Journal 5 - Week 6

Friday   October. 17th   Sunny

This week, I have got my mark for test 1 and it is 38/38. Much appreciated for my TA and professor’s help. I will continue to show my effort. Due to thanksgiving day, our tutorial for this week was cancelled, so I would just talk about contents covered in our lectures in this journal.

Firstly, as usual, I will introduce something new to us in this week. We learned more typical examples for proofs. They are proof about non-boolean functions, proof something false, and proof about limits. All of them are interesting because they have relationships with mathematics. As I have said, I am a Mathematics student, so this really attracts me a lot.

A brief but pretty good slog about proofs which was posted by my classmate:
http://csc165progress.blogspot.ca/2014/10/a-short-week-in-review-more-proofs-and.html

Secondly, I will talk about the most challenged topic for me this week and how I am trying to make it easier. We always pay attention to how to “prove” a statement. However, starting from this week, we will not just “prove” but also “disprove”. For those false statement, we need to negate the whole sentence, and then prove this negation. This new method for proof is kind of difficult for me, so I did a lot of exercises about proof something false and feel much better for now.

Thirdly, I chose one of the proof questions in my exercise, which follows as below.

Q: Prove or disprove: ∃x∈R, ∀y∈R, y > x.

I will not use graph this time, since obviously we need to disprove it.

Let’s negate the whole statement first.

¬(∃x∈R, ∀y∈R, y > x)

⇔ ∀x∈R, ∃y∈R, y ≤ x

Now we can prove the negation.

Proof:

Assume x∈R.

       Let y = x = -1.

       Then y∈R.   # since x, -1 ∈ R, R closed under addition

       Then x - 1 ≤ x

       Then y ≤ x

       Then ∃y∈R, y ≤ x

Then ∀x∈R, ∃y∈R, y ≤ x

We have done!

Again, no tutorial this week, so this is the end of the journal.

Much appreciated for reading my journal.

See you next week :)