Week 7

     This week we learned about proofs by cases, we reviewed proofs, and learned about algorithms.
     First, we were introduced to the concept of cases. This concept allows for us to split the argument into different cases in order to make sense and add structure to the proof. It allows for us to examine the possibilities that lie within the proof, and then actually moving on to proving them. We then did an example of a proof in which cases were used, and found that cases helped in the organization of the proof, as well as helped me visualize all of its components. We then learned about the mathematical prerequisite of uniqueness, which can be denoted with the phrase "one and only one." It is a type of existential quantification which indicates that there is an argument with a certain property.
We then talked about the proof patterns of introduction (smaller statement implies larger statement) and elimination (larger statement implies smaller statement).  We then learned about the different rules for both patterns by negation, conjunction, disjunction, implication, as well as their universal and existential forms. It is important to practice in order to understand and apply these concepts effectively. I plan to do more practice problems in order to better register this new information in my mind.
     We then learned about algorithm analysis and asymptotic notation. We were introduced to this concept with an example of bubble sorting and merge sorting. We checked which of the two aforementioned methods of sorting was the most effective by analyzing their respective algorithms (the process they use to complete the task). Algorithms are of most importance because they determine the running time of a given program.  We also learned about how we do not really care about the number of steps withing an algorithm, but about how the number of steps increases as the size of the input increases.
    Later, we learned about asymptotic notation, did a few examples with the new greek letters introduced for this type of notation, and then learned how to analyse the time complexity of a program. We learned to make this type of analysis through examples with python and linear searches. I found this part of the lecture interesting as an insight into how programs work.
     In the tutorial we did the tutorial exercises, which I was more prepared for solving because I had asked Professor Zhang for help. I understood the material better, but I was not able to complete the quiz question correctly. I plan to ask for help from the T.A. next time in order to have a more complete understanding of the material during the notation.    

Week 6

     This week we learned about cases, quantifiers, and limits.
     First, we talked about non-boolean and introduced the notation to the floor function. The floor function always rounds down a decimal number to its integer. An important thing I took note of was that the floor of a number is not a variable, it is a function. This means that a quantifier cannot be applied to it since quantifiers are only used on variables. Next, we tried proving a statement with the application of this new knowledge about the floor function. We were able to use its definition as an aid to construct the actual proof. I found learning about the definition of floor of x very helpful since using the definition of a function to construct a proof is something that we will constantly use in the future.
     We then learned how to disprove a statement effectively. In order to disprove a statement, you need to prove its negation. It is important to negate the statement correctly to facilitate the 'disproof'. We then did a couple of examples on disproving statements.
     Next, we moved on to proofs about limits, which seemed like an intimidating concept at first. We learned about the use of the greek letters delta and epsilon. Professor Zhang taught us how to manipulate these letters in our favor, and that the best choices of where one might want to take the proof are often encountered when working backwards in the proof. Also, I personally learned that proofs about limits are not necessarily as intimidating as I thought they were.
    I found the content of this lecture easier to grasp than that of the previous lectures. I believe I still need to do more practice with proofs so I plan to go to Prof. Zhang's office hours and doing practice problems from the course website on my own.
   

Week 5

   This week we learned about proof using contrapositive, proofs using contradiction, proof for existence, and proof about a sequence.
    First, we learned how the contrapositive can be used to construct proofs more effectively. Instead of proving what is asked of us exactly the way it is presented, the contrapositive allows us to solve the problem with it's equivalent. This method works well when the reverse direction of the proof is easier to prove than the original. It is a matter of rearranging the problem into the contrapositive and then continue building the proof, This was shown by Professor Zhang through various examples.
   We then learned how to use the contradiction to construct proofs. The contradiction is a variation of the contrapositive. It happens when you say something is and isn't in the same respect. We then did more examples on how to prove with the contradiction. I am not entirely sure how this works in its entirety, and I plan to review the course notes to better understand it.
   We then did some more challenging example. It had to do with prime numbers, and it was interesting because the proof at the end led to contradiction. I then realized how important it was to always be aware of the contradictions withing the proof by assuming the negation of a statement. This helps avoid said contradictions to make a valid proof.
     We then learned about proofs in existential sequences. How we can make proofs basing ourselves on the fact that for an existential statement to be sustained, a single example of it is necessary. How proofs consist of layers and layers within its structure that can be manipulated within the guidelines of certain logical rules to make valid proofs.
    I found this week's material to be the most challenging, and I plan to review the course notes and see Professor Zhang during office hours to better understand what was explained.
    During the tutorial, T.A. Elias solved the tutorial exercises explaining each step. I still had a hard time understanding the entirety of the problem, so when the quiz came along I was not entirely confident. Yet, I believe I did better than expected because of parts of the explanation that I grasped. I plan to ask more questions during tutorials so I can gain a more complete understanding of how to solve the problems.
 

Week 4

     This week, we learned about bi-implications, transitivity, mixed quantifiers, and proofs. We also had a problem solving session. 

     A bi-implication is a type of condition statement where a statement could be combined with its reverse to make a similar statement. I found this part of the lecture somewhat difficult because of the fact that the topic was new and I had no practical experience with it. We did a few examples in negating bi-implications before moving on to transitivity.

     Transitivity occurs in the relation between statements. When one element is in relation with the second, the second with the third, and the third back with the first one. Through an example on the slide, I was able to realize that transitivity is always true when statements imply one another. 

    We then learned about mixed quantifiers, when different types of statements (existential and universal) are combined and how their existence together affects the nature of a statement. 

    Then we got into the part of the lecture which I understood the least. A proof is used to convince someone else of the truth of an argument logically. In order to make a great proof, one must first understand each part of it, which I think was my issue. We then learned to use structures to construct the proofs. Different types of laws were used for the construction of the proofs. I had trouble applying all these laws because I was not sure how to use them effectively. In order to solve this issue, I will go over the laws and attend Professor Zhang's office hours.

     In the tutorial, we went over the tutorial exercises with T.A. Elias. I still did not understand how to make proofs by the end of the tutorial, so I am not confident about my performance on the quiz.

    This has been the most challenging week for me in CSC165 so far, and, like I mentioned before, I plan to go to the Professor's office hours to get the help I need.