Mathematics - 2012-07-03 (page 6 of 6)

Peter Tamaroff

22:00

@DylanMoreland OK. Let me think then.

Well, it is clear.

Dylan Moreland

Haha.

Peter Tamaroff

Since $f^{-1}(f(X))= \{x \in A : f(x) \in f(X) \}$

Dylan Moreland

I agree, though.

Peter Tamaroff

since $x \in X \subset A$ then $X \subset f^{-1} (f(X))$

I guess it is because it is so trivial. I don't know.

@DylanMoreland With what do you agree?

Dylan Moreland

That's it's clear.

Peter Tamaroff

22:08

@DylanMoreland How would you say in "mundanely".

I said: "That is basically saying if f is not onto or one-one, the image and inverse images will "miss" some points of $B$ or $A$."

Dylan Moreland

$f^{-1}(f(X)) = \{x \in A : f(x) \in f(X)\} = \{x \in A : \text{there exists } y \in X \text{ such that } f(x) = f(y)\}$. You take an $x \in X$ and check that it's in there, which it is because you could take $y = x$ in the condition.

Let's see. I think it's true that $f^{-1}(f(X)) = X$ for all $X \subset A$ if and only if $f$ is injective.

Is that the sort of statement you want?

Peter Tamaroff

@DylanMoreland Yeah, I have to prove that later on.

If $f$ is one-one then $f^{-1}(f(X))=X$ and the same for the other. Actually I'm asked to prove the $\Rightarrow$, not $\Leftarrow$.

Jonas Teuwen

@JasperLoy I have replied.

Son of a crack.

user19161

@JonasTeuwen I saw it, geezis.

BenjaLim

22:49

@PeterTamaroff The proof of that proposition is trivial

@RagibZaman The problem is trivial, the trace on one side is zero while on the other it is non-zero.

@JonasTeuwen What is happening with this jordan thing?

Peter Tamaroff

@BenjaLim Prove the other one.

BenjaLim

@PeterTamaroff which one

Peter Tamaroff

$$X \subset f^{-1}(f(X)) $$

BenjaLim

I thought dylan already proved that?

Peter Tamaroff

@BenjaLim Would you prove it in the same way?

BenjaLim

22:56

take $x \in X$

$f(x) \in f(X)$

so we can write $f(x) = y$ for some $y \in f(X)$

Peter Tamaroff

$${f^{ - 1}}(f(X)) = \left\{ {x \in A:f\left( x \right) \in \left\{ {f\left( x \right) \in B:x \in X} \right\}} \right\}$$

N3buchadnezzar

Gahhhh

It hurts my eyes

Peter Tamaroff

@BenjaLim OK.

@N3buchadnezzar The code?

N3buchadnezzar

No the mathtype part.

Peter Tamaroff

Can't we simplify this statement? $${x \in A:f\left( x \right) \in \left\{ {f\left( x \right) \in B:x \in X} \right\}}$$

BenjaLim

22:58

@PeterTamaroff and so $x \in f^{-1}(y)$ by definition of the preimage.

but recall $y \in f(X)$

and so $x \in f^{-1}(y) \subset f^{-1}(f(X))$

@PeterTamaroff get it?

Peter Tamaroff

@BenjaLim Yeah.

BenjaLim

@PeterTamaroff it is almost tautological

Peter Tamaroff

@BenjaLim Yes, that's why I was having a problem in "proving" it.

When I read it I said "well isn't that clear?" but I couldn't prove it.

BenjaLim

@PeterTamaroff Do you have a lot of experience in writing out proofs clearly?

I mean those that like don't involve a lot of algebraic manipulations

Peter Tamaroff

@BenjaLim I wouldn't say a lot, but I have some.

BenjaLim

23:05

@PeterTamaroff Time will come when you have to write proofs like:

A continuous function on a compact set is uniformly continuous

proofs like that don't involve a lot of manipulations

more a lot of words

or say to prove that Arzelà - Ascoli Theorem for $C[0,1]$

user19161

Also @peter sometimes drawing pics helps to prove something. For example, think of sets as ovals and functions as arrows mapping dots from one oval to another.

Peter Tamaroff

@JasperLoy Yes, I think of sets as splash shaped, to make it more fun.

