@FrankScience $f:A\to B$, $X \subset A$, $f(X) = \{f(x) \in B : x \in X\}$

@PeterTamaroff $f(f^{-1}(Y))=\{f(x): x\in f^{-1}(Y)\}$?

@anon That implies $y$ has no preimage, right? (Which can't be)

yes

@FrankScience Yes.

@PeterTamaroff i doubt the paper is published in any reputable journal anyway

@anon Directly, or am I missing an intermediate step?

@Eugene I guess so. For one, it has poor formatting.

it's an old false proof of RH, I remember seeing it awhile ago on MO

@PeterTamaroff So it suffics to prove that $x\in f^{-1}(Y)\iff f(x)\in Y$.

@PeterTamaroff which is the definition of $f^{-1}(Y)$.

@anon yes i know. i posted the related link

@Eugene Ah! That is the one I remember.

@MarkDominus i just caught this. why should $(A^C)^C$ be $A^{2C}$? even playing with the exponential law pun it should be $A^{C^2}$.

@FrankScience So basically, it is trivial.

@PeterTamaroff I know you want to prove it with the method $A\subseteq B,B\subseteq A\,\Longrightarrow\,A=B$.

@FrankScience I actually chose reductio ad absurdum.

if i were a pokemon i'd say "math, math, math" all the time.

@PeterTamaroff No abusing reductio ad absurdum.

@FrankScience ?

and probably be as useless as a magikarp

@PeterTamaroff Use it when necessary.

@Eugene But you can still splash papers and render them unreadable.

@FrankScience I needed it.

@PeterTamaroff Well

@PeterTamaroff why would i do that though?

@Eugene Because you're a stupid magikarp.

@robjohn Are you around?

@PeterTamaroff BTW, I found that observing the law is just a falsehood.

@FrankScience What do you mean by "falsehood"?

@Eugene You're mean. I have to go and cry now. Sorry.

@PeterTamaroff lol.

@PeterTamaroff We should define the law as a procedure, not a stable rule.

@Eugene I can prove it. I couldn't write it down. =D

@FrankScience Go on.

@PeterTamaroff SURE

@Eugene Now I have to prove

@PeterTamaroff TRIVIALLLLLL

Where $\{ X_{\alpha}\} $ is an indexed family of subsets of $A$.

@Eugene LOOOOOOOOOOOOL

@PeterTamaroff prove it for two

then induct

easy

@PeterTamaroff For example, the law of traffic light is active when red light is on. Say, one rule is

@Eugene Yeah, that I what I was imagining. Though it doesn't specify if $I$ is finite. I assume it is!

@PeterTamaroff without induction is pretty easy too actually

@PeterTamaroff $I$ isn't even countable.

R1 Special case? goto R3

@Eugene How do you know?

R2 punish

R3 goto R1

@PeterTamaroff i've taken topology. the indexing set need not be countable.

@Eugene Well, then I can't use induction, can I?

@PeterTamaroff you can use transfinite induction. =P

@Eugene Good

@Eugene But induction is legit for the countable set $\Bbb N$.

@Eugene Something $\omega$ confused me!

@PeterTamaroff See Wiki

@Eugene Induction over what set then?

@PeterTamaroff you needn't use induction. just prove it

@FrankScience I know that is available but I don't have it at hand.

@Eugene OK.

$\bigcup_{\alpha\in I}X_\alpha=\{x:x\in X_\alpha\textrm{ for some }\alpha\in I\}$?

@FrankScience Yes. Why do you ask?

@FrankScience yup

@PeterTamaroff I only know the finite case.

@FrankScience OK. Now you know both!

$f(\bigcup_{\alpha\in I}X_\alpha)=\{f(x):x\in X_\alpha\textrm{ for some }\alpha\in I\}$?

@FrankScience Yes.

I think the first one is trivial.

