Exam Review for Monday March 31

What theorem do I believe is most important?
I believe the whole numerality thing is going to be important for the exam.  You know, cardinality - how to prove a denumerbale set, how to prove an uncountable set.  That was a big part of the last two sections of material that we covered, and I think that a lot of what we will see on the exam will build off of this.

What kinds of questions do I expect to see on the exam?
The same kinds that there always are - for the multiple choice, I believe there will be questions that deal with the definitions of the terms that we are given.  I can see that definitely happening, as it has happened on the last two tests.

And for the other free response questions, I can see them asking us to solve example problems, similar to the ones from the homework and the book.

What do I have a question about?
I would like to review the division algorithm, that was what gave me a lot of trouble.

Particularly problems from section 11.2  Any of those problems would be great to review to help me nail down my understanding of the Division Algorithm.

Sections 11.5-11.6, Due on Friday March 28th

Section 11.5 is about relatively prime integers.  
Two integers a and b, not both 0, are called relatively prime if gcd(a,b)=1.

Section 11.6 is about The fundamental theorem of arithmetic.
Every integer > 2 is either prime or can be expressed as a product of primes; that is n=p1p2...pm  , where p1,p2,...pm are primes.  Furthermore, this factorization is unique except possibly for the order in which the factors occur.

The difficulty for this sections is that there are so many new theorems and corollaries to remember.  That is difficult to keep them all straight and to not get confused about them.  Reading through them is okay, but I fear in my ability to be able to use them effectively later on.  I will need more practice with both of these sections in order to be efficient.  Sometimes it is difficult to read over the proofs, and I feel like that happened to me in this chapter as well...

What I enjoyed about these sections is the new material.  It is neat learning this stuff and seeing how it can be used in such simple ways to prove such complex things.  Maybe it is not that way actually, but it sure feels like that to me.  I really enjoyed reading over corollary 11.18 which says Every integer exceeding 1 has a prime factor.  I know this to be true, but it's cool to see it proved later on with the lemma and the following proof.
-nap

Sections 11.3-11.4, Due on Wed Mar 26

Here I will analyze and give my thoughts on Sections 11.3 and 11.4 of Mathematical Proofs: A transition to advanced mathematics.

11.3 - Greatest Common Divisor
An integer c \ne 0 is a common divisor of two integers a and b if c | a and c | b.

The greatest common divisor of two integers a and b, not both 0, is the greatest positive integer that is a common divisor of a and b.

The whole concept of common divisor and greatest common divisor is something that is not too difficult for me to understand.  I remember learning this as a younger child.  It makes sense - to find the biggest number that nicely divides into two numbers.  That is the GCD.  I kind of like the thought of this.

I believe I will have difficult in proving this - at first.  I can see myself learning and understanding this better.

Here are a couple theorems from the section (typing them up helps me remember them):

• Let a and b be integers that are not both 0.  Then gcd (a,b) is the least positive integer that is a linear combination of a and b.  

• Let a and b be two integers, not both 0.  Then d = gcd(a,b) IFF d  is that positive integer which satisfies the following two conditions:
• 1) d is a common divisor of a and b;
• 2) if c is any common divisor of a and b, then c | d.

I enjoy learning about GCD.  It kind of makes sense to me, which would be why I like it.  I enjoy taking a look at linear combinations - those have always made sense to me.

11.4 The Euclidean Algorithm
This is the algorithm for determining d=gcd(a,b).  It "makes use of repeated applications of the Division Algorithm and the following:

• Let a and b be positive integers.  If b = aq + r  for some integers q and r, then gcd(a,b) = gcd(r,a).

This was difficult to understand at first, but I can see the usefulness of it and the thought process of it by looking at the example.  For example, we want to find the following:  d = gcd(374,946).

We go down the line using the Euclidean Algorithm, and first divide 946 by 374.  374 goes into 946 2 times with a remainder of 198.  We then divide 374 by 198.  This goes in once with a remainder of 176.  We then divide 198 by 176.  This gives us a remainder of 22.  We then divide 176 by 22, which goes in 8 times.  Thus, gcd(374,976) = 22.

I find this whole process fascinating and I look forward to learning more about it.  I did have trouble understanding the last example in the book.  With more practice, I should be able to understand more.

