Sections 8.1 and 8.2, due on February 18th

It's currently 62 degrees in Newport Beach.  For some reason, I find myself in Fashion Island Mall with my computer and my math book, surrounded by palm trees, wealth, and some kids next to me playing games.  What a random, awesome world.

Anyways, sections 8.1 and 8.2

Let's talk about Relations and Properties of Relations.  And just so you know, I'm not talking about relations of people.  If you want to get my thoughts on stuff that's more than just math, go to spanishnap.blogspot.com

What Did I Find Difficult

For this reading, I only found difficult the introduction to the concept of relations.  Particularly, I found the following concepts difficult at first, only because they were so new:

• Let A and B be two sets.  By a relation R from A to B we mean a subset A x B.  

• That is, R is a set of ordered pairs, where the first coordinate of the pair belongs to A and the second coordinate belongs to B. 

• If (a.b) \in R, then we say that a is related to b by R and write a R b.  

I believe that with time and with further learning of these concepts I will come to understand them better.  On page 176, it says that "Although this may seem like a fairly simple idea, it is very important that we have a thorough understanding of it."  I did not really understand what it was saying, but after reading that, I went back and tried to understand it more.  I guess it was difficult for me because it is a new way about thinking of ordered pairs.  

What Did I Find Interesting

What I found interesting about this reading was that which I also found difficult - the thinking of ordered pairs in new ways.  It's neat to think about how 

• "A relation R defined on a set A is called reflexive if x R x for every x \in A."  
• "A relation R defined on a set A is called transitive if whenever x R y and y R z, then x R z, for all x, y, z \in A." 
• The distance between two real numbers a and b is |a-b|.

I can see how these ways of thinking about cartesian products and ordered pairs will be helpful in the future.  I look forward to learning more about them and doing more with them.

Section 6.4, due on February 14th

Section 6.4 The Strong Principle of Mathematical Induction!

What was difficult for me?
The Strong Principle of Mathematical Induction - for each positive integer n, let P(n) be a statement.  If

1. P(1) is true and
2. the implication "If P(i) for every integer i with 1 < i < k, then P(k+1)" is true for every positive integer k, 

then P(n) is true for every positive integer n.

In a recursively defined sequence {An}, only the first term or perhaps the first few terms are defined specifically, say a1, a2, ...., ak for some fixed k \in \textbf{N}.

I don't really understand the definition of a recursive sequence.  The definition in the book is difficult to understand.  I look forward to learning more about this in class tomorrow and understanding how it works.

At first, I didn't understand what the difference between the Strong Principle of Mathematical Induction.  Then I looked online, and went to a website math.stackexchange.com and I found this:

"The only difference between regular induction and strong induction is that in strong induction you assume that every number up to k satisfies the condition that you wish to prove whereas in regular induction you only assume that some integer k satisfies this condition."

Based on that explanation, that is what I understand the Strong Principle to be.  It picks an m and a k, and proves that if m is true, then all values leading up to k are true.  Interesting stuff.

What did I find interesting?
You know, at first thought I really didn't think there was going to be too much difference between the Principle of Mathematical Induction and the Strong Principle of Mathematical Induction.  But after some analysis and thinking through things, I realize that there indeed is a difference.  With normal PMI, you prove a point is true and then prove that the point directly after it is true as well.  With Strong PMI, you prove that a point is true, and then prove that ALL points leading up to another point k are true.

I can see how that would be helpful.  Instead of proving the Domino method with just two dominoes, you can prove that if there is a domino is one spot, and then that there is another domino in another spot, you can prove that all the dominos between your two original dominos will fall.  Now that is cool stuff.

Section 6.2, due on February 12th

Section 6.2 A More General Principle of Mathematical Induction

What did I find difficult?
The Principle of Mathematical Induction
For a fixed integer m, let S = {i in Z: i > m}  For each integer n in S, let P(n) be a statement.  If

1) P(m) is true and
2) the implication   If P(k) then P(k+1) is true for every integer k in S,

then P(n) is true for every integer n in S.

I'm beginning to understand induction, but I'm still having a tough time with it - mainly with the applications.  It makes sense to me to prove that the smallest one is true, and then to prove the "domino effect" is true.  I understand the concept - if one is true, then the next one is true - but it's hard for me to prove that in different cases.

What did I find interesting?
I really like the thought of induction - I can see how it is very useful.  In looking through the results and their proofs, I can see the genius in applying the induction thought process.  I'm excited to continue learning about it.

I really like the simple proof in result 6.9.

"For every nonnegative integer n, 2^n > n.  We prove that the inequality holds for n=0 since 2^0 > 0.  Assume that 2^k > k, where k is a nonnegative integer.  We show that 2^(k+1) > k+1.  When k=0, we have 2^(k+1) = 2 >1 = k + 1.  We therefore assume that k > k + 1.  

By the Principle of Mathematical Induction, 2^n > n for every nonnegative integer n."

I can see how the simple inductions work.  I am nervous to see how I do with the more difficult inductions.  But I look forward to it.

Section 6.1, due on February 10

The principle of mathematical induction.  

What did I find difficult?

A number m in A is callde a least element (or a minimum or smallest element).

