As I said earlier I am going to post the solution for one of A1 questions. I chose the question 3, cause for me, it was more difficult than the others. Well, actually, 4th question seemed at first harder, but when I understood what exactly I needed to prove it  became not that hard. For the third question, I had a little confusions about the partition part of a proof, after going to prof's office hours(which were a looot helpful, by the way) I got how partition worked for this proof, though. So, here it is.

3. P(n): The number of 3-subsets that a set of n +3 elements has is [(n + 3)(n + 2)(n + 1)] / 6
Proof (by Mathematical Induction)
Base case: n = 0. A set of 3 elements has (3 x 2 x 1) / 6 = 1 3-subset. So holds for P(0).
Induction Step: Assume n ϵ N (generic) and that P(n) is true.
    Suppose S is a generic set with |S| = (n + 1) + 3 elements. Now there is some w ϵ S, and we have some subsets of S that do contain w, and some that do not. Say I‾ is the 3-subsets of S that do not contain w, and I+ is the 3-subsets in S that do contain w.
    Number of 3-subsets in S is number of 3-subsets in I+ plus number of 3-subsets in I‾. At the same time, number of 3-subsets in I+ is equal to the number of 2-subsets in a set with |S| - 1elements, since they match. Also, I‾ is equal to the number of 3-subsets in a set with |S| - 1elements.
    Know that a set with n + 2 elements has [(n + 2)(n + 1)] / 2 2-subsets. Using this formula and IH, find that the set with (n + 1) + 3 elements has ([((n + 1) + 2)((n + 1) + 1)] / 2) + ([(n + 3)(n + 2)(n + 1)] / 6) = [(n + 4)(n + 3)(n + 2)] / 6
            = [((n + 1) + 3)((n + 1) + 2)((n + 1) + 1)] / 6 3-subsets
            So P(n + 1) follows.
Since assumed n to be generic positive natural number, ∀ n ϵ N, P(n) ⇒ P(n + 1).
Conclude ∀ n ϵ N , P(n)
 

Hi guys, it is a Friday night, and guess what I've been doing last several hours?? I have been doing my assignment on Statistics lol It seems like there is no time free of studying in UofT for me :) 
I have not been posting anything for about 10 days, because of all of the midterms, assignments, quizes that was going on last two weeks. But now, knowing my mark for the termtest, and two quizes, I actually have an idea about the course.
I start from the quizes, they actually are like free marks. I really do agree that having quiz once a week can help students to keep track of what's going on. I think that sometimes it is not enough time for them, though. But overall I think the quizes are effective and helpful in different ways.
The midterm was not as hard as I thought it would be. So, I was preparing myself for something like an assignment. Each problem seemed to be easy, but, unfortunately, I had a mistake that didn't let me me to get a full mark. From the start of a course I was wrongly assuming that our P(n) includes the Universal quantifier, which, and now I understand why, it doesn't.
The assignmnet 1 was more challenging, but the Danny's office hours was very helpful. I am actually waiting for a mark, so I can undoubtly post a solution for one the questions from assignment :)

Intro

Here comes my first blog-slog! I am actually absolutely new to the world of blogs, but I will try my best to express my thoughts about the CSC236 course. My name is Madina by the way, I am the second year computer science student.You know, there is a feeling inside when you take some course and it seems to be really hard. Time passes, and after all the finals, after your term mark is already up, you realize that it was actually not so hard, sometimes you even think it was easy. That is what almost all the time happens to me. I hope I would have the same feeling for this course, cause this would mean that I really learned something new.
First time I leaned about mathematical induction was in high school. We did the sum of the first n natural numbers kind of proofs. And it seemed really easy. I was not much interested why P(n) ⇒ P(n + 1) worked, but it always did. Now, having more complicated problems, learning that inducton has different kinds, and understanding why it works, induction is becoming more interesting!