Final Exam Review, Due December 10

• Which topics and theorems do you think are the most important out of those we have studied?

I think the most important topic is, frankly, how to prove things! The different methods that can be used (direct, contrapositive, contradiction, induction, etc.) as well as the basic structure of how proofs should look (logic, quantifiers, assuming the premise, etc.)

Also, the whole concept of sets was a relatively new topic for me, and pretty much made everything in this course possible.

• 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.

- How to prove that things are unique. (For example, the intermediate value theorem, proving that there exists one root in a certian interval)

- Review well-ordered vs well-defined and when we use these. 

12.4, Due December 8

Difficult: Well, I do understand how to use these theorems. And I suppose I understood what was going on when they demonstrated the proofs, but on the division and multiplication one, the lemmas they chose to construct were quite.. overly convenient. Like I have no idea how they would do it, they would just prove random stuff, and then it all magically turns to epsilon!

I guess it would be helpful to help understand why and how they would bound f(x) and g(x). Maybe we could discuss what it means in general to have things "bounded".

Interesting: Like I said, it is really interesting and cool to use these proofs once you get through the gross epsilon-delta proofs. But I guess that's what we're here to learn, the gross stuff that makes it all work!

12.3, Due December 5

Professor Jenkins, I had this blog post typed last night, but I guess I accidentally saved it as a draft instead of publishing it. I apologize, but don't worry about changing it in the gradebook, since it was my mistake. 

Difficult: Holy cow, this section was a whole lot to take in!! Given I've never dealt with epsilon before, it was quite easy to get lost in the Greek letters. I understand the logic of what we must prove (I was able to write it using quantifiers! Yeah!) However, the biggest difficulty for me was figuring out how to choose each delta. It makes sense for linear expressions, but once we get to quadratic, polynomial, and rational expressions, I got lost where we were going.

What do they mean by upper bound? They never clarified what that was exactly referring to.

Interesting: This is really interesting, because in my Calculus AB class, to find limits we just looked at graphs and said the limit existed. But now we can prove it!! I feel pretty cool when I win at these.

12.2, Due December 3

Difficult: Awesome, so I understood why we use sequences of partial sums, but will we always need to prove two lemmas to prove convergence? The first lemma to prove {s_n} can be expressed in terms of n and the second to prove that the limit is convergent to a limit? I guess once we have those, we just plug those limits in and the proof is pretty easy.

And, holy cow, it looks tricky to prove a sequence diverges! Maybe it was just because the harmonic sequence is specifically difficult to prove, But those algebraic manipulations didn't seem intuitive at all. Is there a general structure for a proof that a series is divergent? Or does it always just depend on what the series is?

Interesting: Well, I've always used the words series and sequences interchangeably, but now I know the difference! A series is just the numbers in a row, where a series adds them all together! I've also hear of the harmonic sequence before, so it's really cool to finally know what it is. I can't believe it diverges, it seems like it wouldn't! Oh well, my intuition has definitely deceived me before.

12.1, Due December 1

Interesting: Woah, this is a way cool concept! I never thought of a series as a function from the natural numbers to the real numbers. Also, it makes sense that they would have to make a definition for limit, since just getting infinitely close isn't specific enough. Cool calculus proofs and concepts I have never seen.

Difficult: I understand most of the concepts, though I feel epsilon is really important so it'd be good to talk about that one again. I also would have a really difficult time constructing a proof from scratch: why do we need the ceiling function? I also noted that sometimes we assume "for all" and other times we assumed "there exists", so it'd be helpful to discuss the logic behind proving a function is convergent or divergent, or what it converges to.

Question Responses, Due November 25

• What have you learned in this course?

Goodness, what haven't I learned?? I have learned a lot of mathematical notation, logic, and a lot about how mathematicians are able to say the things they do. Things that I once considered trivial, like whether an integer was even, I can now prove in a heartbeat! I have learned how to prove things in a logical and complete sense, and how to prove things that I didn't actually believe were true. (I still can't believe there's only one prime number that's one less than a cube!) I learned a lot about what functions really are, which is pretty cool considering I've been using them now for four years; it's about time I learned what one is! This has also been my first exposure to number theory, which is way interesting and exciting.

• How might these things be useful to you in the future?

Being a math major, I honestly don't know how you would progress without this class. I really did feel like it was an excellent transition into a higher level of mathematics, and now as I take Math 113 and up, I will definitely be able to look at it in a more comprehensive and enlightened way.

