Section 9.3-9.4, Due on February 28th

I'm posting this blog early because I am going to be out of town over the weekend.

What was difficult for me?
A function f from a set A to a set B is called one-to-one or injective. if every two distinct element of A have distinct images in B.

One to one was tough to understand at first, but then I made sense of it!  I understand it that there are unique values for the response, that you can't plug in a 1 and get a 2, and then plug in a 4 and get a 2.  That would make the function not one-to-one.

A function A -> B is called onto or surjective if every element of the codomain B is the image of some element of A.  This was difficult for me to understand.  I will need more help on understanding this.

Bijective or a one-to-one correspondence functions are both one-to-one and onto.  This seems alright to understand, once I understand onto.  The book says, again, "if every element of the codomain B is the image of some element of A."

This website seems to help me out a lot..... http://www.regentsprep.org/Regents/math/algtrig/ATP5/OntoFunctions.htm

What did I find interesting?
I took linear algebra at one point, and I learned all of these things for the first time.  I didn't really understand it at that time. Now I'm beginning to understand it.  Onto and one-to-one are really interesting ways to think about functions.  Learning about all of these different ways to classify functions are helping me understand more about functions.  That's what I find really interesting - finding different ways to understand functions.  I thought I knew a lot about functions, but I didn't really.  Learning about these different types of functions hopefully really help me.

Section 9.1-9.2, Due on February 26th

What was difficult for me?
It was difficult to go back to and understand the implicit meaning of a function.  I've been in so many math classes in my life that I thought I knew pretty well what a function was.  I've always understood it to be kind of like a machine - you put something in and you get something out.  You have an x, you put it in, and you get a y out.  I wasn't sure of this "domain," "codomain," "image," "mapping" vocabulary.  I think this is the most difficult part for me - the introduction to a new vocabulary that deals with something I feel like I have a pretty good grasp on.

What did I find interesting?
I find section 9.2 "The Set of All Functions from A to B" interesting.  It's interesting to look at the set of all functions from A to B by B^(A).  It says in the book that this is a peculiar notion, it is quite logical.  This is the truth.  You take the number of elements in B, and raise it to the number of the elements in A.  This makes a lot of sense.  I'm excited to see how this will play into proofs later on.

Section 8.6, Due on February 24th

The Integers Modulo n.

What was difficult for me?

This whole section was difficult for me to understand.  Most of the time that I read the book, I can understand what is going on.  But this time, I was very lost.  On page 190, I'm confused about residue classes.

I do understand a little bit the "closed under addition" and the "closed under multiplication" parts.  This part seems intuitive to me.  However, I have difficult with residue classes and well-defined.  I am trying to understand these things, and while the reading helps, there is no way that I could prove them.  I hope that with the homework I can improve.  I read the last proof on page 191-192 and it was difficult for me to understand.

What did I find interesting?
First,  I really think the concept of an integer is interesting.  An integer is a whole number - not a fraction, or a part of a whole number, but a whole number.  I think it's really interesting that you can divide a number by another number and get a whole number.  The whole congruence modulo n thing is really interesting .  I really like the closed under addition part of this chapter, as it illustrates an interesting way to think about addition.  The same goes for multiplication.  Equivalence classes are also very interesting - I like to think of them as "solutions for the problem," meaning that these classes help keep the congruence modulo true.  I hope to be able to learn more about these and apply the.

Rod Forcade : Material Lattices

There were black dots.  Then there were green dots.  Then there were blue dots.  It seems that these lattices consist of lines drawn through pegs in a board.

Prior to coming to this lecture, I had no idea what lattices were.  And now, after the lecture, I'm still a little lost.  I do understand that they are like pegs in a board.  They can span a vector space.  They can be used to describe shapes.

But now that I'm here, I'm seeing that this is like linear algebra.  From linear algebra, I remember matrices and the importance of basis matrices.  I think that lattices and matrices are somewhat similar?

In linear algebra, it was very difficult for me to grasp the concept of matrices, with determinants, spanning, etc.  Being back in this lecture brings me back to those days.  It's interesting to recognize and remember those things that I once learned.

I do like the principle of linear combinations.  That seems to make a lot of sense to me.  A lot of this other stuff is over my head, but I know what that is, I'm comfortable with it, I like it.

All and all, these lattices are interesting.  There seemed to be a good turnout at the talk, and the cookies and brownies were delicious. 

Section 8.5, Due on Friday Feb 21st

What did I find difficult?
Congruence Modulo n.  a is said to be congruent to b modulo n, written a = b (mod n) if n | (a-b).
Division Algorithm.
This was a difficult chapter to understand.  Congruence Modulo n.  I understand the attributes of equivalence relations pretty well, but it is difficult for me to see how these congruence modulo n's work with reflexive, symmetric, transitive.  Reading through the example seems simple to me, but I'm not sure how I would be able to do this on my own.  I do look forward to trying this though.

I am also confused in how we should go about defining the distinct equivalence classes.  I realize this is from a few chapters back but I am confused at how this is to work with congruence modulos.  I guess the only way to learn is by doing.

What did I find interesting?
I find it interesting reading through these proofs, especially since they are congruence modulos.  I am still getting to know these congruence modulos, understanding what they're about and how their equivalence classes can be proved.  Walking through these step by step seems to make sense.  I only wonder how I am going to deal when the homework comes around.  I think I find this congruence proofs with equivalence classes interesting because it's a new concept.  I hope to learn more about it and that it can teach me how to think.  I hope to see that this is important in programming - I feel that so much of programming is math.

Sections 8.3 and 8.4, due on February 19th

What did I find difficult?

A relation R on a set A is called an equivalence relation if R is reflexive, symmetric, and transitive.

So, if a relation has all three of these properties, then it is an equivalence relation.

An equivalence class is the class of all members of a set that are in a given equivalence relation.

I found equivalence classes to be relatively simple to understand.  Reading through the example problems did not give me much trouble.  I imagine that as we get more into this, I will come across more difficult applications of equivalence relations.

On the other hand, reading through the properties of equivalence classes was difficult for me.  I understand what an equivalence class is (at least I think I do), but in proving different properties of equivalence classes, I'm having trouble.

What did I find interesting?

I find the concept of equivalence classes interesting.  I'm still having trouble understanding 8.4, so I'm going to focus on 8.3.  It seems to be intuitive that if R is reflexive, symmetric, and transitive then it is an equivalence relation.  For the equivalence class, I like this [a] = {x \in A : x R a}  My understanding is that this consists of all elements that are related to a.  

It made sense when it said, loosely speaking, that [a] consists of the relatives of a.  It also makes sense that the equivalence classes form a partition of the set.

I feel that this is slightly different from other chapters, but I like the way of thinking.  I can see how this will be helpful later on, maybe in proving parts of a statement in order to prove the whole thing true.

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.