10.4, due on November 5

Maybe I just haven't spent enough time with decimal expansions yet, but there are two proofs given in the text that I'm not sure where and how they arrive at their contradictions. One is 10.8 and the other is 10.12. The first is the set of real numbers that is uncountable. I get lost somewhere in the defining of another decimal expansion that helps us reach the contradiction. And then with 10.12, it seems like in 10.11 they state the exact opposite.

The idea of decimal expansions seem neat to me. There was one proof on my homework I did one time where the TA said I should have taken the decimal expansion of the number to prove it was irrational/rational and so ever since I've been excited to learn how.

Blog Post, due on October 29

• Which topics and theorems do you think are the most important out of those we have studied?

For this section I would say it would be functions and the various properties of certain functions (bijective, one to one, onto)

• What kinds of questions do you expect to see on the exam?

I expect to see some proofs regarding these topics. Probably some true false although I'm not a fan. 

• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

I need help with the more general principle of induction. Specifically when we need to use which principle of induction and then how to select a good starting number. Is there a method? or do you just work through a problem to find out which one will work. 

9.6-9.7, due on October 26

This section is straightforward. I have a fairly good grasp of the material. The most difficult parts included finding the inverse of the permutation. Just the way that it's phrased in the book is a little confusing. Other than that was just the bogging down amid the proofs about inverse functions, but they're still very easy probably due to the fact that we've talked about inverse functions since high school.

The neatest part for me was example 9.12 and finding the inverse of that function. The solution was very elegant and in a way that I wouldn't have thought to do. Instead that set up the f of f^-1 and plugged f^-1 into f and solved for f^-1. It's so easy but it was different than my thought.

9.5, due on October 24

The most difficult part of this chapter were the examples at the end of the section. For example, example 9.10 threw me for a loop with showing that the composition was defined. Then I realized that since the input into the next function was a subset of the more or less intended or defined input for the function that that made sense.

Even though it was confusing at first that is my favorite part about this section. Thr functions are like playing frogger. You need to jump from function to function just like lilypads. If you don't go in order it doesnt work. You can't skip from domain to any range you want. But in the example referred to above our jump, or function, landed us in the ballpark that we needed to be allowing us to use the next function. Even though we didn't have every part of that set, the function was prepared to deal with any one element in the larger set B and so it worked for the subset of B

9.3-9.4, due on October 22

One to one, onto, and bijective functions make sense. For me, the waters get murky when we throw in the Real Numbers. Because the real numbers are infinite sets, it gets confusing to me how a function can be one-to-one but not onto or vice versa. It would seem like the domain and codomain in each instance are infinite and so by the same reasons that a function is not onto it would also not be one to one.

Something about proving that something is bijective is just satisfying. It's like, no matter which way I go from domain to codomain I know what's going on. Each element in the domain is unique and the same for codomain. That brings up a practical application question though, would this ever truly happen in a set of data? I suppose if you were measuring the velocity of some constantly increasing object you would have distinct time and velocity data. Is there anything cool that we can use bijective functions for?

9.1-9.2, due on October 19

In section 9.1 (page 198), why does it say that f3 is not a function from A to B because f3 is not equal to A? It seems like the function would never be equal to its sets because the function contains ordered pairs of elements in 2 different sets. So then wouldn't f3 always not be equal to A?

The little history connection was neat. The old definition of a function still holds today. The picture of mapping in the book was very effective in helping me picture what is actually going on. I'm an engineer and prefer things that are more concrete I suppose. It was also neat how functions that we've grown up with make sense with this definiton of a function.

8.6, due on October 17

I got it! As I was writing this it clicked. I couldn't for the life of me figure out how to reduce the equivalence classes down to within 0 through 5. But then I thought about the remainder. Okay...well now that that's settled. Everything else makes sense, but that was the toughest part for me.

I think that the idea of being 'closed' is kind of fun. And then I started thinking about what a set would have to have in it to be a closed under multiplication and addition and realized that that could get big really fast. While that could be tedious, theoretically it's interesting that if we multiply or add two numbers in the set that the product is still in the set. Actually, would we be able to represent that in a finite set? Probably not I suppose since the product of any two integers also can form a product with another integer in the set. Neat.