/sci/ - Science & Math

Oops
>>12182535

>>12190853
The first is certainly more along the lines of what I'd come up with if you ask me.
(Also in Haskell, what's closest to natural transformations are polymorphic functions that fulfill the defining square, and those are defined over abstract a. The situation there might be loser in several ways, though, since all functors, at least in the standard lib, are endofunctors.)

I'm not used to your notation of writing the quantifier at the end, so what you write is a bit difficult to read - for me at least. And you don't seem to be complete anyway, e.g. in the 1. definition you don't end up with the forall a and forall b.
In those 3 lines you seem to repeat one line, and I'm not sure if you properly disallow that two f's with same domain and codomain have different natural arrows.

Anyway, I think you can either go NBG and work with classes better, or you go the Grothendieck way and force C to be just a set in a universe.
The former option is discussed in some length in this book that I came across last year
https://www.amazon.de/Sets-Functions-Measures-Fundamentals-Mathematics/dp/3110550083

Or, you just don't consider sets of nats. Who needs the Yoneda lemma anyway :^)

>>12190956
A bit early there bud, we haven't even hit the bump limit yet

>>12191027

I'm going to rewrite the two definitions, but before taht let me clarify a bit what is going on.

>in the 1. definition you don't end up with the forall a and forall b
I meant "for all morphism [math] f : a\to b[/math] of [math]\cal{C}[/math]".

>In those 3 lines you seem to repeat one line
No, they're similar but there's a difference on the quantifiers. The purpose of the second line was to make sure that every morphism of [math]\alpha[/math] is the left arrow of a commutative square, in contrast with the first that says that there is a square for every morphism [math]f\in \text{Hom}({\cal{C}})[/math].
The purpose of the third line (the one I didn't explicit) was to ensure that every morphism in [math]\alpha[/math] is a right arrow of some commutative square.
I think you can see that there's a mistake here. These last two lines should be combined into one: Every morphism in [math]\alpha[/math] is the left arrow or the right arrow of one of these commutative squares.

>I'm not sure if you properly disallow that two f's with same domain and codomain have different natural arrows
They may have different ones or not, I'm only asking for at least one commutative square for each [math]f[/math]. I don't think I need to ask for different ones, since that's not the case for a identity natural transformation of a single functor that sends every morphism to a single one, for instance.

>>12192067
>since that's not the case for a identity natural transformation of a single functor that sends every morphism to a single one, for instance.
I'm too tired already to think about whether I'm wrong about all possible arrows in a square having to pass through the same two arrows of the natural transformation.

But be that as it may, I don't recall people demanding both doing ZF and also working with categories that are classes. If you want to work in ZF, then e.g. instead of saying "the category of groups", I'm sure you can get far by saying I you consider a set (not specifying that set, leaving room for it to be a large cardinal too, should you later choose to postulate it) where all elements (already in it) are groups. I'm confident that whatever closedness properties of that set you actually need, ZF will do.

For a more educated take, there's that paper by Shulman
https://arxiv.org/pdf/0810.1279.pdf

>>12192103
Sorry, I wanted to edit my post better but ended up picking it in time.

I'm speaking of classes in the Thomas Jech's sense. I first learned about it there, then I read in an appendix about category theory of an algebra book that it was introduced by him.

In short, we think of classes as collections of sets. When it's not a set, we call it a proper class. Proper classes cannot be an element of any class. Other than that, we can do a lot with them as if they were sets, even though they are not objects in ZFC. This is because they are defined by formulas and we can just translate everything onto a bunch of statements over formulas.
You'll find more details in his book, "Set Theory". It's quite short.

So basically the idea is that we can do category theory in ZFC through classes, but we can't climb even further and speak of collections of proper classes. This is a problem when we arrive at the Yoneda Lemma, we'd have to ask the category to be small. Unless we don't want to work with natural transformations as functions, which is why I started thinking about that possible second definition. I said you can see that definition elsewhere because it seems to be how they think of it (when they're not taking components indexed by objects, of course), and that's why they claim that asking the category to be locally small is enough for ZFC. They're wrong anyway, either they should stop addressing components to objects, or the statement is wrong and should ask the category to be small.

>>12192501
Do you read all of them, or just one?

>>12192103
>>12192526
Continuing.

>I'm sure you can get far by saying I you consider a set
I thought about working around without going to huge collections, but apparently it gets complicated.
I want to get to the applications in commutative algebra. For instance, universal properties, like the one of a direct limit. The way it is stated without heavy category machinery only deals with arbitrary isomorphisms (any object satisfying the universal mapping property of the limit is the target of a unique isomorphism from the object representing the direct limit), but these isomorphisms actually assemble into a natural isomorphism that is related to the complete categorical universal property statement. I didn't get to all the details yet, but I've seen a book explaining it and in the process it uses the Yoneda Lemma.

You can prove a bunch of things about modules with direct limits, and they're usually over the category of R-modules for a certain ring R. If you want to prove it for a general R-module M without going up to proper classes, we'd need to find a bigger module having M as a submodule and also being closed under certain operations. Sometimes it seems to get quite complicated. I'm abandoning this approach for now because there's a better alternative, one that would be moving out of ZFC like with NBG or Grothendieck's universes that you mentioned, but better because it has all the "going up" power of the latter (NBG has a ceiling, from what I remember) but is weaker than these universes and carries every theorem in a proper general form to ZFC. In short: You add a constant symbol M to ZFC and an axiom that makes M be like a universe, but way weaker than a Grothendieck's one. M is subjected to the axioms of ZFC too. Call this system ZFC_M. You can show through model theory that every theorem in ZFC_M of the form "[math]\forall x_1, x_2, ..., x_n \in M : \varphi(x_1, \ldots, x_n)[/math]", where the last formula doesn't involve M, gives a theorem...

>>12192550
gives a theorem "[math]x_1, x_2, \ldots, x_n : \varphi(x_1, \ldots, x_n)[/math]" in ZFC.

So if you want to prove something about a class of sets through category theory, you prove it in ZFC_M in that format, where you can go up as much as you want inside M because of it's axiom, then you arrive at a theorem of that format in ZFC_M and use this proposition to get it in ZFC.

Disclaimer: I might be butchering a few things here because I haven't studied logic, set theory nor model theory extensively. But I learned this from a specialist in the area. He gave me a proof of this "theorem transfer" but I didn't even try to read yet since I barely know anything of model theory.


So you may want to ask why don't I just do that and be settled, then. The answer is that I want to know the limits of the pure ZFC approach, I want to know how much all these books are wrong and if it can be fixed (like with that approach of defining them differently).

>>12192550
gives a theorem "\forall x_1,x_2,…,x_n : \varphi(x_1,…,x_n)" in ZFC.

So if you want to prove something about a class of sets through category theory, you prove it in ZFC_M in that format, where you can go up as much as you want inside M because of it's axiom, then you arrive at a theorem of that format in ZFC_M and use this proposition to get it in ZFC.

Disclaimer: I might be butchering a few things here because I haven't studied logic, set theory nor model theory extensively. But I learned this from a specialist in the area. He gave me a proof of this "theorem transfer" but I didn't even try to read yet since I barely know anything of model theory.


So you may want to ask why don't I just do that and be settled, then. The answer is that I want to know the limits of the pure ZFC approach, I want to know how much all these books are wrong and if it can be fixed (like with that approach of defining them differently).

Hi I'm back and still dumb.

