I'm not sure that I know where that comes from or why that matters. Well, it isn't even that I need to know why it matters, but I"m confused by the explanation of where this "generator" comes from.
2. Cyclic Subgroups Generated By "a"
I decided that I like this to be named "cyclic" because it's cyclical!!!!! Well, I guess if it's of infinite order it isn't cyclical, but I like the idea of it being of finite order and being cyclical. I know this is a lame thing to talk about, but that's how I feel. However, I would like to say that while I understand the change of notation for additive groups, I don't like it. I wish it would just be consistant notation for groups and that would make me happy.
I think the hardest part of the reading for me today was understanding "Generators of a Group." I'm not convinced that I'm convinced all the way yet about this concept. Well, I see how, for example {7,11} generate the group U15, like in the example.... but.... I'm not sure that I know where that comes from or why that matters. Well, it isn't even that I need to know why it matters, but I"m confused by the explanation of where this "generator" comes from.
2. Cyclic Subgroups Generated By "a"
I decided that I like this to be named "cyclic" because it's cyclical!!!!! Well, I guess if it's of infinite order it isn't cyclical, but I like the idea of it being of finite order and being cyclical. I know this is a lame thing to talk about, but that's how I feel. However, I would like to say that while I understand the change of notation for additive groups, I don't like it. I wish it would just be consistant notation for groups and that would make me happy.
Tuesday, March 8, 2011
7.2
1. What was the hardest part of the reading for you?
I'm honestly kind of confused by some notation here. Mostly with corollary 7.9 on the part that talks about how "that is, |a|=<|c|..." I'm confused because doesn't |a| mean the size of the group? Like.... shouldn't this be |G| not |a|? That is what I"m most confused about right now.
2. Embarrassed
Well, that was really the hardest thing for me, but as I was looking around for what I could write here, I thought "Find the earliest use of |a| and see if you can figure it out." Well, then I ran into the part where it talks about it and now I'm embarrassed that I didn't see it the first time through. It looks as though |a| is the order of a, which is fine, but I just get so darn confused by the crazy use of notation all over the place! Oh well, I guess I'm just a goober.
I'm honestly kind of confused by some notation here. Mostly with corollary 7.9 on the part that talks about how "that is, |a|=<|c|..." I'm confused because doesn't |a| mean the size of the group? Like.... shouldn't this be |G| not |a|? That is what I"m most confused about right now.
2. Embarrassed
Well, that was really the hardest thing for me, but as I was looking around for what I could write here, I thought "Find the earliest use of |a| and see if you can figure it out." Well, then I ran into the part where it talks about it and now I'm embarrassed that I didn't see it the first time through. It looks as though |a| is the order of a, which is fine, but I just get so darn confused by the crazy use of notation all over the place! Oh well, I guess I'm just a goober.
Sunday, March 6, 2011
7.1 Part 2
1. What was the most difficult part of the reading for you?
I guess that the hardest part of the reading for me was the same thing that was difficult for me in the homework. With these complex operations and definitions of a group, it is difficult for me to figure out if each element has an inverse. I didn't really have a hard time finding the identities of the different groups in number 4, but figuring out the inverses and if they were in the group kind of tricked my head a little, and I think that is the difficult part for me.
2. Shape Symmetry
I really liked the examples they had where they reflected, rotated, etc. the different shapes. I think the reason this really sunk in with me is because I'm currently in Survey of Geometry where we are really looking at those symmetries and such, so having a visual between the shapes and "functions" moving the corners (or whatever) was very helpful for me to understand what is going on in groups (mostly how after you look at the symmetries the you have to end up with the same shape, which really painted a picture for me about being closed under the operation). So.... the moral of the story is that it made me happy.
I guess that the hardest part of the reading for me was the same thing that was difficult for me in the homework. With these complex operations and definitions of a group, it is difficult for me to figure out if each element has an inverse. I didn't really have a hard time finding the identities of the different groups in number 4, but figuring out the inverses and if they were in the group kind of tricked my head a little, and I think that is the difficult part for me.
2. Shape Symmetry
I really liked the examples they had where they reflected, rotated, etc. the different shapes. I think the reason this really sunk in with me is because I'm currently in Survey of Geometry where we are really looking at those symmetries and such, so having a visual between the shapes and "functions" moving the corners (or whatever) was very helpful for me to understand what is going on in groups (mostly how after you look at the symmetries the you have to end up with the same shape, which really painted a picture for me about being closed under the operation). So.... the moral of the story is that it made me happy.
Thursday, March 3, 2011
7.1
(First of all I would like to point out how stupid I am because I totally read 6.3 the other night, and spaced blogging about it. Dumb dumb dumb....)
1. What was the most difficult part of the reading for you?
Okay, I think I am understanding this idea... kind of... but I am really struggling with something pretty basic and silly: What the heck are the elements of this group? Like, in the main example they were using of functions mapping 1,2,3 to 1,2,3, so are the elements 1,2,3? or are the elements the mappings? I understand that the one operation was the "of" function, but what are the elements? If I had to guess, I'd guess that the elements are the mappings, but if that is the case, I think I'm mostly confused about what groups (in general) are or look like.
2. An 'axiom' summary of groups:
i) Closure under operation
ii) Operationally associative
iii) Identity element
iv) Inverse element for EACH element (whatever those are) of your Group.
1. What was the most difficult part of the reading for you?
Okay, I think I am understanding this idea... kind of... but I am really struggling with something pretty basic and silly: What the heck are the elements of this group? Like, in the main example they were using of functions mapping 1,2,3 to 1,2,3, so are the elements 1,2,3? or are the elements the mappings? I understand that the one operation was the "of" function, but what are the elements? If I had to guess, I'd guess that the elements are the mappings, but if that is the case, I think I'm mostly confused about what groups (in general) are or look like.
2. An 'axiom' summary of groups:
i) Closure under operation
ii) Operationally associative
iii) Identity element
iv) Inverse element for EACH element (whatever those are) of your Group.
Sunday, February 27, 2011
6.2 Part 2
1. What was the hardest part of the reading for you?
Maybe it's because the first part of this reading was already covered in class, so that stuff was pretty easy, but I struggled with the First Isomorphism Theorem. I read through it a couple of times, and I still don't really understand exactly what it's saying. Usually when that happens I can understand it by reading the proof, but the proof in the book mostly confused me. It just has so many words!
2. A Rant:
It's silly, and a little embarrassing, but I wanted to scream when I realized they were using "pi" as a function, rather than functioning as the "pi" we all know and love. Why couldn't they use any of the other 24 letters of the alphabet (not counting the commonly used f and g) or any of the other Greek letters at their disposal?!? It's silly, I know, but that really made me stop and think each time they used it. WTF.
Maybe it's because the first part of this reading was already covered in class, so that stuff was pretty easy, but I struggled with the First Isomorphism Theorem. I read through it a couple of times, and I still don't really understand exactly what it's saying. Usually when that happens I can understand it by reading the proof, but the proof in the book mostly confused me. It just has so many words!
