serious-discussion

[river moon]

given any y, the x can be found by just repeatedly applying the formula you have written last

[placid robin]

I know that much, but x is only one of the solutions, as from every root you get a positive and negative result, i. e. x=±sqrt{y}. So when I take that infinite series of roots, what complex results do I get?

Like, let's do it with 2 nested roots:

$$x=\sqrt{y×\sqrt{y^2}}$$
$$x=\sqrt{y×(±y)}$$
$$x=\sqrt{(±y^2)}$$
$$x_1=y; x_2=-y; x_3=yi; x_4=-yi$$

[gilded sluice]

Ah.

I have no idea

[placid robin]

Yeah, see

Thats what my problem is

@river moon

[fair mural]

meow

[hasty leaf]

Woof

[gilded sluice]

oh
I think the answer you're looking for is that the equation is true if x (or y, whatever) is any complex number

[placid robin]

Thing is, x is a complex number. I want to find all solutions of x, when y is a random complex number

I could start with y as a natural number to make it a little easier though

x should have infinite solutions

But I want a pattern within those solutions

[gilded sluice]

for any random complex number y, the equation $x=\sqrt{y*\sqrt{y...}}$ is true if and only if x = y

hmm,

idk why bot isn't invoking

$$test message?$$

$test$

idk

[solid snow]

the bot is currently down

[placid robin]

No, like I wrote it out here, there are a lot of y that fulfill the equation

@gilded sluice

[gilded sluice]

Yes

And there are a lot of complex...

hm

one second

ah im starting to somewhat get what you mean

[placid robin]

Great, any ideas?

[gilded sluice]

No

Because sqrt(25) is 5, it's not plus or minus 5

[placid robin]

Square root is defined to be something that produces the number inside the parentheses when multiplied with itself, so the square root of 25 can be 5 as well as -5

[gilded sluice]

well good luck with that, i like having my sqrt be a 1:1 relation
hopefully someone else can help you

[solid snow]

the real valued square root is defined as $$\sqrt{x^2} = |x|$$

[fathom swallow]

josemom2

[solid snow]

so, only the positive part

otherwise, learn about multivalued functions

[placid robin]

Then let me restate this:

[fathom swallow]

animeonfire

[zealous garden]

The nth root of x is the one with an angle in the first 1/nth of the complex plane

[static loom]

looks like you square it and get x^2=yx

I could see x=0 being possible if it turns out y is small or something and makes it a contraction mapping that converges to 0

[vast wraith]

that server is archived now, but i can dm you an invite to another server with the emoji if you’d like

[tall marsh]

this looks so familiar

[long matrix]

if there are more gifs of that cat sure

just need a few

[vast wraith]

there is only one with this cat but there are other nice ones

[neat lintel]

@tawny hill

[brittle socket]

Hasse diagrams are nice

[wet summit]

What's up?

i'm new in the server

[radiant pivot]

all good, how are you

hey you know, you joined discord on my birthday

[wet summit]

Really happy birthday 🎂

[stray crater]

Can someone help me?

[gilded sluice]

!help

[quasi jetty]

Please read #❓how-to-get-help

[long matrix]

meow

[hasty leaf]

Woof

[neat lintel]

Hi. Just joined the Math discord.

[rotund steppe]

Hello

[gilded sluice]

[long matrix]

[violet coral]

I have the SAT tmrw😓😓

[pure kestrel]

hi

[violet coral]

hi

[neat lintel]

Gl!

[rotund steppe]

good luck 🫡

[rocky shuttle]

<@&268886789983436800> spammed in multiple channels

[cold needle]

banned

sorry how did i miss this wtf

or wait

no they've already been banned but nothing got deleted

interesting

ill just manually doit

[rocky shuttle]

Thanks

[native drift]

cuckulus

cockulus

[jovial ember]

So true

[radiant pivot]

woah! best of luck! it's easy

[split pawn]

Why's everyone eigen-*

[neat lintel]

hmmm

@native drift @rocky shuttle if not for the roles i'd think that booknerd was a boy and eigenzan was a girl

y'all shld exchange personna or stuff idk

[native drift]

i am a boy

[rocky shuttle]

That's why we have the roles.

[abstract maple]

hey i have a question about the help channels

why is it that a lot of help channels are hidden instead of being considered available after a channel is closed?

[bright hill]

the new opm chapter was so cute

tatsumaki is cute when she's angy

[rocky shuttle]

Hi kyuu!

[bright hill]

sup zane :3

you changed your mind about the measure theory group?

[sand ridge]

Can someone come inbox

[static loom]

your email what

[rocky shuttle]

The invitation link expired I could chime in now and then but I'm still not sure whether I can commit to it every week.

Which chapter are you doing rn?

[bright hill]

chapter 2

we did mct and dct

[bright hill]

it'd be awesome if you could get something out of it

[rocky shuttle]

Nice!

[bright hill]

honestly, the hw problems and gomez's problems are fantastic

and I've already talked with john and you won't be joining as a member exactly

more like auditing the group coz i don't think you'll need your hw graded

[rocky shuttle]

Ah

[bright hill]

so you don't have to show up to each meeting

kinda like slurp

[rocky shuttle]

That would be fine actually

[bright hill]

I'll send you another invite then

you should've told me when it expired

[rocky shuttle]

Alright, thanks kyuu

[neat frost]

Don't bully Xander.

[bright hill]

I did not!

[neat frost]

You did so.

[bright hill]

slander

[neat frost]

Libel.

[bright hill]

anyhow

watch my stream

[neat frost]

No.

[rocky shuttle]

[bright hill]

[rocky shuttle]

[cunning compass]

[long matrix]

darq is a triangle

[neat frost]

Good morning Shuri

[rocky shuttle]

xlurp is a rabbit anime girl

[neat frost]

Xannie we're gonna do this real cardinality thing together

[rocky shuttle]

Alright, let's start with fuzzy set theory

[neat frost]

Nono screw fuzzy set theory

I know the basic idea of how it defines stuff

We're gonna start from the ground up

Do everything itself

[rocky shuttle]

Set up the axiom system, then

1. Xlurp is always hungry.

[neat frost]

(ams)
\def\bU{\mathbb{U}}
Okay so I think similar to multisets, the definition of a fuzzy set is a pair $S=(\bU, m)$ where $\bU$ is the universe (a set) and $m\colon\bU\longrightarrow[0,1]$ then you can define the cardinality of $S$ by $\sum_{x\in\bU}m(x)$. This doesn't really do us much good for infinite cardinalities since there's only one infinity.

[neat frost]

But we don't care about infinite cardinalities, since those are boring

[rocky shuttle]

Isn't that the membership and not cardinality?

[neat frost]

m is the membership function

[rocky shuttle]

Oh

[neat frost]

Sorry

[rocky shuttle]

Didn't read that right

No, mb

Ok, so how's the sum defined?

