4.5 - 4.6 & 5.1, due January 30

1. (difficult) The difficult part is wrapping my head around the wording of the proof symbols for sets. It will take a little while to note what is actually being stated by set notation. Also, mixing cartesian products with set algebra is confusing.
2. (reflective) We are finally granted a way to disprove statements with the information in 5.1. It was difficult to just push that option in the back of my mind as we were constantly proving the statements given. Counterexample seems to be the only way something can be disproved, but I have been wrong before about what is available. I might get surprised.

4.3-4.4, due January 29

1. (difficult)  I don't understand result 4.16 and the subsequent proof for it. I read over it a couple of times, and I don't see how the proof arrived at. It seems to take the exact same expression and reworks it to prove itself. Something to the effect of: Prove 5x=y. Well, 5x-y=0, so 5x=y. It just doesn't seem very logical to me.
2. (reflective) The set algebra is tricky. I was honestly hoping that we'd left behind the set notations and intersections, differences, subsets, unions, compliments, etc... But of course we will be using them probably on and off through the course.

• How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
• I have spent roughly one hour on each assignment and one hour on readings and bloggings per assignment. However, the LaTeX assignments usually takes me longer than just writing it. The reading and lectures adequately prepared me for the assignments.
• What has contributed most to your learning in this class thus far?
• reading and writing blog posts about the material
• What do you think would help you learn more effectively or make the class better for you? (This can be feedback for me, or goals for yourself.)
• I often do the homework assignments during class lectures, so I would like to have them done either before or afterward and take that time to listen and ask my questions.

4.1 - 4.2, due January 27

1. (difficult) Modulo is a difficult concept to grasp, especially when the following are equivalent: 

a_= b (mod n)  and n|(a-b). It seems like such a strange way to express an equivalent expression. There must be some way that it is more convenient to use the "mod" in it.

2. (reflective) These mod statements mixed with proofs by cases can be very complex things, in fact the most complex we've studied so far. It's difficult to wrap your mind around at first, but it becomes easier after time is spent trying to really understand what the book is saying about the various statements. It's just going to be another tool in our belt to use while proving results.

3.3 - 3.4, due January 23

1. (difficult) I had a difficult time understanding the proof by contrapositive because it seemed like it was simply the opposite of trivial proof. In other words, the statement is always false when Q is false. I know that is not true, so seeing the truth table to prove P(x)->Q(x)---(~Q(x))->(~P(x)) was helpful.
2. (reflective) It is interesting that WOLOG only applied to the case where x and y were of opposite parity. It totally makes sense that the proofs are the same for simple proof cases where the x and y are of the same parity, etc... It will be interesting to see how complicated cases can get and still be able to use the WOLOG in the proof.

3.1 - 3.2, due January 21, 2014

1. (difficult) Direct proofs can be difficult to get, because sometimes there is only one way to go about proving them, or one proof. If you don't have the right way of seeing the question, then you could get stuck in the proof, or go in a circle. Also, sometimes you don't have the necessary assumptions at hand to know what to do in the situation.
2. (reflective) The last time I did proofs was in middle school in geometry, when proving that two angles were equal by use of many theorems. At the time we would make two columns: one with the logic statements, and one with the theorems being used to prove. We have yet to be introduced to a proper format for writing our conclusions along the way.

Pgs 5 - 12, due January 16

1. (Difficult) The most difficult part of this section is going to be remembering all of the rules when writing for mathematics. Most are intuitive (like when to use each versus every) but some are not so simple (like using that versus which).
2. (Reflective) I write math courses for High School students for BYU Independent Study using a combination of qti and LaTeX. Interestingly enough, we do not have a "Best Practices" document for writing Math questions for the exams. I am considering submitting this section of this text book for  such a document. There are a lot of helpful hints of how to make these questions more readable and understandable.

2.9-2.10, due January 14

1. (Difficult) The relationship between the existential qualifier and the universal qualifier is NOT difficult. However, the negation algebra (if you can call it that) of an equation that has multiple variables qualified with the above listed can get very tricky. Also, grasping the amount of symbols and their definitions in words can be difficult.
2. (Reflective) I was waiting for a list of properties to use to simplify and rearrange logic statements, and I got them in 2.9 (pg. 49). They aren't as extensive as I was expecting, which shows that we may not be using a lot of simplifying and proving one logic statement is equal to another.