Post 42 make up assignment: Focus on Math talk

This talk is titled "Who's number one? from ranking to bracketology".

The speaker talked first about how often we use ranking algorithms every day (e.g. Google). He discussed how predictive methods aren't trying to predict upsets (usually). They're trying to predict the most likely outcomes.

We talked about how we can graph a sport season and rank teams by wins. Winning percentage isn't as good as when we take into account strength of schedule. Strength of schedule, mathematically, is taken care of with linear algebra. Each game creates a dependency between the two teams, expressed as an equation.

12.4, due on December 8

What was the most interesting part of the material for you?
I think it's cool that the fundamental properties of limits are very similar to properties of other mathematical concepts. Being able to use these additive and multiplicative properties will make proofs much faster.

What was the most difficult part of the material for you?
I had a hard time understanding the proof for Lemma 12.26. The use of the deleted neighborhood in the statement kind of confused me, but I'm sure it will make more sense in class.

12.3, due on December 5

What was the most interesting part of the material for you?
The idea of a deleted neighborhood is something I have never heard of before, even in calculus. But it makes sense that we need to use this, so that the domain of the function doesn't technically need to include the value at which we are evaluating the limit.

What was the most difficult part of the material for you?
I have a hard time figuring out what to choose as the expression for delta. Just like the expression for N from before, it will probably come easier with practice.

12.2, due on December 3

What is the most interesting part of the material for you?
It's interesting how the harmonic series diverges. It seems like it's right on the line between converging and diverging. I hadn't yet seen a proof of this fact until now, so that was good to see.

What is the most difficult part of the material for you?
The hardest part of the section was understanding the proof that the harmonic series diverges to infinity. Don't we know that if something diverges, it either diverges to negative or positive infinity?

12.1, due on December 1

What was the most interesting part of the material for you?
I am excited to do proofs about calculus. I loved the calculus classes I took, and it will be fun to more rigorously understand the concepts.

What was the most difficult part of the material for you?
I have a hard time figuring out what to use for N in the proofs. As we do more practice problems it will make more sense.

Responses to Questions, due on November 25

What have you learned in this course?
In this course, I've learned to become more rigorous in my mathematical understanding. In calculus we focused less on rigor, so this class has been useful in that regard. I have become much more precise and articulate when communicating mathematical ideas.

How might these things be useful to you in the future?
The mathematical rigor I've gained from this class will be essential in the upper-level math courses I take in the ACME program. Being able to show something is true, instead of just applying given theorems, gives a much deeper understanding of the idea.

11.5-11.6, due on November 23

What was the most interesting part of the material for you?
I think the fundamental theorem of arithmetic is really cool, because it makes so much sense and makes comparing numbers much easier. Prime factorization is interesting.

What was the most difficult part of the material for you?
I had a hard time understanding the proof for Corollary 15. Using induction for this doesn't make sense to me, because I don't see why we know the base case works.

Pre-Test post, due on November 21

Which topics and theorems do you think are the most important out of those we have studied?
The most important topics and theorems probably include: denumerability, countability, cardinality, the Schröder-Bernstein Theorem, the equivalence of the GCD and the least linear combination of two numbers, and the division and Euclidean algorithms.

What kinds of questions do you expect to see on the exam?
I expect to be asked to prove several theorems about various topics covered in this section. I expect to be asked to show that two sets are of equal or unequal cardinality, both for finite and infinite sets. I need to be able to show that numbers of varying definitions are either composite or prime.

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.
I need to practice the Euclidean algorithm and go over the proof of that method, so I can better understand why it works. I need to go over proving statements comparing the cardinality of sets. I need to go over the proofs for the theorems we might need to prove on the test.

11.3-11.4, due on November 19

What was the most interesting part of the material for you?
It seems like Theorem 11.7 makes it much easier to find greatest common denominators. It seems like it would be easy to plot and find three-dimensional minima for the equation for linear combination. Why didn't we learn this in high school algebra?

What was the most difficult part of the material for you?
I have a hard time understanding Lemma 11.9. I don't see why the Euclidean Algorithm works the way it does.

11.1-11.2, due on November 17

What is the most interesting part of the material for you?
I think it's cool that we get to study number theory. I've always heard from math teachers in high school about number theory but I never understood what it was or had the chance to study it.

What was the most difficult part of the material for you?
I had a hard time following the proof of the Division Algorithm. Specifically, it didn't quite make sense how the book shows that q and r are a unique set of a quotient and a remainder for an integer.

The rest of 10.5, due on November 14

What was the most interesting part of the material for you?
It was fun to read about how the various theorems in this section were proved in pieces and then completely over a long period of time. Also, it makes a lot of sense that the power set of the natural numbers and the set of real numbers are numerically equivalent, so it's cool to see that that's true.

