What was the most interesting part of this material for you?
I think it's interesting that we don't need to actually know what the thing is that possesses the desired property, we can just prove it exists! It's interesting that we know every odd degree polynomial has at least one real number solution, even though we can't always find them!
What was the most difficult part of the material for you?
I had a hard time grasping the proof of "There exist irrational numbers a and b such that a^b is rational." It's difficult because it's similar to a normal case proof but each case deals with a different answer to a^b. That's what made it difficult.
Tuesday, September 30, 2014
Friday, September 26, 2014
5.2-5.3, due September 29
What was the most difficult part of the material for you?
I had a hard time grasping the technical explanation of proof by contradiction. When I read about how to do it in my own proofs, I went back and read the explanation again and it made more sense.
What was the most interesting part of the material for you?
I think it's fun to prove things by contradiction. It's an interesting strategy to have the negation of a statement imply a contradiction. The trick is to figure out what the contradiction is.
Thursday, September 25, 2014
4.5-4.6 and 5.1, due September 26
What was the most difficult part of the material for you?
Sometimes the proofs involving sets are hard to read because they are more compact in form than other proofs. Reading carefully helps me better understand it.
What was the most interesting part of the material for you?
I really enjoy disproof by counter-example. It's fun to wipe out a whole statement with a single example.
Sometimes the proofs involving sets are hard to read because they are more compact in form than other proofs. Reading carefully helps me better understand it.
What was the most interesting part of the material for you?
I really enjoy disproof by counter-example. It's fun to wipe out a whole statement with a single example.
Tuesday, September 23, 2014
4.3-4.4, due September 24
What was the most difficult part of the material for you?
I had a hard time understanding the proof at the end of 4.3 that involves absolute values. I was a little confused by the sequence of equal signs and greater-than signs on the same line, but once I read it the right way it made sense.
What is the most interesting part of the material for you?
I think the section on proofs using sets was interesting. It looks like most proofs involve showing a function of one or more sets being equivalent to another, which is cool to see. Their equivalence can also be shown graphically.
How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
I spend anywhere from 20 to 40 minutes on each assignment. The reading serves to get my feet wet in the content, and then the lecture is where I really come to understand the methodology.
What has contributed most to your learning in this class thus far?
Doing the homework has probably expanded my learning the most, because the interesting proofs presented make me think hard about the problem and really understand the section.
What do you think would help you learn more effectively or make the class better for you?
I had a hard time understanding the proof at the end of 4.3 that involves absolute values. I was a little confused by the sequence of equal signs and greater-than signs on the same line, but once I read it the right way it made sense.
What is the most interesting part of the material for you?
I think the section on proofs using sets was interesting. It looks like most proofs involve showing a function of one or more sets being equivalent to another, which is cool to see. Their equivalence can also be shown graphically.
How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
I spend anywhere from 20 to 40 minutes on each assignment. The reading serves to get my feet wet in the content, and then the lecture is where I really come to understand the methodology.
What has contributed most to your learning in this class thus far?
Doing the homework has probably expanded my learning the most, because the interesting proofs presented make me think hard about the problem and really understand the section.
What do you think would help you learn more effectively or make the class better for you?
Spending a little more time with the assigned reading might help me better absorb the material and come to class already understanding the content.
Sunday, September 21, 2014
4.1-4.2, due September 22
What was the most difficult part of the material for you?
I had a hard time understanding the proof at the end of 4.1. I didn't follow a few steps in the manipulation of the expression for n to make it look like 2a+3b. I reread it and realized that's what was happening.
What is the most interesting part of this material for you?
The most interesting part for me was the section on congruence of integers. I learned about modular arithmetic in high school just so I could get the right answer for questions in science bowl. Now I better understand the mathematical purpose of modulo.
I had a hard time understanding the proof at the end of 4.1. I didn't follow a few steps in the manipulation of the expression for n to make it look like 2a+3b. I reread it and realized that's what was happening.
What is the most interesting part of this material for you?
The most interesting part for me was the section on congruence of integers. I learned about modular arithmetic in high school just so I could get the right answer for questions in science bowl. Now I better understand the mathematical purpose of modulo.
Tuesday, September 16, 2014
3.3-3.5, due September 19
What was the most difficult part of the material for you?
