12.5 due April11

This is my LAST reading assignment for a math class EVER. At BYU. Who knows. Maybe I'll go on to get a masters degree is math someday. For now, though...this is it. What a bittersweet moment.

Continuity is a condition we have been discussing in class. We have had to be pretty careful because you can't say that x is in a certain range that covers the function at a point where it isn't continuous. So now we have defined the three conditions which must be satisfied for a function to be continuous at a certain point. The third condition (to me) is really the only necessary condition. Because if the limit as x-->a = f(x) = f(a), then that implies that the limit exists and that f is defined at a. But I guess we can go step by step if math says so.

Continuity is proving to be harder than expected. I guess because I figured all we had to show was that the limit exists. Which is true, but to match it up with the function is harder. Essentially we need the limit and the point in order to say that they are continuous, and I get it now... Just because there's a point there doesn't mean there were points before that one that lead up to it (the limit) and just because all the points are leading up to it doesn't mean that there's a point there. Fun.

12.4 due Wednesday April 9

It's always a good feeling when the beginning of a section starts with the idea of doing what we just did only we're going to make it easy now.

It makes sense that the limit of two added functions would be the addition of their limits. I say this because if you add two functions together, then the result is the sum of each individual result, which is what you would be approaching through the limit. As for the limit of the multiplication of functions, I would say that this is less intuitive, but we've all seen it before, and it makes sense. I especially enjoyed the case when M=0. I understood that one the first time through.

Probably the most confusing thing for this section was reading about the three deltas. I thought I understood min(1,a(epsilon)) was like bounds...where 1<delta<a(epsilon). But then we all of a sudden have delta=min(delta1,delta2,delta3)...are those three dimensional bounds. So I'm confused. Then in the proof, I'm super confused with how we got to epsilon/2 for the second part. I'm assuming we will derive this in class though...? Hopefully a good assumption.

The good news is that I genuinely believe that this box of tools is going to make math easier. :)

12.3 due Monday April 7

Alright. Office hours with Dr. Doud went well. He writes his proofs a little different than we do in class, and I guess I needed that added dimension. I now know what's going on. I think. And now a play-by-play of section 12.3:

SO 12.3 is just taking what we've learned already about finding limits (from 12.1) and applying it as a limit at n--> a, where a is an integer value, rather than infinity. So we can find a limit for a divergent series. That makes sense, and the proofs appear to be largely the same as before as far as what we're proving. We're proving that if you go out a certain distance (delta...?) from a, then you will always be within epsilon of the value a.  [[I really hope that is the definition of delta.. if it is, then I think I actually am starting to get this stuff!]]

We choose our distance delta the same way that we chose the value N. By solving for f(x)<epsilon. And our new assumption for this section is that 0<f(x)-L<delta. And since we define delta in terms of epsilon, we can basically say delta(epsilon). Things are going well.

For the quadratic (and other) proofs, the only thing was defining the trick that helped us find delta. I could follow it all, but I probably wouldn't have thought of any of that stuff with the lim as x-->3 of x^2 is 9. I hope that in the future the tricks are similar so I can actually find it. ok \qed for now.

12.2 due April 4

So the homework was pretty confusing and I've concluded that maybe I just didn't understand the last section...but here's a shout out to section 12.2...

From my observation, the proof strategy is essentially to take a summation and try to reduce it down to a limit, and then prove it the same as we did in section 12.1. I get how we choose the N and I understand how it works for n > N...I guess I just don't understand what the point of solving for epsilon (sorry, I'm not going to go on a greek letter hunt) is. Maybe I don't understand what E is at all. However, if we can prove the limit of the sequence, we can say that thus we have proved the summation as well.

I'm now going to use my words to try to understand what epsilon is. So. We are trying to prove a limit. If we can show that for some arbitrary integer less than 0, a sequence will equal a certain result less than epsilon, then we have shown that the sequence approaches a certain number. (...?) I guess that makes sense. So then we use the value we chose for N to solve for n and show that the sequence is less than epsilon for that sequence. I really hope that's right.

Now back to section 12.2. I am pretty fascinated that you can turn an infinite series problem into a limit/sequence problem. Are we just supposed to write out the first few elements of the sum and then make an estimation of what the nth term in the set would be, and then take the limit and solve?