BenjaLim

@PeterTamaroff I would say that it has it advantages

but if you want to imagine a closed set

not all closed sets are drawn with a nice thick line around it

for example $\Bbb{Z}$ is closed in $\Bbb{R}$

yet when you draw it out in the number line it does not have the usual "nice shape a closed set has"

Peter Tamaroff

@BenjaLim He, true.

user19161

@BenjaLim So a picture is only an imperfect representation of the concept.

Peter Tamaroff

23:08

@BenjaLim There you're using that if $X \subset Y$ then clearly $f^{-1}(X) \subset f^{-1}(Y)$

BenjaLim

@JasperLoy In commutative algebra there are no pretty pictures

@PeterTamaroff more:

user19161

@BenjaLim Are there even pictures in algebra???

BenjaLim

if $b \in B$ then $f^{-1}(b) \subset f^{-1}(B)$

@JasperLoy don't think so.

Peter Tamaroff

fixed.

user19161

One can't actually visualize groups rings fields the same way as R, R2, R3.

BenjaLim

23:09

well I was talking more of an element being a member of a set

Peter Tamaroff

@BenjaLim Sure.

BenjaLim

@JasperLoy In analysis how do you visualise $(C[0,1],||.||)$?

oh maybe you can.

Peter Tamaroff

@BenjaLim After I get through these formalities, I will get to some diagrams, and commutative diagrams. They are cool.

user19161

@BenjaLim I imagine the interval and some continuous functions defined on it, a fuzzy image in my mind.

BenjaLim

@PeterTamaroff No rush to get to them.

@JasperLoy Try imagining what it means for $C[0,1]$ to be totally bounded with the sup metric :D

Peter Tamaroff

23:10

@BenjaLim It is just two pages ahead =)

BenjaLim

@PeterTamaroff what book is this?

user19161

Of course you can visualise some things in algebra, but my point is that it is not on the same level as in analysis.

Peter Tamaroff

@BenjaLim Introduction to Topology by Bert Mendelson. I'm using it as a complement to Rudin's chapter on basic topology.

Mendelson gives a clearer exposition.

BenjaLim

I thought you would see commutative diagrams in more an algebra text

user19161

@PeterTamaroff That is another classic. Lee's first book treats general topology by the way, though not to a very deep level.

BenjaLim

23:12

how come we are always stuck discussing about books?

Pick one and use it!!

@JasperLoy hey

Peter Tamaroff

@JasperLoy I love this topology book. I really do. Now that I'm getting it, I love it more.

user19161

@BenjaLim I think you will get plenty of those in algebraic topology.

Peter Tamaroff

I'll go and eat some pizza now!

BenjaLim

any suggestions on a good reference for CW - complexes @JasperLoy ?

@JasperLoy Yes I know but if you look at rotman's homological algebra

Peter Tamaroff

@BenjaLim What would we be without books?

user19161

23:14

@BenjaLim I don't really know AT, but my favourite books with AT are these three: Hatcher, Lee and Bredon. No need to look beyond these three.

user19161

Lee's first edition treated simplicial complexes. In the second edition he removed most of that and replace it with CW complexes.

BenjaLim

which lee book @JasperLoy

user19161

@BenjaLim Introduction to Topological Manifolds.

BenjaLim

that is not an AT book

user19161

@BenjaLim It contains quite a good part of it.

user19161

23:17

@BenjaLim Depends on what you are looking for.

BenjaLim

just an intro to CW - complexes

user19161

That's why I listed three for you to pick from.

BenjaLim

hmm

hatcher chap 0

i'll have to look at that again

user19161

And hey only Lee treats the classification of curves and surfaces to such great detail.

user19161

Others either omit this or give a half baked treatment.

user19161

23:20

Bredon is the one which proves many theorems for manifolds.

BenjaLim

@JasperLoy

I ask about CW complexes

my lecturer said that we will be making spaces out of gluing disks @JasperLoy

because

user19161

Is Hatcher not to your liking @ben?

BenjaLim

@JasperLoy I have found it dense so far.

but then again

the calculation of $\pi_1(S^1) \cong \Bbb{Z}$ is not trivial

user19161

Anyway Hatcher is like the canonical text these days.

Jonas Teuwen

@BenjaLim Nothing?

BenjaLim

23:25

@JonasTeuwen what are you talking bout?

Jonas Teuwen

@JasperLoy No problem. The Sky is the Limit.

@BenjaLim What you asked me before about some guy named after some curve theorem?

BenjaLim

well

I saw the starred messages

user19161

@JonasTeuwen There are no limits. The sequence does not converge bro!