11.5-11.6, Due November 24

Difficult: Could you elaborate more on proving the fundamental theorem of arithmetic, specifically the part proving there are unique factorization?

Wow, there were just a lot of theorems and corollaries proved (Eleven new ones in fact!) I followed each proof pretty well, it's now just a matter of making them intuitive and remembering them. And then, of course, learning how to use them. =)

Interesting: I really think it's interesting that the "Fundamental Theorem of Arithmetic" is that all numbers can be expressed as products of primes. What makes it so fundamental? I mean, we've gone five sections previous to it without actually having to prove it. Anyway, just an interesting question I thought of.

Test 3, Due November 21

• Which topics and theorems do you think are the most important out of those we have studied?

I think it's pretty important to understand what it means for a set to be denumerable, and what that implies. Schroeder-Bernstein is really important; it helps get a a lot done. Understanding the relationships between gcd(a,b) and linear combinations is essential to number theory. 

• What kinds of questions do you expect to see on the exam?

A lot of comparing cardinalities of sets, knowing the differences between countable, uncountable, denumerable, infinite, etc. Number theory questions, especially in the free response. Some reproducing of theorems that are essential to know.

• 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.

Number theory problems, especially those that deal with divisibility and gcds. For an example in class, maybe 11.31 in the book (or something similar). 

Math Talk

The Art and Science of Mathematical Modeling

This speech was especially interesting to me because I am considering going into the ACME program, so it was great to hear about a lot of the things this program teaches. I really loved how he was able to to talk about how mathematical modeling can fit into so many different fields: economics, finance, physics, speech, etc.

It was really interesting how he talked about how even something like who survived the Titanic can be modeled. The differences between black-box, grey-box, and white-box modeling was a little difficult to grasp, but I love how there are so many applications to this subject. He made it really applicable when talking about how our human balance is just like a mathematical model, where we make small tweaks to keep us from tipping over.

My favorite quote from the talk: "All models are wrong, some are useful" =)

11.3-11.4, Due November 19

Difficult: So I understand what a linear combination is, it's just difficult to relate the connection between gcd's and linear combinations intuitively. Could we discuss more about linear combinations?

Mostly I have all these theorems all floating around in my head and would love some help making sense of it all.

I do understand the Euclidean Algorithm, however, I'm unsure about the method of undoing it to arrive back at the gcd(a,b)=as+bt part. Perhaps another example of this?

Interesting: This chapter is way interesting!! I was taking four digit numbers and finding the greatest common divisor using this method because it was so handy! I do love the Euclidean Algorithm

11.1-11.2, Due November 17

Difficult: Compared to the previous chapters of crazy sizes of infinities and messy piece-wise functions, these chapters were pretty nice. I would like to see the Division Algorithm proved again; it does have some tricky parts which it uses.

Also, could you clarify the connection between remainders, dividends, and equivalence classes? I understand that remainders and dividends will be in the same equivalence classes (mod divisor), but how does that prove that the equivalence classes are pairwise disjoint exactly?

Interesting: I think number theory is way interesting. In some of the Math Careers talks I've been to, they've talked about finding REALLY big prime numbers, which I'm sure uses a whole bunch of number theory. =)

10.5, Due November 14

Difficult: So, Schroder and Bernstein proved their theorem without Theorem A, and therefore without the axiom of choice? Perhaps could you expand exactly what the axiom of choice is and means. Is it required since Schroder and Bernstein proved their theorem without it?

I actually understood the two functions in theorem 10.19 , as in how they go from numbers in (0,1) --> P(N) injectively and P(N)--->(0,1) injectively. However, I doubt I would ever be able to prove it as they did. However, it does make sense intuitively what they did.

Interesting: I love how the Scroder Bernstein theorem uses concepts and theorems from throughout the whole chapter and previously!! It comes together so nicely!

It's interesting how we use the interval (0,1) for most of the proofs for the cardinailty of the reals. But it always work out, since we proved there is a bijective function between (0,1) to the reals, so whatever is equal to the cardinality of (0,1) is also equal to the cardinality of the reals!

10.5 Part One, Due November 12

Difficult: Holy cow. Theorem 10.17 is confusing me for sure!! I'm trying really hard to think of an example to run this through with, but I cannot think of one... Let me write out one..

