The proof of Theorem 4.21 looks just like a big mess on the page at first glance. Then it still looks that way at second glance. It only looks slightly better after reading. I was getting lost in the proofs and wanted just to say, "Ok, I'll take your word for it." The theorem itself is fine as the commutative, associative, and distributive laws are similar to the same laws we learned as children. De Morgan's Laws are easily accepted and understood. Maybe that is the problem - they are accepted and understood, so I haven't seen them proved before. I think that all of the symbols made the proofs look jumbled and crowded. It was difficult for me to read them straight through.
The section 5.1 on counterexamples seems a bit more like what I've been used to in previous math courses. It's more like solving equations and getting "answers." You look at the statements and you've got to find real numbers or integers which disprove the statements. This was definitely more in my comfort zone as I've found myself in past proofs trying to plug in integers, for example, and calculate the equations to confirm their validity.
Thursday, January 31, 2013
Tuesday, January 29, 2013
4.3-4.4, due on January 30
Why is "absolute value" so hard to understand? Frustratingly, it is something that doesn't come naturally to me in any of the mathematics classes I've ever had. The proof involving the absolute value was difficult for me to understand. I'm not sure if I just get intimidated when I see those absolute value signs | |, but I seem to struggle. Perhaps it was even harder because there were 3 cases and a couple of subcases. We haven't yet read much about those things and it looks like we will be getting into some more complicated or deeper proofs. Should be a challenge.
I find the "Proof Analysis" parts helpful because it offers an explanation as to why the author included certain parts in the proof and gives insight into the logic behind the proof. There was one for Result 4.18 that added much needed clarification. Then the analysis for the proof of Result 4.20 taught me a lot about if/when you need to tell the reader which technique you are using. Without the proof analysis, sometimes, I find it hard to understand why something was done or written the way it was in the book, so they are very helpful.
I've been spending a few hours on each homework assignment. The lectures and reading have prepared me well in completing the assignments. I think that having something due everyday - be it a blog entry or an assignment - has contributed most to my learning by creating a schedule in which I can steadily and effectively learn the material. I appreciate that the professor and TA are available for assistance - this always helps.
I find the "Proof Analysis" parts helpful because it offers an explanation as to why the author included certain parts in the proof and gives insight into the logic behind the proof. There was one for Result 4.18 that added much needed clarification. Then the analysis for the proof of Result 4.20 taught me a lot about if/when you need to tell the reader which technique you are using. Without the proof analysis, sometimes, I find it hard to understand why something was done or written the way it was in the book, so they are very helpful.
I've been spending a few hours on each homework assignment. The lectures and reading have prepared me well in completing the assignments. I think that having something due everyday - be it a blog entry or an assignment - has contributed most to my learning by creating a schedule in which I can steadily and effectively learn the material. I appreciate that the professor and TA are available for assistance - this always helps.
Saturday, January 26, 2013
4.1-4.2, due on January 28
Sometimes I read a proof in the text and I can follow it; that is to say, it all makes logical sense and looks true. However, I sometimes stuggle to understand how it all proved the result. Of particular difficulty was the last proof in 4.1.
Result to Prove For every integer n ≥ 7, there exist positive integers a and b such that n = 2a + 3b.
The two cases proved:
Since q ≥ 4, it follows that q − 3 ∈ N.
Since q ≥ 3, it follows that q − 1 ∈ N.
I still do not understand how this proves the result. So, what does the proof that q − 3 and q − 1 are natural numbers have to do with the result? As the text stated, this is a result of a somewhat different nature. I'm definitely missing something here. I'll try to keep looking at it to make sense of it.
I appreciate that the text and the professor of the course have prepared the student well to understand the material. As I was reading section 4.2, I thought to myself, "What if I picked up this book at the beginning of the semester and started reading from this section. I would really have no idea what is going on." It's true, the language is coming easier to me and I feel like everything has built up nicely to where we are today. The basics on notation and logic that we learned play a critical role in the comprehension of the material in the current sections and those yet to come, I'd assume.
Result to Prove For every integer n ≥ 7, there exist positive integers a and b such that n = 2a + 3b.
The two cases proved:
Since q ≥ 4, it follows that q − 3 ∈ N.
Since q ≥ 3, it follows that q − 1 ∈ N.
I still do not understand how this proves the result. So, what does the proof that q − 3 and q − 1 are natural numbers have to do with the result? As the text stated, this is a result of a somewhat different nature. I'm definitely missing something here. I'll try to keep looking at it to make sense of it.
I appreciate that the text and the professor of the course have prepared the student well to understand the material. As I was reading section 4.2, I thought to myself, "What if I picked up this book at the beginning of the semester and started reading from this section. I would really have no idea what is going on." It's true, the language is coming easier to me and I feel like everything has built up nicely to where we are today. The basics on notation and logic that we learned play a critical role in the comprehension of the material in the current sections and those yet to come, I'd assume.
