Sunday, September 30, 2018

Section 12 due 1 October

I'm not entirely certain that I follow the transition from original statement to the contrapositive assumptions made in example 12.5.

This section just combined previous concepts in new ways. Seems simple enough. I suppose we shall see if it really is that simple in the homework exercises.

This whole course could feasibly pass as a philosophy course. It is more of a course in logical thinking than it is in math thus far. I would love to see this course combined with the material about sets that we are learning in linear algebra.

-Batman

Thursday, September 27, 2018

Section 11 due 28 September

"Proposition 11.6. One of the digits of 2 = 1.414213562 . . . occurs infinitely many times.

Proof. There are only finitely many possible digits {0,1,2,3,4,5,6,7,8,9}
but there are infinitely many decimal places. QED"

It seems to me that this proof isn't complete without at least mentioning the pigeonhole principle. They do address it in the explanation after the proof, but since it isn't included in the proof itself, I am not in convinced that this proof is complete. Is it? If so, why? Would this get full credit on the homework/test?


I don't understand the proof of proposition 11.7.


The formatting is so weird. Sorry if the font randomly changes sizes.

-Batman

Tuesday, September 25, 2018

Section 10 due 26 September

I LOVE set theory! It was a main reason I became a mathematician. I love the idea of defining any set you want and then messing around in the universe defined by that set to see what you can do. I would always ask math teachers in middle and high school how on earth people created new theorems and invented calculus and came up with all of the brilliant math that they did. None of them had an answer for me. When Ken (my professor my senior year) learned that I wanted an answer to that question he began to stay after class to teach me about creating universes through sets and new laws to solve problems and create new math. That is when I fell in love with math and realized I wanted to study it. Now I'm here.

Anyway, as far as something challenging about this section, I am still mixing up the ∩ and ∪ symbols. I just need a good way to remember which is which.  Also, I don't remember learning what a set with a bar over it means. Maybe the negation or inverse or something? We might have learned it and I'm just being forgetful...

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

I feel that the lecture and the reading did prepare me adequately for the homework. I probably spend about 2-4 hours per class on this class.

What has contributed most to your learning in this class thus far?
I like the current format where we read the chapter to gain an initial impression of the content, then review it in class, then practice it on the homework. I think this strategy of reviewing everything several times helps me learn it more thoroughly and help commit it to long term memory.

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.)
I think there are some homework assignments that get immensely repetitive. Even though there are only 6-10 problems, on some assignments most of the problems require the same technique on nearly all ten problems. I understand that lots of practice is beneficial, but after four problems of nearly identical work, I start to get bored. Maybe cut back on identical homework problems so I am not spending as much time on homework assignments?

Your truly,

Batman

Sunday, September 23, 2018

Section 9 due Monday 24 September

Proof by contradiction only works if you are certain that you are working with a scenario that is a boolean. I can imagine a scenario where someone thinks "true" or "false" are the only answers, only to draw a conclusion based on proof by contradiction, when in reality, they are ignoring another possible answer. I do not know how this would play out in practice... maybe in quantum mechanics? I'm not sure if that is even relevant to this. Just a thought...


I would like to hear more of the logic behind the following quote:

"[I]f a proof can be done without contradiction, then that is usually a better option, because you never enter an "imaginary" world where you assume something you are hoping to show is false."

It seems to me that a proof is a proof, right? I don't understand why mathematicians have aversions to certain types of proofs. I have been told that proofs by induction are considered ineloquent and thus should be avoided when possible. Why are some proofs superior to others? "All proofs are equal, but some proofs are more equal than others." :-)


Yours truly,
Batman

Thursday, September 20, 2018

Section 8 due 21 September

Proof by cases was something that I have picked up on through contexts in my math classes last year. It was never something that was taught explicitly. It is good to have written patterns confirming what I already figured out. (Indeed, most of this class has been writing out the rules to things that I only ever figured out from context; in the past I could tell you that something works within the context of a proof, but I wouldn't be able to tell you why. This class is changing that!) WLOG was also something that I picked up on through context. So far, the book also seems to address it only through context. I would love to have a more in depth discussion about when it is acceptable to use "without loss of generality" in a proof, as I feel I only have a vague idea of when to use it.

I want to go over the modulus syntax used in this chapter. I think I would have a better understanding if it were explained to me. The modulus is used in a slightly different way than I have ever seen it be used before.

Tuesday, September 18, 2018

Section 7 due September 19

If a contrapositive restricted to implication statements? Can a contrapositive exist for any other type of statement? If so, how does one go about proving that a contrapositive has the same truth value as the original statement for all statements?

That was my primary thought process reading this section. I thought the section was interesting, but my brain is too tired to form coherent thoughts beyond that. I don't feel well, so I will be going to bed now in the hopes of feeling well enough in the morning to dig deeper into the material during the lecture.

Saturday, September 15, 2018

Section 6 due 17 September

This section alternated between clever tricks for solving proofs and inherently obvious observations about mathematics (see definitions 6.1 and 6.2). Parts reminded me of this xkcd post. However, I liked the solutions to finding whether or not the premise is true. That is not how I would have approached that type of problem. See proposition 6.20.

I learned about the fundamentals of proof at the Prove It! Math Academy camp I went to. It is nice to dive back into the same material. Different parts of proofs are emphasized differently in the textbook, classroom, and math camp. It is intriguing to see the similarities and differences between how all mathematicians approach teaching and solving the same concepts. Also, the camp made the word "trivial" into a running joke (sort of by accident), so I have lots of fun memories associated with that word!

Thursday, September 13, 2018

Chapter 5 due 14 September

I think it is really cool that a set can have a lower bound that is not actually included in the set. It makes me wonder the purpose of lower bounds, though, as -1, -2, 0, etc are all lower bounds for the set (0,1]. If the lower bound can be any arbitrary selection from anything within a specific domain that is exclusive of the domain of the original set in question, what use does the lower bound have?

