Difficult: I did not really find any of the topics in either of these sections very difficult. Although I feel like I understand the Fundamental Theorem of Arithmetic, I feel like it is probably the most difficult concept from this reading. I get what the Fundamental Theorem of Arithmetic means and the logic makes sense to me. However, I think I would like to see some more examples for how to use the Fundamental Theorem of Arithmetic and the various ways to apply it.
Reflective: My favorite part of this reading was the part about perfect integers. It is extremely easy to understand the idea behind this concept. Also, I just think it is really cool how an integer can have the sum of its proper divisors be equal to the number itself. I am curious as to how frequent this occurrence is or if it is more of a rare incident.
Tuesday, March 27, 2012
Saturday, March 24, 2012
11.5, due March 26
Difficult: The most difficult portion of this section actually kind of surprised me. I do not understand Theorem 11.12, like at all. I get that that theorem seems to lead to the term relatively prime, but I do not understand how the theorem is useful or how to apply it to prove anything. Also, the whole part about its converse also being true did not make any sense to me.
Reflective: I liked reading about Euclid's Lemma in this section. I am not entirely confident that I completely understand this concept, but I feel like I at least understand a base. But the main reason why I liked this lemma is because I thought it was really cool how you cannot say that just because a|bc that a|b and/or that a|c. But if a|bc and gcd(a,b)=1 then you can conclude that a|c.
Reflective: I liked reading about Euclid's Lemma in this section. I am not entirely confident that I completely understand this concept, but I feel like I at least understand a base. But the main reason why I liked this lemma is because I thought it was really cool how you cannot say that just because a|bc that a|b and/or that a|c. But if a|bc and gcd(a,b)=1 then you can conclude that a|c.
Thursday, March 22, 2012
11.3-11.4, due March 23
Difficult: The most difficult part in this reading in my opinion was the Euclidean Algorithm in section 11.4. i am not really sure I understand anything about this Algorithm. I get that it is a continuation of the idea behind the division Algorithm but I'm not sure if I really understand that completely yet either... i would like to see lots of examples and applications of both algorithms.
Reflective: I liked section 11.3. This section talked about the common divisor and the greatest common divisor. I liked this part because both of these concepts are simply to understand and I have used before many times. Therefore it made it easier to know what the book was talking about in the rest of the section.
Reflective: I liked section 11.3. This section talked about the common divisor and the greatest common divisor. I liked this part because both of these concepts are simply to understand and I have used before many times. Therefore it made it easier to know what the book was talking about in the rest of the section.
Tuesday, March 20, 2012
11.1-11.2, due March 21
Difficult: The most difficult part of this chapter is the division algorithm theorem presented in section 11.2. This theorem on the surface seemed as if it should be fairly simple to understand. However, after I looked at it and studied, I came to the conclusion that i did not really understand the logic and how the conclusion of the theorem was made. I kind of have a little bit of an idea as to how we get the first part that b=aq+r but I am not sure I understand why the condition that 0 is less than or equal to r which is less than a needs to be included.
Reflective: My favorite part of this section can be found in 11.1 about the primes and composite numbers. I actually don't remember if I have ever hear integers that are not primes be referred to as composite numbers, but in a way, the name makes sense. What I really liked though was Theorem 11.2. i liked this theorem because it reminded me of the conditions for whether or not relations can be categorized as being reflexive, symmetric, and transitive and therefore be equivalence relations.
Reflective: My favorite part of this section can be found in 11.1 about the primes and composite numbers. I actually don't remember if I have ever hear integers that are not primes be referred to as composite numbers, but in a way, the name makes sense. What I really liked though was Theorem 11.2. i liked this theorem because it reminded me of the conditions for whether or not relations can be categorized as being reflexive, symmetric, and transitive and therefore be equivalence relations.
Monday, March 19, 2012
10.5, due March 19
Difficult: In this section, I really did not understand the whole introduction that was talking about the restriction f1 of f to D. I am so confused that I cannot even seem to decipher what it is that I am confused about. I just don't understand what it is saying at all.
Reflective: In this section, i thought that the Schroder-Bernstein Theorem was relatively easy to understand. Because in order for two sets to say that A is less than or equal to B and that B is also less than or equal to A, then it is obvious that we can conclude that A and B must be equal to each other.
Reflective: In this section, i thought that the Schroder-Bernstein Theorem was relatively easy to understand. Because in order for two sets to say that A is less than or equal to B and that B is also less than or equal to A, then it is obvious that we can conclude that A and B must be equal to each other.
Thursday, March 15, 2012
10.4, due March 16
Difficult: The most difficult concept for me to understand in this section of reading is Theorem 10.14. Pretty much I don't really get any part of this theorem. Part of the problem I think is that I am kind of having a difficult time figuring out the principles of something being uncountable. Also I think I still feel sort of confused on what exactly the set 2^A consists of.
Reflective: I liked Theorem 10.15 in this section. It just seemed to actually make sense whereas some of the other theorems I don't understand the logic or anything. But it makes sense to me that the cardinality of a set A would be less then the cardinality of the power set of that set A.
Reflective: I liked Theorem 10.15 in this section. It just seemed to actually make sense whereas some of the other theorems I don't understand the logic or anything. But it makes sense to me that the cardinality of a set A would be less then the cardinality of the power set of that set A.
Tuesday, March 13, 2012
10.3b, due March 14
Difficult: The whole idea that sets (0,1) and R could be numerically equivalent seemed really weird to me at first. I have a hard time with the abstract sense to this proof. However, when I would think about them each as two separate sets, (0,1) and R, and focus on them individually then I understood the logic a little bit better.
Reflective: What I liked about this proof was how it used concepts from calculus to help prove it. I really like calculus and more mechanical forms of math that focus more on arithmetic type stuff. I have an easier time understanding those ideas because they seem less abstract to me. This sense of familiarity helped me understand the idea behind this proof.
Reflective: What I liked about this proof was how it used concepts from calculus to help prove it. I really like calculus and more mechanical forms of math that focus more on arithmetic type stuff. I have an easier time understanding those ideas because they seem less abstract to me. This sense of familiarity helped me understand the idea behind this proof.
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