Questions, due March 31

1. Schroder-Bernstein theorem, continuum hypothesis, division algorithm, euclidean algorithm
2. Gcd questions, comparing cardinality, Denumerability, and infinite sets.
3. Questions like 10.19.

11.5-11.6, due March 28

1. (difficult) It is hard for me to think abstractly about the relatively prime numbers that are introduced here I needed to come up with real examples on my own in order to make sense of the proofs.
2. (reflective) I used to always wonder why the greatest mathematicians loved prime numbers so much, and with the work we keep doing, I can see how these prime numbers are really the building blocks of factorization and many other theorems we are studying.

11.1-11.2, due March 24

1. (difficult) The only difficult part I had was in following the proof of the division Algorithm. I first struggled when the book introduced the value t=r-a, but after some careful looking and reading I understood that. The one spot that I still do not understand is the introduction of, characteristics of , and manipulation of the variables r' and q'.
2. (reflective) I thought it was nice to finally come back and expand on the understandings that we have gained about the integers being able to be described as 2x, or 2x+1, or further as one of the following: 3x, 3x+1, 3x+2. It only makes sense that we come to understand all numbers as being written as aq+r where r<q.

10.4, due March 17

1. (difficult) The gravity of the continuum hypothesis has not hit me yet, and it seems like a statement that says "well, duh."
2. (reflective) It is really interesting that the cardinality of a denumerable set is defined as a letter of the Hebrew alphabet. With mathematics begin mainly written in Greek symbols, I wonder how aleph has survived as being a standard symbol. I'm sure at some point the mathematicians would run out of Greek letters and symbols to use, but Hebrew is a surprising alternative.

10.2, due March 12

1. (difficult) A lot of this section was difficult to understand, beginning with the ideas of Countably infinite, uncountable, and denumerable. The book never explicitly says that these are the same thing, but from what I can tell from the reading, they are. Also, the proof of the theorem "Every infinite subset of a denumerable set is denumerable" is two pages long and very hard to follow.
2. (reflective) I am not quite sure of what use it is to know that a set is denumerable or uncountable. No purpose was given in the text, and I have a hard time finding one on my own. Since I am very tied to practical applications, this may be a difficult concept for me to grasp.

9.6_ 9.7, due March 5th

1. (difficult) Nothing was particularly difficult. I think permutations could get confusing quickly when the sets are much larger.
2. (reflective) I think the idea of an inverse relation is a natural following from learning of functions and relations. It simply follows that there should exist some inverse of them. Not surprisingly, the rules for these inverses also follow the same as the functional inverses that were taught in early math.

9.5, due March 3

1. (difficult) I thought it was frustrating that in this section there was no explicit definition of what it meant to be injective or surjective. I can kind of understand from the examples, but can't be totally sure because of the lack of definition.
2. (reflective) I think it is really cool that we can see the truths and rules of being derivatives and composite functions be proven by the relations and sets that we have used to prove so many things. It just goes to show you how all of these mathematics laws are truths as much as they are anything else.