A Summer of Maths

Slight plug here, I've recently uploaded my notes to (almost) all courses in Part IA, hope they are useful

http://lishen.wordpress.com/2012/07/16/some-of-my-notes-for-the-part-ia-cambridge-maths-tripos/

The IA Groups is lectured by Saxl again this coming year I think, so these notes should be relevant enough.

Yeah, just finished. It could've gone better (i.e. if I can do numbers) but I'm reasonably happy with it. You starting this year?

Question(1):


Btw, I'd put Oxford undergraduate problems in spoilers, so that Oxford people can save them for their actual preparation!


Question(2):

The set $S$ consists of elements $a_1, a_2, ..., a_n$, with $|S| > 2$.

Determine whether $S$ forms a group under $\star$, defined by

i) $\ a_i \star a_j = a_m, \ \ \ \ m = \max(i, j)$

ii) $\ a_i \star a_j = a_m, \ \ \ \ m = |i - j| + 1$

where $1 \leq i, j \leq n$.

I don't think there's a name for that...

Finally got around to writing solutions for your problems. So don't think we wouldn't care, I do appreciate that you post them.

Quick LaTeX-trick: You can use "\quad" and "\qquad" instead of multiple "\"s.

Terribly confusing, and hugely dry, but otherwise an awesome course!
The only book worth getting is probably Sutherland's Introduction to Met and Top.

Terribly confusing, and hugely dry, but otherwise an awesome course!

The only book worth getting is probably Sutherland's Introduction to Met and Top. I don't think you need to know much beforehand, and just stay awake during lectures would be a good tip (you'd surprised how easy it is to let your mind wonder when you start to not understand stuff).

Stick with it. I bliltzed through Analysis and found M&T a shock - once you get used to it and something 'clicks', you'll find M&T a rewarding course. I tend to find that I can either see how to do a problem straight away or its impossible for me - don't know how to plug away at a course like M&T...

A set with a binary operation is a magma....but I'm guessing that you knew that.

Stick with it. I bliltzed through Analysis and found M&T a shock - once you get used to it and something 'clicks', you'll find M&T a rewarding course. I tend to find that I can either see how to do a problem straight away or its impossible for me - don't know how to plug away at a course like M&T...

That's interesting because, in some respects, I found topological spaces to be more intuitive than say, the real line. I mean continuity in this setting (i.e. the preimages of open sets being open) is much easier to grasp than epsilon delta stuff from real analysis. Though, tbf, I don't really know much about either

Footnote: Decipher "Cofu Clobi Ibaatib".

Hmmm, that's interesting, I always found the eps-delta to be more intuitive (not initially though), it also generalises to open balls quite nicely. But I do admit the topological definition is neater.

The topological definition also makes certain proofs trivial/really tasty. For example, if memory serves me correctly, If you want to prove that the composition of two continuous functions is continuous then the epsilon delta proof is pretty messy but in the topological setting it becomes obvious.
Again, I have very limited experience so it's highly likely I'll see things differently in the light of actually having to do university maths.

Are you looking forward to 2nd year?

Question:

Consider a set $S$ and a binary operation $*$ on $S$.

By assuming that $(a * b) * a = b$ for all $a, b \in S$, show that

$a * (b * a) = b$

for all $a, b \in S$.

Yup. So, the assumption is a sufficient condition for an operation to be associative.

Are you saying the assumption implies associativity? This is not true.

Very good, indeed! I like the counter-example.

Prove that any subgroup of $(\mathbb Z,+)$ is generated by one integer or, equivalently, is of the form $n\mathbb Z$ for an integer n.