I've heard of induction in my past math classes - but I've never really understood the principle.  I cannot quantify how many points I've lost on proofs where the principle of induction could be used to show the proof to be true.  I was a little intimated with learning induction, due to my failure to understand it in the past.  That was what I found most difficult - overcoming my past fears of not knowing induction.  Now, this is what I understand induction to be:

For each positive integer n, let P(n) be a statement.  If 

1) P(1) is true and
2) the implication

If P(k), then P(k+1).

is true for every positive integer k, then P(n) is true for every positive integer n.

I understand a little more about what induction means.  I believe with more practice, I will understand it more.

What did I find interesting?

I did some more research on the principle of induction.  This is what I came up with - 

1. Show that it is true for the first one.

2. Show that if any one is true, then the next one is true.

It's interesting - every website I look on to learn more about the mathematical principle of induction teaches the principle using dominoes.  The domino effect is this:

1. the first domino falls.

2. if any domino falls, the next domino will fall

(thanks http://www.mathsisfun.com/algebra/mathematical-induction.html)

This makes more sense!  If the first domino falls, it will cause more dominoes to fall.  Eventually all dominoes will fall.  Understanding this makes the principle of induction hit home.  I mean, what kid hasn't built a line of dominoes, only to knock them down? If I can knock down the first one, then the next one will fall, and the next one and the next one and the next one.  Eventually, all of them will fall!

PreExam Questions, Due on February 7

• Which topics and theorems do you think are the most important out of those we have studied?

As I prepare for this test, I have a couple different feelings about what is most important.  I really enjoy tangible problems - examples that I can see worked out.  I really enjoy direct proofs, contrapositive proofs, contradiction proofs, counterexamples, etc.  I think each of those are truly important.  But I believe the most important thing that we have studied is the syntax for writing sentences.  I think writing sentences using quantifiers (both existential and universal) and understanding how to negate these - that is the most important topic we have covered.  I think this because with a solid understanding of how to take a result that you need to prove and change it into a sentence with P, Q, R, etc. then you can more effectively see and apply other proofs.

• What kinds of questions do you expect to see on the exam?

I expect to see problems that are similar in style, difficulty, and content as to those that we had on the homework.  I think there will be more applications of proofs (direct, contrapositive, contradiction, existence, etc.) because those proofs show that you know how to do a multitude of things.

• 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.

I think before the exam I would just like to go through a handful of problems and talk about strategies for noting how to recognize which proof to use in different scenarios.  I plan on going through all of the homework problems, recognizing patterns of how to go about proving different results.  I feel good about many of the definitions and notation.  I just want to make sure I know how to recognize results, and then how I can plan on proving them.

Sections 5.4-5.5, Due on February 5th

What did I find difficult?
I originally had difficult understanding existence proofs.  The reason for this?  The extent of my proving abilities consisted mainly of contradiction, direct proofs, contrapositive.  You could say that learning existence proofs was like doing a new exercise - it was weird working through them and understanding it, but once I got it, I liked it.  After reading, I think existence proofs are pretty self explanatory.  My understanding of them is that I just have to prove one case - that such a case exists.  I like the story about the mathematician David Hilbert:

"There is at least one student in this class... let us name him 'X'... for whom the following statement is true: No other student in the class has more hairs on his head than X.  Which student is it?  That we shall never know; but of his existence we can be absolutely certain."

Before reading the example, I was strictly going off of the definition of existence proofs.  But this example helps me understand that we just need to prove the existence of something.  I really like it.

What did I find interesting?
Disproving existence statements.  I like the existence way of going about, solving a proof.  I then find it fascinating that you can disprove these existence statement.  I am fascinated with the way set notation works, and how the negation of an existential quantifier turns that quantifier into a universal quantifier.  I really like this way of solving a problem, and proving whether something exists or whether it doesn't exist.  I like that I am able to apply what I learned earlier on in the course and use it in such a way that it makes me understand what is going on.

Sections 5.2-5.3, Due on February 3rd

What was difficult for me?
Proof by contradiction was difficult for me.  C : P ^ (~P) ,    (~R) => C

~R => C is true and C is false, then ~R is false and so R is true.

The syntax and the way of thinking is difficult for me.  It's just a new way to go about proving something.  It's fascinating to learn these different concepts of solving proofs.

However, this one is difficult for me to grasp.  Even reading through the practice problems, I struggled to really grasp what is going on.  I can see the parts and kind of how they fit together, but it would be difficult for me to understand - at least right now.  With practice and with more practice problems I hope to be able to grasp it.

Contradiction-a combination of statements, ideas, or features of a situation that are opposed to one another 

What did I find interesting?
In section 5.3, I really like being able to review the different techniques that I've been introduced to: direct proof, proof by contrapositive, proof by contradiction.  As I think back on what I read, and the homework that I've already done with direct proofs and contrapositive, I can see real progression in my learning of these proofs.

I also find the idea of a contradiction really interesting.  I remember being a kid and learning the definition of a contradiction.  When someone would say something that went against what they originally said or meant, I would smartly reply, "You are contradicting yourself."  I intuitively knew what contradiction was, but now I see what it truly is and how it can be powerful in proofs and in logical thinking.  

I think the thing that I really find interesting about these three different proofs is that you start a proof with the end goal in mind: you know what you want to prove.  There is a table on page 117 that I really like.  It talks about the first step of a proof, and then with remarks/goal.  I think that if I really learn the ins and outs of this table, I will be better able to go through proofs and really solve them better.

I really look forward to that.