After a few reads, most of this section made sense. However I'm still not accustomed to equivalence classes and they're used extensively in this section. For Result 8.7, the alternate proof for proving that 2a + b is symmetric is still throwing me off. I understand how we want to add the two equations together, but I don't understand how we sub in for 3x and 3y. Hopefully we go over that in class.
I enjoy when we incorporate things learned or used previously and here we talked about how an equivalence class can relate to properties of integers namely being able to classify any integer within 3 equivalence classes for a certain relation. Also the proofs of equivalence relations are neat; they're elegant and smart which is fun.
Sunday, October 14, 2012
Thursday, October 11, 2012
Circles, Rivers, and Polygon Packing: Mathematical Methods in Origami by Robert Lang
My biggest problem with the lecture was that I didn't know enough. The last time I did origami I think I was in middle school. Lang just kind of jumped in to the math of origami almost as if we were already well versed in the field. While that's fine and all and I'm sure there's some people there that understood a lot more than I did, it was tough to understand.
However the applications for origami surprised me. I had heard of other engineers solving problems with origami but didn't know how. While I still don't know exactly how, he showed one way that stresses in a structure could be determined with origami. Another neat thing was that he pulled in a little bit of what we talked about in class with the 4 colors for a map. However for origami you must be able to represent the fold pattern with only 2 colors. Another neat thing was that putting a fraction into binary coded your folds that you needed to make to represent that fraction on the paper. Overall the lecture was really neat, the application was cool, and the art was also crazy intense. Who knew origami had some much math involved? Okay....well you probably did, but not me!
However the applications for origami surprised me. I had heard of other engineers solving problems with origami but didn't know how. While I still don't know exactly how, he showed one way that stresses in a structure could be determined with origami. Another neat thing was that he pulled in a little bit of what we talked about in class with the 4 colors for a map. However for origami you must be able to represent the fold pattern with only 2 colors. Another neat thing was that putting a fraction into binary coded your folds that you needed to make to represent that fraction on the paper. Overall the lecture was really neat, the application was cool, and the art was also crazy intense. Who knew origami had some much math involved? Okay....well you probably did, but not me!
8.3-8.4, due on October 12
For whatever reason I had the hardest time understanding why [1]={1,3,6} etc. Once I figured it out I felt really unintelligent. As well was the relation on page 179 for [a]. We had been talking about [a] before but for the second time we returned to the relation a=b. Having forgotten this I was confused as to why the relation that we had just talked about (x in Z such that x R a) was equal to x in Z such that x=a.
Something neat about equivalence classes was being able to predict the equality of [1] [3] and [6]. Since 1,3,6 were all related to each other, then [1]=[3]=[6]. The same went for the other integers as well. Also the use of the properties that we had just learned about was neat too. It provided for clever and elegant proofs, which are always enjoyable.
Something neat about equivalence classes was being able to predict the equality of [1] [3] and [6]. Since 1,3,6 were all related to each other, then [1]=[3]=[6]. The same went for the other integers as well. Also the use of the properties that we had just learned about was neat too. It provided for clever and elegant proofs, which are always enjoyable.
8.1-8.2, due on October 10
Just from the reading, understanding relations was very difficult. Even though I've read some other things too, relations don't exactly make sense. The relation should contain ordered pairs for the examples we're talking about, but it seems to me that normal relations would be after the order of y=3x or some other equation. Yet the relation seems to be definied much more loosely than that. We're given random sets of ordered pairs between two matrices but to me there doesn't really seem to be a pattern in the relation. Also the general lack of examples in this section of the book is kind of disappointing.
Although I'm not sure how they're used yet, I think properties of relations are kind of fun. It's a little bit more finite than a lot of the things we've talked about up til now. The relation either contains these sets of ordered pairs or it doesn't so it's easy to see what properties the sets have.
Although I'm not sure how they're used yet, I think properties of relations are kind of fun. It's a little bit more finite than a lot of the things we've talked about up til now. The relation either contains these sets of ordered pairs or it doesn't so it's easy to see what properties the sets have.
Sunday, October 7, 2012
6.3 - 6.4, due on October 8
A few things were very difficult to understand in this section. A lot of the proofs are very difficult and involve some unintuitive algebraic manipulation and whooplah. The second is that I still don't see the need for the strong principle of mathematical induction as opposed to the normal principle of induction. The book states that it's common for sequences of numbers but I still fail to see exactly why you need to give numerous examples that it's true for initial conditions in order to be able to prove it generally.
These sections were fairly unenjoyable. For me the proofs were very difficult to follow and required numerous readings to grasp their concept. Now that I understand them fairly well, I appreciate them, the minimum counterexample more than the strong principle of induction. It seems like a neat way to prove something but also like you can prove the same thing with normal induction. I assume that it would also be helpful in cases of proofs where contradiction is more convenient than a more direct approach.
These sections were fairly unenjoyable. For me the proofs were very difficult to follow and required numerous readings to grasp their concept. Now that I understand them fairly well, I appreciate them, the minimum counterexample more than the strong principle of induction. It seems like a neat way to prove something but also like you can prove the same thing with normal induction. I assume that it would also be helpful in cases of proofs where contradiction is more convenient than a more direct approach.
Friday, October 5, 2012
6.2, due on October 5
It seems to me like the more general principle of mathematical induction and the initial statement are so close that the book truly didn't need to distinguish between the do. What I think I'm getting is that for the more generalized statement, your base case doesn't have to be one. So why didn't we just learn that when we introduced it in the first place? It's a fairly simple concept. The toughest part of this section was the algebraic manipulation. I feel like the proofs are much more technical because we are trying to symbolically prove a case each time. This leads to a lot of generally tough mathematics!
For what it's worth I did see the benefit of being able to use a different case than the base case (unless I'm off base on that). As well, the well-ordered principle came into play, which was something I was wondering about from last section. It seems like with induction we can basically prove anything dealing with indexed sets or infinite series. So that's cool.
For what it's worth I did see the benefit of being able to use a different case than the base case (unless I'm off base on that). As well, the well-ordered principle came into play, which was something I was wondering about from last section. It seems like with induction we can basically prove anything dealing with indexed sets or infinite series. So that's cool.
Tuesday, October 2, 2012
6.1, due on October 2
So apparently I included comments on this reading in the section where I was supposed to talk about 5.1. So much now makes sense! I was really confused why you had asked us to read 6.1 and then we never talked about it. Induction is a pretty basic principle. The part that I couldn't understand was the usefulness of being well-ordered. It doesn't seem to connect in my mind with induction. If we could talk about it in class that would be great.
Induction is a very cool concept. It's pretty much like a proof by cases on steroids. If the base case is true, and then every other case after that is true, it means that the statement is true! It was also neat the way that Gauss used that to find really large sums. That's a neat aspect of induction.
Induction is a very cool concept. It's pretty much like a proof by cases on steroids. If the base case is true, and then every other case after that is true, it means that the statement is true! It was also neat the way that Gauss used that to find really large sums. That's a neat aspect of induction.
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