"Interesting" being a relative term

The deadline for A1 has passed, and I feel pretty confident about the answers I submitted. I did notice a few interesting discrepancies between the sample solutions and mine though:

1) The sample solutions consisted of fewer than half as many lines as mine, which reaffirms my fear that my proofs are too long. I think this was partially - in question 3 specifically - from my proof taking a circuitous path. In general though, I may have tried to be more precise than was necessary. I think switching to a more, um, prosey (my proofs are anything but prosaic) style cut down on their size a good bit, but I'm still working on cutting out more redundant or obvious stuff.

2) My solution to question 2 was completely different from the sample solution, which I thought was pretty cool. The technique in the sample solution is a bit easier to follow, but I like mine (which consisted of splitting the nth cycle into 2^n-1 pairs, and inserting a pair of new values in the middle of each one) just fine. I'd be really curious to know how many people used my solution, and how many used the other, and if there are even more techniques that work that people submitted.

3) The solution to question 3b made me realized that I had made the careless mistake of conflating "cannot be expressed as a ratio of natural numbers" and "irrational". This didn't actually meaningfully affect my proof, which would apply if I went back in time and replaced the latter term with the former, so I hope I won't be punished too harshly for my slip-up.

Also, no matter how many times I look at the solution to question 3b, I can't make sense of it. I can't decide whether it's more likely that this is the result of a mistake or transcription error on the part of the author, or if I'm just missing something stunningly obvious.

Speaking of missing the stunningly obvious, I almost fell for the pseudo-proof of the six-sidedness of hexagons during yesterday's lecture. It wasn't until after a minute or two that I thought to myself "it sure is neat that he proved that without using any properties of hexagons", and then the inevitable "wait a minute...". I'm embarrassed to admit that I briefly fell for a proof that could easily be adapted to "prove" that all horses are the same colour.

Rethinking my proof style

I was surprised by the sample solutions that were posted recently for Problem Set 1. I had assumed that the proofs we were doing would follow the format from 165 (Assume, indent... then... then... assume, indent... then... then... dedent... then... dedent... then for all...). However, the sample solutions posted seemed much less formal than those from 165, and made much use of prose over symbols. I think I like the idea of writing a proof in paragraph form, given the difficulties I've been having with Assignment 1. In some ways, I've been having more problems expressing myself clearly than in actually understanding and finding a solution for the problems. My proofs to the first two questions look like someone spilled a bowl of alphabet soup on my monitor - way too many subscripts and variables. I think one of the reasons for that was that, if I wanted to discuss a subset of a certain set, without doing anything to the set itself, I would still have to declare and justify the existence of both, whereas if I were doing a more prosaic proof, I could describe the properties of the set and then move on to what interested me, only giving an explicit representation to the things that I was going to use.

If that makes sense.

I'm almost tempted to rewrite my proofs to the first two questions in the style of the sample solutions. I'm loath to erase the <100 lines I've already committed to them, though. I know I'll definitely be adopting a less formal technique for the last two questions, and the second problem set though.

Weeks 1 and 2

"Slog", "Blog", such ugly guttural words. Anyways.

I'm glad to be back. I've been looking forward to CSC 236, because I so enjoyed 165 last year. It's strange - I did pretty awfully in MAT137 last year, but when I'm doing logic and proofs I just feel right at home. Maybe it's because every math class I've taken in my life has focused on "calculations of the grocery-bill variety" (a phrase cribbed from Uncle Petros and Goldbach's Conjecture which I reread over the summer), whereas 165 and 236 have felt like they required a lot more independent thought, and not just memorization and repeated application of formulas and procedures. For that reason, I found solving proofs in 165, and now 236, a lot more satisfying. But maybe that's just me explaining away my incompetence at calculus.

I feel like I've done pretty well with the material during these first two weeks. It's taken me a while to warm up to induction since last year, but at this point I feel quite comfortable with it. The explanation the book gives in section 1.2, reconciling the principles of simple and complete induction with induction as proof technique by expressing the natural numbers for which a predicate is true as a subset of the natural numbers, and then using the principle of induction to show that that set is a superset of the natural numbers (and hence, exactly equal to them).

My only difficulty in solving the first problem set came in the second question, trying to decide the degree of rigour with which I would need to justify a certain statement. Specifically, I had partitioned the set of pairs for {0, 1... n+1}, Dn+1 = {{1, 2}, {1, 3}, {2,3}... {n, n+1}} into Y: {s \in Dn+1, n+1 \in s} and N: {s \in Dn+1, n+1 ~\in s}, and was trying to prove that the cardinality of Y was equal to n, which seemed obvious to me, but somewhat messy to prove. My solution was just to add an appendix to the end of the proof, with what ended up being about a page of writing, elucidating the reasoning behind those two lines, with a note to the marker that they could refer to it if they found the lines alone insufficient.

Wasn't that an interesting story? There are more where that came from. Stay tuned for next week.