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)