$\bigcap_{\alpha\in I}f(X_\alpha)\\=\bigcap_{\alpha\in I}\{f(x): x\in X_\alpha\}$.

alignment is awful.

@FrankScience Try $$\begin{align} y &=& x \cr y^2 &=& x^2 \cr \end{align}$$

@PeterTamaroff The deleted is wrong.

@PeterTamaroff eqnarray is not suggested in AMSMath. instead, use align.

@PeterTamaroff Let me write on the scratch paper first. only imagining the equation is out of my ability.

\begin{align}{l l}

(those are lowercase L's)

@anon and delte one &, ok.

@anon in align, only one & is needed.

@FrankScience why are you telling me?

@anon Isn't this immediate? $$f\left( {\bigcup\limits_{\alpha \in I} {{X_\alpha }} } \right) = \bigcup\limits_{\alpha \in I} {f\left( {{X_\alpha }} \right)} $$

yes

if $X_{\alpha} \subset A$ and $f:A\to B$.

@Peter \begin{align}\bigcap_{\alpha\in I}f(X_\alpha)&=\bigcap_{\alpha\in I}\{y:y=f(x_\alpha)\textrm{ for some }x_\alpha\in X_\alpha\}\\&=\{y:\textrm{for all }\alpha\in I,\textrm{ there's some }x_\alpha\in X_\alpha\textrm{ such that }y=f(x_\alpha)\}\end{align}

@PeterTamaroff so you can try this trick

@Eugene Tell me.

i'll do it for two

let $x \in f(A \cap B)$

then $f^{-1}(x) \in A \cap B$.

so $f^{-1}(x) \in A$ and $f^{-1}(x) \in B$

@Peter $f(\bigcup_{\alpha\in I}X_\alpha)=\{f(x):\textrm{there's some }x\textrm{ for all }\alpha\textrm{ we have }x_\alpha\in X_\alpha\}$.

so $x \in f(A)$ and $x \in f(B)$.

so

$x \in f(A) \cap f(B)$.

ok?

@PeterTamaroff Am I right now?

@Eugene Yeah, but then shouldn't you use transfinite induction if $I$ is uncountable?

@Eugene wrong, should be $f^{-1}(x)\subseteq A\cap B$.

@PeterTamaroff needn't. use the same thing for the uncountable case.

Since you're inducting on $\alpha$.

@FrankScience nope. it's still right because i can just pick one element from the preimage.

@Eugene But do I say "Assume proven for al indexes $\alpha_i \in I$ such that $a_i \leq a_m$ for some $m$. "

@Eugene So you can't use the notation $f^{-1}(x)$, it's a certain element.

@FrankScience whatever.

@FrankScience We understand each other.

@FrankScience it's an arbitrary element anyway so you can just abuse notation

@Eugene Here we go...

@Eugene The left side is for all indexes $\alpha$ the $x$ is same

@PeterTamaroff you needn't use induction. this trick can just be used. period.

@Eugene @Eugene The right hand side is just $x$ could be different but $f(x)$ should be same.

@Eugene Am I right?

@FrankScience yup

@Eugene I don't like taking things by faith! =D

@PeterTamaroff it's not faith. it's rigor

just say

let x in f(bigcap whatever)

so

so f-1x in bigcap whatever

then use the same trick

no induction necessary

@Eugene Oh, OK now I understand what you mean. Duh.

@Eugene Can you show me how many Axioms of set theory is used in that infering?

i'm no good at transfinite induction anyway

In Soviet Russia, notation abuses you.

@FrankScience i have no idea about set theoretical axioms

i know some guy called Mr ZFC says it's ok

@FrankScience You mean Zermelo Frankel axioms?

that's about it

@Eugene LOL

@PeterTamaroff Yes. I find we should be more rigorous.

@PeterTamaroff what's so funny?

@PeterTamaroff So we should point out all the axioms we've used.

@Eugene Not so, just a chuckle.

@PeterTamaroff Just like in Mathematical Logic.