A: {Integers}
B:{Evens, 1, 5}
f:A-->B such that f(x)=2x
ran f is a subset of B, specifically ran f= {Evens}
A-B={Odds, except 1 and 5}
B'={Evens, except 2 and 4 and 8 and 16 and 32... and except 10 and 20 and 40 and 80...}
C=Union of two sets above
D={1,5, and 2,4,8,16,32... and 10,20,40,80...}

So h(x)= {f(x) if x in C, and i_D(x) if x in D

Woah. I think that worked. That is a bijective function from A to B. However, that is a lot of sets to consider and EXTREMELY complicated. There has to be an easier way to think about this... Maybe you can show a less complicated example? (Or worse, they're always complicated...)

Interesting: Well, I find it very cool that I was able to conjure up an example of Theorem 10.17. The Schroder-Bernstein Theorem is also extremely interesting, because it utilizes concepts from four different theorems to come up with an intuitively conceivable fact.

10.4, Due November 10

Difficult: The most difficult part of this chapter was the notation, and understanding a function where you put in sets and get out functions. Because you have to define both of the functions... which just kind of confuses me. We have to define both fS(x) and (S). So for the set A={a,b,c}, would ({a,c})=f{a,c} ={(a,1),(b,0),(c,1)}?

Once all the notation makes sense, it would be helpful to go over the proof that said P(A) and 2^A are numerically equivalent again.

I also didn't understand how to prove that there exists no bijective function from A to P(A). It makes sense to prove it's injective, but the proof by contradiction to eliminate onto confused me.

Interesting: This is so fascinating!! Especially since there are those two sizes of infinity with no infinity in between. Say what?! (Speaking of which, I hope we never have to write that fancy Hebrew letter)

10.3, Due November 7

Difficult: Thanks to a friend, Jarred, I was able to follow and understand the proof that the interval (0,1) is uncountable.  However, if asked to reproduce this proof by memory, it would be difficult to do so with correct notation.  Indeed, it was also quite tricky when we proved that the sets (0,1) and the reals were numerically equivalent. How did they decide whether to use the plus or minus case of the quadratic formula? Their explanation was pretty vague. Are there any other functions that will be difficult to prove onto?

Also, maybe it would be helpful to show other examples where we try to construct a bijective function from two intervals . Like from the reals to (3,4), or from (-2,2) to reals, etc.

Interesting: The proof that the reals are uncountable was really slick!! I can see how that related to the homework assignment with the X’s and O’s. Also, it was interesting and helpful to draw a diagram with the concepts: finite, countable, infinite, uncountable, and denumerable.  It was helpful to see where these overlapped and how these concepts mapped together.

10.1, Due November 3

Difficult: I always find it much more difficult when we start talking about infinities. For example, the introduction of this chapter discusses that the set of all natural numbers has the same cardinality as the set of all the perfect square natural numbers. This statement just blew my head off. I'm with Galileo on this one; I can definitely feel his hesitancy working with such odd things.

The one part I don't understand is how equivalence classes relate to numerically equivalent sets. Instead of a number, like [1], do we now say [A], which are all the sets that have a bijective function with set A?

Interesting: I thought it was neat how the first theorem used collections of sets, relations, functions, and equivalence relations all together. That's an official blast form this course's past!

Exam 2 Review, Due October 31

• Which topics and theorems do you think are the most important out of those we have studied?

Principle of Mathematical Induction and Strong Principle of Mathematical Induction

Reflexive, symmetric, and transitive properties

Theorem: [a]=[b] if and only if aRb

Functions- Domain, codomain, range, surjective, injective, bijective, composition

• What kinds of questions do you expect to see on the exam?

Proof by MPI/SPMI.

Knowing the properties of relations

Prove a relation is an equivalence relation

Proving functions are bijective

• What do you need to work on understanding better before the exam? 

Reminder of SPMI problems

Equivalence class problems 

Well-defined vs Well-ordered

(I've got the hang of recent chapters, it's just about getting those rusty gears from chapter 6 and 8 moving again!)

9.6-9.7, due on October 29

Difficult: Whoa, big concept chapters! A lot to take in! First off, the proof that inverse functions are functions if and only if the functions are bijective was a lot to take in. If we're going to need to prove that on our own, it would be helpful to review it again in class. Also, are rational and linear the only functions we can easily find inverses of? It mentioned we can't find inverses for some polynomial functions, but are there any other ones we can do?

For permutations, do we always use natural numbers that come right in a row? Or could me use B= {2,5,8}, or even B={a,b,c} to find permutations? Also, what do we do with them...? Or just look at them since is it just a concept for now?

Finally, at the end it said "permutations satisfy the the properties of closure." What is a property of closure?