Thursday, January 24, 2013
3.4-3.5, due on January 25
After reading the previous sections of Chapter 3 and doing the homework, the reading of these two sections was easier to comprehend. However, there is one thing that I can see will be difficult going forward: which proof method to use. In this reading we learned of another method - proof by cases. There are benefits and advantages to all the methods learned thusfar. I follow the examples of proofs in the text and agree, of course, that the correct method has been chosen. However, I have a fear that I will not be able to quickly choose the correct method when showing my own proofs, resulting in lost time, mounting frustration, or inability to construct a correct proof.
This fear leads to the part of the text that I appreciated most:
When learning any mathematical subject, it is not the least bit unusual to make mistakes along the way. In fact, part of learning mathematics is to learn from your mistakes and those of others.
This is encouraging to me. I've already made many mistakes in this course. I fully expect to make more mistakes along the way. The thing on which I will focus is learning from my mistakes. I'm certain that I will stumble along the way, but I am still excited about learning the material in this course and, in some strange way, looking forward to making mistakes because I know that I will learn from them.
This fear leads to the part of the text that I appreciated most:
When learning any mathematical subject, it is not the least bit unusual to make mistakes along the way. In fact, part of learning mathematics is to learn from your mistakes and those of others.
This is encouraging to me. I've already made many mistakes in this course. I fully expect to make more mistakes along the way. The thing on which I will focus is learning from my mistakes. I'm certain that I will stumble along the way, but I am still excited about learning the material in this course and, in some strange way, looking forward to making mistakes because I know that I will learn from them.
Tuesday, January 22, 2013
3.1-3.3, due on January 23
I found it challenging to follow some of the proofs. Perhaps it is to be expected to have to reread them in order to make complete sense of them. Once I was able to pause and really think about them, they all made sense - just not at first for all of them. I know that this is essential to the course as we begin to construct our own proofs, so I put a high priority on understanding the fundamentals during the beginning stages of our study of proofs. Following the proofs and comprehending them can be quite rewarding.
I really liked the last part of the last line of Section 3.3:
Also, you might think we used a trick to give the second proof of Result 3.14, but, as we will see, if the same “trick” can be used often, then it becomes a technique.
To me, this was very fun to read. Playing sports all my life, I learned many techniques. In soccer, for example, there were dribbling techniques, passing techniques, shooting, defending, and goalkeeping techniques. I liked how techniques were described as "tricks that can be used often." Looking back, essentially that is what they were. Used often enough, they become techniques - techniques a quality player uses whenever he steps out onto the field. I can see that mathematicians can use many techniques to show their quality in the mathematical field. Scoring a goal can be beautiful just like a theorem can be beautiful.
I really liked the last part of the last line of Section 3.3:
Also, you might think we used a trick to give the second proof of Result 3.14, but, as we will see, if the same “trick” can be used often, then it becomes a technique.
To me, this was very fun to read. Playing sports all my life, I learned many techniques. In soccer, for example, there were dribbling techniques, passing techniques, shooting, defending, and goalkeeping techniques. I liked how techniques were described as "tricks that can be used often." Looking back, essentially that is what they were. Used often enough, they become techniques - techniques a quality player uses whenever he steps out onto the field. I can see that mathematicians can use many techniques to show their quality in the mathematical field. Scoring a goal can be beautiful just like a theorem can be beautiful.
Thursday, January 17, 2013
Chapter 0 - pages 5 to 12, due on January 18
The most difficult part was reading about the difference between "that" and "which." Both example sentences sounded correct, so I was wondering what the point was.
The solution to the equation is the number less than 5 that is positive.
The solution to the equation is the number less than 5 which is positive.
Upon re-reading, I see that there is an important difference between "that" and "which," depending on the question you are trying to answer or the point you are trying to make. I can see how these two words are confused with each other and are often not interchangable. I had to stop and think hard about restrictive and nonrestrictive clauses and how they affect the meaning of the sentences.
I feel that this section was very beneficial. I appreciated that creativity and an attitude of being yourself is encouraged in mathematical writing. Often I find myself holding onto the belief that there is a right answer and a wrong answer in mathematics, that there is a right way and a wrong way - very black and white. I found it interesting that, while there are guidelines and tips one should follow, originality can and should exist in mathematical writing, just as it does in other forms of writing. All of the advice given in the chapter will be of great benefit, especially the suggestions and guidelines. There should be order and accepted principles to follow in mathematical writing, especially when it assists the reader in understanding what you are trying to show, prove, or teach. I'm sure I will return to this chapter to use it as a reference guide in the future.