@FrankScience Just google ZFC

@PeterTamaroff oh. ok. i took like one logic class in my whole life so i know about goose egg about set theory

@PeterTamaroff No

@FrankScience Can you google random stuff there?

I heard China has some interwebs restrictions.

@PeterTamaroff I meant that we should point out the axiom used in each step!

@PeterTamaroff Not roughly saying that ZFC or ZF.

@FrankScience If we know which they are, there is no need in pointing every axiom every time.

It would be a pain in the ass.

that's what my topology teacher said in the beginning of my class

we know that someone out there proved that set theory is ok so we'll just assume that.

@PeterTamaroff I think it's sensible to know, which axiom is used, and which axiom is independent to out proof.

@FrankScience If you're talking about that thing with $f(\bigcup_{i\in I} X_i$, then I think you only need Axiom of Extensionality. You are also using definition of image of a set and definition of union. But that's just definitions, not axioms.

@MartinSleziak Thanks, I'll read.

But I am not sure you should worry about ZFC at this stage. Do I remember correctly, that you're studying some introductory calculus now?

@MartinSleziak Well, let me step into one by one.

@MartinSleziak Your memory is pretty good.

We are talking about $f(\bigcup_{i\in I} X_i)=\bigcup_{i\in I} f(X_i)$?

@MartinSleziak Yes, sir.

@PeterTamaroff same trick

@MartinSleziak We've proved, but somewhat intuitive. Now we should check one by one to point out that only that axiom is needed.

@Eugene I aldready wrote it down.

@PeterTamaroff cool

@Eugene NOw I have to prove $$f\left( {\bigcap\limits_{\alpha \in I} {{X_\alpha }} } \right) \subset \bigcap\limits_{\alpha \in I} {f\left( {{X_\alpha }} \right)} $$

Let me think about it.

@PeterTamaroff i thought i just wrote the proof down???

@Eugene We were talking about $$f\left( {\bigcup\limits_{\alpha \in I} {{X_\alpha }} } \right) = \bigcup\limits_{\alpha \in I} {f\left( {{X_\alpha }} \right)} $$

were you paying attention? look at what i wrote!

@Eugene Oh God, why?

@Peter Just like the discussion yesterday, we have not built up the axioms for the law, and you abused the inferring -- hell.

I'll omit $\forall x$ from now.

i even wrote bigcap instead of bigcup...

Sorry, sorry. Truly sorry, Euge.

EUGENE!

@FrankScience They are built already.

@PeterTamaroff If so, it's unnecessary for you to assume about the hell.

@Eugene Eugene is Eugenio which in turn is Euge. Read it as "Eau-Je" (the Je like a German)

@FrankScience math hell is having to write down peano's axioms everytime i write $1+1=2$.

@PeterTamaroff yes but i am name EUGENE and not EUGENIO

And now just notice that $a\in \bigcup_{i\in I} X_i$ is equivalent to $(\exists i\in I)(\exists a\in X_i)$, which is the definition of union.

@Eugene I'm just fooling around.

@PeterTamaroff i know. so am i

@Eugene I'm "Pedro". But I find it a little dull.

@Eugene But the problem Peter mentioned is a basic problem, in which we need to be very rigorous.

@FrankScience Not really.

@PeterTamaroff ah. why not be Rodriguez? or romanio! sounds cool

@FrankScience isn't $1+1=2$ a basic problem?

"From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." —Volume I, 1st edition, page 379 (page 362 in 2nd edition; page 360 in abridged version). (The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, "The above proposition is occasionally useful."). Principia Mathematica.

@Eugene I prefer Peter (or Piotr hehehe)

@MartinSleziak That is a classic.

I'm too unrigorous, so I want to be more rigorous.

@Eugene So you jumped directly to $\bigcap$ because the first one was so trivial.

@PeterTamaroff more or less...

