SLOG #3

    This week's material is not as difficult for me to understand compared to that of the previous two weeks. The idea of transitivity is quite easy to accept because it is easy to illustrate "if A is a subset of B, and B is a subset of C, then A is a subset of C" using diagrams. I can draw three different circles of varying sizes and call them A, B, and C. Then if A is inside B, and B is inside C, it naturally follows that A is also inside C. The concept of mixed existential and universal quantifiers should have particular orders in a claim was also relatively easy for me to grasp, mostly because it was discussed in my second tutorial for this course.  A specific question asks to translate the English statement "every course has a prerequisite" to symbolic expression. The answer is, for all x in C, there exists a y in C, C(y,x). A student asked whether it would be correct if the order of the quantifier was switched. I must admit I was actually shocked to hear the teaching assistant say "no". To me, the quantifiers originally seemed to act like phrases that set up conditions for a claim, so of course the order in which the conditions are listed shouldn't matter. Then, as the teaching assistant explained in detail of why the following a particular order is important, it became clear to me that there indeed is a difference between "all courses have a prerequisite" and "a course has every course as its prerequisite".

    Although I reflect this week's material was picked up easier than that of the previous weeks, it isn't without any trace of obstacle. For example, I find the process of proof a challenging task for me, mainly because while it is easy to describe, vaguely, in words why some patterns or sets behave the way they do, it is a totally different story when I try to write it out in precise words or symbols. In the lecture, we examined the process of proving the square of an odd number is odd. At first glance, the statement seemed blindly obvious to me, but I struggled to find a precise and convincing way to express why and how that statement is true. The professor had to go through a meticulous, or even cumbersome process of assuming domain, defining natural odd number, defining natural even number,incorporating rules of adding and multiplying natural numbers, and finally substituting a single variable for an operation between two variables to mirror the definition of an odd number, all to prove a very simple statement. I must imagine how much more complex the process would be to prove a not-so-obvious statement, and it honestly overwhelms me.