Math 290 Fall Semester

This section was pretty straightforward, and I remember a lot of the definitions from calculus so it's not anything really new. Series are really easy for me to understand, my only real problem was the formal definition of a limit, it seemed a bit convoluted and complicated. We are also going a bit deeper into series than we did in calculus, and so this may take some time for me to wrap my head around again. But all in all this section doesn't seem awful to me.

This section was wild. I think it may be in part because I am coming back from break and fighting a head cold/flu thing, but this was just so above anything else we have done so far. It's wild that Bernstein discovered this theorem when he was only 19. I think this section and theories will be very useful in the coming tests and proofs, and it's a cool theory to think about, but at the same time it is also very confusing to read at least, and I think i will need to work some examples before I really grasp what's happening.

This section was actually very straightforward for me. It seems logical that the set of real numbers would be uncountable, so my intuition wasn't really broken there. And the proof was very clever and made sense to me. The one thing that I didn't quite get was the piecewise function in corollary 30.5. I didn't understand how the bijection was made and how S-(0,1] played into it at all, but I think it was just because it was phrased a bit differently than what I'm used to.

This section was really just a continuation of section 28, and most of the logic is very similar. I think that once you figure out creative ways to make a set countable, then the rest is just (very loose) intuition, and it's not really to bad. These proofs are kind of tough though, because their structures don't seem as concrete as the ones before them, and therefore they are harder to formulate. The main thing I didn't understand at all was remark 25.9. I did not understand at all how f was injective but not surjective, and also how im(f) was countably infinite. I really need to see it worked out in order to understand this.

this section wasnt terrible, I understood what was going on for the most part. It is a little more abstract since we are dealing with infinite sets. I also didn't really know what the new notation was for, the squiggly N symbol was very weird looking. I think it just need to hear it explained out loud to fully understand it.

Nothing about this section seemed terribly complicated, at it was all pretty intuitive. I've dealt with composite functions in calculus when integrating and differentiating, so that part wasn't necessarily new to me. What seems really hard is keeping track of the domain and codomains, as well as the surjective and injective properties when you compile 3 or 4 sets together to create a function. Other than that, the simple calculation and scratch work doesn't seem that bad.

This section seemed pretty easy to understand as well. Basically you are just checking that each output appears at most once for injectivity and appears at least once for surjectivity. The proofs were a little strange and seemed like they were just stating the obvious, so I'm not sure what roles those play in the definitions. One thing I noticed is that this chapter used the "onto" and "one-to-one" definitions that we learned in linear algebra. I didn't really understand it when I learned it in 313, but it made a lot more sense when I read this chapter. If I had read this chapter before that section in 313, I would have understood that concept much better. I've found that with a lot of concepts in this class actually. I don't know what you have planned for this cohort in the future, but I think it could be very helpful to have 290 as a prerequisite course for this cohort, in order to solidify the mathematical concepts that appear in other math clas...

This section made a lot of sense to me. The definition of function doesn't seem drastically different than what it was in calculus. I could see how it would be a little bit tricky to apply this definition to congruence classes and sets rather than just to real numbers, but I think it's not too bad so far. This more technical definition also helped me understand more in depth about how functions work and are defined. I didn't really understand what the characteristic function meant. I'm sure it will be simple when it's explained in class, but other than that the section wasn't too difficult to grasp.

I think the most important things we have covered in class since the last exam has been induction proofs and proofs about integers and modulus classes. For the most part I feel pretty prepared for it, a lot of it just seems like basic logic applied to numbers, so there isn't a ton of raw computation involved. I expect that most of the proofs will be either induction or equivalence classes and relations and I feel like I have a pretty solid understanding of those concepts. The one thing that I don't understand and wouldn't be prepared for on the test is the binomial theorem. I didn't understand it at all in class and we didn't spend any other class periods on it, so if we could review as much of that section as we can in class that would be the most helpful for me.