@FrankScience Rigor is OK, but only when we go at high speeds that any lump can throw us off the road,

@PeterTamaroff Because they are running on the same way.

Well, obviously the claim $1+1=2$ should follow after you defined positive integers as a subclass of ordinals and after you have defined ordinal addition. It is non-sense that they are teaching people this without proper preparation. ;-)

Now we're on a Sulky.

Now come to the real world.

@FrankScience ?

ref jonas' comment on the right =)

@PeterTamaroff I'll learn basic calculus.

@Eugene Yeah. =)

@PeterTamaroff i disagree with that though. elegance is nice but without rigor it's kind of useless.

@Eugene Well, I think it is understood we're meeting halfway here.

@PeterTamaroff yes i understand.

I think rigor is needed, but it can be counterprodutive to strive for celestial rigor sometimes.

@PeterTamaroff celestial rigor? is that like proving the sky exists before the start of every paper?

@MartinSleziak That is the word. "Overdo"

@Eugene Heavenly?

@PeterTamaroff well only if you believe in it i guess.

Without rigor, could Godel's incompleteness be proved?

@Eugene I was just trying to use a word that would emphasize my point.

@FrankScience : faith and belief can be used to avoid rigour if thats not too embarassing in the given context

@RajeshD Like "I believe $1+1=3$? What do you mean?

@PeterTamaroff i understand

well tahts clearly embarassing for me

@PeterTamaroff it did. seriously did

\begin{align}a&=b\\c&=d\end{align}

now we can prove if something cannot every be proven

:5220701 well, naively I viewed mathematics as something which is beyond any doubt; different from other sciences like physics, chemistry, philosophy. Godel shows that this is not entirely correct.

\begin{flalign*}a&=b&\end{flalign*}

I'm confused

@Eugene How so?

@PeterTamaroff how it's actually done i don't know. refer to my remark about never having done logic

@PeterTamaroff : You mean when someone is teaching a class in real analysis, he should start the class every day with the proof of existence theorem, no the students take it as a belief.

@RajeshD i think peter is saying we don't have to worry about these axioms

@RajeshD If someone asserts a theorem, I assume it is because there is a proof.

I don't take it as a belief, I know that has been proven.

$\LaTeX$ problem:

thats what exactl intend to say, but I start feeling that you guys aree discussing something else

How can I insert text in equations?

\text{This is text}

For example, $\{x: x\text{ is prime}\}$

or use hbox

which is more conventional?

is still don't understand why $(A^C)^C$ should be $A^{2C}$.

For example

@Eugene What I'm reading is that Gödel talks about undecidability, not impossibility of proof.

@PeterTamaroff : Clearly everything in math needs a proof ultimately, no denying that

$\{xy: x\text{ is prime and }y\text{ is prime}\}$

@PeterTamaroff undecidable means we can't figure affirmative or negative no?

Though I'm not saying you're wrong. You're most probably right but I can't read that off a limited Wiki article.

@MartinSleziak seems awful.

@Eugene Precisely.

@Eugene OH FUCK ME

whoooooooooooosh

Maybe I should start reading in my native language for a while.

I read your statement as something else.

@FrankScience What seems awful?

That one: "now we can prove if something cannot every be proven"

BTW Here's from TeX.SE: What is the "correct" way of embedding text into math mode?.

I'll go to sleep now.

@anon Thanks for the help.

@Eugene Thanks for the help.

@MartinSleziak x is prime and y is prime

@MartinSleziak $x\text{ is prime and }y\text{ is prime}$.

@MartinSleziak $xy\text{ where $x$ is prime and $y$ is prime}$

It looks fine in real LaTeX.

Of course MathJax handles some things differently.

I can't believe "poset" is "partially ordered set".

If you compiled such thing with LaTeX you would have the same font in the formula and in the rest of the text.

It sounded so badass.

@MartinSleziak I don't mean the rendering. I think some code like x\text{ is prime and }y\text{ is prime}

I guess this is the reason why MathJax version seems unusual to you.