[neat frost]

Normally

It's a real sum

[rocky shuttle]

[\sum_{x\in\mathbb{U}}m(x)=\sup\left\lbrace\sum_{x\in A}m(x)\vert A\subseteq\mathbb{U}\text{ finite}\right\rbrace]

[fathom swallow]

eigenzan

[neat frost]

Yeah

[rocky shuttle]

I should include my usual macro for set brackets to TeXiT's preamble smh

[neat frost]

Uh huh

If you do it now I can accept it

[rocky shuttle]

I'm also mostly on mobile, so typing LaTeX is very time-consuming

[neat frost]

;-;

[rocky shuttle]

I wonder how DarQ does that

[neat frost]

He's on computer now

[rocky shuttle]

Oh, nice

[neat frost]

Okay so now we have to define functions between two of these sets

Fuzzy sets

(ams)
\def\bU{\mathbb{U}}
So if $S_1=(\bU_1,m_1)$ and $S_2=(\bU_2,m_2)$ then a function between them is $f\colon\bU_1\longrightarrow\bU_2$, where $m_2\bigl(f(x)\bigr)\leq m_1(x)$ for every $x\in\bU_1$.

[neat frost]

Does this definition make sense?

This feels like a confusing definition since it seems like a restriction on m_2 but it's really a restriction on f

[rocky shuttle]

Hmm, I was thinking more about the definition of a function through relations

[neat frost]

Interesting

That may work better

[rocky shuttle]

total, univalent relation

[neat frost]

We could always read about fuzzy sets but I don't wanna

So much more fun to try yourself

[rocky shuttle]

lmao yeah

[neat frost]

Well like first we'd need to define what a relation is

And Cartesian products

(ams)
\def\bU{\mathbb U}
I assume somehow it would be $S_1\times S_2 = (\bU_1\times\bU_2, m_3)$ but how would we define $m_3$?

[neat frost]

m_3(x,y) = m_1(x) * m_2(y)?

[rocky shuttle]

So, a fuzzy function (f\colon A\to B) is a relation (subset of (R\subseteq A\times B)) satisfying [\forall a\in A\exists b\in B] and [\forall a\in A,b,c\in B(((a,b)\in R\wedge(a,c)\in R)\Rightarrow b=c).]

[neat frost]

But these are fuzzy sets

\in is too binary

[rocky shuttle]

ik, I'm trying to somehow relate that to membershipness

[neat frost]

Ah okie

Sorry

[rocky shuttle]

But we should talk about this first

[neat frost]

This makes sense I think

[rocky shuttle]

Ok, just multiplication then

[neat frost]

Like we also should make sure that when m is binary (so the fuzzy set is basically just a normal set), our definitions coincide

This one does

[rocky shuttle]

Good

[fathom swallow]

eigenzan

[neat frost]

m_A <= m_B

This is the same as in multisets iirc

[rocky shuttle]

Ok, so m_R <= m_A * m_B

[neat frost]

Hmmm

[fathom swallow]

eigenzan

[neat frost]

(ams) \def\bU{\mathbb U}
$$ \forall a\in\bU_A,\exists\set{b_i}{i\in I}\subseteq\bU_B:, m_A(a) = \sum{i\in I}m_B(b_i) $$
Maybe?

[neat frost]

Uh

[rocky shuttle]

B instead of N

[neat frost]

Wait but what does \exists b mean here

Like normally it means (a,b)\in R

[rocky shuttle]

Read my correction

I'm going to switch to my pc

This is too cumbersome on mobile

[neat frost]

Right but you used \in

(ams) \def\bU{\mathbb U}
$$ \forall a\in\bU_A,\exists\set{b_i}{i\in I}\subseteq\bU_B:, m_A(a) = \sum{i\in I}m_B(b_i) \text{ and } m_R(a,b_i) = m_A(a)\cdot m_B(b_i) $$
Maybe?

[neat frost]

Hmmm

Wdym?

[fathom swallow]

eigenzan

[neat frost]

Hmmm

Maybe that could work

Wait

How does this make sense? How would we use this?

[rocky shuttle]

So (m_{\lbrace a\rbrace}\leq m_{A})

[fathom swallow]

eigenzan

[neat frost]

This is just saying that m(a)<=m_A(a)

[rocky shuttle]

But I could also be talking nonsense

[neat frost]

For some function m

Idt this makes sense idk

[rocky shuttle]

I think it says m_{a}(u) <= m_A(u) for all u in U

which is something different, ig?

[deep mango]

[rocky shuttle]

lmao ryc

[neat frost]

Right but we're assuming m_{a}(u)=0 for u\neq a, I'm assuming since you wrote {a}

[deep mango]

Ding dongs

[neat frost]

Ryc no judging

[neat frost]

What

[deep mango]

Too late

[neat frost]

Why are we doing dogs?

Dongs

Wtf

Ding dongs

[deep mango]

WHAT?

[deep mango]

fuzzy set theory

[neat frost]

We're trying to extend the factorial but in a way which isn't stupid and analytical

Without reading up on it

[deep mango]

Whatever

Have fun

[neat frost]

Wdym whatever

[rocky shuttle]

This all started because of your (1/2)! = sqrt(pi)/2

[neat frost]

Oh wait

[tiny saddle]

gamma function eh

[neat frost]

(ams) \def\bU{\mathbb U}
$\forall a\in\bU_A,\exists b\in\bU_B: m_R(a,b)>0$?

[tiny saddle]

reminds me of ryc's talk a few weeks ago

[deep mango]

No. It started because you guys wanted to own me

Again

For whatever reason

[tiny saddle]

own what now

💀

[rocky shuttle]

Someone's paranoid

Go play parabox

[deep mango]

Just watch, when my talk recording comes out it will have more views in a day than your "factorial" will ever have

[tiny saddle]

[rocky shuttle]

Sounds reasonable

[neat frost]

So this is totality

But single valueness or whatever funky word you used

[rocky shuttle]

univalent

[long matrix]

ryc the bird

ryc the squawker

ryc the

[rocky shuttle]

rock

[hasty leaf]

Woofer

[sturdy quarry]

Done. You are welcome :)

[neat frost]

You have been hacked!

[gilded sluice]

i swear if i put this in and it's an among us i'm just going to be impressed with how short you got it

dang. no sussy baka today

if you genuinely need help, read #❓how-to-get-help

and actually put your question in chat, don't just say:

[gilded sluice]

(everyone else: this isn't me being rude, this is genuinely the what they said in #1083773127261180014)

[bright hill]

with blood, sweat and tears

I got a lappy tho :3

so I can now type at full speed

[neat lintel]

@neat lintel !help

!help

[quasi jetty]

Please read #❓how-to-get-help

Please read #❓how-to-get-help

[neat lintel]

there you go

[haughty lintel]

Okay but here isn’t the place

Idk why you’re complaining about not getting help in a channel not meant for help

We aren’t trying to be rude

We are just trying to redirect you to a place you might actually get help

I can’t explain that but I ensure you that you won’t get help in discussion two while complaining you aren’t getting help. Maybe try the specific help channels like #help-7｜zen1thxyz . Those tend to be faster

[unkempt torrent]

Does anyone genuinely like combinatorics?

[cinder zephyr]

lots of people

myself included

do you not like it?

