SLOG #2

    I have found this week's tutorial problems to be particularly difficult for me to answer. The last few questions involving translating from English to symbolic expressions have caused so much confusion for me, mainly due to my lack of understanding of when to use existential quantifiers, universal quantifiers, conjunction, disjunction, and implication. For example, the expression "there exists an x in set C, for all y in set C, NOT P(x,y) (for some reason, incorporating symbols in this blog severely butchers the formatting, so I am unable to present a purely symbolic expression; I apologize for the inconvenience) originally appeared to bear two meanings for me. The first one is: there is a course (x) that is not necessarily the prerequisite for all courses, which means it could be the prerequisite for at least one course. My second interpretation was: there is a course (x) that is not the prerequisite for any course. It was after talking with professor Heap that I realized "for all" should be understood as "for each". Therefore the expression contains the second meaning.
    Another problem that I found difficult to crack is stating the truth values for a symbolic expression, whether it is true in one direction or false both directions. In particular, the expression "for all x in D, P(x) is true OR Q(x) is true <==> (for all x in D, P(x) is true) OR (for all x in D, Q(x) is true)." I understood why the statement is true when read from right to left, but I couldn't comprehend why the statement is false from left to right. The main reason for my inability to come to the correct solution was my method of representing the statement. Instead of trying to come up with specific examples in specific domains, I used venn diagrams to substitute the statement. After talking with professor Heap, it turned out that the right side of the statement really expressed three separate diagrams. In other words, it is not to my best interest to compare one diagram to three diagrams and state whether they are equivalent or not. I realized that I should have used specific examples, such as natural numbers. So, from left to right, I should say, "if all natural numbers are even or odd, then all natural numbers are odd, or all natural numbers are even", which is clearly false.