2. A Rant:
It's silly, and a little embarrassing, but I wanted to scream when I realized they were using "pi" as a function, rather than functioning as the "pi" we all know and love. Why couldn't they use any of the other 24 letters of the alphabet (not counting the commonly used f and g) or any of the other Greek letters at their disposal?!? It's silly, I know, but that really made me stop and think each time they used it. WTF.
Thursday, February 24, 2011
6.2
1. What was the most difficult part of the reading for you?
I think the only thing that is becoming a little annoying/difficult for me by now is the stupid notation. Everything else so far seems pretty nice and easy as it comes from everything else we've been talking about. Though, it did make me laugh and annoyed when they talked about the three different uses of "+" on page 146. Mostly I just rolled my eyes.
2. Say something else amazing.
Um.... I mostly don't know what to say since this reading was so quick. I hated the homework for the second part of 6.1. I thought I understood everything, but doing the homework kind of confused me and annoyed me. The end.
I think the only thing that is becoming a little annoying/difficult for me by now is the stupid notation. Everything else so far seems pretty nice and easy as it comes from everything else we've been talking about. Though, it did make me laugh and annoyed when they talked about the three different uses of "+" on page 146. Mostly I just rolled my eyes.
2. Say something else amazing.
Um.... I mostly don't know what to say since this reading was so quick. I hated the homework for the second part of 6.1. I thought I understood everything, but doing the homework kind of confused me and annoyed me. The end.
Tuesday, February 22, 2011
6.1 Part 2
1. What was the most difficult part of the reading for you?
I'll be very specific. I think I'm savvy on most of this reading, but the basic part where I think they were summing some important stuff up on page 140 (at the top) kind of lost me. Maybe it's the notation, or... I don't know. But this "coset" stuff is a bit confusing to me. It makes sense that there would be congruence classes, but.... I don't know. I guess that's just how I feel. It's late.
2. Why is 2+I=[2] a congruence class in the principal ideal 3?
Alright, so 3=0(mod I) since 3=3k in I, and [3]={...-6,-3,0,3,6,...). So, 2=5(modI) and -3=2-5=3k.... I think this makes sense. Wait, does it make sense? I think so, but.... oh whatever. What a pathetic blog.
I'll be very specific. I think I'm savvy on most of this reading, but the basic part where I think they were summing some important stuff up on page 140 (at the top) kind of lost me. Maybe it's the notation, or... I don't know. But this "coset" stuff is a bit confusing to me. It makes sense that there would be congruence classes, but.... I don't know. I guess that's just how I feel. It's late.
2. Why is 2+I=[2] a congruence class in the principal ideal 3?
Alright, so 3=0(mod I) since 3=3k in I, and [3]={...-6,-3,0,3,6,...). So, 2=5(modI) and -3=2-5=3k.... I think this makes sense. Wait, does it make sense? I think so, but.... oh whatever. What a pathetic blog.
Monday, February 21, 2011
6.1
1. What was the most difficult part of the reading for you?
I think that I can follow the concept of ideals pretty well. It seems to make sense to me, and that is good. The thing that I'm struggling with, and it could be my level of focus, is the distinction between the different ideals that the book talks about.
2. Write the definitions of the different ideals:
IDEAL: A subring I of a ring R is an ideal provided that whenever r in R and a in I, then ra in I and ar in I. Note, a nonempty subset I of a ring R is an ieal if and only if it has these properties: i) if a,b in I, then a-b in I; ii) if r in R and a in I, then ra in I and ar in I.
LEFT IDEAL: Like the fourth example on page 136 -- like a matrix that seems to have an ideal for ra but maybe not for ar.
PRINCIPAL IDEAL GENERATED BY C: Let R be a commutative ring with identity, c in R and I the set of all multiples of c in R, that is, I={rc | r in R}. Then I is and ideal. (yeah, I still don't really get this)
IDEALS GENERATED BY C1, C2,...., CN: Let R be a commutative ring with identity and c1, c2,....,cn in R. Then the set I={r1c1+r2c2+....+rncn|r1, r2,....,rn in R} is an ideal in R. (again.... I don't really see it.)
I think that I can follow the concept of ideals pretty well. It seems to make sense to me, and that is good. The thing that I'm struggling with, and it could be my level of focus, is the distinction between the different ideals that the book talks about.
2. Write the definitions of the different ideals:
IDEAL: A subring I of a ring R is an ideal provided that whenever r in R and a in I, then ra in I and ar in I. Note, a nonempty subset I of a ring R is an ieal if and only if it has these properties: i) if a,b in I, then a-b in I; ii) if r in R and a in I, then ra in I and ar in I.
LEFT IDEAL: Like the fourth example on page 136 -- like a matrix that seems to have an ideal for ra but maybe not for ar.
PRINCIPAL IDEAL GENERATED BY C: Let R be a commutative ring with identity, c in R and I the set of all multiples of c in R, that is, I={rc | r in R}. Then I is and ideal. (yeah, I still don't really get this)
IDEALS GENERATED BY C1, C2,...., CN: Let R be a commutative ring with identity and c1, c2,....,cn in R. Then the set I={r1c1+r2c2+....+rncn|r1, r2,....,rn in R} is an ideal in R. (again.... I don't really see it.)
Thursday, February 17, 2011
5.3
1. What was the most difficult part of the reading for you?
I am kind of confused by the idea of the "extension field". If I just blindly accept it, and follow the reading, I understand it's significance and can see why it's important to building complex numbers, but.... the whole a=[x] and plugging that in for each class kind of blew my mind a little. Mostly, watching x^2+x+1 becoming [x]^2+[x]+1 then equaling zero was kind of... crazy.
2. Where I stand
Honestly, I struggled with the 5.1 homework a bunch. I am really not struggling with any of the correlation between Zn and these [f(x)] classes and such, but I struggle to remember all of the theorems and corollaries for everything. Mostly they confuse me. Also, I am kind of having a hard time accepting the fact that [x+1] isn't the same as [2x+2] in Z[x] or Q[x] or whatever. Does that make sense? Why aren't monics important here?
I am kind of confused by the idea of the "extension field". If I just blindly accept it, and follow the reading, I understand it's significance and can see why it's important to building complex numbers, but.... the whole a=[x] and plugging that in for each class kind of blew my mind a little. Mostly, watching x^2+x+1 becoming [x]^2+[x]+1 then equaling zero was kind of... crazy.
2. Where I stand
Honestly, I struggled with the 5.1 homework a bunch. I am really not struggling with any of the correlation between Zn and these [f(x)] classes and such, but I struggle to remember all of the theorems and corollaries for everything. Mostly they confuse me. Also, I am kind of having a hard time accepting the fact that [x+1] isn't the same as [2x+2] in Z[x] or Q[x] or whatever. Does that make sense? Why aren't monics important here?