[rocky shuttle]

[cinder zephyr]

if so that's fine, lord knows there are fields of math (and other areas) I don't like studying and I'm sure you're far from the only person who dislikes combo

[neat frost]

(ams) \def\bU{\mathbb{U}}
So if you have two sets $A$ and $B$ and a relation $R\subseteq A\times B$ the idea is to think of the ``image'' of an element $a\in A$, $R(a)=\bigl(\bU_B, m_{R(a)}\bigr)$ where $m_{R(a)}(b) = \frac{m_R(a,b)}{m_A(a)}$. This is less than $m_B(b)$ and so $R(a)\subseteq B$.

With normal sets, $R(a)$ is all of the elements $b\in B$ such that $aRb$, and if you have another relation $S\subseteq B\times C$ then $S\circ R(a)=S(R(a))$ where $S(X)=\bigcup_{x\in X}S(x)$.

We can define composition similarly, since defining $S\circ R(a)$ for every $a$ defines $S\circ R$ (don't take my word for it). We can define unions, where if $\set{X_i}[i\in I]$ are fuzzy sets in $\bU$ then
[ \bigcup_{i\in I} X_i = (\bU, m),\qquad m(a) = \max\set{m_{X_i}(a)}[i\in I] ]
(For intersections, the max becomes min).
So if we define for $X$ fuzzy set:
[ S(X) = \bigcup_{x\in X} S(x) ]
(This definition may need some reworking because it doesn't use $m_X$)
Then we can define:
[ S\circ R(a) = S(R(a)) ]
This means that
[ S\circ R(a) = \bigcup_{b\in R(a)}S(b) = (\bU_C, m_{S\circ R(a)}) ]
Where
[ m_{S\circ R(a)}(c) = \max\set{m_{S(b)}(c)}[b\in R(a)] = \max\set{\frac{m_S(b,c)}{m_B(b)}}[b\in R(a)] ]
And so if we recall that by definition
[ m_{S\circ R(a)}(c)=\frac{m_{S\circ R}(a,c)}{m_A(a)} ]
Which means that we can define $S\circ R$:
[ m_{S\circ R}(a,c) = m_A(a)\cdot m_{S\circ R(a)}(c) = \max\set{\frac{m_A(a)}{m_B(b)}m_S(b,c)}[b\in R(a)] ]

@rocky shuttle what do you think?

[rocky shuttle]

I like the definition of the image

[neat frost]

The union I'm pretty sure is standard

Could be wrong

Hmmm we may need to change max with sup

Which anyway exists because R is dope

R is so fucking dope I love it

[rocky shuttle]

Now, imagine 3 of it, orthogonal to each other

[neat frost]

Uhhh

I can't

[long matrix]

R is stupid.

[neat frost]

Nuh uh

[long matrix]

yuh huh

[rocky shuttle]

But fuzzy sets, infinite unions and intersections are fine?

[neat frost]

I'm not visualizing anything tho

It's all just writing and whatever

[rocky shuttle]

I agree!

Hate that programming language.

[neat frost]

Probably good to mention that this is indeed a relation

[rocky shuttle]

Wdym? If we take usual sets instead of fuzzy ones?

[neat frost]

Oh I just mean that this is well defined

[long matrix]

slurp is a fuzzy bunny

https://tenor.com/view/bunny-cute-fluffy-tiktok-gif-16625987

[neat frost]

$m_{S\circ R}(a,c) \leq m_A(a)\cdot m_C(c)$ so $S\circ R\subseteq A\times C$

[neat frost]

Boing

And it also correlates with the def of $S\circ R$ when they're normal set relations.

Though we could also define it this way:
[ m_{S\circ R}(a,c) = \max\set{m_R(a,b)\cdot m_S(b,c)}[b\in B] ]
Which may be more natural

[rocky shuttle]

[neat frost]

Again probably should be sup

I hope there's some nice way to synthesize between these two defns

Maybe some way of defining S(X)

[analog tartan]

Why does 1+1=2

[neat frost]

Omg bingo

[haughty lintel]

Let’s go

[rocky shuttle]

Let's continue next time, I'm too tired to think critically

[neat frost]

Okie dokie

Thanks!

[solid thistle]

do yall prefer to shower in the morning or at night?

[bright hill]

fuzzy sets...

fuzzy logic...

[brisk lily]

I guess morning one have more sense

[bright hill]

didn't take you for a crank, zane

[rocky shuttle]

fun fact: my gf did her bachelor's thesis on fuzzy neural networks

I know nothing about it, though

[neat frost]

Wtf

[neat frost]

DarQ whenever someone studies something that the wind told him is stupid: crank

[storm sage]

Nooooooooo not fuzzy things

I was trying to read a paper about fuzzy simplicial sets in data analysis

It was complete nonsense to me

[rocky shuttle]

It's not complete nonsense, just a little bit fuzzy

[bright hill]

[bright hill]

fam

all set theory is stupid, period.

[neat frost]

^

[bright hill]

I did a tiny bit of set theory tho

so it's not completely out of the wind

tho I only did propositional logic so I might give it another chance

[neat frost]

Uh huh

[neat frost]

^

[bright hill]

you can't just keep referencing that message over and over again, slurp

it has to at least make sense in context

[neat frost]

DarQ whenever someone studies something that the wind told him is stupid: crank

[burnt dune]

happy ramadan

[unkempt torrent]

Quite the opposite

I'm the only person in my school who enjoyed it

It was the topic of our first assessment and average was a fail
Too bad I guess

[timber hedge]

is it ok that i gave myself the helper role even tho i can only really help with precalc and below?

[cold needle]

its ok

[fervent pebble]

wsp with u too bro

[vagrant ocean]

This seems like cool news. https://writings.stephenwolfram.com/2023/03/chatgpt-gets-its-wolfram-superpowers/

[plucky coral]

https://cms.math.ca/wp-content/uploads/2023/03/2023CMO-exam-en.pdf what kinda math do i need to study to be able to solve these questions

[fringe summit]

Ayo pass it

[torn willow]

Ask in #competition-math
There's also a discord for this

But I would guess you need combi,NT and geometry

[fringe summit]

This is maybe bad also since it gives a function to the empty set, namely where m_2 is 0 everywhere

[torn willow]

I never realised how biased comp math is toward combi

[fringe summit]

How else do you make bs questions

[torn willow]

Well they also use other sources

Like Euclidean geometry

[neat frost]

We're reworking the definition anyway, but the empty set has a membership function anyway?

(ams) \def\bU{\mathbb{U}}
\hsize=1.5\hsize
So if you have two sets $A$ and $B$ and a relation $R\subseteq A\times B$ the idea is to think of the ``image'' of an element $a\in A$, $R(a)=\bigl(\bU_B, m_{R(a)}\bigr)$ where $m_{R(a)}(b) = \frac{m_R(a,b)}{m_A(a)}$. This is less than $m_B(b)$ and so $R(a)\subseteq B$.

With normal sets, $R(a)$ is all of the elements $b\in B$ such that $aRb$, and if you have another relation $S\subseteq B\times C$ then $S\circ R(a)=S(R(a))$ where $S(X)=\bigcup_{x\in X}S(x)$.