Since there's a basic unit for physical vectors like i, j, and k, is there one for time?
Would that just be the delta symbol?
How plug 3 vectors into time?

>>12192526
>You'll find more details in his book, "Set Theory".
Erm, yes I know what classes are. But good to make sure we read about the same thing I guess.

Again, I don't recall people doing category theory stuff actually actively considering themself to work in ZF while modeling categories as classes. At the very least when you speak of the category of categories CAT, it's clear that the class of all group is not a member of CAT, which would be a big defect of the class approach.
I don't know who they are.

But I'm sure you find it in the Shulman paper and book I cited.

>>12192550
>ZFC_M
If you can gauge how deep your biggest objects are, then you can indeed just take a set in the von Neumann hierarchy. [math]V_{\omega+\omega}[/math] is a model of ZFC and even a topos.
If you use free constructions making use of Replacement, you'll however scale up quite fast.
In case you need all ordinals, which form a class, then you also need more.

>>12192560
>>12192563
>I want to know how much all these books are wrong
I think Colin McLarty is someone who tried to properly research this (e.g. "are ZF universes necessary to formalize the proof of Fermats last theorem, and why not")

If Jech makes a side remark that all you need to do for category theory is classes, then it's fair and good to question that claim.
They are not necessarily experts in all things logic.
You can also find David Hemkins on MO trip over the fact that assuming that ordinals are linearly ordered implies excluded middle, and off-classic-set-theory things like that.

>>12192550 (You)
gives a theorem "[math]\forall x_1,x_2,…,x_n : \varphi(x_1,…,x_n)[/math]" in ZFC.

So if you want to prove something about a class of sets through category theory, you prove it in ZFC_M in that format, where you can go up as much as you want inside M because of it's axiom, then you arrive at a theorem of that format in ZFC_M and use this proposition to get it in ZFC.

Disclaimer: I might be butchering a few things here because I haven't studied logic, set theory nor model theory extensively. But I learned this from a specialist in the area. He gave me a proof of this "theorem transfer" but I didn't even try to read yet since I barely know anything of model theory.


So you may want to ask why don't I just do that and be settled, then. The answer is that I want to know the limits of the pure ZFC approach, I want to know how much all these books are wrong and if it can be fixed (like with that approach of defining them differently).

I want to ask you mathchads something very important.

I want to get into an applied math masters with tracks which are relevant to my undergrad and masters degree. However, neither is properly STEM. I took relevant courses which covered these topics

Undergrad: calculus, linear algebra, some real analysis, calc based probability. Not proof based but theorems were proved in class. more probability, statistical inference, linear regression, discrete time models, binomial trees, monte carlo, black-scholes
Grad: more probability, formal treatment of risk models and time series models in discrete time (think arma, vector autoregression, garch, markov switching, etc), stochastic processes in discrete and continuous time, martingales, markov processes, relevant PDEs (Kolmogorov, fokker planck, feynman-kac), more monte carlo, kalman filter and bayesian statistics

There are big absentees in my track that could get me rejected: proof based real analysis and ODEs. I had no opportunity to get accredited courses in the fields this summer because of the coronavirus. All my knowledge is not certified and self taught.
Almost none of the programs I want require the GRE.

How can I convince the mathchads in adcoms that Im not a scam but I am able to get a masters in app math? What would you like to hear from a non-stem dude to get convinced that he can make it? I know few math people so I ask here too

>>12192574
>But be that as it may, I don't recall people demanding both doing ZF and also working with categories that are classes
>I don't know who they are
Category theorists and research in the area is probably mainly outside ZF.
This ZFC approach is something I've seem in two textbooks and on wikipedia. All of them state the Lemma in the same way, noting the size issues and the "fix".
Since I'm interested in concrete applications rather than going deep into category theory, I got interested in the possibility of not going much further than ZF. If we can't even use Yoneda on concrete categories, then why even bother? Or, if we can apply it without large categories, then why it isn't done? Or maybe it is and I just didn't find it yet.

>If Jech makes a side remark that all you need to do for category theory is classes, then it's fair and good to question that claim.
I think they cited Jech to address his class approach, I don't know if Jech claimed anything.
I'm saying "they" over what I've read in Pierre Antoine Grillet's abstract algebra book, at the beggining of the appendix about category theory. He mentions it as a well known approach.

Thanks for the sources, I'm going to take a look.

>>12192630
>I think they cited Jech to address his class approach, I don't know if Jech claimed anything.
I mean the approach of talking about them in ZFC for convenience, not the approach of using it for category theory. The latter is what I meant about Grillet's comment on it.

Can I learn proofs along the way using Tao's Analysis? I did go over some before quite briefly and they seem tricky but got the hang of it after doing some basic problems (even/odd and induction stuff). Or would it be necessary to spend like a week beforehand just doing basic proof stuff in Velleman?

>>12192630
>I'm interested in concrete applications rather than going deep into category theory
>If we can't even use Yoneda on concrete categories
But you can, really.

Take [math]{\mathbb N}[/math] and apply the powerset operation [math]{\mathcal P}[/math] to it a limit ordinal amount of times. Then you e.g. get something like [math]{\mathcal P}^{\mathbb N}{\mathbb N}[/math]. With [math]V_{\omega+\omega}[/math] you got a cozy set that's a topos, and it has all the groups anybody here has every heard about. The contravariant hom-functor will map back into it.
Don't you think that counts as a concrete category for your purpose? It's very small as far as ZF goes.

Just because you can't use it for the category of ZFC-groups or the category of ZFC-sets, doesn't mean you can't prove the lemma for any category.

Again, if you take [math]\{\}[/math] and apply the [math]{\mathcal P}[/math] to it, say [math]\alpha = (((\omega^\omega)^\omega)^\omega[/math] times. See if you find anything that guarantees you closedness conditions for your objects of interest. The gamble is that finding this out is less work than learning about a new framework.
Then prove theorems about not ZFC-groups and rings, but ZFC-groups and rings or this countably transfinite rank?

*of this countably transfinite rank

>>12192574
>[math]V_{\omega+\omega}[/math] is a model of ZFC
I don't know any of the category stuff you are talking about. But this is not true, Replacement isn't satisfied. For any limit ordinal, V of that rank is a model of Zermelo set theory.

>polynomials continued

So, say you're given a polynomial such as x^2+7x+36, and you're asked to factor it, which means to find its multiplicative factors of the form (x+a)*(x+b) - a and b can be negative btw, so this allows for subtraction. The way you go about this is by satisfying a few constraints. If we recall how distributing works, we see that for whatever a and b we have, we get x^2 +xa+xb+ab. So we want our a and b to add up to 7, and multiply up to 36. The truth in this case is, there are no numbers that sum to 7 and multiply to 36, so no factors as shown above exist (although other factors, such as the polynomial divided by 10, and the scalar number of 10, work, but those aren't as useful for reasons we will see). This is a general rule of thumb you can use, the largest multiplication of numbers that sum to N is N/2. So the largest multiplication of numbers that sum to 7 is 3.5 * 3.5 = 12.25, as 4*3 =12 is smaller. A polynomial like x^2 + 12x + 36 actually works out as (x+6)(x+6) - try it! If you cannot find the numbers for the factors of a polynomial on your own by simply trying what sums and multiplies properly, there is the "quadratic formula," and there is a nice explanation for it in this video: https://www.youtube.com/watch?v=EBbtoFMJvFc

>>12192772
Yeah fair enough, of Z then

He must check if he has all constructions that he needs, I know some free spaces need Replacement