I had a difficult time understanding why the contrapositive of an implication is logically equivalent to the original statement. Going back to the definition of an implication made the contrapositive make a lot more sense. I think of it this way: if the implication is true and the conclusion is false, then the hypothesis must be true to keep the implication true.
What was the most interesting part of the material for you?
It was interesting to see how a proof could be broken into cases and each case solved by a different method. Even when two elements are very close to one another (e.g. 0 and 1e-10), the proof for these separate cases can be wildly different.
I had a difficult time understanding why the contrapositive of an implication is logically equivalent to the original statement. Going back to the definition of an implication made the contrapositive make a lot more sense. I think of it this way: if the implication is true and the conclusion is false, then the hypothesis must be true to keep the implication true.
What was the most interesting part of the material for you?
It was interesting to see how a proof could be broken into cases and each case solved by a different method. Even when two elements are very close to one another (e.g. 0 and 1e-10), the proof for these separate cases can be wildly different.
3.1-3.2, due September 17
What was the most difficult part of the material for you?
I had a hard time understanding the point of the vacuous proof. I feel like it's more important to discuss the conclusion of the implication, rather than on making these irrelevant statements that are true simply by the definition of implications. I'm sure they are just exercises in stretching the definition of a statement in our minds, but I'm open to the possibility of there being wider applications for vacuous proofs.
What was the most interesting part of the material for you?
I think that the method of the direct proof is an interesting one. I thought that was the only way to prove implications, so I'm excited to learn what other techniques there are for proving these statements.
I had a hard time understanding the point of the vacuous proof. I feel like it's more important to discuss the conclusion of the implication, rather than on making these irrelevant statements that are true simply by the definition of implications. I'm sure they are just exercises in stretching the definition of a statement in our minds, but I'm open to the possibility of there being wider applications for vacuous proofs.
What was the most interesting part of the material for you?
I think that the method of the direct proof is an interesting one. I thought that was the only way to prove implications, so I'm excited to learn what other techniques there are for proving these statements.
Saturday, September 13, 2014
Chapter 0, due on September 15
What was the most difficult part of this section for you?
I had a hard time understanding the difference between "that" and "which" as described in this section. The section that describes the difference as being restrictive vs. nonrestrictive made much more sense than the previous explanations.
How does this material apply outside of class?
My major (Applied and Computational Mathematics Emphasis) will require extensive use of mathematical writing, so this content will be invaluable to my progress later on. Writing clearly on a thesis or an article in a mathematical journal is essential.
I had a hard time understanding the difference between "that" and "which" as described in this section. The section that describes the difference as being restrictive vs. nonrestrictive made much more sense than the previous explanations.
How does this material apply outside of class?
My major (Applied and Computational Mathematics Emphasis) will require extensive use of mathematical writing, so this content will be invaluable to my progress later on. Writing clearly on a thesis or an article in a mathematical journal is essential.
Thursday, September 11, 2014
2.9-2.10, due on September 12
What was the most difficult part of the material for you?
In the part of the section that discussed quantified statements with two variables, I had a hard time understanding the definition of the negation of these statements. It seemed like there could have been a chance for other options to appear, where the statement starts with "for all" with one variable and "at least one" for another variable. But after reading over the section again it made sense that those sorts of statements don't fit as a negation for the original statement.
What was the most interesting part of the material for you?
I really enjoyed learning the symbols for the universal quantifier and existential quantifier. Those concepts occur very often in other areas of math, and it will be more efficient to simply write those symbols to communicate in math.
In the part of the section that discussed quantified statements with two variables, I had a hard time understanding the definition of the negation of these statements. It seemed like there could have been a chance for other options to appear, where the statement starts with "for all" with one variable and "at least one" for another variable. But after reading over the section again it made sense that those sorts of statements don't fit as a negation for the original statement.
What was the most interesting part of the material for you?
I really enjoyed learning the symbols for the universal quantifier and existential quantifier. Those concepts occur very often in other areas of math, and it will be more efficient to simply write those symbols to communicate in math.
Tuesday, September 9, 2014
2.5-2.8, due on September 10
What was the most difficult part of the material for you?
The hardest part of this material was understanding the proof that showed that a biconditional statement for P and Q can be stated as "P if and only if Q." After reading it a few times over, I understood the math-heavy words they used to make the proof.
