Journal 6 - Week 7

Friday   October. 24th   Windy

Hi everyone. Most of us have got our mark for assignment 1. The average is very high. Well done! Meanwhile, assignment 2 has been posted. Let’s show more effort. In this journal, I will talk about both contents covered in our lectures and tutorial for week 7.

Firstly, as usual, I will introduce what we have learned this week. Professor taught us proof by cases and then review all the other proofs we have talked about before. Also, we starts a new topic which is “algorithms”. For those who are taking of have already taken CSC148H1 would be quite familiar with algorithms.

Secondly, I will talk about the hardest part this week and how I am trying to make it easier for myself. I think proof by cases is a challenge for me. It is not as straightforward as the methods we learned before. For example, if we are not sure the number we are using is even or odd, at this time, we need to prove in each case. I usually forget to divide them into two cases. To deal with this problem, I did a lot of exercises about proof by cases and feel much better for now.

Thirdly, I chose one of the proof questions in my exercise, which follows as below. Since I have talked about the cases when we can not decide if a number is even or odd, I want to introduce another type of proof by cases question.

Q: Prove: ∀x, y∈R, x⁴ + x³y - xy³ - y⁴ = 0 ⇒ x = y ∨ x = - y

Proof:

Assume x, y ∈ R.

       Assume x⁴ + x³y - xy³ - y⁴ = 0.

              Then x³(x + y) - y³(x + y) = 0

              Then (x + y)(x³ - y³) = 0

              Case 1: Assume x + y = 0.

                     Then x = y ∨ x = -y

              Case 2: Assume x³ - y³ = 0.

                     Then x³ = y³

                     Then x = y ∨ x = -y

              In both cases, x = y ∨ x = -y

       Then x⁴ + x³y - xy³ - y⁴ = 0 ⇒ x = y ∨ x = - y

Then ∀x, y∈R, x⁴ + x³y - xy³ - y⁴ = 0 ⇒ x = y ∨ x = - y

Lastly, our tutorial of this week help us do the complete proof, not just proof structure any more. One of the most interesting one is question 1/(b). In this question, both of the antecedent and conclusion are existential statements, which means when we make an assumption for the antecedent, what we actually assume is an existential statement, not just a number. This is a trick point we need to pay attention on test.

This is  the end of the journal.

Much appreciated for reading my journal.

See you next week :)