>>12192787
>cont'd

If you have ever heard anyone say "find the roots of a polynomial", that means find the x coordinate for which the y coordinate is 0, aka, where the graph intersects the x axis (which lies at y=0). If you have the factors of a polynomial, then you know its roots, because you know that
>y=x^2+12x+36, so
>y=(x+6)(x+6)
You want the x for which y = 0, so you write 0 in place of y as
>0 = (x+6)(x+6)
Then you can just divide out one of the factors and simply have 0 = x+ 6, so x=-6 is the x intercept, and since its the same for both factors, its the only x intercept.
One thing that's very interesting about polynomials is that if they have a root, they have it in the factor (so it goes both ways, roots find factors and factors find roots). The reason it goes backwards is a bit harder to prove, but the idea is that if you have a root labeled "r" then you can turn any polynomial p into (x-r)*q where q is a different polynomial, and since (x-r)*q is still p, the roots of p are still in q (since they're not in (x-r)), so q can be factored again. The reason this is important is because for polynomials like x^2+7x+36, and x^2+1 = 0, the fact that they cant be factorized means they also have no roots, never intersect with the x axis, because if they did, you could find factors. For our polynomial (x+6)^2, if we look at its graph we see it intersects the x axis only at x= -6, meaning its only root is -6, so it must show up twice as (x-(-6))(x-(-6)). We know it shows up twice because x+6*q works, and q cant be anything other than the root -6. We also know because x+6 is order 1, x^1, but we need x^2.

Lastly, the way to find the vertex of a parabola, its lowest point, is to find its intersection points, and since its symmetric, find the middle of those two. Then using that middling x coordinate, find the related y coordinate.

>>12192670
You could try it and tell us
People got into proofs in a lot of different ways

>>12192670
If you want to understand proofs honestly I would just brush up on how logic works, reading a bit about naive set theory would help

>>12192806
Vellemam seems to cover the bases and Tao does too
>>12192797
Ok might update after a week or two

>>12192806
>>12192810
brother proofs are just following an argument. you absoultely dont need naive set theory as groundwork. and the arguments in set theory proofs are not any more elementary or formative either. just read the book and if its too hard read something easier

>>12192544
I'm assuming you read them one by one starting from the first row to the next from top left

>>12192670
I haven't read Tao's book, but in general basic real analysis is a good place to start proof based math. What I'd recommend is going through a book that has a solution set out there for the exercises. This will give you a sense of basic techniques and arguments.

CONJECTURE.
Let [math] (A_i)_{i \in I}[/math] be a family of compact subsets of [math]\mathbb{R}^n[/math], which is descending in the sense that it is a totally ordered chain where [math] A_i \subseteq A_j \Rightarrow A_i \subset A_j^\circ [/math] (included in the interior).
Then the family is countable.

>>12193159
My ideas for a proof would be to use the trick that shows that uncountable sums can't converge. If the sets are super strictly included like that then they must decrease by some [math] \epsilon [/math] in some sense and then that will happen uncountably many times with a decrease of at least a certain fixed amount.

>>12192822
For me, I noticed that the very meaning of certain arguments didn't make sense without first coming to terms with how logic works. It took me a few weeks to really understand what if and only if means. I once made a post on /mg/, about a year ago, where I asked what "for every x in a set" meant, and it was autistic comedy. People without autism might find it less confusing, however.

>>12193159
>>12193183
This is just more or less the separability of R^n.

>>12193159
But this is obviously not true.
Let A_x = [-x, x] for x a positive real number.

Still my favorite math meme of all time. Thank you whoever made it :)

>>12192670
Logic and set theory when taught explicitly poison and retard your mind and stunt your creativity. You will immediately pick up set theoretical techniques and notation from doing actual math. I took analysis and got an A without ever taking an intro to proofs class, did the same in Linear Algebra

>>12192730
I thought something else other than replacement issues.
I think that in terms of size and going down to subsets this is probably enough for most applications, but I'm not sure about other types of closedness.

If I fix a (not too huge) R-module M and consider the functor [math]M\otimes \bullet[/math], it is an endofunctor of the category of R-modules, but it wouldn't be an endofunctor of the category of R-modules with underlying sets being subsets of [math]V_{\omega+\omega}[/math], right? How do you circunvent that? Take all these tensor products and change the underlying sets through bijections with their cardinals? Maybe even impose that in the construction of tensor products.

>>12193227
>but it wouldn't be an endofunctor of the category of R-modules with underlying sets being subsets of Vω+ω, right?
I don't know, I wouldn't be so pessimistic about it.
E.g. if X is in the cartesian closed category, so is X^X and (X^X)^X, and so on, since you just go finite step by finite step and there's no transfinite iterations involved. This is just how in the class of heriteteriliy finite sets (stuff like {{},{{{}},{}},{{}}}), you can take any number of countable powersets and will still have something countable.
Quotient spaces always depend on their coding, but without further thinking, this looks harmless.
What does make problem in the Replacement sense, I think, is transfinite operations such as [math]\bigcup_{n=0}^{|{\mathbb N}|} {\mathcal P}^n X [/math].

Gonna go bully my nerd friend into giving credit for his results to me.
Intimidation skills >>> IQ

>>12193254
PS read the Shulman paper and rethink

>>12193270
you're a good friend

>>12192501
saved.
have a (((you)))

What's the deal with hyperbolic geometry? I'm having trouble understanding what it even means, how the geometry itself is different from a mapping, and how it differs from euclidean

>>12193301
have you tried not being a helpless retard? it tends to help with these kinds of things.

>>12193301
a good way to conceptually understand the difference is basically how straight lines work.
look at parallels. on a spherical geometry, there are no parallel lines. Straight lines always cross.
euclidian geometry, parallell lines always have the same distance between them.
hyperbolic, the distances between straight lines that start parallell will always increase in both directions.
alternatively, angles of a triangle. 180 = euclidian. <180 = hyperbolic. >180 = spherical.
but like this faggot says, >>12193312
read into it, there's plenty of good stuff online with decent pictures.

>>12193312
>have you tried not being a helpless retard?
No, that's why I came here.

>>12193339
I didn't make this post but it's accurate

>>12193320
I tried reading the wikipedia page and making drawings with pen and paper but no avail.

>spherical geometry, no parallels
What about pic related?

>>12193354
Lines of latitude? They are not along great circles, except for the equator.

>>12192501
>tfw when your brain is a smooth manifold

Anyone good in combinatorics can help me with this?

There are A attendants who each take care of B beneficiaries (beneficiaries can be taken care of by more than 1 attendant).

Every pair of beneficiaries have in common C attendants.

How many beneficiaries are there?

For a numeral application take A=11615, B =84, C = 15.

>>12193416
I've posted this in /wsr/ but I doubt there many people who could be of help who browse that board.

>>12193416
AB (B choose 2)=2 (x choose 2) C

>>12193416
There are total AB connections (a,b) where a is an attendant and b a beneficiary.
Now we count it in another way.
For every pair of beneficiaries, there will be 2C connections with common attendants. Summing over all the (x choose 2) pairs of beneficiaries, there will be 2(x choose 2) C connections, but we've overcounted. Since for every attendant, among the B beneficiaries it has each beneficiary is counted B times since there are B pairs with that beneficiary.
Thus
2 (x choose 2) C = A B B
Where x is the total number of beneficiaries.