Tuesday, February 15, 2011
5.2
1. What was the most difficult part of the reading for you?
I think the author knew that this would be confusing, because he kept talking about how it was difficult to grasp, but I don't really understand the significance (or even the idea) behind of having the subsets within the rings.... or whatever that was. The stuff explained in the example on page 126. It made me laugh because he kept saying, "You know, just like that one example on page 126" like that would make the world better or whatever. I still don't really get it.
2. So.....what does it mean? That example that is...
The best that I can imagine is that 0,1 (as opposed to [0],[1]) is closed under addition and multiplication and probably subtraction, so I guess that is why that is a subset. But.... who cares? I don't get it. I'm just not sure if it matters that they only be 0 and 1 or if the subset may also contain another non-0 and non-1 element.... I just don't know. I think I just don't get it because I'm not sure why it even matters. But hopefully we'll see why later.
I think the author knew that this would be confusing, because he kept talking about how it was difficult to grasp, but I don't really understand the significance (or even the idea) behind of having the subsets within the rings.... or whatever that was. The stuff explained in the example on page 126. It made me laugh because he kept saying, "You know, just like that one example on page 126" like that would make the world better or whatever. I still don't really get it.
2. So.....what does it mean? That example that is...
The best that I can imagine is that 0,1 (as opposed to [0],[1]) is closed under addition and multiplication and probably subtraction, so I guess that is why that is a subset. But.... who cares? I don't get it. I'm just not sure if it matters that they only be 0 and 1 or if the subset may also contain another non-0 and non-1 element.... I just don't know. I think I just don't get it because I'm not sure why it even matters. But hopefully we'll see why later.
Sunday, February 13, 2011
5.1
(sorry, I'm feeling pretty sick as I'm writing this, so it'll be short and sweet)
1. What was the most difficult part of the reading for you?
Though there were examples given, perhaps it's because of how I'm feeling, but I'm still not exactly sure what this is saying about polynomial congruence. Well, I guess that isn't true, but the first example of page 122 kind of scared me. I'm going to reread it when my head isn't hurting my soul, but for now I'm just intimidated and scared.
2. Helping to understand
The thing that I think will help me the best is similar to something that I said way back in chapter 1, and that is to always write stuff out when it comes to congruence classes with polynomials the exact same way that I did for the integers.
1. What was the most difficult part of the reading for you?
Though there were examples given, perhaps it's because of how I'm feeling, but I'm still not exactly sure what this is saying about polynomial congruence. Well, I guess that isn't true, but the first example of page 122 kind of scared me. I'm going to reread it when my head isn't hurting my soul, but for now I'm just intimidated and scared.
2. Helping to understand
The thing that I think will help me the best is similar to something that I said way back in chapter 1, and that is to always write stuff out when it comes to congruence classes with polynomials the exact same way that I did for the integers.
Tuesday, February 8, 2011
Exam 1 Prep
1. Which topics and theorems do you think are the most important out of those we have studied?
If I had to guess, I would guess that knowing the different axioms and how they define the different rings seem like they would be the most important thing to know, since it is the foundation of all of the important/non-reviewey stuff we've done. It seems like if I can nail those axioms down, and use them as definitions to help with different theorems and such will be very helpful.
2. What kinds of questions do you expect to see on the exam?
I am kind of imagine that the types of questions we'll see on the exam will be pretty similar to homework questions and questions we've encountered in class.
3. What do you need to work on understanding better before the exam?
I really need to study the 3 proofs from the study guide. I'm mostly nervous about the wording and what past info can/can't be used for these proofs. Similarly, I'm a little nervous about congruence between rings. The idea never worried me, but being able to see the congruence does make me a little worried.
If I had to guess, I would guess that knowing the different axioms and how they define the different rings seem like they would be the most important thing to know, since it is the foundation of all of the important/non-reviewey stuff we've done. It seems like if I can nail those axioms down, and use them as definitions to help with different theorems and such will be very helpful.
2. What kinds of questions do you expect to see on the exam?
I am kind of imagine that the types of questions we'll see on the exam will be pretty similar to homework questions and questions we've encountered in class.
3. What do you need to work on understanding better before the exam?
I really need to study the 3 proofs from the study guide. I'm mostly nervous about the wording and what past info can/can't be used for these proofs. Similarly, I'm a little nervous about congruence between rings. The idea never worried me, but being able to see the congruence does make me a little worried.
Sunday, February 6, 2011
4.4
1. What was the most difficult part of the reading for you?
Well, I'm not going to lie, I giggled when the author of the book seemed to be speaking right to me. The thing that is worrying me the most about this reading is being able o tell when a polynomial is a polynomial in F[x] or when it is the 'rule of it's induced function' as the book says. I'm sure that context will have to play a big role in this, but I just feel programed to treat polynomials as functions, not as...objects... But, as the book says on page 101: "the practice of using x for both is so widespread you may as well get used to it." So, this is me, getting used to it.
2. The remainder theorem
I have to admit that my mind was blown a little bit by this theorem. I was so shocked that the remainder could be determined so easily and as I read the theorem I thought to myself, "How is this possible?!? I want proof!" And the proof was so simple and easy to follow, I just had to laugh a little and say, "How did you not get that before?" Either way, it seems fun.
Well, I'm not going to lie, I giggled when the author of the book seemed to be speaking right to me. The thing that is worrying me the most about this reading is being able o tell when a polynomial is a polynomial in F[x] or when it is the 'rule of it's induced function' as the book says. I'm sure that context will have to play a big role in this, but I just feel programed to treat polynomials as functions, not as...objects... But, as the book says on page 101: "the practice of using x for both is so widespread you may as well get used to it." So, this is me, getting used to it.
2. The remainder theorem
I have to admit that my mind was blown a little bit by this theorem. I was so shocked that the remainder could be determined so easily and as I read the theorem I thought to myself, "How is this possible?!? I want proof!" And the proof was so simple and easy to follow, I just had to laugh a little and say, "How did you not get that before?" Either way, it seems fun.
Thursday, February 3, 2011
4.3
1. What was the most difficult part of the reading for you?
Oh my gosh, maybe it was because we had to do homework right after doing some reading for this section with out a class period to digest, but I feel pretty lost. I'm not confused by the idea that something is "irreducible" or "reducible" but trying to do the homework tonight proved to me that I don't know how to tell for sure when something is irreducible or not. Also, this "associate" thing.... what is that? I don't understand it. I mean, I guess I understand that it is the product of a unit and something else, but I really don't understand much else about it, and that makes me sad.
2. Why I felt lost on the homework
So, I'm specifically talking about 4.3 2 and 9. I really struggled with 2 because I really don't understand associates. I understand being non-zero, I understand monic, and I understand unique. That being said, I don't even understand what I'm being asked to prove in number 2.
