Second Entry - Week 3

The course material has gotten a little more challenging this week. Going into the tutorial, I was a bit confused regarding when in symbolic statements one should use a comma, and when one should use an implication sign (=>). While doing the preparatory exercises for the tutorial, I was under the impression that every statement had to include an implication sign, and was not entirely sure where commas were supposed to be placed. Fortunately, the tutorial gave me a chance to get clarification from my TA about this, and I have a much better understanding of how to write statements properly; the implication sign is used in if/then situations, while a comma is used when making a definitive claim about a relationship between variables. I personally found the answer to question 1f in the tutorial problems,
∀x∈C ∀y∈C ∀y2∈C ∀y3∈C, (P(y1,x) ∧ P(y2,x) ∧ P(y3,x)) => (y1 = y2 ∪ y1 = y3 ∪ y2 = y3)
to be very illustrative of this, and it really helped me to understand the concept. The comma indicates that a claim is being made about a relationship between the four variables, and the implication sign indicates that if the antecedent is true, then the consequent must also be true.

During the paper-folding exercise in Friday's class, Professor Heap's hint that we should "think recursively" really opened my eyes to the problem. Just before, I had been drawing physical diagrams of the paper to try to figure out what in the folding process determined whether a vertex pointed up or down. After hearing that we should think recursively, I realized that there was a distinct, repeating pattern in the position of up and down vertices. Though it is difficult to explain in words, observe the following (kind of awkward) diagram of vertices:

(1st fold)                                D
(2nd fold)                U            D             D
(3rd fold)         U     U     D    D     U      D     D
(4th fold)     U U D  U U D D D U U D   D U D D

The diagram is a triangle, and as you make more folds, the triangle gets larger. Interestingly, you can always build the next level of the triangle by manipulating the previous level. Take the third level of folding, UUDDUDD, and compare it to the second level, UDD. The ending of the third level, UUDDUDD, is the same as the second level. The start of the third level, UUDDUDD, is the second level with the positions and direction of the vertices reversed. The downward vertex in the center, UUDDUDD, remains constant. This is an awfully strange formula, but it holds true for the 4th level of folding; it starts with the third level with positions and direction reversed, UUDUUDDDUUDDUDD, retains the down vertex in the center, then repeats the third line, UUDUUDDDUUDDUDD. I think this is what Professor Heap meant by thinking recursively; by applying the same process to each line, you can produce the following line. Even though lines get longer and longer the more folds you make, these are just larger iterations of the same basic problem. I find all of this a really interesting way of thinking. On a side note, it also reminds me of Pascal's Triangle; the idea of creating the next line of the triangle by manipulating previous ones. I wonder if there could be some relation.

First Entry

After two weeks in CSC 165, I am already noticing that there are very strong connections between the course material, and the material I have been learning in MAT 137. So far, both courses have put a lot of emphasis on universal and existential claims; what they are, and how to prove or disprove them. Often, I have been able to immediately apply what I learned in MAT 137 to the lesson in CSC 165, which has really helped me to understand the material in both courses; back on Monday, I learned about negation in math, which made today's lesson on implications, converses, and contrapositives much easier to grasp. We have also learned about claims related to the empty set, which I think will be very helpful in tomorrow's lesson about vacuous truth. I found the work in the tutorial to be pretty straightforward, so overall, I'm feeling confident in my understanding of the material so far, and eager to see what's next.