The solution to the equation is the number less than 5 that is positive.
The solution to the equation is the number less than 5 which is positive.
Upon re-reading, I see that there is an important difference between "that" and "which," depending on the question you are trying to answer or the point you are trying to make. I can see how these two words are confused with each other and are often not interchangable. I had to stop and think hard about restrictive and nonrestrictive clauses and how they affect the meaning of the sentences.
I feel that this section was very beneficial. I appreciated that creativity and an attitude of being yourself is encouraged in mathematical writing. Often I find myself holding onto the belief that there is a right answer and a wrong answer in mathematics, that there is a right way and a wrong way - very black and white. I found it interesting that, while there are guidelines and tips one should follow, originality can and should exist in mathematical writing, just as it does in other forms of writing. All of the advice given in the chapter will be of great benefit, especially the suggestions and guidelines. There should be order and accepted principles to follow in mathematical writing, especially when it assists the reader in understanding what you are trying to show, prove, or teach. I'm sure I will return to this chapter to use it as a reference guide in the future.
Tuesday, January 15, 2013
2.9-2.11, due on January 16
I found the Example 2.24 in section 2.10 to be difficult to understand upon first reading.
"Suppose that we are considering the set A = {1 , 2, 3} and its power set P( A), the set of all subsets of A. Then the quantified statement for every set B ∈ P( A), A− B = ∅. (2.6) is false since for the subset B = A = {1 , 2, 3} , we have A− B = ∅."
I guess I am still struggling with the principles of sets, subsets, power sets, and empty sets. I had to reread many times the following, B ∈ P( A), A− B = ∅ is false, before I grasped what it was trying to say about the relationship between B and the power set P( A). I understand now, but it was difficult for a little while.
The most interesting part to me was the discussion of how mathematicians rarely use the definition in an "if and only if" statement. To me it seemed a valid characterization of an equilateral triangle. "A triangle T is equilateral if and only if T has three equal sides." However, I learned that this is not a characterization, but rather a definition. An accurate characterization of equilateral triangles was presented, "A triangle T is equilateral if and only if T has three equal angles." I quickly understood the difference and found it to be an interesting point as I look forward to other examples of this in future chapters.
"Suppose that we are considering the set A = {1 , 2, 3} and its power set P( A), the set of all subsets of A. Then the quantified statement for every set B ∈ P( A), A− B = ∅. (2.6) is false since for the subset B = A = {1 , 2, 3} , we have A− B = ∅."
I guess I am still struggling with the principles of sets, subsets, power sets, and empty sets. I had to reread many times the following, B ∈ P( A), A− B = ∅ is false, before I grasped what it was trying to say about the relationship between B and the power set P( A). I understand now, but it was difficult for a little while.
The most interesting part to me was the discussion of how mathematicians rarely use the definition in an "if and only if" statement. To me it seemed a valid characterization of an equilateral triangle. "A triangle T is equilateral if and only if T has three equal sides." However, I learned that this is not a characterization, but rather a definition. An accurate characterization of equilateral triangles was presented, "A triangle T is equilateral if and only if T has three equal angles." I quickly understood the difference and found it to be an interesting point as I look forward to other examples of this in future chapters.
Saturday, January 12, 2013
2.5-2.8, due on January 14
One difficult part of the reading is quickly remembering what the symbols and notations mean. I really have to stop and think about what the statements P=>Q and ~P^Q are saying in English. As time passes, I'm sure my mind will process the new language more quickly. It is very difficult to know the truth or falseness of statements without writing up a truth table. I can see that truth tables are very useful, especially since the following situation is a little confusing:
If P is false then Q is true. This is still a true statement. The realization is that I'll have to trust the truth table. I'll also try to remember the effective example from class. "If you get your pajamas on, then you'll get ice cream." Even if the kids get ice cream without putting on pajamas, the statements wasn't a lie to the kids.
I am really looking forward to mathematical application of this study in logic. An example in the text began with the phrase: "Let R be a mathematical statement that we would like to show is true." It gave me some insight and foreshadowing perhaps on what lies ahead in this course. I anticipate seeing how we apply logic in mathematical proofs. I remember taking a math class in high school in which the teacher asked us to do many proofs. It has been a long time since high school so my memory is a little hazy, but it seems that we used logically equivalent statements in the construction of proofs. I look forward to applying what we have read in these sections later in the course.
If P is false then Q is true. This is still a true statement. The realization is that I'll have to trust the truth table. I'll also try to remember the effective example from class. "If you get your pajamas on, then you'll get ice cream." Even if the kids get ice cream without putting on pajamas, the statements wasn't a lie to the kids.
