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