We can define composition similarly, since defining $S\circ R(a)$ for every $a$ defines $S\circ R$ (don't take my word for it). We can define unions, where if $\set{X_i}[i\in I]$ are fuzzy sets in $\bU$ then
[ \bigcup_{i\in I} X_i = (\bU, m),\qquad m(a) = \sup\set{m_{X_i}(a)}[i\in I] ]
(For intersections, the max becomes min).
This is fine, but if $I$ is a fuzzy set instead of a normal set, we're not really using the fuzziness of $I$, and so we can define for fuzzy sets $I$:
[ \bigcup_{i\in I} X_i = (\bU, m),\qquad m(a) = \sup\set{m_{X_i}(a)\cdot m_I(i)}[i\in I] ]
So if we define for $X$ fuzzy set:
[ S(X) = \bigcup_{x\in X} S(x) ]
Then we can define:
[ S\circ R(a) = S(R(a)) ]
This means that
[ S\circ R(a) = \bigcup_{b\in R(a)}S(b) = (\bU_C, m_{S\circ R(a)}) ]
Where
[ m_{S\circ R(a)}(c) = \sup\set{m_{S(b)}(c)\cdot m_{R(a)}(b)}[b\in R(a)] = \sup\set{\frac{m_S(b,c)}{m_B(b)}\cdot\frac{m_R(a,b)}{m_A(a)}}[b\in R(a)] ]
And so if we recall that by definition
[ m_{S\circ R(a)}(c)=\frac{m_{S\circ R}(a,c)}{m_A(a)} ]
Which means that we can define $S\circ R$:
[ m_{S\circ R}(a,c) = m_A(a)\cdot m_{S\circ R(a)}(c) = \sup\set{\frac{m_R(a,b)\cdot m_S(b,c)}{m_B(b)}}[b\in R(a)] ]
The $b\in R(a)$ condition can be weakened to $b\in\bU_B$ since $m_R(a,b)=0$ if $b\notin R(a)$.\par

@rocky shuttle this isn't exactly the defn from before, but I think it makes more sense because it only "counts" b once (m_R(a,b) m_S(b,c) "counts" it twice)

[neat frost]

I think I should come up with a different symbol for fuzzy unions

A cup with a fuzzy bottom

[rocky shuttle]

Hmm, I see, yeah, that's better than the previous one, I think

[fringe summit]

It has a membership function m if you've got a universe U_empty set aside ofc, maybe even (empty, \bot) as that set

Issue is, what about an empty function so-to-speak to a fuzzy set (U_x, m_x) such that m_x(f) is 0 everywhere

though if m_x isn't 0 anywhere that's gg

[neat frost]

Would that be an issue? This would require that the fuzzy domain is empty and that the image is also empty

[fringe summit]

Well, it'd be empty in the sense that the image has 0 membership

[neat frost]

Yeah

[fringe summit]

but when is (U, m) equal to (U', m') anyway

[neat frost]

Oof I'm sorry, I still don't see the issue? Like the universe doesn't really matter, it just gives you a domain of discourse, the actual set itself is given by m

[fringe summit]

I'm saying there's no real notion of equivalence

and I can map a function from a fuzzy set with m(x)=1 everywhere to one that's 0 everywhere

[neat frost]

Oh I see

[fringe summit]

which is rather unintuitive from the sense of sets

[neat frost]

Yeah okay that's a good point

[fringe summit]

Rather than a < b meaning b->a, it's that a->b

so you

[neat frost]

I mean we're not using that defn anyway but it's a good point on why we shouldn't

[fringe summit]

need uhh m2(f)>m1

[neat frost]

Yeah that makes sense

[fringe summit]

but whoops now empty->A has positive image

or rather, m=0 style empty

[neat frost]

Yeah

[fringe summit]

Idk how they properly do it, and my concerns may be irrelevant

[neat frost]

So like now, as per Xan's suggestion, we're trying to work on it with by defining relations

[fringe summit]

but it's something I saw as a potential issue

[neat frost]

I have no clue either

Im trying to do this without reading up on it at all

More of a fun thing as opposed to a learning thing

[fringe summit]

Well, thinking in terms of a<b meaning a->b, we'd want m_R < m_B or similar yes?

a and b ~= ab

[neat frost]

Yeah

Wait a and b?

[fringe summit]

no that should be inf

hm

[neat frost]

My idea now is to sorta just come up with necessary and sufficient conditions for when a relation defined like this is invertible (I'd have to define the identity fuzzy relation on a fuzzy set A but I think that would just be m(x,y)=m_A(x) when x=y and 0 o/w) which gives bijections (an invertible normal set relation is a bijection iirc) and from there come up with a decent definition of a function. In any case I'm more concerned with bijections anyway

[fringe summit]

So, $\bigcup_{i\in I} X_i$ defined with membership $m_{\cup X}=\sup_{i\in U_I}\inf{m_{X_i}(x),m_I(i)}$

[fathom swallow]

The great Sharp

[fringe summit]

or maybe multiplying

idk

[neat frost]

I did multiplying

It gave a more natural result imo

[fringe summit]

multiplying is like independent probability, infimum is like taking the a<b => a->b

and should be the biggest thing that's less than either in that view

[neat frost]

Tbh I'm not entirely set on what a<b => a->b means here?

[fringe summit]

if a is less than b, you could regard it as truth of a being strong enough to show truth of b

[neat frost]

Hmm

[fringe summit]

rather than independent confidence of membership

[neat frost]

Though shouldn't it be inf sup? Because I'd think the intersection would be whatever the reverse is of this, and it should be less than the union

[fringe summit]

well sup ( min of those two)

but multiplying is pretty sensible if it's regarded as "confidence of membership"

[neat frost]

Well if U_I is a singleton {i} then you get that the union is inf{m_{X_i}(x), m_I(i)}

[fringe summit]

whereas the min(in X_i, in I) is like saying the whole set has concentric rings of "this is the layer we accept is present"

[neat frost]

And the intersection would be the sup instead of the inf of that

Which doesn't really make sense

[fringe summit]

well I'd say you'd have an inf term there regardless

since you're never getting closer to 1 than you already are in I

[neat frost]

So how would you define intersections?

[fringe summit]

$m_{\cap X}(x) = \inf_{i\in U_i}[\inf{m_{X_i}(x),m_I(i)}]$

[fathom swallow]

The great Sharp

[fringe summit]

the outer inf is the intersection, the inner inf is bounding the membership by membership in I

but you also bound by I in union

[neat frost]

So that's just the Infimum of all the $m_{X_i}(x)$ and $m_I(i)$ values

[neat frost]

And if m_I(i)=0 this is 0

[fringe summit]

well each m_X_i and m_i term have the minimum taken, and then the inf is across that

[neat frost]

But that shouldn't be dependent on the universe of I (like adding an element to the universe whose membership is 0 shouldn't affect this I would think), like maybe taking the Infimum over all m_I(i)>0 makes more sense

[fringe summit]

so that they're paired up with the same i index is important

[fringe summit]

so (U, m) ~= (U', m') where m' is never 0, and U' is the largest subset of U where m=m' on U'?