>>12193301
i mean it's kind of tough to understand hyperbolic geometry without understanding very rigorously what a surface is and what a metric on a surface is (riemannian metric). when you do learn these things it is very, very easy to understand what hyperbolic geometry means. it's just a plane where "everything gets further closer when you go downwards." or a circle where "everything get further when you go towards the edge". this is formalized through a metric, which tells you how long vectors are when they start at different points.
on the euclidean plane, no matter what point a vector starts at, if it's the same vector, it has the same length. this is what you change when you want to think about other 2d geometries.

>>12193497
>everything gets further closer

if A is a subgroup of G and B is a subset of A, is B a subgroup of G?

>>12193522
This is possible the dumbest question ever asked on /mg/

>>12193522
Think. Does B have to have the identity in it?

>>12193559
makes sense, thanks.

>>12193514
oops
i typed the wrong thing originally and went back to change it but forgot to remove the other thing
either works but in the upper half plane model it's everything gets further when you go downwards.

>>12193522
no

>>12193522
this is literally "proof: think" tier

>>12193527
Bully not the beginner!

>>12193677
Actually, you are incorrect. The answer is "maybe".

>>12193522
Legendary post.

>>12192569
I don't believe their's a common notation for a time unit vector, people just use x y z t and assume its the standard basis.
>How plug 3 vectors into time?
look up minkowski space

Hot take: Each chapter should end in bibliographical as well as historical notes

Are there good podcasts about pure mathematics?
I only remember a podcast with guests presenting their new math books.

>>12193212
yes, perhaps he was considering only families which guarantee equality between at least some subsets, but even then I'm not sure if true...

>>12193774
>perhaps he was considering only families which guarantee equality between at least some subsets
What the fuck does that even mean?
>>12193702
Spare the rod, spoil the child.

>>12193820
>Spare the rod, spoil the child.
is this ever more than an abusers cope?

>>12193824
>is this ever more than an abusers cope?
It's only abuse if you're a pussy. And pussies don't belong in maths.

>>12193354
lattitudes aren't straight. to highlight the point, if you follow the line near a pole, it will quite clearly look like a circle. longitude lines on the other hand are straight.

>>12193757
I like it when a chapter ends with at least one of those.

>>12193820
Or how they say back home
>who spares the rod hates their child

>>12193824
It's like a literally taking revenge for your own beating as a kid on your own kids.

>>12193831
>And pussies don't belong in maths.
math is essentially entirely populated by pussies.

>>12193835
>It's like a literally taking revenge for your own beating as a kid on your own kids.
Projecting your own mental illness there, tranny. It's basic discipline, nothing to do with revenge. Used to be completely fine and normal until the modern day in the west and look how the kids are turning out.

>>12193844
and only way to raise pussied is to beat them up as children. proving that you do indeed need to beat your children up to be good at maths.

>>12193847
A lot of things used to be fine and normal, but the only finesse and normalcy we should care about those in the algebro-topological context. Please do not use such mean mean words!

POV: You're answer is off by 1

Reminder the more cute the anime girl the uglier the poster behind it.

Is this peak algebra?
https://en.wikipedia.org/wiki/Monster_group

>>12193492
I agree for the left hand side but on wouldn't it be
2(x choose 2) C = 2(B choose 2)A
We want to consider every connection associated with a pair of an attendant

>>12193918
2(x choose 2) C
How many times do we count each pair (a,b)? The answer is as many times as there are pairs with attendant a and one of the beneficiaries b. There are (B-1) such pairs. So it actually should be
2 (x choose 2) C = A B (B-1)

r8

>>12193950
buttplug/10

>>12193950
cool buttplug

Now rotate it around the y axis and calculate the volume

>>12193950
Based.

Have a quick question:
Is a sequence of real numbers still a Cauchy sequence if [math]|x_n - x_m| < \varepsilon|[/math] but [math]m, n \le N[/math]?

>>12194420
Ignore the | after [math]/varepsilon[/math]/

>>12194420
maybe

>>12194423
ugh, I mean [math]\varepsilon[/math].

>>12194420
that would make all x_n trivially equal to each other

>>12194478
Fuck, that is true. Thanks dude.

>>12192787
>continued
Where's the first post?

Well boys, the day was saved again by adding 0. The hero we need, but truly don't deserve.

>>12194708
People always think I'm being facetious when I tell them that adding by 0 and multiplying by 1 are two of the most powerful tools in mathematics.

>>12194808
shut the fuck up faggot

>>12193757
This.

>>12193824
No, it’s a means of filtering retards and discouraging unthinking reactionary problem solving. Retards fuck up more than intelligent students, being abused for their failure drives them out making room and time for more talented students. Naive and untrained students need to learn the hard way when it is important to think deeply and accurately about a problem and it’s solution before embarrassing themselves and their teachers with an idiotic answer.

>>12194808
>People always think I'm being facetious when I tell them that adding by 0 and multiplying by 1 are two of the most powerful tools in mathematics.
everything else is just counting and arrays of numbers

>Xenakis pioneered the use of mathematical models in music such as applications of set theory, stochastic processes and game theory
>Specific examples of mathematics, statistics, and physics applied to music composition are the use of the statistical mechanics of gases in Pithoprakta, statistical distribution of points on a plane in Diamorphoses, minimal constraints in Achorripsis, the normal distribution in ST/10 and Atrées, Markov chains in Analogique, game theory in Duel, Stratégie, and Linaia-agon, group theory in Nomos Alpha (for Siegfried Palm), set theory in Herma and Eonta, and Brownian motion in N'Shima.
what do you think about this?
https://www.youtube.com/watch?v=nvH2KYYJg-o

>>12195061
sounds like absolute garbage to me

>>12195061
>music

>>12195061
It's interesting - music where you can _see_ roughly what's coming.
Remdings me of that Kierkegaard rant.
Also reminds me of whatshisname, that Austrian composer with the horrible life during Nazi Germany, some short name with A or G or so

https://terrytao.wordpress.com/2020/10/04/foundational-aspects-of-uncountable-measure-theory-gelfand-duality-riesz-representation-canonical-models-and-canonical-disintegration/

Bros... I just want to be good at maths.

>>12195709
it's a cool kids only club

[eqn]\sin(x)=\sum_{n=0}^{\infty}(-1)^n\frac{x^{(2n+1)}}{(2n+1)!}[/eqn]

[eqn]\cos(x)=\sum_{n=0}^{\infty}(-1)^n\frac{x^{(2n)}}{(2n)!}[/eqn]

How do I get into algebraic K theory?

>>12195751
learn algebra first, then K-theory

>Read rudin
>Don't grasp anything
>Read rudin again
>Understand all the proofs but can't prove anything
>Am going to read rudin again
T-third time's the charm, right?

I just met Terry Tao at a grocery store and it was one of the worst experiences of my life. I knew it was him but I had no idea how to approach him, so I ended up walking behind him and saying “How's that uncountable measure theory coming along?” as I passed by. He immediately turned around and looked terrified, like I think I really startled him. I thought it’d be funny but clearly it was just cringe so I said “sorry” and walked away. We kept making eye contact in the store, it got to the point where I left early before I got all the stuff I wanted because it was too cringe. A couple minutes after I got in the checkout line Tao got in line behind me, I pretended not to notice but he said “hey” so I turned around, and then he asked me if I was a undergraduate. It was extremely cringe so I just said yes trying to play along but then he immediately walked over to another checkout line without saying anything else. He seemed pretty tense the entire time and I’m not sure what to make of it, did I do something wrong?