For number 9, wow.... Is there an easier way to do this? What I did for parts a) and b) is I wrote down every possible function I could imagine (for part a I wrote down a list of the polynomials with degree 2 in Z(2)) which weren't that many. Then, looking at them, I just eliminated things I knew could be factored. Then, when I remembered I was in Z(2) and should be able to eliminate some more things, I wrote down the two possible factors (x and x+1) and wrote the 3 possible combinations of the two. (x^2, x(x+1), and (x+1)^2) and eliminated those options too, leaving me with the one irreducible thing. Well, I was able to do that for part b, but it took a little more time. Then, realizing that I had to pretty much check 27 different combinations for part c made me tired and I just feel like there must be some easier way to see if things were irreducible or not. I just don't know what it is. Seriously, what am I not getting here.
Oh my gosh, maybe it was because we had to do homework right after doing some reading for this section with out a class period to digest, but I feel pretty lost. I'm not confused by the idea that something is "irreducible" or "reducible" but trying to do the homework tonight proved to me that I don't know how to tell for sure when something is irreducible or not. Also, this "associate" thing.... what is that? I don't understand it. I mean, I guess I understand that it is the product of a unit and something else, but I really don't understand much else about it, and that makes me sad.
2. Why I felt lost on the homework
So, I'm specifically talking about 4.3 2 and 9. I really struggled with 2 because I really don't understand associates. I understand being non-zero, I understand monic, and I understand unique. That being said, I don't even understand what I'm being asked to prove in number 2.
For number 9, wow.... Is there an easier way to do this? What I did for parts a) and b) is I wrote down every possible function I could imagine (for part a I wrote down a list of the polynomials with degree 2 in Z(2)) which weren't that many. Then, looking at them, I just eliminated things I knew could be factored. Then, when I remembered I was in Z(2) and should be able to eliminate some more things, I wrote down the two possible factors (x and x+1) and wrote the 3 possible combinations of the two. (x^2, x(x+1), and (x+1)^2) and eliminated those options too, leaving me with the one irreducible thing. Well, I was able to do that for part b, but it took a little more time. Then, realizing that I had to pretty much check 27 different combinations for part c made me tired and I just feel like there must be some easier way to see if things were irreducible or not. I just don't know what it is. Seriously, what am I not getting here.
Tuesday, February 1, 2011
4.2
1. What was the most difficult part of the reading for you?
Tonight, I got nervous for doing the homework for this section. Nothing really stood out to me as particularly difficult, but realizing that we're probably going to have to use the division algorithm a couple of times to find the gcd for two polynomials worried me. I have a hard enough time dividing polynomials cleanly on my paper so that I can follow it, so having to do it several times makes me a tad nervous, but I guess we just take it one problem at a time.
2. A caution for the future:
Something that worries me, mostly because I know how forgetful I am, will be to convert polynomials to "monics" when calling them the gcd. Hopefully by talking about it in this tiny paragraph will remind me to make sure that the leading coefficient is 1 when talking about it.... I think I can.... But man, I bet finding the formulas (like, the reverse of the division algorithm) is going to suck....
Tonight, I got nervous for doing the homework for this section. Nothing really stood out to me as particularly difficult, but realizing that we're probably going to have to use the division algorithm a couple of times to find the gcd for two polynomials worried me. I have a hard enough time dividing polynomials cleanly on my paper so that I can follow it, so having to do it several times makes me a tad nervous, but I guess we just take it one problem at a time.
2. A caution for the future:
Something that worries me, mostly because I know how forgetful I am, will be to convert polynomials to "monics" when calling them the gcd. Hopefully by talking about it in this tiny paragraph will remind me to make sure that the leading coefficient is 1 when talking about it.... I think I can.... But man, I bet finding the formulas (like, the reverse of the division algorithm) is going to suck....
Sunday, January 30, 2011
4.1
1. What was the most difficult part of the reading for you?
The proof of the division algorithm, definitely. I'm sure it's just because it's kind of late and my brain isn't fully functioning anymore, but I'm going to have to come back to that 2-page proof another day. It makes logical sense to me, but that thing is pretty intense. I also think that reading/understanding it WILL be important, but it sure is scaring me right now. Similarly, the use of "F[x]" Didn't make the most sense in the world to me.
2. Explore F[x]
Okay, I'll start by restating what the book says about F[x]: ...x is given a meaning similar to "[x sometimes doesn't seem to stand for any particular number but is treated as if it were a number in simplification exercises]", then the polynomials with coefficients in a field F form a ring (denoted F[x]) whose structure is remarkably similar to that of the ring Z of integers. So, I'll try to restate that: F[x] is a RING of the polynomials with coefficients that came from a different ring (in this case, F). So, and they may have said this already, but they are saying that F is a subring of F[x], right? And the reason F is a subring of F[x] is that F[x] which will have the same elements (the coefficients) of F will also have X in it? Is that right? I feel like that is what I understood from that, but wow, I'm not quite sure. I guess we'll see in class tomorrow and when I try to do the homework for this section...
The proof of the division algorithm, definitely. I'm sure it's just because it's kind of late and my brain isn't fully functioning anymore, but I'm going to have to come back to that 2-page proof another day. It makes logical sense to me, but that thing is pretty intense. I also think that reading/understanding it WILL be important, but it sure is scaring me right now. Similarly, the use of "F[x]" Didn't make the most sense in the world to me.
2. Explore F[x]
Okay, I'll start by restating what the book says about F[x]: ...x is given a meaning similar to "[x sometimes doesn't seem to stand for any particular number but is treated as if it were a number in simplification exercises]", then the polynomials with coefficients in a field F form a ring (denoted F[x]) whose structure is remarkably similar to that of the ring Z of integers. So, I'll try to restate that: F[x] is a RING of the polynomials with coefficients that came from a different ring (in this case, F). So, and they may have said this already, but they are saying that F is a subring of F[x], right? And the reason F is a subring of F[x] is that F[x] which will have the same elements (the coefficients) of F will also have X in it? Is that right? I feel like that is what I understood from that, but wow, I'm not quite sure. I guess we'll see in class tomorrow and when I try to do the homework for this section...
Thursday, January 27, 2011
...So Far...
1. How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
I have usually been spending 2 to 3 hours on the homework assignments. I work full time, and sometimes I start homework at 11 or 12 at night, so the times that I haven't finished my assignments were because I chose sleep over finishing an assignment. I do feel like the reading and lecture help me prepare for the assignments, but as I stated before -- sometimes I struggle with the creativity of some proofs. A couple of times I've just had to not finish my homework because I couldn't think of an answer or way to do the proof until I had slept on it, but then I'd have to rush to get it done before class (since I work right after class, so I don't really have until 4:00). I'm not meaning to complain, I'm just saying that I struggle with the creativity and sometimes it's helped me to walk away and then come back to it. Ya know?
2. What has contributed most to your learning in this class so far?
This is the first time when I have read the material this consistently. I know it is important to do so before class, and for the first time I've actually been doing it (though all classes say they want you too, I rarely ever have.) I will say that I hate blogging about my reading, but it has made me more accountable for the reading, and that is good. (Mostly, my slight OCD would KILL me if I had a missing section or a section out of order on here.)