[neat frost]

Sure you could do that

I mean yeah that's what I was referring to

[fringe summit]

You could, of course, get an element present with nonzero membership in each set but 0 in the intersection

inf of (0, 1] is 0 after all

[fringe summit]

Aight, lets add a caveat that my supposed union and intersection are for such a 0 removing process

[neat frost]

Yeah that's the case with multiplying too though, so idt we need to worry

[fringe summit]

this is how I kinda see the two different approaches

[neat frost]

Though an issue with both of these defns is that you don't necessarily have $X_i \subseteq \cup X_i$ but then the only way to ensure that I think (ie the minimum set which has this) would just be $\sup\set{m_{X_i}(x)}$ which doesn't do anything with $I$'s fuzziness

[fringe summit]

one's like a tower where picking something in a ring gives everything in that ring & below immediately

[neat frost]

What's the L and X for?

[fringe summit]

just labeling them

[neat frost]

I like your L

[fringe summit]

subset being that m_X_i < m_cup X

[neat frost]

Yeah

[fringe summit]

it's an I

[neat frost]

Lmfaooo

Oopsie

[neat frost]

...

[fringe summit]

yeah that is concerning, but it holds in the case I is a set

[neat frost]

Yeah

[fringe summit]

though the intersection case holds regardless

[neat frost]

Yeah

[proud olive]

First one

[true glade]

mathscr vs mathcal... though the right one still looks kinda mathscr

[neat frost]

Lmfao

[fringe summit]

but also that being the minimum such fuzzy set is the definition of supremum

since the fuzzy set is defined by m, and that's the minimal such m, there ya go

[neat frost]

Right so that's fine when I is a normal set

But the issue here is when I is fuzzy

[neat frost]

Because R(a) is fuzzy here

[fringe summit]

but should it really be identical to the set intuition either

since a fuzzy index set is already odd

[neat frost]

I mean neither multiplying or supinfing are identical to set intuition, they're both sorta odd

[fringe summit]

the multiplication based union also falls short of the X_i < U X

[neat frost]

Yeah

[fringe summit]

but we have that $m_{X_i}(x)\land m_R(x)<m_{\cup X}$

[fathom swallow]

The great Sharp

[fringe summit]

or multiplication

[neat frost]

What's \land?

[fringe summit]

infimum

[neat frost]

Oh

$m_R(x)$ is meant to be $m_I(i)$?

[neat frost]

I think I've seen that notation somewhere

[fringe summit]

yeah but specifically in reference to whatever term you had the relation have there above

[neat frost]

Oh okay

[fringe summit]

specifically, the intuition I'd have is that for the confidence of an element c to be in S(X) should be the confidence that (aRb, and bSc)

[neat frost]

Right I agree with that, but the issue is that confidence can be measured either by infimuming or products, right?

[fringe summit]

well, I can see two methods as such

[neat frost]

The thing is, if you do use the supinf defn you end up getting weird result here (which is ofc not really an issue since) since you get
[ m_{R\circ S}(a,c) = \sup_{b\in R(a)}\inf\set{\frac{m_A(a)}{m_B(b)}\cdot m_S(b,c),; m_R(a,b)} ]
Which has more dependence on $a$ than $c$ which is odd. But that may be because we'd need to rethink how $m_{R(a)}(b)$ is defined

[fringe summit]

I am looking into this a bit more to see usual fuzzy set conventions and in regards to using inf, multiplication whatever

uh yeah you just pick one

[neat frost]

Oh lol

[fringe summit]

if you have some continuity properties (we do in both suggestioins) we get a unique function residuum defined too

@neat frost

so these are the related functions we'd use for union in response to inf, multiplication being the intersection, respectively

The fuzzy intersection is not idempotent in general, because the standard t-norm min is the only one which has this property. Indeed, if the arithmetic multiplication is used as the t-norm, the resulting fuzzy intersection operation is not idempotent.

[neat frost]

I don't know what a residuum or a T norm is

[fringe summit]

T norm is just commutative, associative, and monotonic

[0,1]^2->[0,1]

[neat frost]

Oh okay

I'm looking at the wiki article lol

[fringe summit]

residuum is defined there

also something about continuity for it

[thorn perch]

[neat frost]

Okay so how does this help here?

[fringe summit]

Well these are handy dandy ideas directly relevant to the struggles of the union, and the idea I mentioned earlier about a<b => a->b

[neat frost]

Interesting

But can the t norm handle arbitrary unions and not just binary ones?

[fringe summit]

t does intersections

but the dual of uhh

s=1-t(1-m_A, 1-m_B)

that can do unions

[fringe summit]

see these here

[neat frost]

Yeah I saw, I meant conorm. But in any case can it handle arbitrary unions?

[fringe summit]

might depend on the particular t if it works nicely

[mystic kindle]

Do you know Trignometry I need help with it

[fringe summit]

begone, and read #❓how-to-get-help

[neat frost]

Like I think t norms may be a little out of my depth

I've never heard of them before today

[fringe summit]

it's just some axiomatization of

multiply or inf

or other similar functions

Multiplying or inf are all I'd care about here anyway

[neat frost]

Mmhmm

[neat frost]

So like my main issue with inf (and I'd assume this would extend to any t norm which isn't multiplication) is this. I think it may require some generalization of m_{R(a)}(b) wrt to whatever norm is being used

[fringe summit]

just note that the 1-(1-a)(1-b) as our sup for binary union when we use multiply for intersection

or just a different definition

[neat frost]

So then the sup would become $\sup\set{1-(1-m_{S(b)}(c))(1-m_{R(a)}(b))}$?

[fringe summit]

idk for infinite unions, but for binary ones we need that bit in the middle instead of sup

probability that x is in A or B

using inf for intersections and sup for unions immediately carries over to infinite though

[neat frost]

Wdym? Also where did the 1-(1-a)(1-b) come from? I remember looking at it but now I can't find it

[fringe summit]

if t(a,b) = ab

s(a,b)=1-t(1-a,1-b)

[neat frost]

Oh I see it

Yeah thanks

[fringe summit]

also note that this is P(A u B)

[neat frost]

Yeah

[fringe summit]

if you had P(U A_i), then there ya go

[neat frost]

But then again this is all for the union of two (or more) fuzzy sets indexed by a crisp set

So if the indexing set is also a fuzzy set then don't we still have an issue?

[fringe summit]

yeah idk in that case

[neat frost]

Oof, well thank you very much!

[long matrix]

sluwurp

[hasty leaf]

Woof

[split pawn]

Theres already an imbalance, techno

[near vigil]

hey need

help with a question

can someone tell me how

ow we simplified tan^2x(sinx-1)(sinx-2)
to tan^2x(1-sinx).

[halcyon lichen]

Put x=pi/2 in sinx-2

Becomes -1

-1(sinx-1) = 1-sinx

Also

Ask this in a damn help channel not here

#❓how-to-get-help

[storm sage]

look up umap, it's an arxiv paper

[neat frost]

Hey Eric!