What was the most difficult part of the material for you?
I have a hard time understanding the Axiom of Choice. It seems rather trivial to my understanding, so I'm sure I don't quite understand what the statement means. I'll enjoy talking about it in class today.

10.5 up to Theorem 10.18, due on November 12

What was the most interesting part of the material for you?
I think the Schröder-Bernstein Theorem is cool, because it allows us more flexibility in making comparisons between the sizes of infinite sets. This is very similar to showing sets are equal by showing they are subsets of each other.

What was the most difficult part of the material for you?
I don't quite yet understand the proof for the Schröder-Bernstein Theorem. When we go over it in class, I'm sure it will make more sense. The proofs for the other theorems in this section make sense to me.

10.4, due on November 10

What is the most interesting part of the material for you?
The Continuum Hypothesis is sort of mind-blowing. Even more mind-blowing is the fact that it's been shown to be impossible to prove or disprove. I don't understand how that works, but it would be cool to find out.

What is the most difficult part of the material for you?
I didn't understand the second half of the proof of Theorem 10.15. I got a little confused by all the seemingly arbitrary set definitions used in the proof. We'll talk about it in class and it will make more sense when we walk through it.

10.3, due on November 7

What was the most difficult part of the material for you?
I was a little confused by the first proof in the section that talked about how (0,1) in the real numbers is uncountable. It seems like all we did was give an irrational number as proof.

What was the most interesting part of the material for you?
I think it's cool that (0,1) in the real numbers and the set of all real numbers are numerically equivalent. This is analogous to how the natural numbers are numerically equivalent to the integers. In both examples, it seems like the first set is of a smaller size than the second, but it's not.

10.2, due on November 5

What was the most difficult part of the material for you?
I had a hard time understanding why the crossing of two denumerable sets is also denumerable. It seems like that's kind of like taking infinity times infinity. But after looking at the proof it makes more sense.

What was the most interesting part of the material for you?
I think it's way cool that we can look at numerical equivalence for sets that are infinite. It kind of reminds me of using l'Hospital's Rule to simplify limits with infinity on top and bottom. It seems like you're sort of comparing the "size" of the top and bottom of the limit.

10.1, due on November 3

What was the most difficult part of the material for you?
I don't really understand why Theorem 10.1 is important. I can see how it's important to be able to show that you can create a bijective function between to numerically equivalent sets, but I don't see why we need to know that this relation is an equivalence relation.

What was the most interesting part of the material for you?
I completely understand the part from above now! It seems like the point of the theorem is to show that numerical equivalence is reflexive, symmetric, and transitive! It totally makes sense after reading it again.

Pre-Test post, due on October 31

Which topics and theorems do you think are the most important out of those we have studied?
I think it's most important to remember the regular and strong principles of mathematical induction, because they are from the very beginning of the unit and we may have forgotten the specifics. It's also very important to remember what makes relations equivalence relations or functions. Finally, being able to prove something is injective or surjective is very important.

What kinds of questions do you expect to see on the exam?
I expect questions asking me to prove statements through induction, as well as questions that ask whether some relation is reflexive, symmetric, or transitive. Also, there will be questions about proving statements that deal with the integers mod n and their equivalence classes. We will need to show when a relation is and isn't a function, and whether it is injective or surjective. There will be questions about composite and inverse functions, and maybe a few dealing with permutations.

What do you need to work on understanding better before the exam?
I need to review inductive proofs, and the definitions for relations, equivalence relations, functions, equivalence classes, etc. I need to remember what it means for a set to be well-ordered.

9.6-9.7, due on October 29

What was the most difficult part of the material for you?
I didn't at first understand what the section was talking about when it described how we can create a new function that has the same rule but different codomain, so that it then has an inverse. The example of e^x and ln(x) really helped me understand this.

What was the most interesting part of the material for you?
I liked learning about permutations, because I already learned about them on a surface level in high school and it's interesting to look at them from a more complex mathematical perspective. Permutations here apply in a lot more places than permutations as I learned in high school.

9.5, due on October 27

What was the most difficult part of the material for you?
I had a hard time understanding how composition of three functions could be associative. I looked at an example and it made a lot more sense because I realized it doesn't matter which function you "plug in" to the other function first, because the order is the same.

What was the most interesting part of the material for you?
It's interesting to think about when operations on functions make sense and when they don't. For instance, multiplication and addition make good sense on the set of all real numbers, but when we have an arbitrary set of elements we couldn't say that one element multiplied by another is an element of the set.

9.3-9.4, due on October 24

What was the most interesting part of the material for you?
I liked learning about one-to-one functions because from calculus I remember that the inverse of a one-to-one function is still a one-to-one function (when the functions are on the real numbers). It's cool that we can prove functions to be one-to-one.