Anyone have a recommendation for a good undergraduate (non-measure theory) probability theory text? The course I'm in is using Durrett which is generally good and rigorous, but I feel like I'm missing some of the high level intuition regarding many concepts.

>>12195788
Doesn't durrett use measure theory?

>>12195794
Yeah sorry I wasn't clear. I'm in a measure theory based probability course, but since this is the first probability course I've taken, I feel like I'm often getting bogged down by the details without having the big picture motivation. It feels akin to reading rudin without having taken basic calculus before.

>>12195785
Hahahaha good job now he's gonna have nightmares about you.

>>12195747
Do you even know what those mean or are you just in Calc 2 now and that's the most complicated thing you've seen that you've convinced yourself you understand because smart but lazy, right?

>>12193756
Thank you.

>>12195773
Why do you read it all if you're not understanding it? If you didn't put sufficient effort in the first read through, yes, you are reading it a third time, then probably a 4th to be able to do a good enough amount of problems given, then a 5th to be able to apply it elsewhere and a 6th after you forget the material in a couple years because you didn't internalize it.

>>12195851
>Why do you read it all if you're not understanding it?
During my first time, I did understand most of it, but rudin, at the time for me, was too terse. In my second reading, I did understand every single proof of his. The difficulty lies in actually applying those proofs and solving exercises, as I am self-learning for the purpose of amusement. I found dummit and foote to be much more palatable due to its encyclopaedical nature.

>>12195061
Peak fucking composition
A location of a set of emotions I have yet to experience before
Thank you

Is tcs the funnest area?

>>12192670
A more introductory text on proofs is probably a good idea. Haven't read Tao's text, but I'd recommend How to Prove It or Reading, Writing, and Proving prior to diving in (assuming no/minimal experience), or something along those lines. I'm also a Rudinfag, and I loved that text

For our American and travel oriented friends:
>2021 GRADUATE STUDENT TOPOLOGY AND GEOMETRY CONFERENCE
https://gstgc2021.iu.edu/

>>12196056
>Statement of Inclusiveness
>Policy on Harassment and Discrimination
do americans...

>>12195785
I like the one with "huh. huh." more

>>12190956
How do you get good at real Math? I could probably do calculus, Linear algbra, probability and statistic just fine but i get filterd by the first chapter of baby Rudin

How do I internalize proofs and stop being a brainlet?

>>12196151
drop LEM and write algorithms

>>12196150
Increase your IQ to at least 130. There is no satisfaction in mathematics for brainlets.

>>12195168
Nah, this would be a really good horror movie backing track. It's the sort of thing I hear when I'm doing analysis

>>12196169
>Increase your IQ by at least 130
more like it

>>12195844
it's the taylor series expansion of sine and cosine and it has an infinite radius of converengence, is there anything more to it?

>>12196300
no, he's just acting tough because he's already in calc 3

Yup, I'm thinking it's category theory time.

>>12196435
What are those?

>>12196469
Take a guess.

>>12196511
Medication

why is algebra so boring? I'm taking a grad algebra course this semester and I swear I spend 80% of my time just trying to interpret the notation and what it actually means. Analysis is way more fun and gets to problem-solving quicker.

>>12196530
It's the exact opposite.

>>12196530
it's mostly a "useful, but not very interesting in their own right" type of subject
get into algebraic number theory or algebraic topology if you want to see the usage

>>12196530
100% subjective opinion

I saw Terry Tao at a grocery store yesterday. I told him how cool it was to meet him in person, but I didn’t want to be a douche and bother him and ask him for photos or anything. He said, “Oh, like you’re doing now?” I was taken aback, and all I could say was “Huh?” but he kept cutting me off and going “huh? huh? huh?” and closing his hand shut in front of my face. I walked away and continued with my shopping, and I heard him chuckle as I walked off. When I came to pay for my stuff up front I saw him trying to walk out the doors with like fifteen Milky Ways in his hands without paying.

The girl at the counter was very nice about it and professional, and was like “Sir, you need to pay for those first.” At first he kept pretending to be tired and not hear her, but eventually turned back around and brought them to the counter.

When she took one of the bars and started scanning it multiple times, he stopped her and told her to scan them each individually “to prevent any electrical infetterence,” and then turned around and winked at me. I don’t even think that’s a word. After she scanned each bar and put them in a bag and started to say the price, he kept interrupting her by yawning really loudly.

>>12196643
what a lil faggot

>>12195709
Same man

>>12196150
Unpopular opinion here but baby rudin is a pretty subpar book for intro analysis. His green book is much better. Try Abbott or Pugh for analysis.

Tired of not wearing glasses. I can't see anything and it pisses me off. I need to get really close to the screen to read a sentence. Fuck my eye doctor, I'm gonna wear them all the time now.

>>12196547
>get into algebraic topology if you want to see the usage
Working out some groups related to the 5 dimensional sphere sure is a "use" ...

I mean nothing at all against algebraic topology, but this is circular thinking. If you like abstract algebra or algebraic topology, that's great. To say that the one language game gives use to the other is not a great justification.

Any good references for proofs that I can look at while taking analysis? I intuitively get a lot of the things I need to prove but I can't always figure out the right way to write it down in a mathematical proof

>>12196779
I don't think you need a reference, that sounds pretty normal. Taking analysis is a good way to get better at writing proofs

>>12196779
Spam the triangle inequality my son.

>>12196792
This and Cauchy-Schwarz are everywhere.

What area of pure mathematics has the potential to become applicable/lucrative in the real world over the next 10 to 20 years?

>>12196932
I hear triple integrals is about to become big

>>12196932
Arrays of numbers.

Have you ever taken an exam for somebody? I made the mistake of taking a multivariable calc exam for somebody and I cant live with myself. I dont deserve mathematics.

>>12196988
If you did it for money go kys but if you did it just for the sake of math its okay.

>>12196932
I want to ask the exact opposite of this question.

What area of pure mathematics is the purest and literal zero applications. In future if someone asks me what are you working on, I want my answer to be "honestly I don't even know what I'm working on it's all just notation to me". So what should I study lads?

>>12197080
https://en.wikipedia.org/wiki/Abstract_nonsense

>>12197080

https://en.wikipedia.org/wiki/List_of_large_cardinal_properties

>>12197080
Number theory used to be the answer to this, but it was ruined.

>>12197068
I had a good time taking the exam, it was a lot of fun

>>12197080
Continuum theory?

applied math for work, theoretical math for leisure.

>>12197269
This.

>>12197125
>tfw no inner model for supercompact

>>12197269
Math for work, anime for leisure.

>>12197299
anime rots your testosterone

>>12197080
>What area of pure mathematics is the purest and literal zero applications.
Formal Verification of mathematics
but you could argue that it can be used to debug software

>>12197345
i know, my doctor said it goes well with my hormone replacement therapy

>>12197372
Post chin

>>12197363
Formal verification is far away from being unrelated to application. Consider strong type theories for small high security code (airplane sensor control, smart contract, etc.)

>>12197382
what chin?

>>12197425
type theories are only for super duper enormous homos

>>12197465
that's where you wrong, fren

If I have two sets [math] A_1, A_2[/math] with functions [math] j_i: A_i \to A_1\sqcup A_2[/math] and a set [math] Z [/math] with injective functions [math] f_i: A_i \to Z[/math], is the function [math] g: A_1\sqcup A_2 \to Z[/math] necessarily injective?

>>12192501
Is the OCW any good?; I really enjoy Kühnel, but as far as introductory courses go, am I missing anything?