[neat frost]

Okay so an issue with this is that there is no identity relation then. Like for $I\subseteq A\times A$ to be an identity, it would need to satisfy $R\circ I=R$ and so
[ R\circ I(a,b) = \sup_{a'\in I(a)}\set{\frac{m_I(a,a')\cdot m_R(a', b)}{m_A(a')}} = m_R(a,b) ]
But since $m_I(a,a')\leq m_A(a)\cdot m_A(a')$ since it's a relation this is less than $m_A(a)\cdot m_R(a',b)$, and so if $m_R(a,b)$ is the maximum value wrt $a$ this is less than $m_A(a)\cdot m_R(a,b)\leq m_R(a,b)$ and there's no reason for an equality (no reason for $m_A(a)=1$).

If we somehow get the denominator to be $m_A(a')^2$ then we could use $m_I(a,a')=m_A(a)^2$ when $a=a'$ and $0$ otherwise. But the issue is finding a natural reason why the denominator should be $m_A(a')^2$.

[neat frost]

potentially
[ m_{R(a)}(b) = \frac{m_R(a,b)}{m_B(b)^2} ]

[neat frost]

No this isn't well defined

[storm sage]

heyyyyy

[neat frost]

What happens if
[ m_{R(a)}(b) = \frac{m_R(a,b)}{m_A(a)\cdot m_B(b)} ]

[neat frost]

This works

Nice

This is a much better defn

Fixes a few issues

[rustic ether]

Can someone tell me about it answer

[neat frost]

!help

[quasi jetty]

Please read #❓how-to-get-help

[rustic ether]

@neat frost yes

[neat frost]

Read the message.

[neon elk]

Wanted to know, can I also ask physics problems in the help section?

[rocky shuttle]

That's fine, but you'll get better response in a physics server.

[neon elk]

Understood. Do you know any physics server?

[hexed gazelle]

check #old-network

[neat lintel]

discord.gg/physics

[fringe summit]

This is not very descriptive

[onyx ferry]

guys can you help me solve this problem?

[storm sage]

https://arxiv.org/abs/1802.03426

[fringe summit]

I did eventually find it

[zealous girder]

hi guys

[rare falcon]

hello there

[vapid wasp]

696969696969+876543212345678987654345678=7654323456789098765445678o

[rotund steppe]

Ban

[neat lintel]

!chill

[storm sage]

hahahaha

I guess it was sufficiently descriptive!

but maybe not efficiently

this is like a nonconstructive existence proof

[urban vigil]

☭

[split pawn]

Discussion 2

which never became so alive

[gilded sluice]

thinking about that wendigo guy

he asked question in #1021175428326633542 and nobody answered him

so he went to #serious-discussion to ask for help instead
(the only answers he got were !help obviously)

but i wonder if they ever got their help

[brisk lily]

People here always failed to apply first aid (joke)

[split pawn]

!volunteers

[quasi jetty]

Helpers are just people volunteering their time to help you. Be polite.

[gilded sluice]

im not sure how this is relevant

like, i already knew-

oh

it's relevant for the next time someone appears

makes sense

[neat frost]

Is the wendigo guy you?

[gilded sluice]

no

[hasty leaf]

Hi

[neat frost]

Hey technie!

[deep mango]

Techno tanuki is the wendigo guy?

[hasty leaf]

No lmao

[neat frost]

In #bots do ,list

Make sure to do this in #bots so you don't add so much spam

[fringe needle]

that moment when you finally get an idea for how to solve a problem youve been struggling on for hours as you go to sleep, and forget all about it in the morning...

[eager reef]

I usually send a voicemail to one of my friends where I „save“ that idea

This has happened to me a lot

[halcyon lichen]

jus use voice record

before u go to sleep

ez

[haughty lintel]

I love when my roommate spends all night gaming so I get 6 hours of horrid sleep

[long matrix]

brain record coming soon !

[formal fractal]

[halcyon lichen]

6 hours is a lot lol chill out

[eager reef]

it's not

[long matrix]

its kinda average

or below

[halcyon lichen]

its a joke why yall getting baited

[brittle socket]

Yeah 6 hrs seems a little tiring to deal with

[halcyon lichen]

xDD

[brittle socket]

[hasty leaf]

Sensitive plant

[neat frost]

toxic sleep measuring

[long matrix]

plants r always asleep

when theyre awake they roam around and eat u

[hasty leaf]

Wait catnip is plant...cat weakness

[long matrix]

nu

[hasty leaf]

Catnip here ...kitty kitty

[long matrix]

[bright hill]

it definitely didn't seem like one

it was either a particularly bad joke or you're bailing out

[tall minnow]

schrodinger joke

based on whether people call you a douchebag retract your statement and call it a joke

[surreal kindle]

Go back to chill stop talkin here

[balmy wyvern]

hi

[cyan glacier]

whoops

[brittle socket]

Lmao calling Shuri meow in a serious convo is funny ngl

[rocky shuttle]

Oh I think you're right uwu >.<

[neat frost]

.<

[gilded sluice]

[long matrix]

tsumiki

[tiny saddle]

LMFAO

[long matrix]

Youve already been warned by a mod

keep the crap and gifs in #chill

[neat frost]

Yeah go to chill shuwi

[surreal kindle]

Yeah get outta here!

[worthy galleon]

I NEVER STUDY MATH
can i share my info?

[patent quartz]

I'm literally going to cry

[proud olive]

y?

[patent quartz]

🤬🤬🤬🤬

So my friend

Was supposed to be here today at hell

[proud olive]

you copied your friend's hw?

[patent quartz]

Andddd

I have a presentation in science

I TOLD HER ALL WEEKEND

I'm gonna scream my head off

[proud olive]

rip

[echo tundra]

[autumn verge]

[gilded sluice]

[tiny saddle]

Heya @velvet dagger wanted to inquire abt 2 of Yr reviews, specifically the one on LA and AA

That OK with u?

[bright hill]

you should prolly just ask

https://dontasktoask.com

[tiny saddle]

Kk, I was a bit worried if they actually minded and didn't want to be inquired on it

For the LA list, what is meant by "computational/low brow" and "abstract" when u were talking abt the LA books? I'm looking for an LA book which covers most topics of LA (unlike LADR etc which cover mostly R and C fields), but don't progress too quickly either (shilov and Grenub). Som.. Artin or Hoffman and Kunze?

As for Algebra (abstract, I assume? The extreme bottom pin), similar to LA, I'm looking for a book which covers most of AA. I was thinking of Dummit and Foote, but some say it's boring, and earlier peeps like delerick recommended rotman what are Yr thoughts?

