I really should've posted this on Friday, since that was March 1st, but my ability to keep track of dates (or lack thereof) seems to have completely ruined that possibility. Slowly, my SLOG is becoming more inactive, especially since I forgot to post after reading week, which most people seem to have done. For the record, I spent it watching Hey Arnold. I wasn't productive in the slightest.
I do believe I've given up on attempting to solve the problem from a few weeks ago (February 15th), since my mind has totally died out. Which is a shame, but maybe one day I'll try and revisit it. Maybe at some point in time when I'm not trying to desperately avoid assignments and midterms. Speaking of which, progress on assignment 2 has just been splendid.
Since the due date hasn't passed, I'll keep this spoiler-free and completely ambiguous. About 80% of my problem with a2 was "Am I even writing this properly?" and doubting myself. I honestly like proofs, but I'm horrible at writing them. I can put down the basic structure and figure out the logic mathematically, but I tend to create my own variables for temporary use and at that point, I have no idea if I'm supposed to add a line noting that "k∈ℝ" or if it's fine if I just use it in the definition of, let's say, z. For example "z = ki" (for the sake of it, let's just say i was already declared as some number). Do I really need a whole line saying "Let k∈ℝ" or "∃k∈ℝ, z = ki"? And if I do, which form do I use? My sister and I decided to omit the line for now, but I'll probably post this on Piazza after I upload this post. Also, if my whole statement that I'm trying to prove is an existential, where I don't even need an "Assume" statement, do I have to indent at all? It looks extremely strange, but I don't see how I would indent it.
For the record, question 6 seems to be the hardest for me to prove, since I'm not sure if an assumption I'm making is even valid. Oh well, part marks for structure!
I would probably write that as
ReplyDelete"Let k∈ℝ such that z = ki" (Be careful though, you have to prove that such a 'k' exists if it's not obvious! For example, if I say let k∈N such that 7/2 = k, then such a k can obviously not exist).
If you have an existential you might still have to indent depending on the rest of the sentence. Basically you should indent whenever you assume something. This can be assuming a variable is a generic member of its domain (universal quantifier), but can also be assuming the antecedent when trying to prove an implication.