3. What do you think would help you learn more effectively or make the class better for you?
Something that is a difficult line to walk is the repetition between the reading and the lectures. Sometimes I find myself getting pretty bored because as the lecture is happening I keep thinking to myself, "I already read and understand this." I'm sure the repetition is helpful when it comes time for doing the homework, but when a lot of the examples used in class are the same as in the book, it can just feel tedious (which is usually why I stop reading in other classes). However, that being said, I'm sure that that repetition and some of the questions asked in class (that I usually consider stupid, since I actually did the reading the night before) do help me when it comes time for the homework. Does that make sense?
P.S. Sorry this is a little late. I was having internet connection problems tonight and just got it up and going...
I have usually been spending 2 to 3 hours on the homework assignments. I work full time, and sometimes I start homework at 11 or 12 at night, so the times that I haven't finished my assignments were because I chose sleep over finishing an assignment. I do feel like the reading and lecture help me prepare for the assignments, but as I stated before -- sometimes I struggle with the creativity of some proofs. A couple of times I've just had to not finish my homework because I couldn't think of an answer or way to do the proof until I had slept on it, but then I'd have to rush to get it done before class (since I work right after class, so I don't really have until 4:00). I'm not meaning to complain, I'm just saying that I struggle with the creativity and sometimes it's helped me to walk away and then come back to it. Ya know?
2. What has contributed most to your learning in this class so far?
This is the first time when I have read the material this consistently. I know it is important to do so before class, and for the first time I've actually been doing it (though all classes say they want you too, I rarely ever have.) I will say that I hate blogging about my reading, but it has made me more accountable for the reading, and that is good. (Mostly, my slight OCD would KILL me if I had a missing section or a section out of order on here.)
3. What do you think would help you learn more effectively or make the class better for you?
Something that is a difficult line to walk is the repetition between the reading and the lectures. Sometimes I find myself getting pretty bored because as the lecture is happening I keep thinking to myself, "I already read and understand this." I'm sure the repetition is helpful when it comes time for doing the homework, but when a lot of the examples used in class are the same as in the book, it can just feel tedious (which is usually why I stop reading in other classes). However, that being said, I'm sure that that repetition and some of the questions asked in class (that I usually consider stupid, since I actually did the reading the night before) do help me when it comes time for the homework. Does that make sense?
P.S. Sorry this is a little late. I was having internet connection problems tonight and just got it up and going...
Tuesday, January 25, 2011
3.3
1. What was the most difficult part of the material for you?
Something that worried me during the reading was something that was addressed during the reading. I am pretty sure that I understood the concepts as they were explained in the reading, but it made me nervous to think about HOW to figure out what f was. It reminded me of those word puzzles in the newspaper that I suck at so badly -- you know, the one with the letters all mixed up and one letter stood for another letter and you have to figure out the code to get the message? It seems to me that that is exactly what is going on here, and I am so bad at those puzzles. But, the good thing is that it seems like it won't always be necessary to define f, but rather to understand if your two fields/rings are isomorphic/homomorphic or not.
2. Remembering injections and surjections
One issue I've had that go back years and years is remembering what it is to be injective and surjective. Like, I understand the two different definitions, and I know the two different words, but I struggle to remember which is which. So, tonight I am trying to think of a way to remember which is which. As far as I can remember, an injection is the 'one to one' thing where each element of R is going to be 'transformed' to an element of S (f(R)=S). The way I plan to remember that is by imagining "1-1" which looks like an "I" sideways, so "I"njection means "one to one". Then, a surjection (or 'onto,' which still sounds so stupid to me) means that each element of S comes from exactly one element of R. In other words, if it is surjective it would not be true that f(3)=5=f(2). The lame way I plan on remembering that is by remembering that a surjection doesn't look like this: f(S)=U=f(R). You know? If I knew how on this blog, I'd just write a "does not equal" sign and remember that as being a surjection, but we have to use what we've got.
Something that worried me during the reading was something that was addressed during the reading. I am pretty sure that I understood the concepts as they were explained in the reading, but it made me nervous to think about HOW to figure out what f was. It reminded me of those word puzzles in the newspaper that I suck at so badly -- you know, the one with the letters all mixed up and one letter stood for another letter and you have to figure out the code to get the message? It seems to me that that is exactly what is going on here, and I am so bad at those puzzles. But, the good thing is that it seems like it won't always be necessary to define f, but rather to understand if your two fields/rings are isomorphic/homomorphic or not.
2. Remembering injections and surjections
One issue I've had that go back years and years is remembering what it is to be injective and surjective. Like, I understand the two different definitions, and I know the two different words, but I struggle to remember which is which. So, tonight I am trying to think of a way to remember which is which. As far as I can remember, an injection is the 'one to one' thing where each element of R is going to be 'transformed' to an element of S (f(R)=S). The way I plan to remember that is by imagining "1-1" which looks like an "I" sideways, so "I"njection means "one to one". Then, a surjection (or 'onto,' which still sounds so stupid to me) means that each element of S comes from exactly one element of R. In other words, if it is surjective it would not be true that f(3)=5=f(2). The lame way I plan on remembering that is by remembering that a surjection doesn't look like this: f(S)=U=f(R). You know? If I knew how on this blog, I'd just write a "does not equal" sign and remember that as being a surjection, but we have to use what we've got.
Sunday, January 23, 2011
3.2
1. What was the most difficult part of the reading for you?
Let me start by saying 2 things. 1: I know I'm cutting it close. It's 11:48 and I am trying to hurry. 2: I realized the day after typing my last blog how silly I was for not catching that a "field" was mentioned in the reading for the class period before. I guess I was just sleepy and not thinking as well as I should have.
Anyway, to answer this questions, I'll say that something that concerns me for homework and a test is how much detail I need to worry about for these proofs. There were several times in the reading (like in proving Theorems 3.3, 3.4, 3.5...etc) when I was thinking to myself: Duh. It seems like these proofs are stating such silly and basic things with a lot of detail to get from point A to point B. I guess I can see why they are using that amount of detail, but I get worried that I won't know what I can use and what I shouldn't use when it comes to my proofs.
2. What can you do to overcome the thing you mentioned above?
I'm truthfully not exactly sure. I guess, as I've mentioned a couple of times in this blog, the important thing is to write down 1) the hypothesis, and the information you know because of that, 2) Your objective, or the thing you're trying to get to, and 3) fill in the stuff in between making logical connections that, though they seem obvious and silly, are important in proving your case. Doing this more on the homework will help me prepare for the test, so I'm going to try to be more detailed and a little more careful when it comes to stuff that I fell is silly.