[peak tide]

i don't know Grenub, but i'm familiar with Shilov and H&K. If you think Shilov progresses too quickly, you probably won't like H&K either, it's kind of the Baby Rudin of LA

[tiny saddle]

ic

[peak tide]

the book by Friedberg/Insel/Spence is my generic recommendation to most people who want a mathematical (as opposed to computational/application-oriented) intro

[void tundra]

FIS was so dry tho

[void tundra]

You don't yet have the mathematical maturity to start abstract algebra

[tiny saddle]

Ofc not but just wanted to get the books out I'm looking to have them printed for my personal use

[royal cradle]

you can start abstract algebra with no maturity

its relatively disjoint from everything else except set theory i suppose

then again you can ignore that and still read the book

[void tundra]

and not learn anything

yup

[royal cradle]

no you will learn

[void tundra]

you're going to need at least a little set theory

[royal cradle]

yes

[void tundra]

so he should be doing that first

not much

[royal cradle]

yes

[void tundra]

I'd say you can probably learn the required set theory in about 1-2 months

[peak tide]

most LA books are kinda dry.. i'd love to know of an exception

[void tundra]

Honestly I wish I knew

I've been trying to find a non dry LA book too

[tiny saddle]

[royal cradle]

jim hefferon linalg

thats fun but incomplete

[cinder zephyr]

it is

but I liked it alot

[wooden flax]

Shilov

Computing shit on day 1

[tiny saddle]

Istg tho when I saw determinants on the 3rd page

I was like "OK wait what"

[peak tide]

i like his treatment of determinants, but it's weird to do them first... it seems like that might have been the style in Russian LA once upon a time. There's another Russian LA book whose author I can't recall offhand, which is an intro to LA but assumes that the reader already knows determinants and does all the basic linear equation solving using them

[tiny saddle]

Mhm

[bright hill]

you need proof writing lel

[bright hill]

hubbard for sure

@void tundra

it's really good if you're also interested in multivar anal and it's super beginner friendly

it's the first math book I ever picked up

tho most of the LA you're gonna be doing there is over the reals

and fairly biased towards the stuff you need for calc

[little vine]

I'm an Axler stan for LA.

[little vine]

Tfw no eigenvalues

[brittle socket]

Or spend a year like me

[brittle socket]

Sean rec'd morton curtis to me last time but im not sure how good or bad it is. He liked it though

Told me it has stuff like exterior algebra in there for defining determinants lol

[tiny saddle]

Based on mod's tier list in books channel (pins), it's ok

[brittle socket]

In other words, my algebra

[brittle socket]

Yes I have read Dami's reviews countless times already.

[tiny saddle]

[acoustic slate]

Is there some higher level math that’ll improve your ability to solve IMO style geometry questions that isn’t just spamming IMO style geometry questions?

[tall minnow]

just the ability to notice what depends on what

not any high-level maths but just staring at things

and it will all be useless past high school

[acoustic slate]

I see

Being good at what type of competition math problem is most useful outside of competition math?

[static loom]

useful? competition math? outside?

[acoustic slate]

Being?

[tall minnow]

number theory and combinatorics obviously continue into actual maths

[acoustic slate]

Word

[tall minnow]

euclidean geometry stays as brain teasers

[acoustic slate]

Word 😦

Idk why I’m even upset about that

[tall minnow]

only basic geometric ideas do continue getting developed

[acoustic slate]

I’m actually so bad at those problems

Just find them interesting

[tall minnow]

heh I was the best at Euclidean geometry

but college maths is just much more interesting

[acoustic slate]

Oh yeah? Name every angle

[tall minnow]

when you actually learn stuff that matters

[acoustic slate]

What’s your phd in?

[tall minnow]

do a search

[acoustic slate]

What

[tall minnow]

I'm not in the mood to answer something countless times

[acoustic slate]

Lol

[halcyon burrow]

got a yield on for in case with 12% in equation linear regression

[fervent pebble]

if im interested in what now

[velvet dagger]

Multivariable analysis

😛

[neat lintel]

windows users when no shebang

[dry knot]

So, I have to "test" my understanding of the topic and by that I designed a little riddle. Basically I have this yellow surface and this blue curve that is imbedded into that surface. Now I have to mathematically construct new surface that is tangent to this surface and it sits on a blue line that should represents normal curvature of this blue line on this yellow surface
BUT im too stressed out do do things that I once loved doing it 😭

[split pawn]

I gotta watch that Mythbusters video

on

shockwaves

[neat lintel]

HIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

[rocky lake]

hi

[cyan goblet]

when do people usually meet with professors to talk about a directed reading course?

i have a professor who already they'd supervise a directed reading course for me

we agreed on diff geo, but nothing specific

im reading Lee's intro to topological manifolds rn, and will probably try to start Lee's intro to smooth manifolds over the summer

so should I meet with him around the start of next semester to let him know where I'm at?

[jovial ember]

You should just talk to him and work that out with him directly

[cyan goblet]

oh yeah true

very good idea

[mellow imp]

I usually reach out via e-mail and/or office hours

I'd second @jovial ember , more communication is better

[neat lintel]

Chmo

[dapper current]

maybe erlangen style group theory or like harmonic analysis?

[native drift]

hi

[bright hill]

sup

[tiny saddle]

sup

[rocky shuttle]

sup kyuu

[bright hill]

gud

[rocky shuttle]

[long matrix]

Q

[hasty leaf]

R

R lang

[kindred glacier]

hello

[radiant pivot]

anybody into functional programming?

[granite kelp]

I am learning Haskell

[radiant pivot]

mee too, recently got into functional programming

I'm in search for better resources

[granite kelp]

I found these two books helpful:
Learn You a Haskell for Great Good, free to read online.
http://learnyouahaskell.com/

[storm sage]

haskell <3

[granite kelp]

And This one:
https://nostarch.com/learn-physics-functional-programming

[radiant pivot]

oh I'm reading that actually

[grizzled grove]

What's the point of using Haskell?

Compared to python

[dapper current]

the different features/aspects of it. Like it being statically typed, being able to use linear types, monads, and how the program is structured, and other stuff

[distant raft]

@solid snow derivative of 2x

[solid snow]

!help

[quasi jetty]

Please read #❓how-to-get-help

[fading moon]

nobody cares if you know calculus/linear algebra/probability theory, they only care if you know Java, backend, frontend, deep learning etc

[rose dock]

my internship certainly cared that I knew abstract algebra and number theory

[torn willow]

Java is a very low bar tbh

Like knowing how to program is like to knowing how to read and write

You absolutely need more than "how to read and write" for anything

[cyan goblet]

i will take note of this when i apply to math research positions

[alpine kindle]

I don't think a criminal psychology position will care about backend development experience

[warped creek]

It’s a good thing python and maple exist

Idk much about C. Maybe down the line

[torn willow]

Well you should be able to fetch data from a database, I think

[fading moon]

it cares about math?

[torn willow]

Well I assume you need some kind of social network analysis stuff in criminal psychology

<@&268886789983436800>

[fading moon]

why do you @ moderators?

do you think I broke the rules or what?

