9.3-9.4 due February 28th

1. The most important thing we have to discuss is that my tickets for Calculus the Musical have been reserved. BOSS.

2. In regards to the reading, I am excited that I understand what the horizontal and vertical line tests we all did in elementary algebra actually were telling us. The vertical line test is how we can tell (from the graph) that a graph is a function. We can know for certain that what we're seeing is a function, because every value of the set A is assigned exactly one value of the set B. The horizontal line test is how we can tell (from the graph) that a function is one-to-one. We can know for certain that it is one-to-one because every value of the set A is assigned exactly one unique value of the set B. Fascinating.

3. My weak understanding of these function things basically just tells me that bijective sets are the sets formed when the cardinality of A and the cardinality of B are the same, such that every value of A has a unique value B. This makes the function of A and B linear,  onto and one-to-one. I can see that this would be useful, as there are many things in mathematics that depend on weather or not something in linear...and now we can know by simple Math 290 mapping! (Or we could just do the horizontal/vertical line test. Whatever works.

9.1-9.2 due February 26

1. You know that feeling when you've find out you've been lied to all your life? That's the feeling I just had as I read about functions. You'd think that in the 3,896 (give or take) classes I've had where we discussed functions, I would've figured out that f(x) meant that "x" was simply mapped onto a function. I'm kind of embarrassed that my mind was so blown over this fact. Also I realized why the vertical line test works for testing to see if something is a function. BECAUSE EVERY x MUST HAVE A VALUE, AND IT CAN ONLY HAVE ONE ASSIGNED y VALUE! (insert fireworks of excitement here.)

2. Well now, I hope I realize that cardinality and absolute value are not the same thing. When I originally read that the cardinality of B^A is the cardinality of B raised to the cardinality of A, I read it as absolute value signs, and was about to rip the page out of the math book. The good news is that, as usual, the problem was user error. This chapter basically just reworded something we already knew for one dimension, and put it into two dimensions for us. Rather than working with the elements of sets, we are working with the sets themselves. Fun stuff.

8.6 due February 24

Well this was exciting...we got the teaser in class about how we could add and multiply sets, and today I got to read about them. I actually did the reading over an hour ago and didn't blog about it immediately because I wanted to think about it some more. I felt like there was a jigsaw puzzle starting to put itself together in my mind. Not sure if it's all the way done fitting together...probably not.

I actually figured out that the equivalence classes of Zn was the set of {[0],[1],[2],...,[n-1]}. I had postulated this idea in some form during lecture last time, and then today doing the homework, I did a problem where we were dealing with modulo 5 (n=5) and observed that we were going to have 5 different possibilities for equivalence classes. That's not to say that we will have 5 every time, but we can only have 5 possibilities because you can only have a remainder of 0,1,2,3 or 4. After that it will repeat. This information seems so useful...hopefully it still feels useful later and I can figure out how to use it.

8.5 due February 21

1. I can't quite figure out what's different from what we've been doing and what I read about today. I must be missing the mark. However, I had never heard of the Division Algorithm, and it sounds scary...so I don't know if I'm actually excited to learn about it in Chapter 11. From what I understand, the algorithm is just that every integer can be written as a sum of a quotient times an integer, with a remainder added on. Seems easy enough. But recently I've been feeling like saying "seems easy" a lot, and then all of a sudden, I am highly confused. Not good. I've been pretty grateful for the homework this unit...I have felt pretty confused on almost everything until I've sat down and chugged out a few problems.

2. Equivalence classes. No idea what they have to do with anything. But they are sweet. I wish there were infinity classes like this and like linear algebra where I could just do abstract math and never understand the relevance. (I do want you to know though that Ryan and I have made quite a few "If...then..." statements and analyzed them in our personal lives.)

Anyway. From the reading, it appeared that the number of unique equivalence classes is the same as the divisor. (That's what we call the mod __ number, right?) Anyway. Maybe if we could talk about some applications of these modulo questions, that would help me. I love them though.

8.3-8.4 due February 19

1. The bad news about the reading today is that I didn't understand hardly a word of it. I was pretty excited when we defined an odd number as the variable y that makes the statement x=2y+1 even. I chucked thinking that it's funny to find an odd by looking for an even. So. Equivalence relations. What I understand is that an equivalence statement is one that is reflexive, symmetric and transitive. Even after the reading, I'm still unsure of the role of the equal sign in the equivalence relation. That's how confused I was. I'm excited for the lecture and maybe rereading the section when I'm not half asleep.

2. Equivalence classes are fun. I don't see the purpose for them, but I'm starting to see a theme here...the less I understand the purpose of something, the more I like it. Perhaps in the pre-moral life I chose to be one of those people who loves abstract math. BUT. I liked thinking of the equivalence classes as families. It made it semi-easier to understand. Now I know there are things in the equivalence class that are not equivalent to each other, but ARE equivalent to a larger set, A.

8.1 - 8.2 due February 18

1. The good news about todays reading is that it wasn't about mathematical induction, so it seemed really easy and read almost like a novel. (Those mathematical induction sections were like reading molasses, even when I understood them.) But it was still fun to understand what some of the past exam questions I saw while studying for exam 1 were talking about. I did try to see an application for all of this, and can't quite seem to find one. I'm hoping that one will come in a later reading section, or from the lecture in class.

2. Exciting times in this chapter...Implications for the win! The most interesting thing by far was seeing the transitive property at work, making a statement vacuously because the hypothesis was always false. I feel like a missing piece of math finally put itself together in my head. I love it when everything I've learned starts to put itself together. The transitive property has made sense before, but I love that this section gave it a new dimension, showing that it works for sets as well.