Interesting: Well, the one thing I do like about permutations is that they are easier to write than listing out all the commas and parentheses. I also thought it was interesting how we can make all functions inverses by cutting off the codomain that we don't use, but I guess that's kind of like what we did for the inverse trig functions.

9.5, Due October 27

Difficult: For some reason, it always throws me off that g ∘ f is g(f(x)), since I always think that the letter that comes first should be the function we do first, so that kept throwing me off but I think I've got it figured out now.

Also, it started talking about how ran A only needed to be a subset of B' for (g∘ f):A--> C to be defined. Well B' does NOT mean the derivative of B, correct? Because that would be MEGA confusing.

Interesting: At first, this concept was totally tricky. However, it made a lot more sense with the nice picture with circles and arrows, along with the examples of finite sets. I also never knew that composition could be associative, yet not commutative. I thought that was fancy!

9.3-9.4, Due October 24

Difficult: Alright, now I've got a lot of concepts floating around in my head that I need to work on solidifying. So we have three main special relations that we've learned: one-to-one, onto, and bijective. I understand most of them visually, like on a graph, but I need to understand them conceptually now. So, if we have to prove something is bijective, we must prove it is both one-to-one and onto? Several examples of proving these two proofs would be helpful.

I didn't understand completely about why theorem 9.4 is false for infinite sets, but I assume we don't need to completely understand that yet, since we haven't discussed infinite sets with these properties yet.

Interesting: It's interesting thinking about these properties with finite sets. I can always picture a graph on the real Cartesian plane to fulfill the properties, but when we start talking about equivalence sets, it's more difficult to picture/conceptualize, but easier to prove.

9.1-9.2, Due October 22

Difficult: So let me make sure I have this correct, the difference between codomain and range is that the codomain is all possible values of y, while the range is the only ones that get used? What does the symbol that is an o with a vertical slash through it mean? I just think it looks nice, but does it actually mean something specific? Also, if B={1,2}, would it be bad to represent the set of all functions from A to B by 2^A? (Since B is supposed to be {0,1} according to the last line of chapter 9.2)

Interesting: There is definitely a lot of interesting notation/vocabulary used and learned, such as image and mapping. I also thought the evolving definition of the word "function" over time was interesting as well, though the really old ones are really quite abstract and difficult to understand.

8.6, Due October 20

Difficult: So this all sounds great and dandy, I'm just not so sure about when sums or products are not well-defined. I mean, I guess the example in the book at the end of the chapter just completely changed the definition of what multiplication (redefined the function) and then called it not well-ordered... Sounds like cheating to me. Haha

But it would be great to go over proofs of well-defined. But from what I see, It looks like we just added another layer. Well defined --> Equivalence Classes --> Relations --> Modulos--> Divisiblity and then all back out!

Interesting:  Man, we use modular arithmetic all the time! Clocks is a great example, both minutes and hours are mod 60! Days of the week work in mod 7, but I guess days in a month gets messed up because it's (mod 28, 30, or 31). But still cool nonetheless!

8.5, Due October 17

Difficult: The most difficult thing to do is to figure out what to do with the symmetric property. Reflexive and transitive seem to happen pretty similarly, but proving symmetry seems to be a little more difficult. It's helpful when you show us writing the skeleton and why you write what you do, so examples like that will be awesome.

Interesting: I thought it was pretty tricky when they showed that not every modulo relation a(congurent)b (mod n) has n equivalence classes. Like a^2(congruent)b^2 (mod 3) only has two distinct equivalence classes. It doesn't make sense intuitively, but it's pretty interesting and makes sense  when you prove it and such.

8.3-8.4, Due October 15

Difficult: Hold on here, so remind me, can you have the same element twice in a set? I'm assuming not, so {3,2,1+1} is bad and should be written as {3,2}? Otherwise , Theorem 8.3 would not make sense since there would be multiple, equivalent sets in the set P (the partition). Or maybe we just don't count them twice, maybe could you remind us about that?

Again, like a lot of things in this class, there are a big handful of concepts to master that will take practice. So examples rock.

Interesting: It is way cool that we can call things equivalence relations if they are reflexive, symmetric, AND transitive. I went back through the examples we did in class and realized that mod's are reflexive, symmetric, and transitive, so it's an equivalence relation! Which means... we're probably going have to start proving more of those soon.