Jonas Teuwen

@JasperLoy Or multiple ones. It is not Hausdorff Bro.

@BenjaLim I wouldn't think about it.

user19161

The Jordan Curve theorem?

BenjaLim

23:26

ok

user19161

By the way, there is the generalised Jordan curve theorem in n dimensions.

user19161

There is the related Jordan-Schonflies theorem, again in one and n dimensions as well.

user19161

The one dimensional case is used to prove the triangulation theorem for curves.

Jonas Teuwen

No Tresspassing or thou shalt be triangulated.

user19161

This Jordan-Schonflies theorem is so special that no book in print I know of proves it.

BenjaLim

23:34

@JonasTeuwen still coming to ANU?

Jonas Teuwen

@BenjaLim Yes.

BenjaLim

@DavidWheeler

user19161

mathdl.maa.org/images/upload_library/22/Ford/…

user19161

MAA has an article on it.

BenjaLim

@JonasTeuwen I got my mark for analysis yesterday :D

@JonasTeuwen

@JonasTeuwen There was a lot of scaling

@JonasTeuwen Take a guess

user19161

23:35

Oops, sorry I meant it is used to prove surface triangulation not curve triangulation.

BenjaLim

@JasperLoy have you studied AT

user19161

@BenjaLim Not really bro, just taken a look here and there...

BenjaLim

ok

I am enrolling in it for credit

@JonasTeuwen hey

Jonas Teuwen

@BenjaLim Cool! :-). What is the scale?

BenjaLim

why

Jonas Teuwen

23:38

@BenjaLim Hmm, don't know but it is pretty damn good.

BenjaLim

@PeterTamaroff The first non-stop flight from BA to sydney landed yesterday!

@JonasTeuwen It is because your marks get pushed up

Peter Tamaroff

@BenjaLim What?

user19161

@ben Why so secretive?

BenjaLim

@JasperLoy marks are triviao

@PeterTamaroff I think I will make a plan to go to argentina

at the end of the year

user19161

@BenjaLim Therefore, there is no need to delete them. QED.

BenjaLim

23:40

@JasperLoy why?

Peter Tamaroff

@BenjaLim Cool.

BenjaLim

@JonasTeuwen How can I claim that 99 is not trivial when I don't understand a lot of the several variables material?

user19161

@BenjaLim When one is non-attached, there is no need to delete something.

BenjaLim

@PeterTamaroff If I go I will tell you before hand

@JasperLoy I do not want to give the impression of being boastful.

Peter Tamaroff

@BenjaLim Do you have a place to stay?

BenjaLim

23:40

no

I know I have one if I go to mexico city

Jonas Teuwen

@BenjaLim Don't think too much about it. You have done well, congratulations.

user19161

@BenjaLim Well, people can think whaetever they want. I am always misunderstood too.

BenjaLim

@JonasTeuwen thanks if I get a congratulations from a sifu I am happy

Peter Tamaroff

@BenjaLim Why do you want to visit Argentina?

BenjaLim

@JonasTeuwen sifu = en.wikipedia.org/wiki/Sifu

Jonas Teuwen

23:41

@BenjaLim I was googling already! 8-). Thanks!

BenjaLim

@JonasTeuwen sifu usually refers to like a kungfu master or something

user19161

@BenjaLim I was wondering why the h was omitted.

BenjaLim

@JasperLoy cantonese

user19161

I speak Mandarin not Cantonese bro.

BenjaLim

@PeterTamaroff for fun

never been to south america

@PeterTamaroff I will learn spanish before I go

@JonasTeuwenv Or you can imagine sifu as being a jedi master

like qui gon jin

Peter Tamaroff

23:43

@BenjaLim Hahha maybe you should.

user19161

@BenjaLim Or a Belieber.

BenjaLim

@JonasTeuwen or sensei

@JasperLoy I don't like JB now

Jonas Teuwen

Thanks :-).

BenjaLim

he seems all

@JonasTeuwen like this: youtube.com/watch?v=3PycZtfns_U

user19161

@BenjaLim JB also stands for Johor Bahru!

BenjaLim

23:47

@JasperLoy I think 17 year old girls are always like emo/indie and shit

wassup with all this man

@JasperLoy

Peter Tamaroff

@BenjaLim There is a high school some blocks away from mi uni

It's an arts school.

Everybody looks weird.

Everybody-

Transcript for

$Mathematics$ Mathematics