I will most likely be taking a trip to office hours with your friend tomorrow.

12.1 due April 2

So that test was super fair, I thought. Then I came out and found out I did way worse than I thought I did. In fact, I found out I did 10% worse than last time. So....hopefully we don't repeat this pattern on the final.

Regarding section 12.1...I've never heard of the ceiling of x. Is it the same as what something converges to, but just in different notation? I like this idea of limits. I never liked it until recently, but while we were doing number theory, I couldn't help but think about limits. Turns out the real numbers are awesome (even though they  make proofs way harder.) I suppose that us having a point that numbers converge (or diverge) to is useful. Perhaps later in the chapter I will fully understand how this relates to calculus.

Why do I never remember the rules for inequalities. Don't you think I should know these things by now? I didn't realize that if you take the inverse of both sides, you have to switch the sign. Learned that in an example. If that's not true, then I'm officially confused.

I think my confusion for this section boils down to the fact that I can follow the proof, but I have no idea what we're proving or how the proof really proves anything. Usually problems of this variety solve themselves in lecture and when it's not midnight.

Again, nice to see you tonight! Hope your scouts enjoyed the show!

Thoughts, but mostly a question.

While studying for the test today, Ryan and I had a question.

How is it possible that (N) X (N) is countable, but 2^N is uncountable...? Because, isn't 2^N just (N) X {0,1}? That is all.

Studying for Exam 3 due March 31

Well, honestly I haven't spent a lot of time studying for the test yet. I will, though. I've been doing a lot of studying as we go though, because this unit has been quite hard for me (as you know). Based on what we've been doing in class and on homework, I would say that the most important things from Chapter 11 would be finding the gcd and the smallest prime factor. This is easy enough to be tested, but hard enough that we would have to think in order to do it. These seem like valid multiple choice questions. For Chapter 10, I would expect much simpler questions than the ones that we did on the homework. I might be wrong here, but some of those homework questions were ridiculously hard and long. So...simplistically, I think we could show what sets are denumerable and which are uncountable. We probably are going to be accountable for the definitions as always.

Hopefully I will do alright on the exam this time around. Monday-Wednesday are by far the worst days for me.  >.<

Also, I had tickets to Calculus the Musical and totally mixed up the times and missed it. My weekend was ruined by this. I hope you got to attend and enjoy it for me.

11.4-11.5 due March 28

My thoughts on the "relatively prime." My first observation was that we are trying to get the linear combination to add to one. This was unique to me because the definition of a prime is that its only factor is itself and one. So in this case of relatively prime integers, we are trying to get their greatest common factor to be one...even though they might be composite numbers. So I guess from what I understand, "relatively prime" essentially means that the linear combination is prime. I love how the text states, "This is extremely useful." That is a biased statement, and I'd like to see us tell the English majors how "useful" that statement is.

Next thought...I at first was pretty fascinated by the idea that every integer exceeding 1 has a prime factor. Well. That was interesting until I remembered that 1 is a prime number...and everything as 1 as a factor. So, does this corollary even tell us anything? These are the things I wonder about.

And these divisibility rules are pretty fantastic. Why did no one ever tell me that if I summed the digits in a number and it was divisible by three that the number itself was divisible by three? I feel like this information would have caused me much less stress in math over the years. It also would have saved paper because I wouldn't have done so much long division.

The proofs in this unit are really hard to figure out how to do. I wish I could get along with prime numbers. Can we go back to doing even and odd proofs? I liked those. That is all.

11.3-11.4 due March 26

So I want to say first off that this post is going to be essentially worthless...but not because I didn't try to do the reading. I just didn't understand the reading from last time, which made the reading for today pretty close to impossible to understand. I tried to do the homework, and it was equally confusing. I am so nervous about this unit. The last few homework assignments have been terrible and I haven't hardly figured a single problem out without the help of the back of the book, lecture, or another student. So that's where I'm at. Hopefully my work schedule will cooperate and we can have some bonding time at office hours soon.

Linear combination proofs look like the proofs we were doing earlier in the semester with "Every number greater than 3 can be written as a combination of 2x+y=n. I would be lying if I didn't tell you that I just barely remembered what a linear combination is...and DUH that's what we were doing earlier.

