Reflections, Due January 27

• How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?  I spend about 1-2 hours on each assignment.  Yes they help, except on the really tricky proofs.  I need to stop procrastinating them so much so that I can really think about them sufficiently and not be pressed for time.
• What has contributed most to your learning in this class thus far?
• I would say that Lecture and Homework has contributed the most, but part of my benefit from lecture comes from me reading the chapter before-hand so I already know the questions I have and what to look for.
• 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 think that I need to study the definitions/proofs a little bit every day, or even three days a week so I am not cramming so much for the test.

3.3, Due January 25 2016

1) Can you please explain what it means to be injective, surjective, and bijective?  I forgot this.  Also, I guess we can just think of "changing" the name of all the elements as putting the elements through a function?
2) How is it advantageous to us to recognize that there is an isomorphism or homomorphism from one set to the other?

3.2, January 22

1)  Is there a way we can prove that there always exists an inverse matrix (well actually one doesn't always exist) but that the determinant must not be 0 in order to find it?  I also don't understand how the 3.9 proof is proving it.
2)  I am so used to just using the fact that the numbers I work with are part of a ring so I can perform all of these operations, but I haven't thought about how we need to prove all of these theorems in the first place, before we can even consider using them for practical use.

3.1 Due January 20

1) So basically an integral domain is when there are no zero divisors?  Why do we have to specify that 1R does  not equal 0R?  And for a field to exist, why do we need to specify that when ax=1R has a solution, a does not equal 0?  I didn't really see how the proof is sufficient for Theorem 3.2.
2)  I feel a lot more confident with rings and it is cool to see how they are defined in so many different ways.

3.1, Due January 15

1) What exactly is an axiom?

So is a congruence class a ring? It seems to have the same properties.

I DO NOT GET THE ADDITION AND MULTIPLICATION TABLE on pg. 43

2) I guess I just don't really understand what makes a ring distinct or different from things that we have learned before because it has the same properties.  It sort of seemed like they put a bunch of random stuff that I have learned together in this chapter and I'm still not really sure what a ring is, but hopefully it will make more sense tomorrow in lecture.

2.3, Due January 13th

1)  When a and n are not relatively prime, I guess I don't really understand why the equation ax-b may have no solutions or several solutions in Zn.
2)  I guess it makes sense, since we use the Euclidean algorithm to find the inverse of a certain number mod(n), that we would also use that to prove that the inverse always exists.  Simple enough, but when I first saw it I wasn't exactly sure as to how we would prove it.

2.2, Due January 11th 2016

1) When it talks about the properties of multiplication in modular arithmetic, it gives names of different rules (i.e. closure for multiplication, associative multiplication, distributive laws, etc.).  However, it doesn't give a name for #11: ab=0, then a=0 or b=0.  Is there a name for that?  And why does this property hold for Z5 and fail for Z6?
2)  It's interesting to see that even exponents in Zn are the same, but I guess it makes sense since multiplication and addition are the same.

2.1, Due January 7, 2015

1) In the first paragraph when they say, "two integers a and b are equal if their difference is 0", why did they have to include: "or equivalently if their difference is a multiple of 0?"  How could it be a multiple of 0 instead of just 0?
What is disjoint again?
2) I had never really thought of thinking something is 0 if the difference is 0 but it totally makes sense: if a-b=0, then a=b.  Then since nmod(n) acts like 0 in the modulus, it would only make sense that two numbers are congruent if their difference is n (now here I can see why they would have to specify if their difference is a multiple of n.  Now I understand why we said a is congruent to b if a-b=nk.
I understand equivalence classes so much more clearly now!

1.1-1.3, Wednesday January 6

1) I didn't understand how the proof of Lemma 1.7 really proved that (a,b)=(b,r).  Especially when they are using different values for the common divisor (c in the first, and e in the second).

2) I never thought about the fact that different numbers can have multiple prime factorizations depending on the negatives.  I always just considered the case when there are only positive numbers.  I also find the way that they prove some things to be very interesting and enlightening once I stare at it for a second, but I am just hoping that I can learn to do it on my own.

Blog 1, Wednesday January 6 2016

• What is your year in school and major? 
• Super senior, Math Education
• Which post-calculus math courses have you taken?  (Use names or BYU course numbers.)
• 290, Linear Algebra, Differential Equations, Multivariable Calculus, Cryptography, Geometry proofs
• Why are you taking this class?  (Be specific.)
• Required for my major, want to become better at 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?
• The worst math professor that I had was very unorganized and very uninterested in the students.  Made lots of typos on the exams, and unclear in lecture.
• The best math professor that I had was very organized, kind, clear, and extremely passionate about the subject.
• Write something interesting or unique about yourself.
• I will run a half-marathon in April
• If you are unable to come to my scheduled office hours or the TA's scheduled office hours, what times would work for you?