-nap

Sections 11.1 and 11.2, Due on Monday March 24th

What was difficult for me?
I understood pretty well the divisibility property of integers.  I understood what prime and composite numbers are.  That part wasn't too difficult for me.

What was difficult to understand for me (but will become easier with more practice) is the division algorithm.

The Division Algorithm: For positive integers a and b, there exist unique integers q and r such that b= aq + r and 0 < r < a.

I will need to practice with this chapter to understand it.  I look forward to that.

What was interesting to me?
I remember learning to divide for the first time as a little kid.  I think I was in 3rd grade.  It's really cool to now look at the division algorithm and put together how it makes sense.  I enjoyed reading through this section, and even though there were parts that were difficult for me to understand, I enjoyed the section overall.  I like learning the deeper meaning of simple things that I once learned.  It's really neat.

Section 10.5, due on Friday March 21

This blog will be on the second half of 10.5 - the Schroder-Bernstein Theorem.

This is a very short section.  We will talk about the Axiom of Choice.

Theorem: The sets P(N) and R are numerically equivalent.

Corollary: The sets 2^N and R are numerically equivalent.

What was difficult for me?
This section was not difficult for me.  It was rather simple after the discussion in class yesterday.  The Axiom of choice states For every collection of pairwise disjoint nonempty sets, there exists at least one set that contains exactly one element of each of these nonempty sets.

I enjoy this section, and it is not too difficult for me to understand the theorems.  The only thing that is difficult for me is the proving of the theorems.  And since this is a proofs class, I need to focus more on this.  I enjoy knowing the proofs and seeing how they work - it is the thinking through them in the moment that is difficult for me.  I look forward to learning more about how to prove the theorem and corollary above.

What did I enjoy most?
I enjoy counting the cardinality of sets. While at time it is counterintuitive (an infinity can be bigger than another infinity, but they are still numerically equivalent), it does make a lot of sense and is quite fun to work with.  I enjoy showing that there are one-to-one functions and the difference in numerical equivalence.  I find this interesting.

Section 10.5 up to Theorem 10.18, Due on Wednesday March 19th

The Schröder-Bernstein Theorem

What was difficult for me?  Nearly everything.  Well, at first.

Then I took a step back and really thought about what this theorem was saying.  This allowed me to clear my thoughts and think about what this was really asking.  It's really simple.  If the cardinality of set A is less than or equal to the cardinality of set B, and the cardinality of set B is less than or equal to the cardinality of set A, then the cardinality of set A must be equal to the cardinality of set B.  That's the only thing that makes sense.  You can't have two sets that are BOTH less than each other.  This theorem actually makes a lot of sense when you sit back and think.

The most difficult part of this reading was the reading leading up to the various theorems (lemma 10.16, theorem 10.17).  It was difficult to read through these.  But, it all started making sense once I took a step back.  

What was the most interesting for me?  I like the simplicity of this theorem.

There is a part of the reading, Theorem A, that says "For every two sets A and B, exactly one of the following occurs: (1) |A|=|B|, (2) |A| < |B|, (3) |A|>|B|."  This contains a lot of truth.  When comparing two things, either one is greater and one is less, or they are equal.  There is no disputing this.  This is always true, even though we might not have the information to prove that this is true, meaning, we aren't sure about the characteristics of the sets.  When we know about the sets, we can tell some important things about them - what is greater, lesser, or if they're equal.  It will now be interesting to prove this.

Section 10.4, Due on Monday March 17th

Section 10.4 - Comparing Cardinalities of Sets

Two nonempty sets A and B have the same cardinality if there exists a bijective function f : A -> B.

Theorems:

• For every nonempty set A, the sets P(A) and 2^A are numerically equivalent.
• If A is a set, then |A| < |P(A)|.

What was difficult for me?

I learned the definition of the cardinality of a set earlier this semester.  It was a foreign concept to me, but know I understand it.  The definition, for those of you that don't know, is below.

The understanding of the cardinality is not too difficult, but the proving of the difference of cardinalities of sets has been difficult for me to understand.  I understand that you have to show a bijective function, which is simple enough.  It's the in between steps (explaining, next steps, etc.) that are tough for me.  Those are the parts that I mess up on and I will need more practice before I master these concepts.

What did I find enjoyable?

I like the enjoyable nature of cardinality - it's counting!  I like counting.  When I was a kid I would count things.  