The well-ordering principle has got to be the most confusing thing I've learned so far in this class. I don't understand its usefulness at all...and yet the book keeps using it...I feel like there's going to be a test question about it and I'm going to totally blank.

Have I mentioned to you that I dream in proofs? Yes. And the other day, the temple ceremony was coming into my head in the form of mathematical proofs. Something about this class has me pretty excited, because it has made my brain function differently.

10.5 - The End due March 21

So...mind blown...

I am overwhelmingly fascinated by the fact that we can define the cardinality of an uncountable set. And that we can do so simply! I already knew that we could call it "c," but the defining of 2^A being 2^aleph-null...it just seems so simple. Well, I suppose that nasty looking aleph null symbol doesn't make it look too simple, but it is!

The main problem I have with these proofs is that they seem to all be proving the theorem. (They are.) I just don't understand how I'm going to figure out how to do these problems without the infinite hints of the example. For instance. In the proof of Theorem 10.19, I don't think I would've even known how to define the sets without the help of the example. To put it simply, I am must more scared of this test than I was of the previous two tests.

What exactly is the benefit of knowing that the cardinality is less than another cardinality? I see great benefit in knowing that if (in the language of LaTex) A\leq B and B \leq A, then A=B. But what if the only thing you can do is prove A \leq B? Then we can assume that A is not equal to B, but that doesn't tell us that it's denumerable, does it? Questions, questions...

10.5 up to Theorem 10.18 due March 19

So what we are trying to do here is take a function that isn't bijective, and make it bijective by changing the domain. If we have a function that isn't one to one, we can change the domain to make it one to one, and therefore bijective. That sounds like it could be very useful. It's almost as if you could change an uncountable set to be a denumerable set. Dreamin' big here.

As for the thing that confused me and made my brain spin around, The proof of theorem 10.17 has me pretty lost. In my logical attempt to understand the theorem itself, I have determined that the identity function works. That's all good. So we're just proving that it works for proper subsets. Therefore, the set B' must be a proper subset of B? Ok. I continued reading through it, and after a study of the proof, I think I actually get it now. There were just a few too many variables for me to understand it the first time through. But it makes sense.

10.4 due March 17th

IT'S MY BIRTHDAY! YES!

I read today about infinity. And double infinity. It was kind of like reading about Chuck Norris.

I think I have always figured that there are different sizes of infinities...now I just have proof of it. It makes sense. I guess the main thought I have is that my favorite Chuck Norris joke is,

"Chuck Norris counted to infinity. Twice."

Now I see that that is not nearly as impossible or impressive as it sounded in my youth. You see, if he was counting to infinity on a denumerable set, then it's not impossible. All he needed to do was include all the natural numbers, and he'd be fine. To make this more impressive, I've decided to rephrase the joke to,

"Chuck Norris counted a set greater than the uncountable set of the real numbers. Twice."

There we go. Now that is impressive. I am glad to see that there's no limit to infinity. I would be concerned if there was. The Cosmos (TV show) had an animation of an ancient philosopher's idea of the universe as Cupid standing on the edge of infinity and shooting off an arrow. There are two choices for what happens to the arrow, 1) it continues on forever, or 2) it hits a barrier and doesn't continue, from which he can now stand and shoot another arrow. In either case, it goes to show you that he wasn't actually standing on the edge of infinity, nor will he ever be. It's boundless.

And that's my thoughts for today. I need to go to bed.

10.3 due March 14 (HAPPY PI DAY EVERYONE! This class doesn't do anything with pi, but still.)

1. So I'm not sure what my problem is regarding this proof (0,1) being uncountable. I just don't understand how b1 is not in the set (0,1)...and if it's in the set (0,1), shouldn't it be mapped to? Maybe I just don't understand the way that we defined the function. From the way I am looking at it, there are only 9 elements in the function...why? I don't understand. I wish I could come to office hours tomorrow to discuss this, but I cannot. Perhaps I will come on Monday.

2. On the bright side, the proofs after that all made sense, because they are largely the same proofs as section 10.2, only some are contradiction-style. I can handle that. Once I wrapped my head around infinite sets (as much as you CAN wrap your head around them) all the proofs about them became essentially logical. If you have an infinite set, then it will always be infinite, even if you subtract elements out of it.

