Proof and Disproof of Big-Oh Statements About Polynomials

On week 9, we learned how to prove and disprove big-oh statements about polynomials. To disprove an big-oh statement, the trick is that we need to negate the statement and find an n which  consists with B and c. In some case, we have to use limit and L'Hopital's Rule in order to pick a correct n.

Big-Oh

On week 8, our professor introduced the concept of Big-Oh. However, I missed Monday's lecture because I went to a wrong classroom. I noticed I was in a statistic class after class began. I was too shy to get out. At least, I understood what that professor said and I thought I learned that kind of things in data management course in  high school. I was smart enough to understand the annotated slides on course website, but I was pretty sure that I missed something on Monday. That might be very important...

Inference and Counting Lines

On week 7, we summarized some rules of inference. I was familiar with them because I had used them for several weeks. After that, our professor introduced some basic sorting strategies and explained how to count lines or steps with his example of linear search.

I did not use the way I learned to count steps on the quiz because the numbers were small and I thought to count directly was more efficient. But I wouldn't do it again anyway because I did not get full mark.

Proof by Case and Disproof

On week 6, I learned how to split a statement and prove it step by step. It is an useful technique to solve some problems. Later on, we covered the concept of disproof. It was about proving a claim false. To do so, we first negate the claim, then we prove the negation true. We only had 2 lectures due to the Thanksgiving Day.

Proof and Floor was Introduced

On week 5, our professor started introducing proof. Proof in this course had rigorous structure. It was easy to remember and pretty straightforward, but some other concepts came in and raised the difficulty. For example, floor had the definition consisted with three basic tools. They were the only ones we could use to proof something related to floor.  One of these tools was an implication, meant that it had a contrapositive. Sometimes this contrapositive was required in our proof. Whenever I was stuck I would think about this. And it always took a long time to solve this kind of proof by using these tools. On the assignment 2, there was a statement which looked like epsilon-delta implication and included floor. It took me a whole weekend to do the proof. Unfortunately, It was a wrong statement and I proved it true. On the test 2, I was unable to prove two floor statements.

Gotta review it very hard.

Mixed Quantifiers and Epsilon-Delta Implication

On week 4, I learned mixed quantifiers. The order of quantifiers is important.
∀ a ∈ A, ∃ b ∈ B, a+b = 5 and ∃ b ∈ B, ∀ a ∈ A, a+b = 5 are different. But if quantifiers are same, then two implications have same meaning.  Epsilon and delta were also introduced this week. They bound x-axis and y-axis respectively. Although I had learned the concept in MAT137, epsilon-delta implication still confused me a little bit. Anyway, both courses covered it and this really helped me understand it.

Learning How to Write Proofs

This week, proof was mentioned quite a lot in both MAT137 and CSC165. My professor of MAT137 taught us how to use proof as a tool for learning mathematics. Unfortunately, I struggle to write or even to understand a proof.  The main reason might be I did not do a lot practice related to proof in high school. Therefore, it is not surprised that I feel uncomfortable with it. In the past few days, I was seeking a way to overcome the difficulty. I experienced some memorizing and tried to recall it few hours later. But it was so hard and stressful to remind the process of proofs. Maybe I did not truly understand each proof, or maybe it took far more time and require more effort than what I expected. However, a professor inspired me during the lecture. It was all about the way of thinking. The reason that I forgot how to prove what the professor had proved at the end of each lecture is because I was not using a easy and efficient way of thinking. If I think geometrically when I write a proof, I can get more comfortable with some calculus problems. Since images are visualized in the head, it is more likely to have a clear thought to prove something. Anyway, it always takes substantial effort to master math.