8.1-8.2, Due October 13

Difficult: The most difficult thing was definitely the transitive property. I was familiar with all three of these properties, but not with relations. It was okay when I had an explicit set of points to look and see if it was transitive. However, when it became more vague, such as |a-b|<1, then it became much more difficult to visualize. The most helpful thing would probably be working on examples to get these concepts down. I feel, just like we did with sets, we're going to learn these concepts, and then have to prove them to the maximus in upcoming lessons.

Interesting: I'm a really visual guy, so when I had to see if a set was transitive for specific, explicit elements of a set, I would just picture the ordered pairs bumping into each other. (They weren't allowed to rotate, but they could move and bump other elements in the set) If the right side of one bumped into the left side of one and they matched, then the middle two numbers would disappear and leave me with a new ordered pair. That ordered pair had to be present, or else the set was not transitive.

I know that's kind of goofy, but it helped.

6.4, Due October 10

Difficult: This is most certainly the most difficult lesson I have read so far. I just don't get why we are using i... that inductive step is just throwing me off! So is all we're doing is more base cases so we can assume more about that k is a bigger number? I thought we already did that...

When you throw four variables in there, (a, k, i and n) then these proofs get so much harder to follow. I can't even understand this concept well enough to make a sensible question about it: it's all just a blur...  I watched a YouTube video about strong induction, but it still didn't make that much sense at all. Tomorrow will be filled with lots of questions and examples hopefully.

Well, I guess one question, why are recursive sequences always shown in sets? Is that just... how they are?

Interesting: I'm sure this will be much more interesting when I can understand what this is saying.  Recursive sequences, however, are very interesting. Like the Fibonacci sequence and how it is EVERYWHERE in nature. That's pretty cool stuff!

Math Careers- Financial Engineering

I thought it was really interesting how highly he thought of a math degree and how much math intersects other fields of learning. They explained the role of mathematics in their job really well; they talked about how derivatives, multivariate statistics, and differentials played into their buying and selling. They also used extrapolation to predict how prices will raise or lower over a set amount of time. Trends over time are much more accurate when the math behind the algorithms are more sound. Being a financial engineer requires much innovation, critical thinking, and creativity.

It was difficult to understand all the financial terms they were discussing about markets, pricing, and modeling. But it was still interesting nonetheless.

6.2, Due October 8

Difficult: One concept I wan't sure on is when do you want to use induction as opposed to other proofs? We learned some of the basic key indicators of when to use direct, contrapositive, and contradiction proofs. Are there similar components of proofs requiring induction? Is it any time you would have to do an infinite number of proofs?

Also, the example with proving the cardinality of a power set is 2^n was tricky to follow, so an example in class would help a lot on that one.

Oh, and do we need to write "By the Principle of Mathematical Induction..." at the end of each of our proofs?

Interesting: I am really glad we are learning induction, as it is one of the main proof techniques we use in our Putnam class. I find it interesting that we are able to do so many different types of proofs using induction. From even/odds, to divisibility, to sets, they all work out!

Exam Review, Due October 3

• Which topics and theorems do you think are the most important out of those we have studied?

I personally think the most important topics are the three kinds of proofs and the axioms. As long as you are able to remember those, you should be able to prove theorems if you dont have them explicitly memorized

• What kinds of questions do you expect to see on the exam?

I expect to see a lot of proofs using the principles of sets we learned in chapter one. This is the best way to test comprehensive knowledge, because you have to know both the information on sets and the strategies to conduct a proper proof.

• 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

The biggest thing I need to work on, (after taking the practice exam) is not to mess up on the little things and make sure I have all the definitions down soundly. I would like to see a couple of proofs involving sets (them being equal and subsets), and problems with congruence and modulo stuff.

5.4-5.5, Due October 1

This post would make more sense to reverse the difficult and interesting headers

Interesting: I thought these two chapters would be simple. I just have to find some value for which R(x) is true, and "wa-lah!", a solution exists. But it's much more difficult when you don't know the solution but still have to prove one exists. This was actually pretty cool, that we could say "I don't know what the answer is, yet I know there is one." I really liked David Hilbert's quote about hair:

"There is at least one student in this class . . . let us name him 'X' . . .for whome the following statement is true: No other student in the class has more hairs on his head than X. Which student is it? That we shall never know; but of his existence we can be absolutely certain."

Difficult: Though I found this concept interesting, it is also what I found difficult. I understood how they proved that "There exists irrational numbers a and b such that a^b is rational." However, I have no idea how they decided to use sqrt(2) and would have never thought of that proof by myself.

I also have a hard time understanding proving there are unique solutions. So from what I understand, you basically take two different elements and prove that they HAVE to be equal?

