8.5-8.6, due on February 22

1. 8.5 doesn't seem too bad, kind of just a review of old material, but showing how it is an equivalence relation. 8.6 had me totally lost, though. There were so many new terms, like residence classes, closed under addition or multiplication, and well-defined. Actually most of it made sense after rereading it, but I'm still not quite sure what a residue class is.

2. I thought it was interesting to read that I will see congruence modulo n again, because I had never heard much about that before taking this class, and now it turns out it is some important mathematical concept! Strange.

8.3-8.4, due on February 21

1. I missed Friday's lecture and forgot to do the reading, so I thought that it would be really hard to catch up. Luckily the stuff I missed was about relations, which don't seem to difficult so far. I think what I will have the most trouble with is 8.4, which introduces equivalence classes. Equivalence classes seem to be a lot like sets, which are hard to prove stuff with. Just like with proofs of sets, I understand the examples in the book, but if I tried to do it by myself, it would be really hard.

2. Equivalence relations are actually kind of interesting. It's cool that we can prove things about a relation, without actually having to know what that relation is.

7.1-7.3, due on February 15

1. The first two sections didn't seem very difficult. 7.1 didn't really introduce anything except the term "conjecture," and 7.2 was just like a review of stuff we learned earlier. 7.3 didn't seem difficult to understand, but I think it will take some practice before I am able to easily determine whether to prove or disprove something. It was easier when I knew where I was supposed to be going, because I know that I at least got to the right answer, even if I got there the wrong way.

2. 7.1 has been my favorite section of the book so far. It amazes me how much has been discovered in the math world, but how much there still is to learn. It's weird that right now it seems impossible to prove some conjectures, but we can't even imagine the progress that math, science, and technology will make in the future. Just thinking about how much we have progressed in the past 100 years astounds me.

6.3-6.4, due on February 13

1. These sections both seem really hard. I understood 6.3 for the most part, but I think I will have trouble doing the related problems. 6.4 didn't really make sense to me. In the example on top of page 149, it said that since k+1 is greater than or equal to 3, it follows that: and then a bunch of stuff that followed, but I didn't understand how any of that followed from the previous statements.

2. I think it's interesting that we are doing so much stuff with induction and recursive series. I remember I really liked recursive series in algebra II, but something must have changed between then and now haha.

6.2, due on February 10

1. I didn't really see many differences between this section and 6.1. Was the difference that now, instead of the base case being 1, the base case can be any number that is the least element of a well-ordered set?

2. I thought that the examples for this section were interesting, because they were a lot different from the examples in 6.1. For example, I didn't think about using induction to prove something like 2^n > n. That seems tricky.

6.1, due on February 6

1. Mathematical induction made a lot of sense to me, but only after I thought about it for a while. It seems like it will be pretty easy to set up the proof, but then actually working through it and grouping and rewriting terms to make it work for k+1 will be the difficult part.

2. I think that probably the most important topics we have learned about have to do with direct proofs and proofs by contrapositive, because so many of the other concepts have stemmed from this knowledge. I am expecting to see a lot of questions on the exam that have to do with this. I also expect to see some tricky definition-type questions in multiple choice. I think the sections I had the most trouble with were the ones that involved set proofs, so if I had to choose something to see more examples of, it would be that.

5.4-5.5, due on February 3

1. These sections don't seem too bad. The only new thing we have to do is in 5.5 where we figure out how to turn the proof into something that we can prove using the methods we already know.

2. I feel like 5.4 is the opposite of disproving stuff. It's probably going to take a little more thinking, because it's not too hard to find examples of why something doesn't work, but it's a little harder to find the example of when something is true.