For g), apparantly the answer is i) but it seems to me they're all the same.If |b'| is a finite subset of an infinite set, then |b| must be an infinite set as N is infinite. Is there any difference between saying "b is infinite" and "|b| = infinity"?
For k), apparantly the answer is i) as well, but I still don't understand the notation. Could someone explain it through?.
[assuming ' = complement]
g) ii), iii) contains the set of odd #s, but i) does not
As for your question, no difference.
In this case, a complement of a complement is the original set, so ii) and iii) are immediately equal. The set of odd #s is infinite -> in both ii) and iii). Its complement is the set of even #s, which is also infinite -> not in i).
The complement of an infinite set is not always finite.
k) is tricky. i) = {0}; ii), iii) = empty
i) Ilambda0 contains 0, for any lambda > 0. Thus, no matter how many of these sets you intersect, 0 will always be an element. Thus, the final set is not empty.
But for g), how can you say ii) and iii) constitute the set of odd numbers? Couldn't they be even numbers? Or any infinite random subset? Now I think about it, ii) and iii) could be different, by being different subsets of N.
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Anonymous2008-08-29 10:34
No sorry ignore that last part about ii) and iii) being different, I was being stupid.
>>4
For g) The set of odd numbers is an element of ii) and iii), but is not an element of i).
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Anonymous2008-08-29 16:07
But it doesn't say anything about odd numbers.
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Anonymous2008-08-29 19:08
I worked out why i) in g) is different now.
ii) and iii) describe "The set of all subsets of the N that are infinite". This set could include the set of odd or even numbers, for example.
i) on the other hand describes "The set of all subsets of N that are infinite, but have a finite complement". This would disallow the set of even numbers from being in the set, as the complement (the set of odd numbers) is infinite.
what is this a general set theory class? or are you breaking into upper division classes? if you find this stuff interesting, you can take a class in general (point-set) topology. or the more 'boring' route of axiomatic set theory.
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Anonymous2008-08-30 7:54
I'm just reading up on topics from a pretty shallow book before uni starts. I'm going into the first year of maths at Cambridge.
Yeah. For example at trinity you get supervised by trinity fellows and grad students. Our fellows and grad students are much better than your fellows and grad students.
For example I've been supervised by people like Ben Green, Imre leader, tim gowers.
Also it's self perpetuating. All the mathematicians at trinity are better, so you learn more from your peers (Although don't hand around with maths students, they're autistic).
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4tran2008-08-31 12:14
ITT fagoots don't know what they are talking about.
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Anonymous2008-08-31 13:27
>>17
Big deal. I'm sure whatever the staff, regardless of their fame, they at least understand undergraduate material just fine. Does a PhD in maths teach GCSE/A level maths any better than someone with just a masters? No.
I couldn't give a shit about my peers or their aptitude either.
Sounds like you're just have a superiority complex.
I've been supervised by the guy who SUPERVISED Tim Gowers. Wayne Boucher the legend XD. He said that Cowers wasn't too bright when he was taking him, lol.
Anyway, yeah you do have a bit of an inferiority complex (>>19 had it the wrong way round). Though the best students tend to be at Trinity (bar e.g. one guy in my year, Clare), on average the people you work with will be roughly the same at any college.
>>14
I heard once that Korner is DoS at Tit Hall and you don't have to do example sheets in the first year. Nice choice.
It's not inferiority or superiority, just better at applying to colleges. There's no reasons not to go to trinity anyway, best (and cheapest) rooms, grant money coming out of your arse, One of the largest student bodies, centre of town.....etc.
Maybe I'm confusing him with another professor, but Korner's a dick.
>>19
Not so at all. Maybe you'll pass the exams just as fine, but you'll end up with a shallower understanding of the finer points of the material and it's implications in the wider scheme of things.
Oh and, not sure how true this is for non-pure mathematicians, but in almost all the subjects I revised for the exams this year exam material was written by trinity professors, which is a real help when they're giving you revision supervisions.
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Anonymous2008-08-31 18:23
Trinity should detach from Cambridge and only teach mathematics.
>>22
| There's no reasons not to go to trinity anyway
I can think of a couple. You get not only the people who are amazing at maths, but also the people who *think* they're amazing. I.e. the massive nobs. I got a good first this last year, which has put me above a good number of Trinity students (including Congo boy lol), all without having to suffer the smarmy arrogance of people I've known from my school who've gone to Trinity.
Also, I hear the work ethic is pretty heavy; I know people probably do varying amounts but it seems like a lot of Trin students work themselves skinless in the third term. I don't want that kind of pressure, doesn't work for me.
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Anonymous2008-08-31 20:50
>>22
And it's Korner who decided that his first years didn't have to do examples. I wasn't saying anything about the man himself. I wouldn't pretend to know him, but I've read some of the material on his homepage and he writes very lucidly and intelligently, which endears me to him.