One more thing, I really struggle with what we can and cannot assume about irrational numbers. Can we assume that any rational multiplied by an irrational is irrational? What about when we add or subtract? Or do we have to prove those?

5.2-5.3, Due Sept. 29

Difficult: I really like proof by contradiction, but I had a really difficult time following the proof : Prove that sqrt(2) is irrational. Since it cannot be written as an implication, do we have to use contradiction? Or can we make an implication, like: "If sqrt(2) is irrational, then there are not two integers for which a/b=sqrt(2). (And then use proof by contrapositive... but I guess that wouldn't be much of a proof...) Another thing I wasn't sure about is when the book quotes and uses theorems from previous chapters. In our homework, when we do that, do we need to cite which theorem we used? Or can we just say "A previous theorem said that since xy is odd, then either x or y is odd."?

Interesting: I really found the example about the three prisoners with dots on their heads really interesting. After I read it and thought about it, I could have sworn the third prisoner just guesses. But as it explained how the prisoner used contradiction, it really clicked!

4.5-4.6 and 5.1, Due Sept. 26

Difficult: The thing that I found (and find) the most difficult is proving things slowly and logically. I look at most of these results and say "Oh yeah, of course that is true." But I find difficulty when I have to explain each step along the way. I need to understand very clearly what I am allowed to assume at what times. I guess that all goes back to definitions, but I have a hard time not showing all the steps because I don't even realize that I'm doing them. Maybe review what is allowed to be assumed when going between set notation and "x is an element of" notation.

Interesting: I really love and understand counterexamples (5.1) a lot more than the material Chapter 4. I guess it's because it's much easier to find one counterexample than have to account for all x in the domain. But I guess counterexamples don't prove anything, they just prove things incorrect. Which I guess is a proof?

(P.S. I found I mislabeled my last blog post 4.1-4.2 instead of 4.3-4.4, so I went back and changed it. Hopefully you read the post and understood I read and blogged about the write sections on Wednesday, I just typed the title incorrectly)

4.3-4.4, Due September 24

Difficult: Woah, I totally got lost when they were trying to prove |x+y| (< or =) |x| + |y|. That proof just really through me, because they would change the = to a > throughout the proof, and seeing as they considered this a pretty important proof, maybe could you go over this one or one similar in class?

When I read these chapters, it's so difficult to apply what I learn. Like when I was reading through the subsets proofs, it made sense what they were doing, but I don't (currently) have the ability to come up with it on my own. My understanding is really hazy, but then as we talk about it in class, it becomes a little clearer, and then as I start doing my homework, it becomes a lot clearer. This cycle has become really prevalent and useful throughout these chapters. So what I'm saying, I get what it's saying, but it's going to take some lecture and practicing around with before I will get it.

Interesting: It's great that we can start using all real numbers. The proofs before were great, but they were slightly limiting because you could only use integers.  Expanding the domain definitely has a lot more applications to real world situations. Off the top of my head, I can think of a lot of engineering situations where the domain would be all real numbers, so it’s important to be able to work with this set.

--How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?

I probably spend two-two and a half hours on each homework assignment, including the prereading, blog, and homework. The prereading and lecture ABSOLUTELY help a lot, as I said above.


--What has contributed most to your learning in this class thus far?

I think the biggest thing is having good solid examples, and then learning by doing. I can watch people do proofs all day long, but until I actually sit down and start doing it, I never really master the concept. Since I can't do that until after class, having solid definitions and examples gives me a great base to refer to when I need help.

--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.) 

This is mostly a goal for myself, but I would say increasing my diligence. A lot of the time I will get discouraged because I can't figure things out right away, or that I have to just sit there and think without writing anything down for a couple of minutes. Even though I often feel "stuck," those are often the moments where I grow the most, through trial and error. I really don't have any negative feeback for you, I love your class structure; it really works with me.

4.1-4.2, Due September 22

Difficult: A lot of section 1 was pretty basic, but could you go over how to do the "divides" and "does not divide" symbols in Latex? I don't know if you just use the absolute value sign, or if that'll mess up Latex. Also, example (result) 4.8 in the book made sense, but the proof didn't seem intuitive. It didn't seem direct or contrapositve, so maybe could you show an example like that in class? (For every integer n>=7, there exist positive integers a and b such that n=2a+3b.)

Congruence was a little tougher. These will probably just take a lot of practice and learning the definitions to get those proofs through smoothly.

