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.

9.1-9.2, due February 26

1. (difficult) One difficult thing about this section was that with all the talk of the function mapping the coordinate points in the relation, there was not a single graph given to illustrate the point. Therefore, I must assume that the actual graph matters very little to the purpose of the section. It would have been helpful in illustrating I think.
2. (reflective) It's pretty interesting that the number of sets from A to B is |B^A| = |B|^|A|. It's pretty cool that there is usually a way to calculate these things that would be insane to try and count from just writing down all the possibilities.

8.6, due February 23

1. (difficult) The explanation of the equivalence classes and their operations was super confusing at first, have [2]*[3]=[0] for Z_6, but it definitely makes sense now. The book does a good of job of explaining, especially with its likening the operations to a clock that resets at 12 'o-clock to essentially be a zero hour.
2. (reflective) The idea of being well-defined seems to be of little/no use, or have little application in real life. I would really like to have more application for these things in real life scenarios.

8.1 - 8.2, due February 17

1. (difficult) The most difficult part of the reading to understand is the use of the words Relation, Domain, and Range that departs from any other mathematical and english usage that I have ever used or been familiar with. It is actually very frustrating to try to understand what is being described when the words mean things that my mind does not immediately understand.
2. (reflective) I can't see any application yet for a relation to take on the properties of being reflexive, symmetric or transitive. Is there going to be a way to shortcut to figure that out? Because checking each value to be sure its inverse is also in the set seems to be a waste of time.

5.4-5.5, February 4

1.(difficult) Everything was pretty straight-forward. However, the proof the book gives of Result 5.24 is barely follow able. I could not see where the basis for the proof came from. It was confusing to say the least.
2.(reflective) There is one sentence that would have made all of the difference in my initial interactions with proofs: "In a proof, we are not required to explain where we got the idea for the proof". I often felt like I had to explain where my knowledge of the beginning of the proof come from. This is KEY information!

5.2-5.3, due February 2

1. (difficult) The most difficult concept I ran across in this reading is the in depth look at proof by contradiction. On the surface, and in the problems that they gave as examples, it looked fairly easy. But, after the story problem example, I was thoroughly confused. It seems in the story problem as if the proof by contradiction was shaky, and not totally convincing as a proof. I began to wonder if this is often the case with Proof by contradiction?
2. (reflective) I've  been looking for a chart like the one that is given in 5.3 to help a person to decide what course to take with their proof. I was actually contemplating making a chart myself, but this will be of use instead.