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!