I would take 1+1=2. 

Then 2+2=4.  

Then 4+4=8.  

8+8=16.  

16+16=32.  

32+32=64.  

64+64=128.  

128+128=256.  

256+256=512.  

512+512=1024.  

1024+1024=2048.

I JUST DID THAT FROM MEMORY. That's what I find interesting about cardinality and counting.  Now it will be fascinating to learn more about comparing cardinality.  It can't be that much harder than counting, right?

And, I also think it's interesting that cardinality and cardinals have nothing to do with each other.

Section 10.3, due on Friday March 14th

What did I find difficult?
The concept of uncountable sets was difficult for me to understand at first.  When I began reading the chapter, I thought I felt good about it all.  I thought "you know, I understand what an infinity set is.  When you have an open interval, you can get infinitely closer to the number at the end of the interval without ever reaching it.  But, when I started getting to the proofs, things got more difficult for me.  I understood that an open interval is infinite, but I was struggling with the proofs of it all.

I will need more experience with solving proofs, and will need to hear something think out loud when solving a proof showing that a set is uncountable.  With that, I should be better at understanding this material.

What did I find interesting?
The whole idea of uncountable sets is fascinating.  My whole life I've been dealing with finite sets, things I can wrap my head around.  Now, things are getting a lot bigger a lot faster than I ever thought possible.  It's exciting.  Reading through the proofs are really cool.  Proving that something is infinite is a way cool thing to think about.  I just lack the faith that I can do it myself right now.

It's an interesting to think about - infinity.  Something so big that it never ends.  It keeps getting bigger and bigger and bigger or smaller and smaller and smaller.  The other day, I let a balloon go, and it floated up and up and up and up.  It got smaller and smaller and smaller.  I stood riveted, watching it float away.  I loved watching it.  It got to the point where I feared blinking would prohibit me from finding it again.  It kept getting smaller and smaller before it disappeared behind buildings.  This is what infinitely smaller is for me.  I'm excited to be able to prove that.

Section 10.2, Due on March 12

10.2 Denumerable Sets

Definition - a set A is called denumerable if |A| = |N|, that is, if A has the same cardinality as the set of natural numbers.

A set is countable if it is either finite or denumerable.  Countably infinite sets are then precisely the denumerable sets.

A set that is not countable is called uncountable.  An uncountable set is necessarily infinite.

What was difficult for me?
The whole concept of countably infinite was difficult for me.  I always thought that being infinite meant that you couldn't count it.  But I guess it is.  The best example that I understand from it is the sands in the sea or the stars in the sky - there is a limited amount of them.  Understanding this concept this way made a lot more sense.

Reading through some of the theorems showed that this section will just be understanding what certain words mean and when to use them.  With practice, all will be well.

What did I find interesting?
I love this way of thinking.  It's really neat to think like a mathematician - to think of thinks as countable, countably infinity, or uncountable.  I also had never heard of the word denumerable before.  I'm excited to start using that in my daily vocabulary.  I found it interesting to look at the figures they used in the book to see how to classify groups of objects (sets).  I'm just fascinated by this way of thinking.

Section 10.1, Due on Monday Mar 10

Numerically Equivalent Sets
Two sets A and B (finite or infinite) are said to have the same cardinality, written |A| =|B|, if either A and B are both empty or there is a bijective function f from A to B.  Two sets having the same cardinality are also referred to as numerically equivalent sets.

What did I find difficult?
You know, the one theorem 10.1 was difficult for me.  I had a tough time understanding how numerically equivalent sets played into the theorem.  The definition of numerically equivalent sets seems pretty easy to me, but I'm not sure if I'm actually understanding what these sets are.  I believe I will need more practice and explanation on what these sets are.

What did I find interesting?
I love the idea of cardinality.  Google defines cardinality as "the number of elements in a set or other grouping."  I think it's a natural thing to do - to count the number of elements in a group.  It's a very simple thing to do, but it shows an important property of a group.  The number of elements tells you a lot about what you can do with that set or group.  I'm excited to learn more about this.

Due Friday March 7th

• Which topics and theorems do you think are the most important out of those we have studied?
• From what we have studied, I would say that the basic proofs of direct, contrapositive, and contradictory would be the most important.  I feel that we still see these kinds of proofs often, no matter what kind of proof we are doing.  I feel like I have a solid grasp on these concepts, but will need to be doing some reviewing before I feel solid.