[hasty leaf]

Maybe the convo was getting a bit tiring and annoying.... this Math vs Programming is like Meow vs Woof

Why not like both

[fading moon]

well I dont have interests to continue it as well

just view it as a rant from a foreign student who's trying to find a job in the local job market

[hasty leaf]

Ok to be practical but there are many jobs that require math or even programming jobs that require math... well music isn't that useful but it makes life better.... doesn't math improve problem solving skills in general and hence maybe useful to programming

[fading moon]

yeah you're right, shouldnt talk about programming here

math is helpful for everything

[hasty leaf]

No you can

Many programmers in this server for some strange reason

[torn willow]

Good luck

[torn willow]

I have a feeling it's because they just like found interesting math in a programming project

[surreal sapphire]

pretty sure that almost anyone below the age of 30 has programmed at some point

[elfin ridge]

<@&268886789983436800>

[foggy mason]

Hey, I was wondering what works for you guys while studying for an exam in a proof heavy course? I've taken a bunch of pure math courses this year and ive had a rough time with almost every midterm or final I have written haha. I have another set of finals coming up in 3 weeks and I was wondering what the best way to study would be. Usually I read over the notes or textbook to make sure I know all the concepts, then I try to do some practice problems. Would you say I should spend more time practicing and less time reading or vice versa?

[surreal sapphire]

practice more

do old exams if possible

[zealous lantern]

One study trick I do is I read the textbook and skip over proofs to save time unless they look fairly short

And then for each theorem/lemma/definition/etc I will come up with the simplest possible problem that I can use the theorem to help solve

Then it helps me to remember the theorem and learn when to use it

For example:
Let’s say the theorem is
the derivative of f(cx)= cf’(cx)

Then for my simple problem I will say what if f(x) = e^(2x) what is the derivative? Then I work it out to be 2*e^(2x)

[alpine kindle]

skipping proofs is a bad idea

[zealous lantern]

And if you want to get an even better understanding you can try to prove it

[long matrix]

its just an applied approach

u can go back to them later

[zealous lantern]

Not if it takes 24 hours a week to do all that for one class

[tiny saddle]

unless yr into mathematics and the actual reasoning behind it, that's fine tbh

[alpine kindle]

they are there to show you techniques

[tiny saddle]

like, chemists, biologists and whatnot won't really care, they don't rlly have the time, they just need to know that it works

[long matrix]

there are some proofs u shouldnt skip

[alpine kindle]

yeah but if you're doing a maths course

[long matrix]

but for a first readthrough

there is nothing wrong with skipping

and coming back later

[tiny saddle]

mhm, true that

[alpine kindle]

the key is coming back later

look at them at some point

[zealous lantern]

But if you are taking an easier class where they only give like under 10 theorems per week then maybe it would be possible to read the proofs

[long matrix]

u cant do all of math, no proofs, no.

some theorems will fail to make sense

Its a bit silly to blackbox "identities are unique in a group"

for example

idk

[alpine kindle]

wdym
this is incredibly difficult

[long matrix]

astounding

[zealous lantern]

I was really talking about proofs that can take like 15 minutes or MORE to read

If it is a short proof then that would help you remember and understand the theorem a lot in a short amount of time

[long matrix]

u should at the very least eventually skim the ideas

that are needed to show the theorem

so u have a kind of map

in your mind

of what theorem relies on what

this is fairly important in general

[prisma swallow]

Don’t you just hate the applied calculus courses?

[bright hill]

not if you know what you're doing

[vocal roost]

säs

[open aspen]

Skipping proofs of technical lemmas is Chad energy

Once you sort out the details it's clear

(Don't sort out the details)

[long matrix]

<@&268886789983436800>

[bright hill]

yes!

proving certain functions are measurable is cancer, for example

and most of the time, I don't think the proof brings much insight other than familiarity with the definitions

[tall minnow]

literally you have to do something very hard to break measurability

anything you can describe with countable operations is measurable

[gentle gorge]

ive got a question for anyone whos interested, i'm definitely missing an obvious answer to this but is there any function that can return 1 for all values except itll give the output of 0 at 0

my first idea was the derivative of the absolute value function |x|/x but thats undefined at zero

[solid snow]

a piecewise function

[gentle gorge]

i mean yeah fair but

itll be a bit complex to do that

especially since its all inside of a parametric equation

[brisk lily]

|sign(x)| /sarcasm

[gentle gorge]

huh

ohhh

sorry i dont use tone indicators so i thought sarcasm was a denominator

lol

[brisk lily]

I mean, I sure it isn't an answer you are looking for, so that's why i used sarcasm

[gentle gorge]

yeah

real quick question and this is gonna sound dumb probably

what does sign(x) mean?

[solid snow]

-1 for x<0, 0 for x=0 and 1 for x > 0

[gentle gorge]

ohhh

just another way to write absolute value got it

yeah

thats not what im looking for

but

i think its definitely part of the answer

[marble delta]

No

[gentle gorge]

wait yeah

youre right

slope of absolute value plus 0 at x=0

i think thats possible to use

yeah this might actually work for what im doing lol

thank you

[solid snow]

the function is going to be piecewise in the end anyways

[gentle gorge]

okay wiat

thats helpful but i just realized theres a whole other issue now

if anyone of you wants to know what im actually doing just dm its gonna be a long explanation lol

[solid snow]

Why not just ask your original problem

[gentle gorge]

well actually that fixed part of it

it just created a secondary issue now

[solid snow]

https://xyproblem.info

[gentle gorge]

fair enough

here

[brisk lily]

Don't mind about it much, i sure this has mistake

$\lim_{a\rightarrow\infty}{\sqrt[a]{x}}$ /hj

[fathom swallow]

Quicksilver

[onyx ginkgo]

Are you looking for an equation? Because I'm thinking just of an If Then Else if on a calculator.

[gentle gorge]

An equation

If you want I can dm you my whole conundrum

[onyx ginkgo]

It might help, but it seems pretty straight forward.

[hollow onyx]

i have one question

In which case can I use L'Hospital's rule

[fiery talon]

if u get indeteminant form

use it

makes ur work easy

[radiant pivot]

indeterminate form of expression, 0/0

[radiant pivot]

no absolute function and signum function are different

[surreal sapphire]

this isnt true

there are some more requirements for lhopital

[fiery oriole]

High key considering signing up for a hackathon

[tiny saddle]

both functions must be differentiable

[surreal sapphire]

Consider $x\mapsto \frac{x+\sin x \cos x}{(x+\sin x \cos x)e^{\sin x}}$

[fathom swallow]

Lochverstärker

[torn willow]

Why not just 1/e^{sin x}

[surreal sapphire]

that isnt of the form infinity/infinity?

[rocky shuttle]

We want to have the f(x)/g(x) form

[torn willow]

Ok this is tricky

You can't just factor out that way

[surreal sapphire]

anyways, show that if you take derivative of numerator and denominator and compute the limit for x to infinity, you get a finite result

[neat frost]

What is this an example of?

[surreal sapphire]

but for the original expression and x to infinity the limit does not exist

why does this not contradict lhopital?

[torn willow]

||Because you can't find a open disk where denominator's derivative is nonzero everywhere||

[neat frost]

What are you guys doing?

[torn willow]

Ok I don't think that works

[neat frost]

Well it's not about a disk

[neat frost]

For any c there exists an x>c where g'(x)=0

So every disk around infinity ig

[surreal sapphire]

you figured out why it doesnt work

well

"disk" is weird because we are considering limit to infinity

but yeah ||3. doesnt work, this is the most subtle way for lhopital to break||

it remains to show that the limit of the original function does not exist

and that the limit of the derivatives does