What was the most difficult part of the material for you?
I didn't get the definition for surjective functions at first because I couldn't remember the relationship between codomain and image. I think an easier definition for a surjective function is when the codomain is equal to the range.

9.1-9.2, due on October 22

What was the most difficult part of the material for you?
For a minute I didn't understand the difference between a codomain and a range. A codomain is the set of objects that could be in the range (like all real numbers for f(x)=x^2).

What was the most interesting part of the material for you?
It's cool to think about the set of all functions from one set to another. Usually when we talk about functions we talk about functions with infinite domains. But when the domain and codomain are finite, it actually does make sense to talk about this, and talk about the length of it as well.

8.6, due on October 20

What was the most interesting part of the material for you?

Understanding when an operation is well-defined seems like an interesting topic, because it's interesting how the definitions of addition and multiplication we've used are well-defined. It seems intuitive at first, and the proofs help it to make more sense.

What was the most difficult part of the material for you?

I don't understand why alternative operation definitions would be useful, especially when they aren't well-defined. The example at the end of the chapter seems random and not useful.

8.5, due on October 17

What was the most difficult part of the material for you?
At the very end of the section I had a hard time understanding why not all equivalence relations in terms of the integers mod n have n distinct equivalence classes. The example helped that fact make a little more sense in my mind. Maybe all linear congruence mod n relations have n distinct equivalence classes?

What was the most interesting part of the material for you?
I think it's interesting to study a new definition of congruence modulo n that uses our understanding of relations. I think it brings a deeper understanding of congruence than we had before.

8.3-8.4, due on October 15

What was the most interesting part of the material for you?
Theorem 8.3 was interesting to me because that the equivalence classes of R make up a partition of the set R is defined over. It's interesting that equivalence relations have that property.

What was the most difficult part of the material for you?
I had a difficult time understanding why equivalence relations exist that don't exactly have an equation that describes them. I suppose using the definition that we have is the most useful.

8.1-2, due on October 13

What was the most interesting part of the material for you?
This section is cool because relations seem to be a very abstract idea of a function in 2- or 3-space that we deal with in calculus. Now we consider a more general idea; it's interesting to think about how two sets are related at the most basic level.

What was the most difficult part of the material for you?
Understanding what made a relation transitive was difficult in the same way that understanding the truth table of an implication was difficult. For a relation to be not transitive, we have to show an example where (x,y) and (y,z) are elements of R, but not (x,z). So it's actually a stronger statement to say a relation is not transitive than it is to say that it's transitive.

6.4, due on October 10

What was the most interesting part of the material for you?
I thought the proof at the end of the section was interesting, because I didn't quite see how the writer could see where to go with the proof. At the end, it suddenly made sense, and now I understand how to think about these kinds of proofs.

What was the most difficult part of the material for you?
At first I didn't see why we would ever need to use the Strong Principle of Mathematical Induction. But until now I've only seen exercises where it was simple enough to get from the assumption to the conclusion of the induction step. I didn't realize there would be proofs that would require more assumptions to reach the conclusion.

6.2, due on October 8

What was the most difficult part of the material for you?
Sometimes it's difficult to understand how to show that if P(k) is true then P(k+1) is true. There's a certain methodology to knowing how to start these problems that will take practice.

What was the most interesting part of the material for you?
I think it's cool that we can prove things in math both with deduction and induction. I've always thought of experimental science as the only inductive science, and theoretical science and math as only deductive. This is obviously not true.

6.1, due on October 6

What was the most interesting part of the material for you?
I think it's interesting that the set of natural numbers cannot be proven to be well-ordered. I wonder if there is a proof showing that a proof of this is impossible.

What was the most difficult part of the material for you?
These proofs by induction generally contain much more algebra than other types of proofs, so it's easy to get lost in the algebra. I need to make sure I follow each step carefully and correctly as I prove things by induction.

Questions, due on October 3

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

The most important topics include the methods of proofs (direct, contradiction, contrapositive). The basic definitions of sets are also important, as are truth tables and understanding when statements are equivalent.

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

I expect some questions about set operations, truth tables, statements, biconditionals, and proofs by various methods.

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.

I need to be sure to remember theorems like De Morgan's laws, so I can convert statements to simpler form in proofs.

My math question:

Prove that π is irrational.

5.4-5.5, due on October 1

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.

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.

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.

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?

Spending a little more time with the assigned reading might help me better absorb the material and come to class already understanding the content.

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.

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.

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.

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.

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.

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.

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.

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.

Introduction, due on September 5

1. What is your year in school and major?

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.

6. If you are unable to come to my scheduled office hours, what times would work for you?

The office hours currently available will be just fine.