I think that's what I've enjoyed the most about linear algebra and about this class is the fact that the things we learn aren't necessarily hard or new. The concepts we learn here add a new dimension to what we already know, and make us think in a new direction, rather than just teach us another trick. Theoretical math is so fun.

6.4 due February 14 (VALENTINE'S DAY)

1. Math is my Valentine.

2. So basically the difference between the Strong Principle of Mathematical Induction and regular Mathematical Induction is that for the Strong Principle, you are verifying that the statement is true for every single value from one to k, rather than just for k. The definition make it almost sound like an induction within an induction. (Twilight Zone music)

3. Ok. It took me a minute to figure out that recursive relationships were referring to the individual parts of a sequence, not to the sum. Once I figured that out, things dramatically improved as far as my understanding of the proof. I think I'm going to feel better about these proofs after seeing a couple done in class. I think my confusion is just coming in why we need to (or want to...?) prove k-1.

6.2 due February 12

1. The most difficult part of the reading by far for me was understanding the proofs. The concept seems to work really well in my head. I understand the concept of induction: We have to prove the first value, then we assume it for an integer k and prove it for an integer k+1. Simple enough. But the book was right, the algebra is where people get confused. I just don't know when we're switching from RHS to LHS. I think I've always struggled a little bit with inequalities. Those greater/less than signs get me every time.

2. Well I was pretty excited when I did the reading and understood something about proofs with sets on the first time through. It's pretty awesome to see proofs that I already understood and knew well to be proved further. As far as I can see, induction is a fancy way to prove that things are true for every value. It's a nice way to feel really secure about theorems and be 100% sure they are true. I like it. I hope that I actually do understand it.

6.1 due February 10

1. I LOVE INDUCTION. I have always loved mathematical induction, but these proofs were just awesome. So simple and yet mind blowing. Who knew it could be so easy to prove something for an "every element" set. I loved it. Probably the hardest thing for me so far was trying to think of what sets have a least element and which don't. It seems easy, but there were a couple that I guessed incorrectly. So my analysis of it is that if there isn't a set end point, there can't be a least number. So open ended sets (that either end with infinity, negative infinity or they approach a value but aren't equal to it) have no least value. I take it that because of this, we will mostly be working with the natural numbers.

2. So this well-ordered set business was pretty fun. It reminded me of the test! When you make a subset of the natural numbers, one of those sets is going to have to be open ended and go to infinity. So it can't be well-ordered. Hooray for seeing an way that a test question could be expanded to have a part b!

My only question is this: Why do we need to know that the implication works for P(1)? I get that we can assume the hypothesis (direct proof) and so assuming for P(k) is ok, but I am confused about why we need to know that P(1) works. It's nice to know, I just don't see the value.

Studying for the Exam

I would say the most important things we have studied are proofs involving sets (which are still a little rough for me) and then just basic proof strategies. I expect that there will be a few proofs, one of each kind that we have studied, and one of them will probably be really hard. I have had lots of fun experiences with the BYU math department and their history of "one hard question" on the exam. I usually am part of the majority who misses that question, but I have a goal to not miss it this time.

Before the exam I just need to focus on chapter 4. Proofs involving sets and proofs involving divisibility of integers have been the hardest for me, so tomorrow morning and then before I take the exam, I plan to do quite a few of those.

The true/false section always seems to stump me because they put questions that your first instinct isn't always right. I think it would be very helpful to me if we had the key to the practice exams. Would that be possible? Hopefully I will do all right on this exam, because I have to take it tomorrow after class. Thanks for all you do, Dr. Jenkins!

5.4-5.5 due February 5

1. The example proof of Result 5.24 was super confusing to me. In fact, I still don't think I get it. But I do understand existence proofs, I think. They are basically the opposite of counterexamples. We prove that there's some element for which the statement is true, and thus, the proof is true. So this works only for proofs that state "there is some element" or something like that. I like these proofs, I think. It's nice to know that there's some proofs that you can prove simply by looking at them and seeing that some value for it would work.

2. Unique. What does it mean to be unique? My parents always told me that I was unique, because there is nobody who is exactly like me in the world. So what does it mean to the math 290 world? It means that there is only ONE value in a certain set that has a certain property. So, the examples in the book illustrate that this doesn't mean you are only working with one variable, it means you're working with one value. (This is probably going to confuse me. Wish me luck.) So you can use multiple variables, you just then have to prove that they are equal to each other, thus really making them the same value.

5.2-5.3 due February 3

1. I realized today that I didn't know what the definition of an irrational number was. A brief date with google though, and I am all set. I really enjoyed reading about proofs by contradiction. It's nice to know that there is a way to prove a proof by showing that it doesn't work. (I feel like a lot of times I'm better at proving that it doesn't work than that it does.) I was especially excited about the application of the three prisoners story problem. Sometimes it's hard for me to see application for the things we do in the this class (which is fine, because I'm a much more theoretical person anyhow), but I did enjoy seeing an application and feeling really smart for knowing how to solve a real life (sorta) problem.

2. This was my favorite section I've read yet. The hardest thing, by far, for me is just coming up with the idea of what to do. The math isn't ever hard for me, and I'm starting to get good at remembering the definitions and theorems, but I don't know how to recognize what to do. However, being able to see that there are three different proof strategies we can use, and that sometimes all three of them will work, will hopefully make it easier for me to come up with the solution.