Interesting: It's really cool, because I often thought about the concept of congruent/modulo numbers, but never knew that they had a name in mathematics. It's also really cool that you can prove really obscure things that I would have never guessed were true using these proofs. The even/odd proofs were pretty obvious, but I never knew that if x isn't a multiple of three, then x^2-1 is always a multiple of three. Just really interesting!

3.3-3.5, due on September 19

Difficult: I have a few questions from reading it over. In the text, it says "The sentence x (is an element of) Z is commonly not written in the proof because it is stated in the result." When I went in to talk to the TA, he specifically said to ALWAYS define all of your variables before using them in your proof, even if they were defined in the result. Could you address this in class?

And I am also still struggling with getting started. If I sit down to a result, it's hard for me to determine where to start. Should I use trivial, vaccuous, direct, contrapositive, proof by parts, or a combination of all of these? I'm sure as we continue, the list will only get more and more complicated. What is a good place to start? (Perhaps only practice will tell)

Also, when you get to the end of the proof, do you have to say "This statement is true because I just proved it using contrapositive." Or, does your reader have to be smart enough to know that you just proved that using contrapositive?

Interesting: It's so great, because I realized the value of partitions! When you use the "proof by case" method, you have to prove it for the entire partition of the set. If you don't include all values in the set, (a pairwise disjoint) you may run into errors. I like how everything we're learning is stacking onto each other really great.

3.1-3.1, due on September 17

Difficult: Oh, now we're getting to the meat of this course! So, I'm not really completely certain what a "result" is versus a "statement"? I can tell they are worded slightly differently, but is the word "result" used specifically for proofs?

Also, it says, when proving vacuously (not sure how that's pronounced), that we pick an arbitrary element from P(x) and prove that it satisfies Q(x). But doesn't that just prove that the statement is true for that element? How does that prove the rest will satisfy the statement as well?

Interesting: I am excited to start doing proofs! I find it interesting how they never take anything for granted. I mean, they even started with proving if a number was even or odd, using only three base rules. It's so fascinating how pretty much everything we know in math, no matter how simple or complex, is derived from a proof. If you think about a Calculus proof, it probably is based upon several algebra proofs, which are based upon other proofs, which could be based on others. It's like a family tree of proving stuff in here!!

Shout out to Michael Te

Chapter 0, Due September 15

Difficult: Wow, this was quite a unique chapter, and something I've never really connected mathematics with. The first difficulty I had was putting this knowledge into context . Having never written a mathematical paper before, it was a lot of new information to soak in at once. It's hard to imagine what kinds of things I would write in a mathematical essay, or what I would prove or accomplish. I've never done the type of research I do for a writing class for a math class before, so this whole chapter was an introduction to a totally new concept for me.

I also wasn't sure by what it meant about "that" vs "which". I've never really consciously had to decide which one was correct and which one wasn't, so it would be useful to hear some more examples on what the rules are.

Interesting: This whole chapter was interesting to me!! I've never thought consciously about the word choice used in mathematics, and the different rules for writing and communicating mathematical ideas. It's really awesome, because think about how hard it would be to learn math without good textbook authors. And those textbook authors learned math from other mathematical writings, so it's like a constant chain of keeping mathematical truths alive and flowing, When my Writing 150 teacher says that writing is used in literally every subject, I now have no disagreements at all.

2.9-2.10, Due Sept. 12

Difficult: There are a couple of questions I have from reading the text. Are there only two ways to use quantification, specifically the universal quantifier and the existential quantifier? I'm also still not quite clear on distributing negations to these statements. It gets confusing because you have to change both the quantifier and the statement? Is that always the case? Otherwise, these statements can be very handy, and I can tell why they will be important in proofs.

Interesting: These past two units have been so helpful as I have been taking Math 391R. When I walked it there on the first day and saw all of their Greek letters and notation, I pretty much died on the inside. Everything went over my head and I had no idea what they were talking about. But as I have been learning all this notation, I have been slowly catching on to what they're saying. Not close enough yet to be proving abstract formulas like some of the juniors and seniors in that class, but closer than before. And hey, that's all I can ask for, getting better each day!

2.5-2.8, due on September 10

Difficult: The thing I found most difficult were compound statements. There are a lot of theorems to memorize, and it's sometimes difficult to remember what is legal and what is not. But, I guess it all comes back to definitions of the statements. For example, I know it's proven, but it doesn't just "click" to instantly say that P => Q and (~P) V Q are logically equivalent. I have to stop, use a truth table, and then use examples to help more fully understand the concept. However, truth tables seem to save the day every time!! They are so handy and make a lot of sense! So hopefully we can keep using those.

