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.