What has been killing me is remembering that there are theorems for these proofs, so I don't have to write everything out! I did a problem for this last assignment and it took quite a lot of space. When I checked my answer in the back, there was a theorem. Oh well, at least I know how to prove it. Hopefully.

10.2 due March 12

I figured I would do the blog entry early this week since the homework for Wednesday is from 10.2. Aaaaand I forgot to do the blog for today for section 10.1.

1. By far the hardest thing about these denumerable sets is that you have to try to comprehend the idea of two non-equal sets that are related to each other in some way to have the same cardinality. The example in class today with the hotel rooms was helpful though. It's taking me back to my linear algebra days when I had to attempt to comprehend the n-th dimension. It's just counter intuitive to think that you can "count" an infinite set.

2. I am just loving that everything we've learned is packed together in this one section. We've got subsets, cardinality, types of proofs, induction, least elements, well-ordered sets, the identity matrix...it just never ends! I've never been one to understand application, but I always enjoy seeing it, and since this class is so fascinating to me, I will probably put in the time to understand it. My newest idea for comprehending it is imagining a graph that goes from [0,infinity) that has a range from (-infinity, infinity). This is entirely possible, and would have to be a function, and it would have to be one-to-one and onto.

Studying for the second midterm

• Which topics and theorems do you think are the most important out of those we have studied?
• Well I would have to say that the most important thing is proofs about functions. I say this because I learned a ton when I was learning about them about what functions really are, which changed my entire perspective on math. I also had the hardest time understanding it, so it's most likely the most important thing.
• What kinds of questions do you expect to see on the exam?
• I would say problems that look like the homework. The homework for the most part is very appropriate. The difficulty is exactly as it should be, and it covers the expanse of the material. Historically, the homework I've done for math classes is far harder than anything we ever need to understand at that level, and so it's frustrating and we don't profit from doing the homework at all. But this homework prepared me very well for the last exam, and I would expect the same for this exam.
• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.
• Currently I am focusing on SPMI proofs, but I also need some perfecting in the f-circle-g proofs. It's better than it was when I came to office hours, but I haven't reworked any problems from that section yet. That's my plan for when I finish this blog post. On that note...over and out.

9.6-9.7 due March 5

1. The inverse function. It makes sense that it needs to be both injective and surjective. A function has to have every value of the set A as the first coordinate, so naturally if not every value of the set B is in the range of the function, then when you take the inverse and switch the coordinates, then it wouldn't be a function anymore. So it has to be surjective. And it has to be one-to-one because all the points have to be unique and distinct, whether it's (a,b) or (b,a). The book shows a few examples that are easy to find by algebraic manipulation...so what do we do if it's not possible to find the inverse by manipulation? Is there a way? I don't know if we're responsible to know how because the book didn't teach us how, but it would still be interesting to know.

2. So a permutation...I feel like I've heard about these before. I forgot what they are. So a permutation is just a function of A-->A that is one to one? So basically a subset. That sounds good, but the use of it is not totally known to me. I seem to have this problem all the time. I know how to calculate something, but don't know why I would want to do it. Other than the format that we can write it in, I don't quite know what the difference between permutations and everything else we've been looking at is.

9.5 due March 3

This is a shout out to that awesome moment when I do the reading and then forget to blog about it before falling asleep. I was dreaming about proofs and definitions of functions...does that count for anything?

Calculus and math 290 meet...KAPOW!!! That was exciting. I think I forgot how to do integrals and derivatives though, so hopefully it doesn't come back full force. I did get excited seeing the derivative formulas that I had forgotten though. So it's really true that it was a study of composition functions that lead to the formulas for derivatives? I am still failing to see quite how that works...(are we supposed to understand it? Or was it just some "food for thought"...) I also believe that the dot has a few too many functions these days and I just know that I'm going to forget one in the testing center and accidentally multiply a composite function.

Lastly, what exactly is a "corollary"?  I know what a theorem is, and I figured that Corollary 9.8 was actually just a theorem, but now I'm not so sure. I suppose I will find out in lecture tomorrow. \qed

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.

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!!