• What kinds of questions do you expect to see on the exam?
• I expect there to be the same kinds of questions on this exam that there were on the previous exam.  I felt like I was adequately prepared for the last exam, but did not perform as well as I wanted to.  I must've been taken by surprise by the material of the test - even though it was very fair.  I believe that if I am to study the material, and go over the homework, I will do well on this exam.

• 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.
• For the exam, I believe the questions involving modulo n are going to be the most difficult for me.  I think back on the homework assignments that we had that involved modulo n, and those were the most difficult for me.  What I would like to review are the different methods and ways to think through solving a modulo n problem.

Due March 4th, Chris Danforth

When I first heard Chris Danforth speak, the first thing I did was go and follow him on Twitter.  Anyone of his brilliance is sure to continue producing quality work, and I look forward to following him and his work.  https://twitter.com/ChrisDanforth

I found the topic fascinating.  He wants to cycle through data and be able to track how happy/sad the people are feeling.  It was so interesting to see the process which they thought about solving the problem... The data they wanted to use consisted of song lyrics, Google books (books that have been scanned and read into Google) and tweets.  This was fascinating to see.  They then went through and had to set the dial on what were happy words, what were sad words.  I found this entire process to be fascinating.  They used a service called Mechanical Turk on Amazon, where they paid people to go through and rank words.  This was fascinating to me.  He kept using the term "set the dial" on the experiment, meaning he wanted to come up with a robust way to classify the data.

Once the data was collected, he got rid of the noise (the,and,of,it) in order to amp the signal!  The signal and the noise.  Great book by Nate Silver I believe.

Overall, I thought the analysis and the computational techniques were incredible.  And plus Chris Danforth was such a fascinating individual.  Going to this talk made me want to learn more about scraping data - and it made me want to start doing cool research and investigate things that interest me.

Section 9.6-9.7, due on March 5th

What did I find difficult?
Chapter 9.6 is about inverse functions.  I had more difficulty with this section that I did with 9.7.  Prior to reading this chapter, I did understand that the inverse of a point is just switching the x and y term.  For example, if you have (x,y), then the inverse relation of that would be (y,x).  Reading through the proofs led to difficulties in understanding how to go about proving the inverse relations.  I will need more practice in solving these problems.

Chapter 9.7 is about permutations.  We are learning about permutation testing in my Nonparametric Statistical Methods class.  I find them very interesting.  From the book, a permutation of (or on) a nonempty set A is a bijective function on A, that is, a function from A to A that is both one-to-one and onto.  For 9.7, I was confsued by the illustrations of 213 about the composition of any two permutations.  I thought I understood that, but the visual given with the matrices was difficult for me to understand.

What did I find interesting?
Like I said before, we are learning about permutation tests in one of my stats classes ( 435 nonparametric with Dr. Grimshaw).  From an earlier probability class I have taken at BYU I have learned how to count the number of permutations.  I like the drawings and visual explanations that they used in the book on page 212 to show the ways to represent the permutations of {1,2,3} in a 2x3 matrix.  

Section 9.5, Due on Monday March 3

What do I find difficult?
When I looked at the visual for the composite function of g o f in Figure 9.2, I got really nervous.  Holy smokes that looks difficult.  At first glance, it looks like a composite function is a function within a function.  Or a combination of functions.  If you run one function, the output of that function ends up being the input of another function.  That's what I understand it to be.  If it is really that simple, then I think I got it.  But if somehow the functions interact and relate other than that, I'm going to be confused.
With associative functions, h o (g o f) and (h o g) o f are associative if these two functions h o (g o f) and (h o g) o f.  This seems to be pretty straight forward.  I look forward to learning more about this.


What was interesting to me?
I've always found linear combinations interesting. When there is a combination of functions, I think it provides for interesting opportunities and situations.  The composition g o f, defined by (g o f)(a)=g(f(a)) for all a \in A.  I think this is a very basic concept, and I look forward to learning more about it.  I think we can do some interesting things with it, in finding different functions and looking at the interesting way that things interact.  I think we're going to find a fascinating way that functions interact, and I'm excited to learn more about that.

Viva Espana.