study guide /due February 1

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

proof of a ring, and integral domain, and field (differences and similarities

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

prove that something is a subring, prove that something is a field, prove that something is an integral domain, show something is an isomorphism and show something is a homomorphism

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.

show the differences and similarities between a field and a field of quotients.

9.4 Due January 29

Difficult)
Lets go with all of it.  Notation, reasoning and logic of a field of quotients...deffinitely struggled with that idea.  Also realized that I need to read a lot more proofs relating to integral domain and fields to become confident on the subject.  I tried doing the homework early, and got stumped on a few.


Exciting/Relational)
I see how this crazy notation is useful in describing a field of quotients, but I'm so stumped right now on the concept that I fail entirely to understand or apply the reasoning.  I appreciated the comment that thenotation is useful in helping avoid the assumption that everything works as we are used to it working...because the proof was very tiresome.

(reflect) Due January 27

How long have you spent on the homework assignments?  about 1-1.5 hours on average

Did lecture and the reading prepare you for them?  YES!
What has contributed most to your learning in this class thus far?  Trying to understand everything before coming to class
What do you think would help you learn more effectively or make the class better for you? (This can be feedback for me, or goals for yourself.)
I value reading other people's proofs the most, and seeing the approach to writing the proof and the thought process.  I learn best with examples of proofs, and trying to reason why each step was important.  I really appreciate the detailed explanations of proofs in class and reading through extra examples outside of class!!!

Thankyou!

This is my favorite math class thus far!

Focus on Math Extra Credit

A mapping is Surjective if and only if the codomain is equal to the range.

A mapping is Injective if every distinct element of the domain leads to or gives a distinct element in the co-domain.

 A bijective function is a mapping that contains both properties of injective (one-to-one) and surjective (onto).

These properties build to form other identities and tools for graphing in computer programming, and image definitions in computers.  Last semester I was able to write a python program that opened a file of a picture of text, and converted the picture into a character on the computer using the mapping relationship properties and linear algebra to define vectors and matrices.  By taking a linear transformation on the data sets, and isolating the eigenvalues of the matrix which represents the image, I was able to generate code that enabled the image to be captured as text.

I am working on developing an application that will allow you to take a picture of grocery receipts and automatically balance your checkbook.  I plan to convert this application into a tracking measure to also tabulate the rounding error that grocery stores take on individual transactions, and use it to ask for a reimbursement from stores that I frequent over the year!

I see how valuable the properties of injective, and surjective mappings are as a foundation of a lot of different mathematical tools!  I hope to continue to learn about these principles and grow in understanding so that I can continue to explore the possibilities of application of math in my daily life!

3.3 Due January 25

Difficult)
Why is it essential that for theorem 3.12 that for parts 4 and 5 that R ( or the domain of the mapping) is a ring with Identity and that f (the mapping) is surjective?  I know that a ring with identity is a ring that has a multiplicative identity such that 1a=a=a1 for all a in R, and that surjective means that for every element in the co-domain, there exists at least one element in the domain that maps to it.

For us to be using these to show that the mapping of the domain's multiplicative inverse is defined as the multiplicative inverse of the codomain we cannot show this exists without the mapping being surjective...this idea seems really straightforward now that I put it into words, but initially it was stumping me why it was necessary for the mapping to be surjective.

Awe-Inspiring)

It intrigues me how injective and surjective mappings are so essential to mathematical deffinitions.  In differentiation, and rings, and sets, and topology...and probably more, but i've used them in all of those classes fo far.  It makes me think that would be a good research field.  It seems these two things are diverse and all encompassing...the only distinction between isomorphism and homeomorphism.

I took and ACME class that pretended to teach these things, but they make so much more sense in the context of rings as an introduction that they do as a brief section in a tolology chapter.  I really enjoy the way this book introduces an idea, and then repeats it in different examples and un-examples.  This may very well be my favorite math book.

3.2 Due January 22

Difficult)
I'm struggling to see the value and usefulness in not having a zero divisor, or similarly in having an integral domain.  I need to review and re-apply the understanding of the integral domain so that I understand it's application better.
After reviewing a lot of different websites, I'm still stumped...I was getting caugth up on the definition of the O and 1...no not binary, but the additive identity O and the multiplicative identity 1
The integral domain is dependant on the existance of both and that they are not the same.  There are some rings for which the additive and the multiplicative are the same.  The integral domain is a distinction that forces these to be diferent, or that 1 does not equal 0.


Inspiring/Relational idea)
It looks like a unit is a field that doesn't commute. but that there is also a unique solution to au=1 instead of simply having a solution for any a in the ring.  The unique solution is a very powerful tool in defining the tool if subtraction,  Very useful way to manipulate rings that are not commutitive!!!

