The second assignment was difficult but rewarding for me. I had a very easy time with the last questions--they were of the sort that I referred to in an earlier post, in which you start with one or two variables that are given certain equations, and multiply/manipulate them to show that some dependent variable (e.g. the square of the first one) is equal to another related equation. Of course, I may have done them incorrectly, but from where I stand now, I would say that I was unchallenged by them.
The first questions, which depended on the floor function, were a good deal more difficult. The fact that we had to decide for ourselves whether or not we thought the statements true made them a good deal more difficult. The first time I sat down and worked on the assignment, in fact, I spent the whole the time staring at the definition of the floor of x and at each of the statements involving the floor of x and simply trying to wrap my head around them. I jotted down notes and wrote out each statement in basic English, and i found myself having to write the statement down in English again and again to refine my understanding of it. After quite a while, I had several pages of notes on every statement and exactly how it might or might not be true. They had started out as obscure and baffling, but after this long process and after I had nailed down my understanding of them, they became quite fascinating and very understandable.
The thing I had the most trouble with in the assignment was the statement
[for all] x in R; [for all] e in R+, [there exists] d in R+, [for all] w in R, | x - w| < d [implies] |floor(x) - floor(w)| < e
which I have written here with the limited abilities of this given word processor. I hope it is legible.
The most difficult part, for me, was that d could depend on x and on e, but it could not depend on w. I had a whole solution sorted out in which d was equal the absolute value of the differences between x and floor(x) and w and floor(w), but I had to start over when I realized that d could not have x in it. Luckily, it worked out (I hope, given that I have not got the assignment back yet) because I was able to show that this absolute value was less than one, so I could use a series of inequalities in which d was equal to one but I could still fit my absolute value involving the differences of x and floor(x) and w and floor(w) in. With this little trick, I think I got through the assignment okay.
The second test is tomorrow, and I feel pretty comfortable that the assignment prepared me well for it. We shall see.
***Amendment***
I feel pretty stupid, reading over this post. Of course, my answer for question 1.2 of Assignment 2 was completely wrong. I was thinking I might delete this post in light of that, but I have decided to let it all hang out and simply view it as a lesson. I have a theory on how it all happened.I remembe very distinctly my first night working on Assignment 2. I did not understand several of the statements at first, and so my biggest challenge was to try to wrap my head around them. That I did; slowly but surely, I began more and more to see what the symbols really meant, and to intuit it.
Sadly, that is where I stopped. I was so exhilarated by having finally begun to see the meaning that I didn't bother to see that process through to the end; I simply developed a rough understanding of it (as in, "This has to do with the relationship between the difference of the floors and the difference of the numbers; let's think about that!") and then so congratulated myself for making some progress that I did not think even more deeply about the meaning of the statement. If I had, I would easily have realized that there were some nuances I was fatally missing when I considered it.
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