>>12190956
What is the most abstract thing you consider worthwhile?

Is it worth double majoring in CS and Applied Math if I'm going to go to grad school after? I'm currently doing my BSc in Applied Math.

>>12197734
Do a minor instead.

>>12197675
freedom

>>12197608
i =1, 2?
This is a confusing question. So j are the canonical maps? Then no

>>12197608
No. [math]A_1 = \{ x\}, A_2 = \{ y\}, Z = \{ z\}[/math], then your injective functions into Z will be identities but take x and y both to z.

>>12197759
nobody has red i's

>>12197372
Why don’t you back those big words up and come be my housewife (male).

>>12197755
Yes sorry. i= 1,2. The [math] j_i[/math] are the inclusion maps.

>>12197759
Is the disjoint union the problem here or is this also not true for all coproducts (at least the ones that make sense here)

Any advice to deeply understand proofs in math papers?
I can follow the logic and verify if they are true. But I don't know how the authors come up with such proofs in the first place.

>>12197890
It boils down to the following fact: if [math]g \circ f[/math] is an injection(/a monomorphism), then [math]f[/math] is an injection(/a monomorphism), but this implication is not invertible in general. Because of this, even with the inclusions into the coproduct being injective/monic, there is no guarantee that the induced morphism will preserve the injectivity/monicity. Dual version stands for products.

>>12193522
This is literally too simple to explain.

>>12197949
It literally isn't.

>>12197928
Hmmm ok. I guess I have to think about how to do my problem for a little while longer. I really wanted this to be true.

>>12197983
What is the problem you are trying to solve? Or is it a secret?

>>12198014
I'm trying to prove a version of this statement using free products.

>>12197911
It’s called having an IQ above 130 just give up now midwit.

>>12198040
It would be true for surjections.

why are complex functions so aesthetic /mg/?

>>12198166
https://imgur.com/r/math/NjSDk

Worried about not finding a job.

>>12198333
Go into data science.

>>12198333
actuary is easy money

>>12198390
the few people I've known who did actuary work say it's a very stressful job...

>>12198392
Yes, and?

>Get masters in math
>can only get a part time job at a scam college
>Math skills atrophy for two years
>job fires me so they so they don't have to pay unemployment when the Pandemic starts (thanks Texas)
>look into getting into a PHD program
>First question: What field do you want to specialize in?
>I have no real interest in any of them
I think I'm ready for the noose boys

>>12198539
Richardson?

>>12198539
Instead of just trying to straight up find the field you're interested in, look ate everything thing you KNOW you don't like. Then, once you've narrowed it down to a short list of subjects you don't hate, maybe ones you even like, then start exploring them. Spend a decent amount of time with each of them, and explore their intersections. Try popping in to some online seminars, download some high level textbooks if you can find them, look at research on arxiv. If all that fails, look at the universities that interest you, and see what the profs are researching. See if anything piques your interest.

>>12198539
do it

>>12196779
Learn to add by zero and multiply by one.

>>12198438
the money seems like shit. why not do quant finance or tech if you're going to sell out?

>>12197949
cringe. why even bother replying. answer: yes, if B is a subgroup of A.

>>12198564
how much math do i need to know for a gf like that?

>>12198539
Just Do something in Algebra/Cryptography you could probably turn that into a job after with a bit of programming skills.

>>12198539
should just do something in engineering. all you have to do is do math excpet the equations are given, only difference is you can get a job a lot easier. i worked a summer job and a guy i worked with also had a masters in math, in a factory working minimum wage.

>>12193379
Smoothness in mathematics has nothing to do with smoothness in real life

>>12198950
cringe

>>12193522
The post that saved /mg/.

Terence Tao

Has anyone read pic related?
I picked it up today, but I feel like I have to work through a 100 pages of definitions and generalization of undergrad linear algebra and topology before I get to the good stuff.

>>12199496
Just skip to the good stuff and go back when you need the basics. Be honest with yourself though.

Undergrad here. Finishing my degree majoring in pure soon and am thinking about a PhD, maybe after honours. What are the most exciting fields of math in the world right now? I.e. what should I aim my study towards? Open to absolutely anything.

>>12199594
Just spew definitions from algebraic geometry or homotopy theory which you memorized without understanding until you get accepted to grad school at Rutgers.

>>12199605
Are you in Rutgers? I’m an undergrad there.

>>12199615
Nah I'm just making stuff up and definitely not talking trash about any real person. This is an artistic work of fiction, etc.

>>12199568
>jump to chapter 2
>First sentence is "let X be a k-scheme"

>>12199656
Sounds like maybe the preceding basics aren't so fucking basic for you then.

>>12199677
I didn't mean I don't know what a k-scheme is, I meant I was happy that things are getting a bit more interesting.

>>12199110
I was told that Algebra and Topology are extraordinarily unprofitable subfields, was I lied to?

>>12199680
Oh okay, sorry I was a little bitchy. I misread your tone.

Q (set of rationals) is countable
R (reals) is uncountable
R is constructed of dedekind cuts of Q, so if you were to take the cut A = {p^2 < 2 or p < 0: p belongs to Q} and B = Q \ A, that is describing root 2 (an irrational) but the sets A and B only have rationals and from this pops out an irrational number? Doesn't that mean that every irrational number is approximated by two rationals? How the fuck then are irrationals uncountable? I know the proof of this (R is uncountable) using nested intervals but this seems really strange. Can someone help a retard out? Doesn't this seem counter intuitive? Am I wrong somewhere?

>>12199687
No, not at all. No one uses algebro-topological wankery in applications. People will try to convince you there's some of kind of boolean algebra or finite field stuff behind computers, but they're just jealous that only calculus is ever applied to build real stuff.

>>12199692
>every irrational number is approximated by two rationals
What precisely do you mean by this?

>>12199702
because the dedekind cut is the set of all values of Q before the cut, and it's sandwiched between the rest of Q above the cut no? Like in my example for root 2:
A = {p^2 < 2 or p < 0: p belongs to Q} and B = Q \ A
A has all rationals below root 2
B has all above
root 2 is sandwiched between A and B right?

>>12199717
askyourself what 2 rationals 'approximate' root 2

>>12199717
How would you use that to show that the irrationals are countable?

>>12199759
becuase we would have an injection R -> Q^2. He wrong tho

>>12199754
I'm really not sure. I am having trouble understanding how R comes from Q. If Q are rationals where do irrationals pop out from?

>>12199759
I would just assume that if every irrational can be approximated by two rationals you could maybe form a counting scheme, but maybe it is harder than it sounds like >>12199770 said.

So overall, do irrationals just pop out of whatever cut you want? They seem to just arbitrarily appear. Should I just move on?

>>12199792
>Should I just move on?
No, because you clearly don't understand what Dedekind cuts are.

>>12199792
OK wikipedia seems to have the answer:

>The important purpose of the Dedekind cut is to work with number sets that are not complete. The cut itself can represent a number not in the original collection of numbers (most often rational numbers). The cut can represent a number b, even though the numbers contained in the two sets A and B do not actually include the number b that their cut represents.

>For example if A and B only contain rational numbers, they can still be cut at √2 by putting every negative rational number in A, along with every non-negative number whose square is less than 2; similarly B would contain every positive rational number whose square is greater than or equal to 2. Even though there is no rational value for √2, if the rational numbers are partitioned into A and B this way, the partition itself represents an irrational number.

