CSC165week4 SLOG:

Review of the quiz#2 question:
    Let C be the set of courses, and P(x,y) mean x is a prerequisite for y .  CSC108 and CSC463 are courses in C.     Write the following symbolically:
    CSC 463 has a prerequisite for which CSC108 is not a prerequisite.
    My wrong answer was:
         ¬∃y∈C,∃x∈C, P(CSC108, y) ∧ P(x,CSC463)
    Basically, I misunderstood the question. My answer would be right if it says 'CSC 463 has a prerequisite and CSC108 is not a prerequisite' . Now I find out that the thing  I am really struggling with is to understand the statement. The phrase 'for which' definitely has a different meaning to 'and', and I knew it, but I still make such mistake. 
    The right answer is:
           ∃x∈C, P(x,CSC463) ∧ ¬P(CSC108,x)
    If I see such questions next time, I will first rewrite the statement into a more simple and more straightforward sentence. Take this question as an example, there exists a course which is a prerequisite of CSC463, and CSC108 is not a prerequisite of the course. In this way, I will have a smaller chance to misinterpret the statement in the future.
 
Lectures and assignment:
    In this week's lectures, I learnt the concepts of proof. I found proof to be very interesting, since it help me to have deeper comprehension about some mathematical rules. For example, the square of an odd number is also odd. I know this rule many years ago, but I only memorize it ,and never think about why this rule is true.

    Based on what I learnt about proof this week, I set up a proof question:
    The square of a even number is even, prove:
    ∀n∈N,  n even -> n^2 even
 
    My solution:
        Assume n∈N #n is a natural number
             Assume n is even
             Then ∃x ∈ N, n = 2x  #definition of n even
             Then n^2 = (2x)^2 = 4x^2 = 2 * 2 *x^2  # some algebra
             Then ∃y ∈ N, n^2 = 4y = 2 * (2y)  # let y = x^2 and we know x^2 must be positive
             Then ∃z ∈ N, n^2 = 2z  # let z = 2y
             Then n^2 is even  # by definition of n^2 even
         Then n is even -> n^2 is even. #from above
         Then ∀n∈N,  n even -> n^2 even  # conclusion

    (P.S.: I cannot ensure this is 100% correct. If you find some errors, please let leave some comments.)