w-V

APOSTABOUTWEEKFIVE-OHHIMARK

Alright. Week five in the course was pretty cool and great.

``There was a mid-term!''

And then there was some stuff about about proofs and proof structures. I don't really have any brilliant or lucid insight into proof structuring and writing out, however I will say that graph paper makes dealing with the course-requested indenting structure much easier for hand-written proof (structures).

I as I see it, the only really salient point to talk about or even think about is pre-proof structuring. More specifically, how to know when and where to attempt direct proof, or contraposition, or contradiction, or induction or whatever. Some observations, true or otherwise:

1. Proving something that we intuitively think of as ``obvious'' is often easier by contradiction, for example, that there an infinitely many prime numbers.
2. Trying to make claims about integers and their squares (as we've done a few times in the lectures), it is best to start with the integer itself and any conditions on it and then move towards making claims about its square. The set of integers is not closed under taking square-roots and so we must be cautious with how we handle these numbers.
3. Induction is best (possibly even necessary) when we are looking at an indeterminate number of objects. For example, if we say ``the finite intersection of open sets is itself open'', we wouldn't want to take every possibility for any open sets and look at their intersection. It is enough to prove that the intersection of any two open sets is open. From there we can sort of reason through ``adding'' more sets through intersection one-by-one and knowing that these intersections will be open because we are intersecting a open set from intersecting two elements with a new open set. Another example is the uniqueness of polynomial decomposition up to scalars (units) where we cannot say how many irreducible polynomials that we might ``pull out'' of a polynomial.
4. If you try to apply rigorous proof (and/or proof structure) in regular conversation, your friends will probably get mad at you.

I guess also about making proofs, we must be cautious about what we take to be ``assumed knowledge''. For example, we are asked to prove:

If we know analysis or calculus or topology or something, we could evoke the intermediate value theorem. We would not be finding an example to prove an existential condition, but would instead be directly applying another existential truth to this example. Obviously, without knowing this theorem, it is easier to find a single example than to prove the entire intermediate value theorem, but I certainly have a knee-jerk reaction to this posed question to evoke my entire body of knowledge of calculus/analysis, including any and all theorems about polynomials, limits, and continuous functions. But, I'm pretty sure that I'm not supposed to do that. It is better that I should demonstrate these things with minimal assumed knowledge, perhaps just guess-and-check methods (in this case) with only some algebra so that I can use the standard operations of addition and multiplication. Probably.

That's about all that I have to say about that.

Thanks for plig-a-plog sliggin' my slog,

J