So they (irrationals) don't pop out, they are just represented it seems. I guess I was assuming that they would be generated or take on something more concrete. I guess I've never thought enough about what irrationals are until this. Thanks for the help.

>>12199831
Does wikipedia's explanation seem right? See >>12199834
The cuts can represent a number even if they are not in either cut

Brainlet here, can you use ∀ to denote "for each"? what's the difference between "for each" and "for all"?

>>12199845
they're are the same anon

fuck springer and fuck all publishing jews
the book i ordered has finally arrived and it's absolutely shit quality
the book looks like they were running out of ink, and the print looks low dpi
letters look jagged and smaller letters are literally missing parts, for example in one place the expression [math]2^2^k[/math] occurs and half of the [math]k[/math] is missing

>>12199845
No, for each is ∃ (E for Each)

>>12199845
∀ means there exists.
For each is ∃.

>>12198383
>import sklearning potential

get in on the grift with me boys

>>12199845
Don't listen to the other 2 anons, they're trolling.
∀ means for all x, there doesn't exist (hence the upside down A for all)
For each is E.
∃ means for every all of x.

>>12199845
for each x in A means for each element in the collection A.
for all x means for all x no matter where it is, there's no specified collection.
>can you use ∀ to denote "for each"
We use ∃

>>12199879
>>12199868
∃ denotes "there exists", no?

>>12199884
Not exactly, but close. ∃ is reverse E, which stands for "every".
∃ x can be interpreted as "for every x that there exist" (you don't want to quantify over the x that don't exist.

>>12199884
Shut up retard. It means for every. I bet you are some midwit European.

>>12199889
In classical logic you only use ∃ and ∀, but in nonstandard logic you also use E which is just like ∃ but instead of saying "for every x that there exist" it's "for every x no matter if it exists or not". In standard mathematics you only deal with objects that exist so you don't see E often there.

>>12199838
Yes, but make sure you understand this: Each real number is by definition one of these cuts. They're not just being represented, the cuts are literally what the real numbers are. The problem is that you've internalized the real numbers as being "points on a number line" but that is actually not a precise mathematical definition of anything.

Jesus christ, forget I asked.

>>12199908
I will not my son

I don't want to be a giga-faggot here but I went to reddit and got a straight answer instantly. Suck my nuts retards.

>>12199963
you're the retard if you don't already know basic logic notation

>>12199963
ok cowboy

You guys are mega faggots because you're all uppity bitches who don't understand reality and how to live. I had a dream of Goku and he said 4chan is full of faggots. He said 4chan faggots are just as bad as normies. Jesus said so too.

>>12199963
Based.

>>12199963
Ok midwit. Enjoy being filtered out of calc II later.

>>12200239
Why are you on 4chan?

I'm curious for those of you at Zoom university– how are your professors doing assessments?

>>12200292
Mine has us strip down and show our bodies on camera before the test so they know we're not cheating.

>>12196543
This, the notation in analysis is fucking cancer.

>>12200269
To discourage other people from going to graduate school so I have less competition.

>>12200292
One of my classes has all take home open notes exams, another has the entire grade as homework, and another just watched us take it over Zoom.

>>12200292
Only based on homeworks, we will have to do a final test, however.

>>12200292
There's a 24 hour window where we can take the exam, but once you start it has to be completed within the allotted amount of time. There's no proctoring and it's open notes.

The other ones are using HonorLock.

>>12200292
Most are just "available at X time, turn in by X day, open note, open internet, pls no talk to classmates" some have a timed portion to be due during class then it's the same few days deadline

The only proofs i can do of the top of my head are root 2 irrational and the thing with a prime and only having to check primes less than the square root.


How much of a brainlet am i?

>>12200509
Yes.

>>12199687
Pure algebraic topology possibly yes but the point is you can still make a lateral move with that knowledge into software/applied cryptography.

>>12200327
some squiggly marking taking up doubble vertical estate here
d has been reporpoised here it is fore differencingness
and no delimiters here just pretend the dx is one also the big squiggle
oh rember how x has been repurpoised now u also have to imagine multiply without deliminters there now in blank spaces but the book dos not explicitly say is unritten rule see

yo wtf is this solving for
sqrt(1 + 2i)
(x + yi)^2 =
x^-2 - y^2 + (2xy)i
So xy = 1
Then
x^2 - y^2 -2xy = -1 = (x-y)^2
So this should not be solvable with reals but thats obviously nonsense. where did i fuck up

>>12200509
how may primes are there
sum 1 to n is n(n+1)/2
lagrange theorem
answers on a postcard by tomorrow buddy

>>12199884
you're being trolled

very funny people here today

>>12197080
basically anything. I still don't understand how people say "this has application in real world such such such", it doesn't. no enginoob's gonna use a formula that is asimptotically faster by a 0.00000000000001% but 1000x times harder to understand.

>>12197675
the foundamental theorem of algebra

>>12200317
>tfw couldn't do an assessment because i didn't pass the penis inspection because i could hide something in my foreskin

>>12199103
Be undergrad and study.

>>12197675
fourier series

>>12200166
pls reddit fag include me in the screenshot.

>>12200815

>Doing calc 3.
>professor gives ugly question with ugly decinal answers.

Seriously, fuck this shit! , the answer should always be a nice round integer so you could check your work, the derivatives should be symmetric and exploitable.

every math exam should be doable with no calculators. you get what i mean? fuck enginnering.

>>12200865
>doesn't use fractions

>>12200865
are you an engineering student? prof is literally preparing you for your job

>>12200782
>implying pure math research leads to faster ways of computing things
Math today is just about the math people invented yesterday, Galois killed math in 1832.

>>12200898
You have to enter it as decimal and the fractions aren't nice either. This shit triggers my ocd, there are unspoken rules, ya know. Like every math competition should have the year as part of the questions.

Someone has a fucking clue of how Reid's proof of Noether normalization works? No link on internet that explain why his proof is correct, I already found a typo that gave me headache for a day. Is there a proof of Noether normalization lemma that doesn't rely on Nagata's proof and that doesn't use the hypotesis of inifinitude of the field?

Any good online reference material for math induction and strong induction?

Sell me on Groethendiek.

>>12201130
his diek was very groethe

>>12201071
Post the proof

>>12201098
If you're looking for a book that argues against it, see

https://web.math.princeton.edu/~nelson/books/pa.pdf

Assume x is a set with one element. Is there standard notation to denote the element of x?

>>12201161
[math]{x}[/math]

>>12201178
Thats just x. I want the element of x.

>>12201178
Classic Nikolaj question.

>>12201181
[math]x \in x[/math].

>>12201161
a \in x

>>12201181
Sets are written capitalized, so you have [math]x\in X[/math] as your element

>>12201161
no

>>12201193
>Sets are written capitalized
Everything is a set.

>>12201187
>>12201181
[math]x \in {x}[/math]

>>12201204
shhhhh don't tell him too soon you may pop his head

>>12201161
>>12201203
or maybe [math]x = \{ *\}[/math]

>>12201144
Here it is the full statement plus the horrible lemma https://imgbox.com/gallery/edit/qbPk1e13j9/E7dqPTqL4Ciadhpw, the part in red is what I don't understand, how is it monic and in [math]y_n[/math]? There are [math]y^*_i[/math] in [math]G[/math] and such. It can't be integral over [math]k\left[y^*_1,\dots,y_{n-1}^*\right][/math]

>>12201161
\bigcup, i.e. as a single term, use [math]\bigcup x[/math].