How can this material apply outside of mathematics?
Biconditional statements are a very important logical principle, even outside mathematics. It's important to understand the difference between a biconditional statement containing P and Q, and P and Q being logically equivalent. Using English versions of these logical connectives to explain meaning and logic can help a writer become more logically persuasive.
The hardest part of this material was understanding the proof that showed that a biconditional statement for P and Q can be stated as "P if and only if Q." After reading it a few times over, I understood the math-heavy words they used to make the proof.
How can this material apply outside of mathematics?
Biconditional statements are a very important logical principle, even outside mathematics. It's important to understand the difference between a biconditional statement containing P and Q, and P and Q being logically equivalent. Using English versions of these logical connectives to explain meaning and logic can help a writer become more logically persuasive.
Saturday, September 6, 2014
2.1-2.4, due on September 8
What was the most difficult part of the material for you?
The hardest part to understand for me was understanding why a conditional statement with a false condition and a true result is still a true statement. It made more sense when I read the example situation with the teacher and student.
How can this material apply outside of mathematics?
Logic is obviously the foundation of most human thought. Understanding how conclusions are drawn logically allows us to better communicate ideas and opinions to other people.
The hardest part to understand for me was understanding why a conditional statement with a false condition and a true result is still a true statement. It made more sense when I read the example situation with the teacher and student.
How can this material apply outside of mathematics?
Logic is obviously the foundation of most human thought. Understanding how conclusions are drawn logically allows us to better communicate ideas and opinions to other people.
Thursday, September 4, 2014
1.1-1.6, due on September 5
What was the most difficult part of the material for you?
The most confusing part of set theory for me so far has been dealing with the empty set in problems. I've been stuck on multiple problems where the empty set has been an element of some other set, and finding things like the power set or different subsets of that set has been a difficult task. After a bit of study, I figured out that the best way to handle this is to replace the "∅" with "{}", so I can remember to consider that empty set as a regular set that is an element of the greater set.
What is the importance of the material outside of this class?
The idea of set theory is a very powerful idea, both in math and without. The organization of things is important to every field. The set operations we study in this class can be applied to sets found in other areas. We've all seen Venn diagrams that organize not just math, but history, and many other subjects. Intersection, union, and complementing are all operations that are useful to organization of all things.
The most confusing part of set theory for me so far has been dealing with the empty set in problems. I've been stuck on multiple problems where the empty set has been an element of some other set, and finding things like the power set or different subsets of that set has been a difficult task. After a bit of study, I figured out that the best way to handle this is to replace the "∅" with "{}", so I can remember to consider that empty set as a regular set that is an element of the greater set.
What is the importance of the material outside of this class?
The idea of set theory is a very powerful idea, both in math and without. The organization of things is important to every field. The set operations we study in this class can be applied to sets found in other areas. We've all seen Venn diagrams that organize not just math, but history, and many other subjects. Intersection, union, and complementing are all operations that are useful to organization of all things.
Wednesday, September 3, 2014
Introduction, due on September 5
1. What is your year in school and major?
6. If you are unable to come to my scheduled office hours, what times would work for you?
I am a freshman here at BYU and I am majoring in the Applied and Computational Mathematics Emphasis with a concentration in Computer Science and a minor in Physics.
2. Which calculus-or-above math courses have you taken?
I have taken calculus 1 and 2, multivariable calculus, ordinary differential equations, and linear algebra.
3. Why are you taking this class?
I'm taking this class for a few reasons. First, math is something I've always been interested in and I think this class will greatly improve my understanding of basic concepts in math theory. Second, it's a prerequisite for a large number of classes that are required for my major.
4. 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?
My favorite math teacher to this point was my high school AP calculus and AP physics teacher. He really instilled in me both a love for the power and beauty of math, and an appreciation for the value of putting mathematical thoughts on paper clearly. I feel like I have benefited immensely from the impact this teacher had on my mathematical thinking.
5. Write something interesting or unique about yourself.
I love to run long distances, and I feel like I perform better in other areas of my life when I live an active lifestyle. For me, the longer the race, the more comfortable I am with pushing the pace and running to my potential. I also enjoy running because of the analytical aspects of the sport; I love being able to quantify performance in such a variety of ways.
The office hours currently available will be just fine.
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