Tuesday, 10 April 2012

M338 TMA2 away

Well for better or worse (probably worse) TMA02 is away. I have no real confidence that I understand what is going on. However for interest here is a summary of the questions and my response. Part 1 had to calculate d(x,y) a so cakked distance function  for various x y

Question 1 a distance function  is defined and we have to show that it is a metric ie we have to show
M1 d(x,y) >= 0 with equality only if x = y
M2 d(x,y) = d(y,x) ie d(x,y) is symmetric
M3  Finally show that d(x,z) >= d(x,y) + d(y,x) (ie, the triangle inquality)
The metric had 2 forms depending on whether or not given two points in R^2 (x1,y1) (x2,y2) y2 = x2 or not
First part just asked  you to show  that you can apply the definition correctly think I did OK on this.
Second part show that the metric satisfies M1 and M2 of the definition of a metric
M1 is simply that d(x,y) >= to 0 with equality only if x = y. A bear trap here is to neglect to show that if
d(x,y) = 0 x = y think I managed to negotiate this sucessfully. Again this seemed relativiely sttaightforward
(In what follows e means element of )
Then one had to show that given a,b e R^2,  that |a1-b1|+ |a2-b2| < d(a,b) again this seemed straightforward
But this was supposed to be a hint to be used in proving the triangle inequality ie to prove that d satisfied M3
I seemed to prove that M3 was satisified directly without using the hint so there is a nagging doubt that I missed some cases.
Then one had to sketch 2 open balls of d for various d(x,y) an open ball is such that d(a,r) < r for a e R
where a is the centre of the open ball, This is a generalisation of the definition of an open interval
This was relatively straightforward although conterintuitive for the second ball. I came to the conclusion that there was no open Ball centered on (1,1) with radius 1/2 which satisfied the second condition of the metric,
logic tells me I'm correct but my instinct doesn't. Antyway I couldn;t fault my logic so it stands for better or worse.

It was then desired to show that d(x,y) was not metrically equivalent to the Euclidean metricd2(x,y)  The euclidean metric is simply the disance between two points Here I totally failed. The definition of two metrics d1 and d2 being equivalent is that it is impossible to find two real numbers m and M such that

                       md1(x,y) < d2(x,y) < M d1(x,y)

for all x,y e R^2 and we were supposed to use a solution to the first part as a hint to show this. Unfortunately all I could demonstrate was that given d2 and d1 it was indeed possible to find two number m and M such that the above inequality was satisfied. I'm looking forward to Alan's solution to put me right (watch this space).

So for question 1 reckon I've got about 75% of the full marks at least I've explained my confusion

The first part of Question 2 introduced a definition of some open sets and we had to show that they formed a topology on R the first part.
A topology T on a set X is a collection of subsets of X satisfying  the following axioms
T1 the topology must include the empty set and the set  X
T2 the topology T must be such that any two intersections of the subsets of T must also be an element of T
T3 the topology T must be such that any unions of subsets of T must also be an element of T

For finite toplogies it is sufficient to demonstrate this for any two subssets of X which are elements of T. 

The elements of T are called open sets of X, their complements are closed sets of X. 
At first sight the definition of an open set of X may seem totally arbitrary but T2 and T3 guarantee a form of closure. 

 I reckon I got this correct. The second part introduced a new definition of left continuity and we had to show that a specific function was left continous. I think I got most of these parts correct,
Finally we had to show that given a function is both left continuous and monotonically increasing that the funcition was indeed continuous on R.

The background to this is that, so far the purpose of M338 has been to generalise the M208 definition of continuity to reformulate it in terms of whether or not the inverse of f(a) gives rise to an open set of X. As my mastery of the various terminology is still a bit hazy I'm not convinced I got anything resembling a perfect answer to this part of the question.

Question 3 tested whether or not you understand basic definitions such as closure, boundary, exterior interorior or the 'density' or 'non density' of certain finite sets given a topology T defined  on those sets

As there were no explicit examples (unlike M208) this involved a) understanding some basic definitions and b) how to apply them. I think I got there in the end but was a bit of a struggle to say the least. Also when defining a map from a set X to X if the function exclude a certain element a then the inverse of f(a) = 0 the empty set, But this is still part of the topology of T. So the ambiguity when testing for continuity is whether or not f(a) has an inverse mapping onto the topology, I decided that it did as 0 is still an open set of T but I can see arguments for the other point of view. So again confident that I got about 75% of this question correct.
Question 4 involved sketching certain sets finding their exterior and interior and boundary and deciding whether or not they were open or closed or neither. I think I got most of this correct.

So overall I might be pushing the boundary between grade 2 and grade 1 for this one but if I get above grade 2 that will only be due to the genoristiy or otherwise of Alan's marking. In terms of really understanding the subject I'm still only on block A2. I neeed over the next 4 weeks to get to grips with A3 and A4 before moving onto block B which admittedly looks a bit more straightforward.  So yes I'm being pushed out of my comfort zone, but as my mate Neil said I wanted a 'brain fuck' and I think I've got this in spades. Obviously my confidence has been quite shattered and as a caution for those who think this is the ultimate in pure maths, it should be noted that a course in Topology, along the lines of M338 usally forms a second year course in most brick universities. Anyway I've eaten most of my main course so for the next two weeks, I'll be having my pudding of MST324 I can't wait.

Obviously I will have to try and consolidate my extremely limited understanding of the first part of M338 as well so it will be like eating porridge with salt as opposed to sugar. Still KBO (keep buggering on ) as Churchill is alleged to have said.

1 comment:

  1. Hi Chris,
    Thanks for the comment. It was heartening, in a perverse kind of way, to see you had as much a feeling of bashing your head against the wall on this, as I did.
    I am rather convinced that block B will be a walk in the park in comparison. Let's hope so. I'm already starting to feel sense of impending doom about block C though. The only option is study, study, study, I guess.
    Good luck with the result!