4.5-4.6 and 5.1 due January 31

1. The proofs in section 4.6 were super hard to follow. Not because the math was complicated, or because I couldn't follow it, but because there was so much math on one line. It was hard for me to go from step to step, even with the words "then" and "and" separating the statements. I don't think I'm allowed to critique how the authors of the book about proofs write their proofs, but I probably would have made the proof go onto multiple lines so it was easier to follow.

2. Is the "equal" sign equivalent to "if and only if"...? I am guessing that it is, since the proofs with just an equal sign are proving both the statement and the converse.

I have been wondering how we would write proofs that are usually true, but there are some values for which the statement is false, and I really like the format that was introduced in section 5.1. of (excuse the latex writing...) x \in {R} - {set for which the statement is false}. It makes it so that there are many more useful theorems in the world (such as the tan^2 (x) + 1 = sec^2 (x) example.

This coming week I am going to have to make a list of all the vocabulary and rules that we have for this unit. I need to be able to remember the Laws and the Cartesian product (and everything else we've been learning...) for the test.

4.3 - 4.4 and additional questions due January 29

1. Proofs involving sets had my brain going a little wild. I found it was a lot easier to simplify the proof in my head to lots of little proofs (lemma style.) The textbook is so easy to read and to follow, and it basically supported my thought process. I also didn't realize just how careful we have to be with switching operators until this section. I have to really have my guard up to know if we're proving the converse or proving by contrapositive, both of which are completely different and can yield some pretty...wrong...results.

It's hard to do proofs with greater than and less than symbols. I am overthinking the changing of the sign and making the equations equal.

2. In learning about proofs for real numbers, I couldn't help but notice that they are pretty much the same as proofs for integers, only this time the two cases are positive and negative, verses even and odd. The proofs involving sets are the same as proofs for integers, there's just different operators. I was excited to see the proof of result 4.20 though, where the entire thing pretty much was done through wordy observations. It was pretty easy to follow, if you can think in terms of set operations.

And a few answers:

1. The homework takes me about 2 hours, usually. Sometimes longer, sometimes shorter. I feel pretty prepared for them, and for once in my life actually feel like the homework questions are testing us on our knowledge of the material, not trying to show us how many exceptions there are to the rule. I have liked it a lot so far.

2. My learning has most been benefitted by the reading, unfortunately. I don't like to read textbooks, so I'm always looking for a reason to not read them. But I have benefitted greatly from the reading. It makes it so that the lecture is just to answer the questions I have, and then I can do the homework fairly easily. This is a pretty useful textbook.

3. I think probably the best thing for my learning would be if I would be smart and get ahead on the homework again. Last week I fell behind, but before that, I was always working on the homework before I went to lecture, so I knew what my problem was, and could use the lecture to fine tune my own learning. Hopefully this weekend I will catch up again.

4.1 - 4.2 due January 27

1. Well today I was fascinated by the divisibility proofs, and I wanted to try typing up some problems in latex. Well, that lead to a long and wild goose chase, in which I never quite found the answer to what the latex code for "does not divide" is. The current answer I'm using is, "\nmid" but it doesn't look quite the same as what I see in the book. It is the symbol that was recommended by google and detexify, though. (If you have time, could you let me know what code you'd like us to use for that symbol?)

In other news, I stared at Result 4.8 and it's connected proof for quite some time trying to figure it out. It's nice to have an example that puts so many of the things we've learned together in one example. We used the definition of evens and odds, proof by cases, ranges that aren't "the integers," and complicated vocabulary. It was a fun one. I'm am officially terrified of such a thing showing up on an exam though.

2. The world of proofs gets more interesting day by day. Today I realized just how many variables we can have in an equation and still prove it true or false. I then got to imagining the possibilities of what we could prove using lemmas. We can get some pretty complicated equations and prove them by plugging in variables for the equations themselves and lemmas and cases...the truth values of everything can be known!

The format for the congruence of integers felt really familiar to me. Then I realized that it was because it reminded me of the way that logarithms are written. I kind of doubt there's a connection, but it's probably going to help me remember it.

3.3 - 3.5 due January 24

1. So is a Lemma is just a means to an end...? I didn't realize this. There is a proof that used a lemma to simplify the proof. Although there was an alternate way to solve the problem, the lemma made it a lot easier to follow and see what was happening. The alternate way was a fancy trick, but it was also one of those tricks that I never remember while I'm taking tests.

2. I am just loving all of these even/odd proofs. I can't imagine that they're going to go on forever, but I love them. So much beautiful algebra. It's pretty rewarding to get to the end of these proofs and see that you actually accomplished something. I think my most favorite thing about it is that you get the right answer, and then it's just a matter of proving your way there. I'm not too excited about having to decide if the proof is valid or not, because that's not the same as having the right answer and just having to figure out how to get there. Proofs are so much less evil these days. I love it.

3.1- 3.2 due January 22

1. Probably the most difficult part for me was just understanding what was in the proof analysis. I read through the proofs, and I realized that a lot of assumptions were being made. I guess that's what the proof analysis is for...explaining the assumptions. I am predicting that I have a semi-hard time figuring out what assumptions are obvious and what assumptions need to be stated.

2. I didn't realize that proofs don't have to be orderly steps in a table. I guess this is what I get for doing proofs in 8th grade geometry and then never doing one since. They seem so simple and look so clean! I am excited about this new tool of proving odds and evens using 2*k and 2*k + 1. I feel like this would have been very useful during the last 22 5/6 years of my life, had I been smart enough to know it. I am realizing now that the reason proofs are written so cleanly and simply is for people like me.