Note that the union "[math] a\cap b [/math]", unlike the intersection ([math] a\cup b [/math], existing via the Separation axiom), is not actually set up in terms of any two sets [math] a [/math] and [math] b [/math] directly, but instead requires you to first form the pair and then build the union over it:
[math]a\cap b = \bigcup \{a,b\}[/math].
That is to say, union is flattening.
So, funny enough, if you don't adopt the axiom of pairing, you can't prove that the union of any two sets exists as a set.

If you want to be exotic, you can also use Hilbert/Bourbaki notation
[math]\iota y. y\in x[/math]
Namely, they had this choice principle where given a predicate P, you can get an element that fulfills it, if any, as
[math]\iota y. P(y)[/math]
But don't actually do that.

>>12201184
What did you mean by this

edit: mixed up cup and cap

>>12201161
\bigcup, i.e. as a single term, use [math]\bigcup x[/math].

Note that the union "[math] a\cup b [/math]", unlike the intersection ([math] a\cap b [/math], existing via the Separation axiom), is not actually set up in terms of any two sets [math] a [/math] and [math] b [/math] directly, but instead requires you to first form the pair and then build the union over it:
[math]a\cup b = \bigcup \{a,b\}[/math].
That is to say, union is flattening.
So, funny enough, if you don't adopt the axiom of pairing, you can't prove that the union of any two sets exists as a set.

If you want to be exotic, you can also use Hilbert/Bourbaki notation
[math]\iota y. y\in x[/math]
Namely, they had this choice principle where given a predicate P, you can get an element that fulfills it, if any, as
[math]\iota y. P(y)[/math]
But don't actually do that.

>>12201184
What did you mean by this

To give an example, I want to write an expression for the unique x in the interval (-pi/2, pi/2) such that sin(x)=1/5 squared. Something like
(T(sin^-1{1/5} cap (-pi/2, pi/2)))^2
Where T picks out the element

>>12201223
do you mean arcsin(sqrt(1/5))?

>>12201221
>>12201223
So that would be
[math]\left( \bigcup \left( \sin^{-1} \{ 1/5 \} \bigcap (-\pi/2, \pi/2) \right) \right)^2[/math].
Looks like garbage!

>>12201223
For a case where what the trigonometric anon suggested doesn't fit, you could in principle do

[math]\iota z. \exists x\in \left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right). \big(z=x^2 \land \sin(x)=\tfrac{1}{5}\big) [/math]

which I've still seen in some type theory texts.

But in almost all cases, you just want to use natural language.

>>12201246
>Looks like garbage!
Indeed

>>12200815
i'm a phd student already...

>>12201297
you're fucked then

next edition is dedicated to penrose. I'm hitting the bed so don't fuck it up /mg/.

Fuck I finally understood how the proof of Goodstein's theorem (that the sequence terminates) works. I feel retarded for having read it 3 times and not getting the point until now.

>>12201297
actual original poster, there is no way to get the girl. However, depending on what you choose to study, you may get to become the girl

anyone got a good ode textbook? I got a midterm tomorrow and the textbooks/class notes/lectures are absolute shit. Ty in advance!

>>12201361
Arnold

>>12201297
You can be my girl

>>12201363
4chins ate my prime symbol.
[math]\text{Arnol}'\!\text{d}[/math]

>>12201375
and now it won't process my math tags.

>>12201383
Its okay, I appreciate the response. This is miles better than what I had before.

>>12201375
But I think there was no good reason not to go with just \text{Arnol'd} in the first place

I hate statistics.

For every [math]k\in \mathbb{N}[/math], why is there a monomorphism [math]\phi: F_k \to F_2[/math] where [math]F_k[/math] is the free group generated by k generators?

>>12193522
Holy fuck lmao

>>12201933
why not try and define it. Start with k=3.

>>12202112
Let [math] F_3 = F({x,y,z}), F_2 = F({a,b}) [/math]. I was thinking of doing something like [math] F(x) = a^2, F(y) = b^2, F(z) = z^2 [/math] but I don't really know how to generalize it past this point.

>>12202162
Sorry, I meant [math] F(z) = ab [/math]

>>12202162
How about [math]\varphi(w) = a^3, \varphi(x) = a^2b, \varphi(y) = ab^2, \varphi(z) = b^3[/math] when you have 4 generators?

>>12202197
Ok. I thought about it and I think I have a way of extending it to any natural number k. It's clear to me that these are obviously injective. However, 1) it's not clear to me that these are homomorphisms. 2) Why are we allowed to define these homomorphisms only on the generating set for the free group?

>>12202252
Basically, you just take (x+y)^n and forget all the coefficients. To address your points:
(2) every element of a free group will be a word consisting of the generators. If we know how the generators are mapped, then we can define a new function using that information. For example, let's go with your idea for the 3 generators. We have a function [math]f \colon \{ x, y, z \} \to \{ a, b\}, x \mapsto a^2, y \mapsto ab, z \mapsto b^2[/math], and we want to use this to define a function [math]\varphi\colon F(x, y, z) \to F(a, b)[/math]. We do this the following way: [math] \varphi( x^{k_1} y^{m_1} z^{n_1} x^{k_2} \cdots x^{k_p} y^{m_p} z^{n_p}) = f(x)^{k_1} f(y)^{m_1} \cdots f(z)^{n_p}[/math] (notice that the function [math]f[/math] can actually be anything here). This gives a well-defined function.
(1) Using the thing above, it is easy to see that [math]\varphi (gh) = \varphi(g) \varphi(h)[/math] for all elements.

>>12202307
Correction, [math]f\colon \{ x, y, z\} \to [/math] can be any function and [math]G[/math] can be any group. In your case, however, you have those generators like that and you want to use them.

>>12202312
Correction number 2, [math]f\colon \{ x, y, z\} \to G[/math].

>>12202307
Thank you for making it clear. Quick follow-up question, why does this fail when we try and define a monomorphism between [math] F({a,b})[/math] to [math] F({a})[/math]? Or rather, for any k, why are we not able to define a monomorphism between [math] F({x_1, ..., x_k}) [/math] to [math]F({a}) [/math]?

>>12202389
Because we only have one generator for the codomain. What is the free group with one generator isomorphic to? Believe it or not, it is actually an infinite cyclic group! Now, suppose we any number n greater than one of the generators on the domain side. Either we would map [math]x_1, x_i, 2 \le i \le n[/math] to the same element of [math]F(a)[/math] and fail immediately, or we would have some [math]g, h\in F(a), g\neq h[/math] we would choose as the images, respectively. Now, since it is infinite cyclic, we have [math]g = a^k, h=a^m[/math], and so [math]\varphi (x_1^m) = f(x_1)^m = g^m = a^{km} = h^k = f(x_i)^k = \varphi(x_i^k)[/math], and this is also a failure to be injective. Sorry for the late reply, I tried to get some sleep.

>>12199496
>I picked it up today, but I feel like I have to work through a 100 pages of definitions and generalization of undergrad linear algebra and topology before I get to the good stuff.
nothing specific to you, but why does every wannabe autodidact think you treat textbooks like novels
Hell, beyond just textbooks I've seen otherwise intelligent people flounder about like idiots with the simplest of tasks when they're out of their element rather than spend 5 seconds looking up how to do something properly. These are the same people that as kids probably bitched and moaned about having to learn formulas and facts because they could just "look them up when they need them".
I wonder if elementary or middle schools should include education on how to self educate.
sorry for the short rant

>>12201572
That would be an apostrophe not a prime symbol.

New >>12203008