Let me start by saying 2 things. 1: I know I'm cutting it close. It's 11:48 and I am trying to hurry. 2: I realized the day after typing my last blog how silly I was for not catching that a "field" was mentioned in the reading for the class period before. I guess I was just sleepy and not thinking as well as I should have.
Anyway, to answer this questions, I'll say that something that concerns me for homework and a test is how much detail I need to worry about for these proofs. There were several times in the reading (like in proving Theorems 3.3, 3.4, 3.5...etc) when I was thinking to myself: Duh. It seems like these proofs are stating such silly and basic things with a lot of detail to get from point A to point B. I guess I can see why they are using that amount of detail, but I get worried that I won't know what I can use and what I shouldn't use when it comes to my proofs.
2. What can you do to overcome the thing you mentioned above?
I'm truthfully not exactly sure. I guess, as I've mentioned a couple of times in this blog, the important thing is to write down 1) the hypothesis, and the information you know because of that, 2) Your objective, or the thing you're trying to get to, and 3) fill in the stuff in between making logical connections that, though they seem obvious and silly, are important in proving your case. Doing this more on the homework will help me prepare for the test, so I'm going to try to be more detailed and a little more careful when it comes to stuff that I fell is silly.
Thursday, January 20, 2011
3.1 Part 2
1. What was the most difficult part of the material for you?
The most difficult part of this reading for me was the part that talked about subrings and subfields. I'm still not exactly sure what the difference is between the two. I mostly think I got confused with the example on page 49 where it says "Z is a subring of the ring Q of rational numbers and Q is a subring of the field R of all real numbers. Since Q is itself a field, we say that Q is a subfield of R." It could be because I'm not exactly sure what a "field" is that they are referring to here. Is it like a vector field? I'm not sure. I'm also confused why they aren't just saying that Q is a subring of R..... you know? I'm confused.
2. Find out what the heck a "field" is here, and write about it :)
I found a site on the internet that said the following: "A field is a ring, such that for any a that is not equal to 0, there is an element b that is inverse to a with respect to multiplication: ab=1." So, I'm guessing that since in Q we could say that a=2, and then b=1/2 and they are both in Q and ab=1. So, I guess that answers my questions, but I don't know why they don't just call it a ring and make my life easier.
The most difficult part of this reading for me was the part that talked about subrings and subfields. I'm still not exactly sure what the difference is between the two. I mostly think I got confused with the example on page 49 where it says "Z is a subring of the ring Q of rational numbers and Q is a subring of the field R of all real numbers. Since Q is itself a field, we say that Q is a subfield of R." It could be because I'm not exactly sure what a "field" is that they are referring to here. Is it like a vector field? I'm not sure. I'm also confused why they aren't just saying that Q is a subring of R..... you know? I'm confused.
2. Find out what the heck a "field" is here, and write about it :)
I found a site on the internet that said the following: "A field is a ring, such that for any a that is not equal to 0, there is an element b that is inverse to a with respect to multiplication: ab=1." So, I'm guessing that since in Q we could say that a=2, and then b=1/2 and they are both in Q and ab=1. So, I guess that answers my questions, but I don't know why they don't just call it a ring and make my life easier.
Tuesday, January 18, 2011
3.1 Part 1
1. What was the most difficult part of the material for you?
Oh man, I can tell that we're about to venture into something pretty darn complicated. You can tell just by the set up that this is headed somewhere scary. The thing that is the most difficult is being able to remember the different axioms and which part of the definitions they belong to. I feel worried that I won't be able to just look at something and know that it is a "ring," a "commutative ring" or a "ring with identity". I know that I'll probably have to write down the different axioms, similar to what's in the book, and just work off of that until I get the hang of things, but I guess I'll just wait until that homework assignment to know for sure.
2. How are you going to keep it straight?
Because I'm nervous about venturing into this unknown world or 'rings' and whatever, I've really been trying to read things carefully, and, on my own, think of examples/counter examples to what they are talking about. For example, as I was reading and they were talking about the set E (even numbers) and how it was a commutative ring, I stopped a moment and asked, "would the set of odd numbers also be a ring?" and doing this helped me kind of run through the axioms and stuff. I also asked, "would this be a ring with identity" before the book even talked about it! (yea me!) and knew it couldn't be, because 1 is odd. Similarly, before the book even mentioned matrices, I asked myself about what would be a ring but not a commutative ring, and instantly I thought about how most matrices are not commutative. So, I felt excited when the book later talked about it. So, the moral of the story is: the best way to keep things straight is to ask yourself a lot of questions, and not move on until you know the answer. You know?
Oh man, I can tell that we're about to venture into something pretty darn complicated. You can tell just by the set up that this is headed somewhere scary. The thing that is the most difficult is being able to remember the different axioms and which part of the definitions they belong to. I feel worried that I won't be able to just look at something and know that it is a "ring," a "commutative ring" or a "ring with identity". I know that I'll probably have to write down the different axioms, similar to what's in the book, and just work off of that until I get the hang of things, but I guess I'll just wait until that homework assignment to know for sure.
2. How are you going to keep it straight?
Because I'm nervous about venturing into this unknown world or 'rings' and whatever, I've really been trying to read things carefully, and, on my own, think of examples/counter examples to what they are talking about. For example, as I was reading and they were talking about the set E (even numbers) and how it was a commutative ring, I stopped a moment and asked, "would the set of odd numbers also be a ring?" and doing this helped me kind of run through the axioms and stuff. I also asked, "would this be a ring with identity" before the book even talked about it! (yea me!) and knew it couldn't be, because 1 is odd. Similarly, before the book even mentioned matrices, I asked myself about what would be a ring but not a commutative ring, and instantly I thought about how most matrices are not commutative. So, I felt excited when the book later talked about it. So, the moral of the story is: the best way to keep things straight is to ask yourself a lot of questions, and not move on until you know the answer. You know?
Thursday, January 13, 2011
2.3
1. What was was the most difficult part of the material for you?
I think the most difficult thing for me with this reading is going to be the different techniques for proving. As I mentioned before, I'm bad at proofs, and there is a creativity required that I don't really have. So, thinking about proving 2.11 makes me want to puke. Also, keeping the different theorems and corollaries makes me nervous when it comes to these prime integer modules.
2. What was helpful in this reading?
I found the two basic techniques for proving statements involving Z_n kind of helpful. I guess we'll see how the homework goes.
I think the most difficult thing for me with this reading is going to be the different techniques for proving. As I mentioned before, I'm bad at proofs, and there is a creativity required that I don't really have. So, thinking about proving 2.11 makes me want to puke. Also, keeping the different theorems and corollaries makes me nervous when it comes to these prime integer modules.
2. What was helpful in this reading?
I found the two basic techniques for proving statements involving Z_n kind of helpful. I guess we'll see how the homework goes.
