Math 290 Fall Semester

The ideas in this slideshow are pretty simple but were a good reminder for me on how to write coherently. It also helped me to better understand the syntax of mathematical writing. I could always tell that my writing was a bit "off" but I wasn't sure why or what was causing the discrepancy between what I was reading and what I was writing. I also enjoyed the quotes about writing from the different authors.

The hardest thing for me to understand about this section was the addition and multiplication of congruence equivalence classes. I just don't understand the concept or the computation behind it. Does it mean that you add the values together, or the congruences, or the sets? I mostly understood how the congruence classes are formed, like 1,5,9,13 are all in the same mod 4 congruence class. But beyond that I didn't really understand the section, so I think it will help when I hear it explained in person.

The hardest part of this section for me to understand was how to go from partitions back to an equivalence relationship. I think the idea is just to work backwards, but the proofs were a bit confusing to visualize. Also, the concept of transversals was totally lost on me, the equations got ugly looking quickly and I didn't really understand how they were being calculated. I know what the definition is, I just can't wrap my head around the computation and the "floor" of x. Other than that it wasn't too bad. The jigsaw puzzle analogy helped a lot and was easy to visualize, and the idea of not overlapping, no empty sets, and covering all elements of A made total sense to me, which I think will help me understand the more confusing parts of the chapter.

For the most part this section made sense, it seems to just be further defining relationships on sets. Classifying the relationships on sets was a little bit confusing because it's a bit abstract, but I think I got the gist of it. You are essentially seeing which elements in a set have the same relationships and defining those relationships. I'm not super familiar with the definition so I'm not quite ready to write a proof on relation theory, but for the most part it makes sense.

This section seemed pretty straightforward, it was actually enjoyable to read because a lot of it is just simple logic which was my favorite previous section. The only things that were hard to wrap my mind around were the new definitions, especially anti-symmetric and symmetric. It's just a little bit abstract for me at the moment, so if you could do some examples in class where we decide if a relation has those properties then that would be the most helpful.

This section seemed pretty straightforward to me. A little hard to visualize, at first, but just like with section 17 I think that I will get it as soon as it is drawn on the board for me. It's cool to see how each of these properties and algorithms relates to each other. I'm not entirely sure what the applications are, but I was always fascinated by how numbers and counting are related.

For the most part it made sense, however in the division algorithm I wasn't sure what r' and q' are and where they came from, it might just be a simple notation but I didn't really get that. The Euclidian algorithm was a little unclear but I think if I read it again and have it explained to me and demonstrated in class then I'll understand it better. It seems like a much easier way to find the GCD of two integers and very easy to program as well.

This section honestly didn't make much sense to me. I understood how to calculate k choose n, but after that I was completely lost. I understood how to set up Pascal's triangle, but I didn't understand how the binomial theorem related to that. I think I just need to hear it explained out loud, that's typically how I learn math better. The induction doesn't seem terribly complicated itself, it's just a basic induction proof. Once I wrap my head around the calculation a bit more then I will be prepared to do it on my own.

The problem with the cities seemed pretty straightforward until they started dividing the cities into sets. I'm not sure why they did that and I got really confused after that. The square problem was also a bit tricky, I understand it visually but inductively I don't really understand that proof. Strong induction seems pretty straightforward otherwise, you just get to assume more in your premise which will help you later on in your proof. The book used strange examples but I think once I hear it explained out loud it will make more sense.

I don't quite understand the point of having multiple base cases, it seems like for the most part you would just be able to start wherever you want to anyway and not need to prove multiple base cases. I realize this is probably just a lack of understanding, but I also don't know how to identify where I would need to prove multiple base cases. Being able to start at a different number as your base case is helpful because not all statements are true for P(1), so it seems very handy that you would be able to start elsewhere for your inductive proof.

This section was a little tricky to understand, but I think I got the gist of it. One thing that I had a hard time with was warning 13.8. I understand cognitively that P(k) cannot equal a number or a part of a statement, but I'm not sure how I would prevent myself from potentially doing that and writing an incorrect statement. I think once I have someone explain to me in English what P(k) actually represents then I'll be able to visualize the concept better and know what I'm writing. Induction seems pretty straightforward and handy for proofs. Basically it says if something works for a natural number, then and then it works for the next natural number, you can replace the first natural number with the second one, and prove the third once by using the already proved definition, and so on. All in all it's a pretty useful concept.

I think the most important topic we have studied have been how to outline and write proofs. They are simple enough for the most part, but they have helped me to think more deeply about topics that I have understood but maybe not understood fully, and they have also helped me to think more abstractly to understand things that I haven't understood before. On the exam, I'm mostly expecting to see proofs and set theory questions. I think there will be a few logic questions but they will mostly be tied up in set problems and in formal proofs. I would like to see a few more examples of rationality proofs, like the ones from homework 11 and 10. Those were a bit harder for me to understand the theory behind the proof so seeing those written out would help me out a lot.