Chapter 0, pages 5-12 due January 17

1. Well, today's reading officially made me terrified to write proofs... To think the whole meaning of a sentence can change with one word! (I think that's what they were trying to tell me in all those grammar classes...) I think the most difficult thing for me today was just reading and knowing that I need to keep track of so many important phrases and not mix them up.

2. Now that I've done the reading, I feel like I know the secret now as to why you can actually be assigned "reading" in a math text book, and then proceed to read it, despite the fact that you're reading about numbers. I enjoy how clean everything looks when it's written properly. I probably never would have noticed how much better things look when written professionally, if it weren't for the examples contrasting the two. It makes a huge difference to do something so simple as start a sentence with words rather than symbols.

2.9-2.10 due January 15

1. I understood and followed everything in the reading, I am just struggling with the application of why we would care that there is some value for which a statement is true or false. Don't we want to specify what that value is? The only reason I can see for why we would use this statement is so that we can know that an open sentence can be solved. Luckily, so far the lectures have been helping me to see some kind of application...hopefully this will be no exception.

2. This is exciting! Math is starting to make so much sense now! I loved reading about the proofs of the associative law, distributive law, commutative law and De Morgan's laws. I started to see some form of application. Without working too hard, we can tell if two statements are going to be true or false. I was also pretty grateful that while I was reading all the complicated examples, instead of feeling overwhelmed, I felt like the big jigsaw puzzle was being put together quite nicely. I think I'm getting the hang of all these symbols.

PS: I hope your vacation is going well and isn't too tainted by still having to read our blog posts.

2.5 - 2.8 due January 13

1. The hardest part to understand today was logical equivalence. I'm still not entirely sure what the application of it really is. On the bright side, the theorems did make sense. I wonder if I'm confused between logical equivalence and tautology. I understand that tautology is when the statement is always true regardless of the truth value of the individual parts. Logical equivalence, from what I understand, is two statements that have the same truth value no matter what the truth statements of their individual parts. After some research on google, I've gotten closer to understanding. According to millersville.edu, two statements are considered logically equivalent if those two statements are a tautology. So, any two statements can be logically equivalent, if their truth tables are the same for every combination of truth statements of their individual parts. This is going to take some further thought and some application before it's solid.

2. I am actually pretty fascinated by all of this. Who knew this would be so interesting? Not me. Anyway. I found the most interesting thing to be the idea of biconditional statements. It's pretty excellent to have an if and only if statement, because then you don't get the question wrong if you mix up the parts. I also secretly (well, not secretly anymore) loved reading the proof of biconditional statements being logically equivalent. I'm still working on a real life application for all of this though. It looks so nice on paper...there's got to be some good way to apply it all!

2.1-2.4 due on January 10

1. The most difficult part of today's reading for me was the truth tables. I found myself having to double check that I was following the correct variable and statements. Prior to this class, I think I read through things very quickly and didn't seem to pay too much of a cost for that. But this class is so far not allowing me to be hasty in the reading. There are so many statements and variables and symbols to follow! I loved reading about implications though, and keeping the statements to a simple 'P' and 'Q' helps me to understand.

2. The most interesting thing to me today was reading about implications. I had never thought that if you have a pair of statements that aren't both true, the implication can still be true. If one of the statements turns out to be false, the implication can still be true, because it wasn't proven false. This idea kind of blew my mind, because I had previously always thought that for an implication to be true, both statements had to be true.

1.1-1.6, due on January 8

1. The most difficult part of the reading for today was section 1.4 about Indexed Collections of Sets. I must have forgotten how to do series problems. I was able to figure out and follow the text only when I wrote out the example problems on my own sheet of paper and wrote out each step. There are also so many different symbols, that I struggled to remember which symbol was which and what the problem was asking. Once I accurately differentiated between ∪ and ∩, I came out with the right answer. I am going to have to make some flash cards with all the symbols on them so that I can know what the text is saying when I read it without a key. It's hard to follow the math and words when there are so many symbols! Hopefully this gets easier with time.

2. What I liked most about todays reading was that everything I read about could be visualized. I'm not much for learning through visuals usually, but I was able to understand the concept of sets so much better with the Venn Diagrams. I thoroughly enjoyed seeing how many different ways something can be written and how many smaller subsets can describe one great whole. This might be a little far out, but in terms of religion, it reminded me a lot of how God is like the Universal set, and we are all individual subsets under him. We, as his subsets, have our identity made up of parts of his Universal set, yet we are all individual and unique.

Introduction, due on January 8

Hi. I'm Dylan Lambert. I'm a girl.

I am a senior here at BYU, and am majoring in Physics Teaching. I have taken Math 113, 313, 314 and 334. I am taking Math 290 because it's the last class for my math minor. I want to have a math minor because if I become a high school teacher, it will be easier to get certified to teach math if I already have the math minor. 

The most effective math professor I've ever had was Dr. Wayne Barrett, who taught my Math 313 Linear Algebra class. He was very personable and explained things very clearly. He worked swiftly, but was certain that we were following him and that we could keep up. When it came to doing the homework, he had office hours in the math lab, which made it easy for us to work together and to get our questions answered. It helped that he had a heart of gold, too.

As for the interesting things about myself, the first was stated in my first line. My parents named me after a male, Irish poet, named Dylan Thomas. It didn't seem to matter to them that I was a girl. I just returned from serving a mission to the New York Utica mission, which is the completely rural part of New York. It was wonderful, but it is good to be back.

I will be able to come to your office hours on Friday, but I have a class Monday and Wednesday at that hour. I would love it if you had office hours on Tuesday, as I do not have class that day.

Thank you very much for reading! I am excited for a fun semester of math!!