Tuesday, January 11, 2011
2.2
1. What was the most difficult part of the material for you?
Because I feel like I have a pretty good grasp on classes, this section felt pretty natural to me. There is one thing that does make me nervous, and it is the "new notation." I feel worried that they are dropping the brackets and by saying 4+1=0 should be enough indication to know we're in Z_5. I kind of understand wanting to make it simple, maybe it's too much work to write [4]+[1]=[0], but you don't even need to press the shift button to make the bracket, so I think that if there is too much back and forth between integers and classes I'll get annoyed and probably make silly mistakes.
2. What is the most interesting to you?
I'm not going to lie, I think the addition and times tables are pretty fascinating. At first I was curious why they were different sizes between Z_3, Z_5 and Z_6, but then I laughed when I woke up and realized why. So, I like that they are "limited" and not infinite, and I think the patterns are pretty interesting. But also, I love the idea of additive and multiplicative inverses and how they translate from integers into classes here.
Because I feel like I have a pretty good grasp on classes, this section felt pretty natural to me. There is one thing that does make me nervous, and it is the "new notation." I feel worried that they are dropping the brackets and by saying 4+1=0 should be enough indication to know we're in Z_5. I kind of understand wanting to make it simple, maybe it's too much work to write [4]+[1]=[0], but you don't even need to press the shift button to make the bracket, so I think that if there is too much back and forth between integers and classes I'll get annoyed and probably make silly mistakes.
2. What is the most interesting to you?
I'm not going to lie, I think the addition and times tables are pretty fascinating. At first I was curious why they were different sizes between Z_3, Z_5 and Z_6, but then I laughed when I woke up and realized why. So, I like that they are "limited" and not infinite, and I think the patterns are pretty interesting. But also, I love the idea of additive and multiplicative inverses and how they translate from integers into classes here.
Saturday, January 8, 2011
2.1
1.What was the most difficult part of the material for you?
I feel like doing Congruence and Modular Arithmatic is like speaking Spanish for me. I understand the rules of Spanish, I can speak it and understand a lot of it (I served a Spanish speaking mission), but it takes me a little longer than speaking and listening to English. I understand this section, but when it comes to doing these types of problems, I feel like I have to take a little extra time to write the congruence whatever into the equation that I can understand a little better. Also, the 'classes' aren't completely clear to me. Again, I kind of understand what they are, but it takes me a little extra time to process the problems. I think it will be helpful to talk about this stuff at the beginning of the semester so hopefully I can see why this concept is useful at all, because I still don't get it.
2. A little more clear this time around....
The last (and first) time I learned about these modulo things I struggled understanding what was going on. Reading though this section, I was able to visualize things a little. For example, for mod3, I imagined a long number line where starting at zero there was a mark for every number that was a multiple of 3. (...,-6,-3,0,3,6,....). Then, if I think of [2] in congruence modulo 3, I imagine those same intervals of 3 shifted to the left 2 numbers (...,-4,-1,2,5,8...) which is why [2]=[-1] in congruence modulo 3. Does that make sense? It makes sense to me.
I feel like doing Congruence and Modular Arithmatic is like speaking Spanish for me. I understand the rules of Spanish, I can speak it and understand a lot of it (I served a Spanish speaking mission), but it takes me a little longer than speaking and listening to English. I understand this section, but when it comes to doing these types of problems, I feel like I have to take a little extra time to write the congruence whatever into the equation that I can understand a little better. Also, the 'classes' aren't completely clear to me. Again, I kind of understand what they are, but it takes me a little extra time to process the problems. I think it will be helpful to talk about this stuff at the beginning of the semester so hopefully I can see why this concept is useful at all, because I still don't get it.
2. A little more clear this time around....
The last (and first) time I learned about these modulo things I struggled understanding what was going on. Reading though this section, I was able to visualize things a little. For example, for mod3, I imagined a long number line where starting at zero there was a mark for every number that was a multiple of 3. (...,-6,-3,0,3,6,....). Then, if I think of [2] in congruence modulo 3, I imagine those same intervals of 3 shifted to the left 2 numbers (...,-4,-1,2,5,8...) which is why [2]=[-1] in congruence modulo 3. Does that make sense? It makes sense to me.
Thursday, January 6, 2011
1.1-1.3
1. What was the most difficult part of the material for you?
Well, this reading assignment was probably easier than any future assignment we'll have, since 2/3 of it was already covered in class. Also, it seemed pretty basic because it was the first reading. However, as someone who just managed to not fail Theory of Analysis and is pretty nervous about this class, I will point out what I think will be difficult for me from here on out. I, like a lot of people, really don't like proofs. I imagine that the proofs will be the hardest thing for me in this class, and the reason why I find them difficult is that they are so darn imaginative. As I was reading these sections, and after Wednesdays class, I just kept thinking to myself, "Who would think to do it that way?!?"
For example, Theorem 1.11 just seems so basic to me. Of course every number is prime or made up of a unique combination of primes. And, as I follow the book it all makes sense, but I don't think I would ever think to write that proof using a contradiction (mostly because I am afraid of contradiction proofs).
The creativity astounds and intimidates me. Sure, when a proof is shown to me I can usually follow it, but I am so bad at thinking of them on my own. Even if I think I might have a good idea to write a proof, I get very intimidated by the wording of it, and usually mess it up in that way. So, I think that is what is going to be hard for me with this material in the future. Being intelligent and creative enough to be able to write reasonable proofs.
2. What can I do to overcome my Creative Proof Shortcoming (CPS)?
My best friend is in a basic algebra class at UVU and was asking me for help with proof by induction. I always try to help when I can, and was nervous when he asked me to help him with proofs. However, I remembered something about proof by induction that I shared with him. I remember sitting down with him and saying, "Okay, write what you know," which was usually your basic n formula. Then I told him to write what he wanted his equation to look like at the end (the n+1 formula). Then, I explained to him that the trick was to get from the n formula to the n+1 formula using only what you know to be true. I told him that the key was that you can't look at that as a 'restriction' but rather as a guideline showing you how to write the proof. Since the n formula was given, I explained that he would definitely be using THAT formula somewhere in the proof, and it would usually be somewhere in the middle. The hypothesis wasn't restricting him, it was pretty much giving him the answer. I realize that this is very similar to how I need to be thinking to be successful at writing proofs. Not that it will always be as easy as induction proofs, but hypotheses (?) are worded very carefully, and if I actually use the information given, getting from point A to point B should be easier, because the tools are usually given, you just have to put them in the right place. So, whenever I get stuck, I will do the same thing and ask, "Where am I? Where do I want to go? And what have they already given to me to get me there?"
Well, this reading assignment was probably easier than any future assignment we'll have, since 2/3 of it was already covered in class. Also, it seemed pretty basic because it was the first reading. However, as someone who just managed to not fail Theory of Analysis and is pretty nervous about this class, I will point out what I think will be difficult for me from here on out. I, like a lot of people, really don't like proofs. I imagine that the proofs will be the hardest thing for me in this class, and the reason why I find them difficult is that they are so darn imaginative. As I was reading these sections, and after Wednesdays class, I just kept thinking to myself, "Who would think to do it that way?!?"
