Math 290 Fall Semester

The proof for Proposition 12.1 seemed a little confusing, because it the proposition was a little too self obvious. If S is a subset of T, then wouldn't it be automatic that S is also a subset of S union T? And how can S be a subset of itself? It's just a little confusing when you have an implication that is the conclusion of another implication. The proofs otherwise seem very straightforward, and for the most part you can work contrapositively or directly, which is helpful because those are the two most simple ways to write out proofs.

The hardest part for me to understand was the proof for proposition 11.7. I don't quite understand how exactly that proved the statement, and it was a bit confusing because it wasn't necessary to know which case was true. In the other proofs we have done with cases, you have to know that each case could happen, and then prove that no matter what, each case would yield the same result. Proofs by examples and counterexamples makes more sense to me because it seems more intuitive to just give an example to show if something works or doesn't work, rather than just use completely arbitrary numbers and variables. With these kinds of proofs all it requires is looking at the statement and figuring out if it "looks" right in a sense. It's more logic based, rather than computational.

1. I had a hard time with the proofs and the meaning of tautology. The proofs were easy before because it was mostly just using the definition of divides, congruence, and even/odd. It's a bit trickier now because now I have to use the definition of sets, subsets, intersections, unions, and complements. I'm just not sure what I need to define or explain in a proof to be able to get full credit, and the flow of the proof is a bit more unclear now. 2. I'm starting to connect the dots on how everything we have learned so far is coming together. At first it seemed a bit strange that we spent so long talking about sets, then jumped to logic, and then started working on proofs. But now I see that it's all intertwined with each other. Other questions; I usually spend about an hour or hour and a half on each written homework assignment. The reading and lecture both helped me understand the homework better, but I would say the lecture was more helpful. I understand a concep...

I really enjoyed listening to Dr Schnieder speak on 9/20/18. He is a very captivating speaker and made the subject matter more interesting. His method of introducing the subject was good too. He started with the history of number theory as it related to prime numbers in order to illustrate the concept for his song. Including the sieve in his music was fascinating too; I really doubt I would have picked up on that just listening to the song myself. It really showed me how math plays an instrumental (no pun intended) role in music and how our minds really crave that numerical perfection that music offers.

1. The premise that we are working in an "imaginary world" was a bit strange to wrap my head around. I understand it well enough when working on basic even/odd proofs, because you are just assuming the statement is false and then working to find a contradiction, but with the more complicated ones, such as the rational number proof in the textbook, it was a bit difficult because I don't know what I am looking for in a contradiction. The lowest terms contradiction seemed a bit contrived and I'm not sure if I would be able to locate something like that in a different example. 2. It's nice to have another way of writing proofs because I can think of examples that would be very difficult if not impossible to prove by using only direct and contrapositive proofs. Even though this method is a bit less concrete and difficult to understand, I can see how it would be useful in proving more difficult statements.

1. The modulo concept did not make a lot of sense to me. In the Python lab I learned that modulus just means taking the remainder after division as the answer, but I did not grasp how this was working in the reading or how it was being computed. Once someone else explains it to me in English I will probably understand it better. 2. The definition of absolute values seems critical in these proofs. I already knew what absolute value meant, but it now has a deeper meaning seeing how I can use the equation |x|=-x for all x<0. using this in both directions will help with many proofs in the future.

1. The section where it wanted us to find out what statement the proof was proving was confusing, I had a hard time when there were two variables x and y, and understanding which variables corresponded to what. 2. The idea of contrapositive proofs made sense to me otherwise. In some cases it's much easier to assume that the conclusion is false and therefore the premise must be false, rather than trying to do a direct proof that the premise is true. It seems like this will be a useful tactic to remember when writing other proofs in the future.

1. The most difficult part of this section for me to understand was example 5.15. Is S  a set, or is it a range of numbers? How can you take x=0 or x=-3 to determine if it has a lower bound if the range is (0, 1]? And wouldn't the lower bound be 0? Why does it not have a least element? The whole concept of lower bound and least element was a bit foggy for me to understand. 2. I think that it's very handy to remember that when you are negating a compound statement, all you have to do is change the quantifier and the sing of the statement P. Understanding logical equivalences really helped me cement this idea and see how to "distribute" a negation of a statement.

1. The hardest part about this section was translating mathematical statements from symbolic language to normal English, and vice versa. For the most part the notation makes sense, and I don't have a problem with it, it just requires a lot of out of the box thinking. Translating from symbolic to English is easy enough because I can derive the meaning, but I feel like I would have a much harder time trying to translate a statement. 2. It's cool to see the connection between the logic side of math and the computational side. I wasn't quite sure how they would connect to each other but when you plug in certain values of x for P and Q, then it makes sense and gives a very cool and important application to the truth values of statements.

1. The most confusing thing in this section was the part where they explained how to build a truth table. It seems intuitive enough on its own; you just take every possible truth value for P and Q, determine if each compound statement including P and Q is either true or false, and then compare the resulting truth values. Where I got lost was where it explained the efficient way to organize it by rows until you had 2^n rows. I didn't get what it was saying. If I could see someone else doing it then I might be able to understand it better. 2. I thought the table of equivalences was fascinating. It's really cool how they can distribute, and the proofs made total sense to me. It's good to know as well because this will come in handy later when I am inevitably dealing with incredibly complicated statements because I can look at these generic examples and be able to efficiently determine if certain components are true or false.

Section 1: 1. The most difficult concept to me was understanding the empty set. On example 1.26, when defining the power set of U, it listed the empty set both by itself and in brackets, and I thought that it already had brackets implied because it itself is a set, so is that saying the power set of U has the empty set as well as a set containing the empty set? I also am having a tough time remembering how complex numbers work, but I think that is mostly due to the fact that I have not used them in several years. 2. What I found the most interesting in the reading was the section about unions, intersections, differences, and complements. I thought it was really cool how the Venn Diagrams illustrated how you can calculate cardinality of a set based on the complement and the other set. It seems like that will really be built on as I learn more about sets. Section 2: 1. Most of this section seemed clear to me, but there were still a couple things that are unclear. With the ex...

My name is Benson Merrill. I am a sophomore at BYU and have been a student since Summer 2015. I enjoy running, playing ultimate frisbee, hiking, playing video games, being with friends, and watching movies. I returned from my mission in England almsot a year ago. I do not currently have a declared major, although I am considering Mathematics. At the very least I am planning on getting a minor in mathematics and seeing where that takes me as far as a major goes. I took Calculus AB and BC in high school and Math 113 at BYU as a refresher once I returned home from my mission. I'm taking this class and this cohort because in the past I have done well in and enjoyed my math classes, and I was interested in seeing where a degree in math could take me, especially since I still didn't know what I wanted to major in. The coding and Python aspect of the cohort also caught my attention because I had a lot of friends say that it was a great major, so I will be looking into that as well. ...