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.
Friday, September 26, 2014
Friday, September 19, 2014
SLOG #1
This week has been both exciting and confusing for me. I very much enjoyed learning and absorbing new information in class, but was occasionally perplexed by the unusual concepts. I say "unusual" because the concepts differ from everyday common sense. Specifically, I had difficulty grasping two ideas discussed in lectures.
The first idea is verifying universal claims. The notes explain "to verify a universal claim, show there is no counter-example". The explaination looks great at first glance. If I want to verify all the coins in my wallet are quarters, I should look through them and find no penny, nickel, dime, loonie, or toonie. Simple enough. However, I could not accept the idea that empty sets consist of no element, therefore no counter-example, resulting in a situation that all universal claims are true. If I applied that to the coins-in-my-pocket example, it would be equivalent of claiming, with correctness, "all the coins in my pocket are quarters", despite my pocket being empty. In order to wrap my head around this I had to carefully look over the lecture notes once again, and understand that "to verify a universal claim, show there is no counter-example" is not just stating the obvious. It is defining a method of verification.
The second idea is implication. In particular, the lack of causality in "if... then..." statements and the "vacuous truth" aspect. I must admit that I was completely baffled when I read the notes "If you eat your vegetables, then you can have dessert.... in logic, we allow the case where you don't eat your vegetables and still eat dessert to be consistent with the implication." I kept wondering, for that specific example, how one can be allowed to have dessert without eating vegetables. Only after reading some course notes pertaining to "vacuous truth" did I begin to accept the above concept, that regardless of the consequent, so long as the antecedent is false, the implication will ring true due to the empty (true antecedent) set's property of belonging to any set (true set for consequent).
The first idea is verifying universal claims. The notes explain "to verify a universal claim, show there is no counter-example". The explaination looks great at first glance. If I want to verify all the coins in my wallet are quarters, I should look through them and find no penny, nickel, dime, loonie, or toonie. Simple enough. However, I could not accept the idea that empty sets consist of no element, therefore no counter-example, resulting in a situation that all universal claims are true. If I applied that to the coins-in-my-pocket example, it would be equivalent of claiming, with correctness, "all the coins in my pocket are quarters", despite my pocket being empty. In order to wrap my head around this I had to carefully look over the lecture notes once again, and understand that "to verify a universal claim, show there is no counter-example" is not just stating the obvious. It is defining a method of verification.
The second idea is implication. In particular, the lack of causality in "if... then..." statements and the "vacuous truth" aspect. I must admit that I was completely baffled when I read the notes "If you eat your vegetables, then you can have dessert.... in logic, we allow the case where you don't eat your vegetables and still eat dessert to be consistent with the implication." I kept wondering, for that specific example, how one can be allowed to have dessert without eating vegetables. Only after reading some course notes pertaining to "vacuous truth" did I begin to accept the above concept, that regardless of the consequent, so long as the antecedent is false, the implication will ring true due to the empty (true antecedent) set's property of belonging to any set (true set for consequent).
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