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.