I am really looking forward to mathematical application of this study in logic. An example in the text began with the phrase: "Let R be a mathematical statement that we would like to show is true." It gave me some insight and foreshadowing perhaps on what lies ahead in this course. I anticipate seeing how we apply logic in mathematical proofs. I remember taking a math class in high school in which the teacher asked us to do many proofs. It has been a long time since high school so my memory is a little hazy, but it seems that we used logically equivalent statements in the construction of proofs. I look forward to applying what we have read in these sections later in the course.
Thursday, January 10, 2013
2.1-2.4, due on January 11
Overall, the reading in this section of the book was straightforward and easier to comprehend than the material in Chapter 1. One challenge is all of the definitions presented to the reader. Now we learn of disjunction and conjunction, negation, open sentences and domains, just to name a few. Also, with all of the new symbols, I find it difficult to keep the notation straight without reviewing and realizing, "Oh, yes, that is what that means in English." The challenge presents and exciting opportunity to learn all this class will offer.
The reading was interesting. I like logic and believe I am a logical thinker. Of particular interest was Example 2.6 in the section about The Implication. I found it to be an effective real-world example of an if-then statement. I could relate to the example because I have been, possibly like many other students at one time or another, in a similar situation. The author in the book presents the example: "If you earn an A on the final exam, then you will receive an A for your final grade." It really helped me understand the four different possibilities in the truth table: 1. P and Q are both true. 2. P is true and Q is false. 3. P is false and Q is true. 4. P and Q are both false.
The reading was interesting. I like logic and believe I am a logical thinker. Of particular interest was Example 2.6 in the section about The Implication. I found it to be an effective real-world example of an if-then statement. I could relate to the example because I have been, possibly like many other students at one time or another, in a similar situation. The author in the book presents the example: "If you earn an A on the final exam, then you will receive an A for your final grade." It really helped me understand the four different possibilities in the truth table: 1. P and Q are both true. 2. P is true and Q is false. 3. P is false and Q is true. 4. P and Q are both false.
Tuesday, January 8, 2013
1.1-1.6, due on January 9
The most difficult part of the material for me is empty sets. I find it challenging to understand them conceptually. I feel that it will take some added focus and exercise in comprehending them and their importance. Also the notation is new to me and will take some getting used to; although, I'm sure it will come naturally as I become more engaged with the material in this course.
I find that it very interesting that this is mathematics. I know that it truly is mathematics, but it reminds me that many people I know believe that mathematics is arithmetic. The material is analytical and logical. I really appreciate that aspect of the mathematics. I believe this will be a very useful skill to sharpen as I progress in my academic career and also in my career as an actuary, which is exciting to me.
I find that it very interesting that this is mathematics. I know that it truly is mathematics, but it reminds me that many people I know believe that mathematics is arithmetic. The material is analytical and logical. I really appreciate that aspect of the mathematics. I believe this will be a very useful skill to sharpen as I progress in my academic career and also in my career as an actuary, which is exciting to me.
Introduction, due on January 9
I am a junior and my major is actuarial science.
I have taken Math 112 and Math 113.
I am taking this class to gain a deeper understanding of mathematics with hopes that it will prepare me better for future mathematics classes and all types of learning in my life. This class can potentially contribute towards a mathematics minor.
The least effective math teacher I've had was very hands-off, requiring little from the students. I felt that many of my classmates and I would have learned more if we were required to give more effort and challenged to fully comprehend the material. The most effective math teacher I've had really cared about the students, I believe, and encouraged learning the material while not focusing so much on studying just to pass a test. This helped strengthen a desire within me to embrace learning and master the subject matter.
I am a returning student after being away from BYU for 8 years. My freshman year at BYU was in 1998. I decided I wanted a career change and to finish my undergraduate degree, so my wife and our 3 children moved from Denver, CO to Provo in August 2012 so that I could finish my degree in pursuit of a more satisfying career.
I have no conflict with your scheduled office hours. The indicated times work for me.
I have taken Math 112 and Math 113.
I am taking this class to gain a deeper understanding of mathematics with hopes that it will prepare me better for future mathematics classes and all types of learning in my life. This class can potentially contribute towards a mathematics minor.
The least effective math teacher I've had was very hands-off, requiring little from the students. I felt that many of my classmates and I would have learned more if we were required to give more effort and challenged to fully comprehend the material. The most effective math teacher I've had really cared about the students, I believe, and encouraged learning the material while not focusing so much on studying just to pass a test. This helped strengthen a desire within me to embrace learning and master the subject matter.
I am a returning student after being away from BYU for 8 years. My freshman year at BYU was in 1998. I decided I wanted a career change and to finish my undergraduate degree, so my wife and our 3 children moved from Denver, CO to Provo in August 2012 so that I could finish my degree in pursuit of a more satisfying career.
I have no conflict with your scheduled office hours. The indicated times work for me.
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