I understand the greatest/least element and upper/lower bound thing when I read it on paper, but I think it would sink in better and I would have a more full understanding of it if I were to hear it explained. That was probably the hardest part - connecting the meaning behind the phrases to the bigger picture. I struggle with that when I am reading more so than when it is explained.

Tuesday, September 11, 2018

Chapter 4 due September 12

I would like to clarify the syntax in the following statement to ensure that I understand.


"As an example, if we let x and y be variables with the domain R, and let P(x,y) be the open sentence
P(x,y) : x>y,
then the sentence x R,P(x,y) is not a statement, because it still has a variable that has not been specified or quantified (namely y). In order to make it a statement, we need to evaluate y; we could say y R, x R, P (x, y), which means that
For all real numbers y and for all real numbers x, it holds that x > y."


Even though the author stated that "x and y [are] variables with the domain R," because this is not explicitly stated in the claim itself ("∈ R,P(x,y)"), it is not a statement. I am assuming that the aforementioned thing in parenthesis is the only thing I am supposed to be looking at when evaluating the statement, rather than look at the paragraph as a whole. Is that correct?


Also, I was taught not to start sentences with math symbols when writing formal math things. However the book does so: "P implies Q. P only if Q. Q if P." What is the proper standard?

I am wondering what practical application that this section has. I'd imagine that it is more of a thought exercise than an actual useful tool in most cases. However, I have read formal and relevant proofs that did include statements similar to the ones about which we are learning. However, I have not seen them in nearly as much depth as the book is providing.

Monday, September 10, 2018

Chapter 3 due September 10th 2018

Honestly, the hardest part is memorizing all of the symbols and the specific details behind what each symbol means. The concepts themselves aren't particularly difficult.  It is hard for me to keep track of the meanings of ∧ and ∨ because they are so similar in shape.

Understanding logical statements might be a polarizing ability (as in, most people are very/decently good at it or very bad at it). It isn't particularly difficult for me, but I have friends who struggle immensely. It makes me think that some people's brains are hardwired differently than others'. In an English class I took we were examining riddles and logical statements. The teacher and most of the other students had to examine each statement in great detail and still ended up confused, despite spending exorbitant amounts of time on each problem. The teacher ended up coming to me for help when he confused himself to the point where he couldn't help the others. The other students who understood the statements easily were all very good at/interested in math. Maybe the type of people math attracts are also the type of people who have an easier time thinking logically.

Thursday, September 6, 2018

Section 1 due on September 7

I had never heard of power sets before. I thought our discussion about them in class was enlightening because it made me realize that the number of elements in a power set is much greater than the number of elements in the original set. However, I do not see an immediate use for power sets. I am eager to see their relevance in the world of advanced mathematics.


Assuming I have a correct understanding, I absolutely adore cardinality because it makes combinatorics possible. I learned about combinatorics at a summer math camp I attended and it was a fascinating subject.

Section 2 due on September 7

I do not understand the point of attempting to graph the unions between sets. Is it meant to be purely a visual representation of the concept of unions, or will graphs of unions play some significant role later on? I'd imagine it is probably the former, though I find that visual representations can provide an alternate way of viewing problems that can prove enlightening.


I find it very exciting and fascinating when I learn math that provides context for simpler math I learned at another point in time. For example,

"The set R2 = R × R is called the Cartesian plane. We view elements in this set as points {(x,y) : x,y R}."

This is a perfect example of the way the math curriculum simplifies math to make the basics easier to understand. Sometimes I think it is a good idea to simplify the content. After all, why would you learn about sets before you can add? Other times I wish I had been exposed to the "big picture" of math much earlier. When I was a senior in high school I began to witness this bigger picture behind math for the first time. That was ultimately what convinced me to switch from my plans to be a physics major. It makes me sad that so many of my peers will never see the bigger picture because they did not progress far enough in math. It makes me think that we need to drastically rethink the primary education math curriculum to ensure that all students have had enough exposure to math to truly understand whether or not they want to pursue it. Most of the math I learned up until my senior year of high school were merely tools for exploring more complicated subjects; they didn't even address the meat of the subject. (How on earth are you supposed to phrase that sentence correctly? Is "math" inherently singular or plural?)

Introduction due on September 7

Hi!

I suppose you would probably consider me to be a sophomore. However, this is my first year of college. I am majoring in pure math.

I have taken calculus 1, 2, and 3 (multivariable). I have also taken a differential equations class that was supplemented with some linear algebra topics (eigenvectors/values and basic matrix math).

I chose EMC2 because I want to get to know other mathematicians so that I can begin developing a professional network of mathematical colleagues. Also, it is nice to have friends who understand my math jokes!

My math professor my senior year of high school ultimately convinced me to be a math major (I had originally planned to major in physics). He answered my questions in as much detail as he could, and was even willing to stay after class for 45 minutes to teach me more about a theory we had touched on in class but would not be delving further into. He helped me to understand the mathematical community and the sorts of things that I could pursue if I chose to study math. He also gave me the resources to learn more and put me in touch with several experts so that I could get additional opinions. He never once pressured me into switching to math; he was immensely supportive when he learned I wanted to study astrophysics at the beginning of the year.
Tl;dr: He thoroughly answered my questions about 1) the math we were learning in class, 2) math I was learning on my own, and 3) what it would mean for me to study math in the long run. He was my friend who simply wanted me (and worked to enable me) to learn as much as I could about what I thought was interesting.

I am a swimmer. I do not like brown bananas. I only like bananas when they are still slightly green, just about to turn yellow. I like being outside and weightlifting. I love the Enderverse by Orson Scott Card.

I can come to office hours as currently scheduled.