For example, Theorem 1.11 just seems so basic to me. Of course every number is prime or made up of a unique combination of primes. And, as I follow the book it all makes sense, but I don't think I would ever think to write that proof using a contradiction (mostly because I am afraid of contradiction proofs).
The creativity astounds and intimidates me. Sure, when a proof is shown to me I can usually follow it, but I am so bad at thinking of them on my own. Even if I think I might have a good idea to write a proof, I get very intimidated by the wording of it, and usually mess it up in that way. So, I think that is what is going to be hard for me with this material in the future. Being intelligent and creative enough to be able to write reasonable proofs.
2. What can I do to overcome my Creative Proof Shortcoming (CPS)?
My best friend is in a basic algebra class at UVU and was asking me for help with proof by induction. I always try to help when I can, and was nervous when he asked me to help him with proofs. However, I remembered something about proof by induction that I shared with him. I remember sitting down with him and saying, "Okay, write what you know," which was usually your basic n formula. Then I told him to write what he wanted his equation to look like at the end (the n+1 formula). Then, I explained to him that the trick was to get from the n formula to the n+1 formula using only what you know to be true. I told him that the key was that you can't look at that as a 'restriction' but rather as a guideline showing you how to write the proof. Since the n formula was given, I explained that he would definitely be using THAT formula somewhere in the proof, and it would usually be somewhere in the middle. The hypothesis wasn't restricting him, it was pretty much giving him the answer. I realize that this is very similar to how I need to be thinking to be successful at writing proofs. Not that it will always be as easy as induction proofs, but hypotheses (?) are worded very carefully, and if I actually use the information given, getting from point A to point B should be easier, because the tools are usually given, you just have to put them in the right place. So, whenever I get stuck, I will do the same thing and ask, "Where am I? Where do I want to go? And what have they already given to me to get me there?"
Wednesday, January 5, 2011
Introduction
- What is your year in school and major?
I'm a super-senior... I've been here for 8 years and I'm not even a doctor. Instead I'm still an undergraduate studying Math Education.
-Which post-calculus math courses have you taken?
Foundation Mathematics (290), Elementary Linear Algebra (313), Multivariable Calculus (314), Ordinary Differential Equations (334), Theory of Analysis (341)
-Why are you in this class?
So I can graduate and finally get out of school!!!! Oh yeah, and change the lives of the kids who I'm sure are dying to know what I know about abstract algebra...
-Tell me about the math professor or teacher you have had who was the most and/or least effective. What did s/he do that worked so well/poorly?
Oh man, honestly it's hard to even tell them apart at this point. I don't feel like I've had any stellar math professors, but I did have an incredibly negative experience with one. For my Multivariable Calculus class I remember specifically that I worked my butt off that semester, studied any chance I got, did my homework and never missed a class period. I have never complained about any grading issue, and I don't even mind getting bad grades because it shows exactly what I need to improve on. However, on one test I studied particularly hard on I got a 0/30 points on a specific question because I copied 1 number down incorrectly at the very very very beginning of the problem. I did all of the steps correctly (it was a 3 part question that was 10 points for each part) and know that I knew what I was doing, I just happened to copy down "15" instead of "13" and got zeros on all 3 parts. I decided to talk to the professor (the first and only time I've ever done that in my 8 years here) to see if he could work with me since I thought zero points was a little extreme. He wouldn't even agree to set an appointment with me. He just looked at the test and said, "You got the answer wrong, and there's nothing I can do about it." That really ticked me off and I stopped caring the rest of the semester. Luckily I did well enough for the first half of the semester that my sour attitude didn't completely kill my grade by the end. Anyway, I felt the teacher didn't acknowledge my general efforts and didn't care about me as a person and that was poor teaching to me.
-Write something interesting about yourself
Well, on Christmas Eve Eve (Dec 23, 2010) around 2 AM I was admitted into the hospital for an emergency appendectomy. It wasn't the worst thing ever. I was released on Christmas Eve and since I am a full time manager at FYE (a movie store) I got to miss the two busiest/crappiest days of the year. I would do it all over again next year if God would only bless me with a second appendix.
-If you are unable to come to my scheduled office hours, what times would work for you?
Since I work full time and come to school part time, I don't think I have ever in my entire life used a professor's office hours, so though they aren't the best for me, I really don't think I'll ever use them, so no worries. If I'm in dire need of help, I'll contact you and maybe we can schedule something.
The End.
I'm a super-senior... I've been here for 8 years and I'm not even a doctor. Instead I'm still an undergraduate studying Math Education.
-Which post-calculus math courses have you taken?
Foundation Mathematics (290), Elementary Linear Algebra (313), Multivariable Calculus (314), Ordinary Differential Equations (334), Theory of Analysis (341)
-Why are you in this class?
So I can graduate and finally get out of school!!!! Oh yeah, and change the lives of the kids who I'm sure are dying to know what I know about abstract algebra...
-Tell me about the math professor or teacher you have had who was the most and/or least effective. What did s/he do that worked so well/poorly?
Oh man, honestly it's hard to even tell them apart at this point. I don't feel like I've had any stellar math professors, but I did have an incredibly negative experience with one. For my Multivariable Calculus class I remember specifically that I worked my butt off that semester, studied any chance I got, did my homework and never missed a class period. I have never complained about any grading issue, and I don't even mind getting bad grades because it shows exactly what I need to improve on. However, on one test I studied particularly hard on I got a 0/30 points on a specific question because I copied 1 number down incorrectly at the very very very beginning of the problem. I did all of the steps correctly (it was a 3 part question that was 10 points for each part) and know that I knew what I was doing, I just happened to copy down "15" instead of "13" and got zeros on all 3 parts. I decided to talk to the professor (the first and only time I've ever done that in my 8 years here) to see if he could work with me since I thought zero points was a little extreme. He wouldn't even agree to set an appointment with me. He just looked at the test and said, "You got the answer wrong, and there's nothing I can do about it." That really ticked me off and I stopped caring the rest of the semester. Luckily I did well enough for the first half of the semester that my sour attitude didn't completely kill my grade by the end. Anyway, I felt the teacher didn't acknowledge my general efforts and didn't care about me as a person and that was poor teaching to me.
-Write something interesting about yourself
Well, on Christmas Eve Eve (Dec 23, 2010) around 2 AM I was admitted into the hospital for an emergency appendectomy. It wasn't the worst thing ever. I was released on Christmas Eve and since I am a full time manager at FYE (a movie store) I got to miss the two busiest/crappiest days of the year. I would do it all over again next year if God would only bless me with a second appendix.
-If you are unable to come to my scheduled office hours, what times would work for you?
Since I work full time and come to school part time, I don't think I have ever in my entire life used a professor's office hours, so though they aren't the best for me, I really don't think I'll ever use them, so no worries. If I'm in dire need of help, I'll contact you and maybe we can schedule something.
The End.
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