@FrankScience If it renders OK why do you care what the code looks like?

@FrankScience Well, since I'm used to it, id does not look awful to me.

@MartinSleziak Now nested $...$ seems supported.

Which is a good thing.

$$\int\limits_a^b {f\left( x \right){\text{d}}x} $$

I guess it would be good if most things worked in the same way in MathJax and LaTeX.

\text{$x$ is a prime and $y$ is a prime}

The text is somewhat main, and math typeface of $x$ and $y$ is more unimportant, so \text first then $...$ inserted.

One example in the src of Concrete Mathematics by Graham, Knuth and Patashnik:

$$W&=\sum_{n=1}^{1000}[\hbox{$n$ is a winner}]\cr$$

Source code of Concrete mathematics is freely available online? D.E.Knuth is really a generous man.

@MartinSleziak I don't know whether it is legit.

IIRC I have seen TeX-source of TeXbook somewhere.

I have printed out the TeXbook long time ago, but I've only read a few pages.

@MartinSleziak See here. Is it legit to download?\

I would think so. (Considering that DEK is from Stanford himself.)

ok I'll have to go

See you later!

@MartinSleziak Bye, but taocp is not downloadable.

@anon Are you in downtown?

what?

downtown

not quite

@JasperLoy Why can't we compile it?

@JasperLoy I found that I cannot compile CMath, too.

@JasperLoy Who says? Don Knuth?

@JasperLoy Is it said in readme?

@JasperLoy Oh, see.

@JasperLoy TeXbook is not a FREE book.

@JasperLoy I look upon this: free documentations

@JasperLoy Read this

@JasperLoy And how about selling freeware

@JasperLoy Exactly no. But I look up serious, although I cannot be usually.

@JasperLoy When the things are happening, I might not be serious, but I think I should think seriously when the things ends.

@JasperLoy BTW, have you cell-phone with android?

@JasperLoy Shall I remove Ubuntu from one of my computer?

@JasperLoy I read this Nearly all GNU/Linux operating system distributions add proprietary packages to the basic free system, and they invite users to consider this an advantage rather than a flaw.

@JasperLoy It's killing freedom.

I think I should change ubuntu to debian, though the first distro seems easier to use, but containing non-free automatically, but debian doesn't.

@JasperLoy Oh, I use awesome window-manager and fbpanel.

@JasperLoy Any distro.

@JasperLoy Are you using vim?

I use vim to write programs, $\LaTeX$ or something else.

@JasperLoy Learnt a lot, but no I'm familiar with.

@JasperLoy For example, Pascal, C, C++, Ruby, Scheme, Common Lisp, Haskell, OCaml.

@JonasTeuwen Well, if you work at my department, you will more often say "mathematics is about being rigorous"

E, Ilya. How are you? Long time no something.

@Gig: hi, I'm ok. Was busy with work, that's why I'm rarely here

how are you?

Not bad, thanks.

What actually is the definition of rigour? Does it have a (rigorous oops proper and unambiguous) definition ?

@RajeshD yes. Don't state what you didn't prove. When you prove that, justify each step, each part of the proof. Shorten only those parts of proof, which you have done at least 10 times as an exercise. After you proved something, try to find a counterexample to this statement of yours.

thanks @Ilya That was a wonderful definition. I agree that if there is no rigour then it isn't proper math at all.

But at the same time I should emphasize that there is something called informal discussions in math where most of the importance is given to the intuition, and these kind of discussions/exchanges are not only important but also highly desirable.

who does algebraic topology here?

@Ilya Hey there, stranger!

@robjohn: hi!

@Ilya We're getting ready for fireworks here tonight (in about 18 hours)

@robjohn in 40 minutes there should be a news from Geneva about Higgs' bozon

@Ilya or we'll all be sucked into a black hole

either of two, certainly

today I'm in the black hole mood :)