Interesting: This unit is all together really interesting, I love studying logic, it's so applicable to anything! We were talking in my Honors class about great questions, and with great questions requires great logic. You have to make sure that any argument you make, especially in rhetoric or persuasive writing, is logically solid to avoid fallacies.

We discussed some of these concepts in geometry, but not as in depth. All I remember from then is that if the statement is true, than so is it's contrapositive. In addition, if it's converse is true, then so is it's inverse (or visa versa).

2.1-2.4, Due September 8

Difficult: The most difficult part for me to understand is a lot of the notation. I am used to reading P(x)=2x+2, but is that equivalent to P(x) : 2x+2? I know the first one is an equation, but is the second one a set, or just an expression? Or also a function?

I understood a lot of the vocabulary, but I am still not sure why P => Q can be read "P only if Q." Suppose P was "You are happy." and Q was "You clap your hands," then the conditional statement is "If you are happy, then you clap your hands".

Well, now I just typed up and deleted a whole paragraph about why that didn't make sense... but then after thinking about it, it does. I guess it's just tough to understand right off the bat all those six ways of saying conditional statements, particularly "only if", "is sufficient for", and :"is necessary for"

Reflective: This whole reading section brought me back to geometry class where we did a lot of proofs. I remember the inverse, converse, and contrapositive of statements and remember learning about "iff" (if and only if). This chapter, logic, is really interesting because it doesn't just apply to math, but easily correlates to any subject you are learning, or life in general. Doctors, lawyers, and computer technicians use it all the time. If the bone is fractured, then we will put it in a cast. We use these statements, determine their truth value, and connect them to our lives multiple times per day.

1.1-1.6, due on September 5

Difficult: The most difficult part of this chapter for me was definitely learning, reading, and getting used to the notation. For example, the symbol ∈ looks very much like ⊂ which also looks like ⊆, yet each symbol has much different meaning and is read differently. In addition, I have always been familiar with the different sets of numbers (Real numbers, rational numbers, complex numbers, etc.), but I never knew that each had a distinct notation as well: ℂ, ℕ, ℚ, ℝ, and ℤ. However, these are minor technicalities that can be mastered with practice.

I also had a hard time understanding 1.4, about the collection of sets. It's tricky when you can write a collection of sets in so many ways, and I was so confused about which is correct.That was until I reached example 1.18, which was an excellent problem that answered many of my questions. It was also sometimes overwhelming when you have sets inside of sets inside of more sets, but I suppose that's just the nature of this chapter, to push the boundaries of what we can do with sets.

Reflective: The part about this chapter I really thought was cool was when it would tie the sets back to something I was familiar with. Two observations I thought were neat were that the Euclidean plane could be described as by the product of ℝ x ℝ. Also, that you could give the set of ALL the points on a line, like 

{(x,y) ∈ ℝ x ℝ : y = 3x - 5}

We could also define intervals, open and closed, using sets (neat-o!).

(Professor Jenkins, could you talk in class one more time about what the length of these blog posts should be. I feel like what I have written is a thoughtful reflection of the material I have studied, but I would hate to be docked consistently for not having long enough posts simply because I wasn't aware of the expectations. Thanks!)

Introduction, due on September 5

Good day Professor! My name is Tyler Mansfield, and I am section 003 of your Math 290 class. I am a freshman at BYU currently majoring in math, but I am also thinking about an ACME or Stats major. I have taken Math 112, though it was back in my junior year of high school, so my math gears might be a little rusty. I am taking this class to start testing out the waters of deeper mathematics, so to speak. It's a requirement of both a Math and ACME (and probably Stats) major and the people on the Putnam team say it will really help me with my proofs.

I really haven't had a lot of math teachers; I had the same one throughout all of high school. But what made her extremely exceptional was that she was very clear what we as a class were expected to know. Our assignments were graded and returned to us before the test so we could see what we had "aced" and what we need to work more on. It's impossible to know if you are doing a new concept correctly or not without some sort of feedback from a friend, professor, or TA. (The latter two are usually more reliable)

I am unique because of my strong love for people and desire to serve and uplift others. I love helping out; I often tutored many of my friends in math. I spent 2.5 weeks on a humanitarian trip to Suva, Fiji to build bathrooms and supply a school with necessary school supplies. I always try to be approachable, friendly, and happy.

Your office hours are great for me. Thanks for all you do!!

Yours Truly,
Tyler