3.1 (part 2) Due January 20

Difficult)
Theorem 3.1 along with the example preceding it are a new concept to me.  The context of a field or a ring made out of a cartesian product is confusing to me and I need a lot more explanation for it to make sense...
Are we doing the cross product?  FROM WIKIPEDIA I copied the definition.  Would have been nice in the book.
That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Interesting or question about the reading)
I am curious.  The reals are a ring, and also a field, and the quotient numbers are a subring of the reals, and also a subfield.  Where the integers are a subring of the quotient numbers, proving a subring is actually easier than proving the superset is a ring.  We only need to show closure under addition and multiplication, the additive identity, and the additive inverse.

3.1 Due January 15

(Difficult)
The abstract idea of a ring is going to take some getting used to...
The 1R (ring with identity) was a difficult concept to understand and the examples were very helpful.  I also needed the un-example, and I think a few more un-examples would be a helpful addition to the book.  I searched around to try to get a better understanding.

Interesting)
Modular arithmetic and equivalent classes are a very powerful lead-in to the ring identities.  I tried reading about rings before semester started and was confused, but after reminding myself about equivalence classes, and primality I see that it was a great way to set up for introducing the concept of a ring in the definition of systems in mathematics.

2.3 Due January 13

Difficult concept:
theorem 2.8 had a challenging proof concept map.  I spent 15 minutes working through the proof in order to catch onto the steps in the proof.  I really needed the illustration techniques explained in order to better write and formulate a proof.


Reflective:
the set of all congruence classes modulo p (where p is prime) is a powerful statement and I can see a lot of application for applying this to number theory.
I spent some time to memorize the theorem 2.8
let p > 0 then p is prime => for each a!=0 in Zp, ax = 1 has a solution in Zp =>
Whenever ab = 0 in Zp then a=0 or b=0 => that p is prime.
it can also be states that each of the three points of the theorem are equivalent for  p > 0.

2.2 Due January 11

I think the idea of addition and multiplication on congruency sets is fascinating and powerful, but I would like to see some more examples on operations and verify for myself how this works and if it defines a metric space or has some relation to a specific metric space.

I enjoyed the table and the application in hands-on application and verification of sets.  I look forward to working on the problem sets and challenging my understanding on the reading to show myself what I missed.

(I forgot about the blog and quickly did a skim reading to to the blog.  I'll update it after working on the problem set for the homework.)

2.1 Due January 8

I am curious if these concepts apply in the reals or complex numbers and not only in the integers?

Question answered.  Only on integers is modular arithmetic defined.

The definition of [a]=[b] was a difficult concept to pick up and I appreciated the examples in the book that explained how the definition applies for the congruency relationship of a congruent to b mod n.

I enjoyed writing out and solving the proof using symmetry, transitivity, and reflexivity.  I also liked the example of proving that a-b=nk, b-c=ln, then a-c=mn.  But I still want to check and prove if k=l=m...?

1.1-1.3 Due on January 6

The most difficult concept in this reading section is the understanding of the definitions of the divisor, and the application points of prime numbers.  I had to do a few simple example problems to convince myself of the fact of the Fundamental theorem of Arithmetic to verify that the statement was true.  It helped to notice that the definition of p being prime is true if and only if -p is also prime!

I really enjoy the Euclidian algorithm, and using it to find (a,b) or the greatest common divisor.  It is also very useful in knowing and applying understanding for prime numbers.  I'm excited to use these concept to describe modular arithmetic and building on these principles.

Introduction Due on January 6

Introduction Due on January 6

What is your year in school and major?
I am a junior majoring in Mathematics

Which post-calculus math courses have you taken?  (Use names or BYU course numbers.)
112,113,313,314,334,290,341,344,345,320,321(acme junior year first semester)

Why are you taking this class?  (Be specific.)
Requirement for graduation.  and to become more proficient in analysis proofs

Tell me about the math professor or teacher you have had who was the most and/or least effective.
What did s/he do that worked so well/poorly?
Least effective was Dr. Bell--I stopped attending and taught myself.  Didn't have structured lesson plans and was hard to follow his train of thought.
Most effective was Dr. Jarvis--Helped clarify small details and made complex theorems clear and concise.

Write something interesting or unique about yourself.
I am a father of 4 children, and studying to become a seminary teacher.  I ahve had two internships as an object-oriented programmer, and have another this coming summer with Intermountain Healthcare.

If you are unable to come to my scheduled office hours or the TA's scheduled office hours, what times would work for you?
I have cs 235 at the same time that office hours are held.  I can attend office hours anytime Tuesday/Thursday, or MWF: before class any time, or after 4pm.