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Anonymous2008-08-31 21:51
>>26
I think you mean it endears him to you. I'd question the usage of "endears" here though, as I find its connotations a bit too lovey-dovey for this context.
I do hope you didn't get these literary habits from his "lucid and intelligent" writing.
The trick is not to assosciate with the majority of the maths students socially.
From what I can gather though 90% of the maths students hang out in their own little nerdy clique, and since the top ones don't actually have to work at all, none of the rest do.
I seem to remember in our first year one of the best people spent the entire third term drunk and as a result only got 18 or so alphas (only was his own choice of words, not mine).
I know. Though I'd like to think of a 1 from Cambridge as something of an acheivement. It amuses me how Mr. Olympiad can care little enough to drop like that.
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Anonymous2008-09-03 17:53
>>32
And I thought I was the one with the complex...
Cong's actually a nice guy. Sure he has obvious issues, but he tries quite hard to acheive a normal sort of social interaction.
I respect that a lot more than most of the other mathematicians attitude.
Meh, the pure was quite hard this year I think, although there was a gift of a conformal mapping question. I'm pretty certain cong's spent his summer doing some work for Imre on some combinatorics research, so I don't think he's too bothered about the 2.i
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Anonymous2008-09-04 18:45
>>33
Combinatorics, cool. He seems like that sort - innovative and new thinking on simple-looking concepts like Leader et al.
I agree that pure was quite hard, thank fuck I internalised some standard Quantum and Markov techniques. I don't remember there being an easy conformal mapping of the kind they've been having for the past four or five years though...
The reason it was a gift is because it was on the Complex methods question, as opposed to the shared question. Meant complex analysis students got a 4th pop, which was rather unfair.
I couldn't be arsed to learn anything remotely applied/applicable, I deserved my 2.i. Was very pissed off with the Linear algebra though, 3 of them I answered totally apart from the very last bit, which was a bitch on them all. I had a word with the person who wrote them, Ben Green, he wasn't very sympathetic...
I'm totally with you!! I was annoyed by the Linear Algebra too, looking at my breakdown I only managed 3 betas in it, hah. On most of the past years' papers I had managed at least 2 alphas overall. I picked up my marks in GRM, CATAM and Anal II.
Ben Green is the young upstart who took the year below us for M&TS, right? Seems like he was intent on making the questions pretty obscure!
I do remember there being a mapping in Comp Methods now. I'm not sure why I didn't attempt it... did it have a bit that wasn't very standard?
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Anonymous2008-09-05 9:28
>>36
It started off with a question about the logarithm, which may have turned you off, but it just asked you to suitably define the logarithm the map C - { (0,-inf] } to some slice of the complex plane.
Yeah, ben green was that guy, he proved that their are arbitrarily long arithmetic sequences of primes amongst other things. Nice guy as well.
Compared to past papers I did abysmally. Somehow I even managed to mess up that question that was literally "Prove all norms are equivalent in R^n", and I still have no idea how. CATAM was an easy 4 alphas though.
I should have spent the time to learn something bankable like optimization or Markov chains, but fuck it, it's this year that counts.
That probably did turn me off. I remember spending some time the days before the exam going through Priestley and annihilating any conformal map q's I could find, just for the extra marks. At the time I was convinced I'd lost the chance for a first.
I like Ben Green... I went to his little talk about Ramanujan and loved being able to flick through the problem book he taught himself mathematics from.
I do want to go for Part III. Still uncertain about shooting another first, but the interest and enthusiasm is there. I know everyone says that you shouldn't really go for it unless you know you want to head into PhD and research zones, but I wouldn't see a huge problem in indulging in another year of fascination before exposing myself to the real world.
Have you started on the new Catam projects? I'm not sure whether to start early (now) or wait until I know a little more about what I'm really doing.
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Anonymous2008-09-05 15:47
I did the first program on one or two of them. Like prime seives, and euclidean alogrithm for polynomials in a prime field.
From the looks of them they're all pretty self explanitory, but as you say, I'm wary to jump straight in.
Also I'm pretty shit at programing, so I'm gonna wait till I can ask people who know some questions about how to do shit.
As much fun as it is doing it all with basic loops, I tend to lose track of my variables. One of my programs last year was over 1000 lines long. I never could be arsed to write functions.
Then again I read up on galois theory but couldn't make total sense of the program question, differences in notation and stuff. Don't read any book that say's it's going to follow Galois' methods, it's almost unreadable and pretty pointless. Extension fields is where it's at.
I've already bought Stewart's =[ I'm making my way through but it's annoying having to skim sections - on stuff anyone that reads it should already know - in case he introduces a method or notation that I haven't seen before.
How can you not be arsed to write functions, but be prepared to go through the trauma of recalling which variable does what, presumably write similar loops over and over and have to face a monumental pile of compiling errors?