help-23

[chilly condor]

Wait nvm thats right

[thin bridge]

looks sus for a question

[severe pond]

it isn't x^2 - 100x?

are you sure it's 100x^2

[chilly condor]

i did type it wrong actually but

[thin bridge]

ok, so not the original

[chilly condor]

ye

sorry

[thin bridge]

always post the original especially if people ask for it

anyway let's focus on the core property here
a(b - c) = ab - ac

do you have any issue with that?

[chilly condor]

im gonna be honest i dont really know what that means in this context

im shit at math i must admit

im sorry

[thin bridge]

this is the distributive property

[chilly condor]

gotcha

[thin bridge]

when you multiply into a sum/difference, you multiply into each term

[chilly condor]

yeah

[thin bridge]

which is what
a(b-c) = ab - ac
saya

[chilly condor]

i know the distributive property i just dont know how it relates to this problem

[thin bridge]

distribution is the process of going from
a(b-c) to ab - ac
common factor factoring
Is the process of going from
ab - ac to a(b-c)

The same property is being applied

[safe radish]

@chilly condor Has your question been resolved?

[safe radish]

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[fossil coral]

im confused

[fossil coral]

for d it says answer is semi eulerian

but the network is neither eulerian or semi eulerian

[median sand]

What, isn't it neither?

[safe radish]

@fossil coral Has your question been resolved?

[safe radish]

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[formal wolf]

i don't understand what is wrong with my exam board why do they treat a standard high school a level like it's a university admissions test 😭

[formal wolf]

(i) was fine but

I0 = e - 0(...) = e, I1 = e - 1(e) = 0, I2 = e - 2(0) = e, J2 = e/2
r = 2 implies that sum is equal to 1/2
but (1/e)(1+(-1)^2(J2) = 1/e + 1/2
so 1/e + 1/2 = 1/2
?

cant even get the freaking base case correct

[arctic raven]

careful, it's indexed for only positive n

[formal wolf]

oh

[arctic raven]

so your base case is actually n=1

[formal wolf]

base case is n = 2

it says n>=2

[arctic raven]

uhhh so true

[formal wolf]

what you mean is for the In bit in (i)

that is n >= 1 sure

but then how do i find I1

because the definition i got in (i) is not very useful

i mean i could manually work out the integral again

because it was just the integral of (x^n)e^x iirc

but idk if i want to do allat again

[arctic raven]

you could probably use the recurrence relation of (i)

it ought to be true that I1 = e - I_0

[formal wolf]

yes but then what is I0

[arctic raven]

same as I_n with n=0, just int_1^e dx

bounds

or directly you could do IBP on ln(x)dx

[safe radish]

@formal wolf Has your question been resolved?

[verbal cloud]

You can set K_n = (-1) ^n J_n if that makes the telescoping easier to see

[split kayak]

Notably, the I only go down to I_1 = 1 > for reasonable purposes

[safe radish]

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[sturdy raven]

how can i give a specific bijection f:A∪B -> {1,2,...,a+b} if all i know about A and B is that they are finite ? the broader context is in the image attached to this post

[split kayak]

More importantly, you have that A and B are finite and disjoint. Aka no element in A can be in B and viceversa.

For the first assumption that is

[sturdy raven]

i'm not sure how that helps here. do i use a piecewise definition for f based on whether x is an element of A or B ?

something like that ?

wait, that's not quite right i think, what if b is smaller than a

that makes even less sense when i think about it more, writing it like that doesn't tell me what element x gets projected onto anyways

[split kayak]

Tbh im writing a proof for it but its a bit of magic wand waving

im not happy with it

Resumes to:

Show that there is a f that maps injectively from AuB to Z
and there is an inverse f⁻¹ that maps bijectively from a restriction of Z to AuB

We call the restriction rZ for the sake of ease of writing

Now, from the definition of disjoint sets, we can show that there is some partition of rZ that divides the set into those elements who map into A and those who map into B

Now, we know that |rZ| = |AuB|

[crude bear]

It's a typo, instead of b should be a+b

[sturdy raven]

something like that could work ?

[crude bear]

This was correct

Just in the last line replace b with a+b

[split kayak]

And that |rZ_a| = |A|, |rZ_b| = |B|
|rZ_a| + |rZ_b| = |rZ|

Still, a bit convoluted

[fathom jewel]

One question, aren't the sets arbitrary or why do they start from 1, 2 etc.

[split kayak]

I assume it its some form of arbitrary counting of elements

[fathom jewel]

yeah ok cause it would be silly with the case A n B = {}

[split kayak]

I dont think so really cause you cant precisely assume that |AuB| = |A|+|B|

[sturdy raven]

that's the definition for finite we were given

[split kayak]

Really weird way to define finite sets.

[fathom jewel]

Kannst du die deutsche version posten

[sturdy raven]

i wanted to put the whole thing in english before as well but the formating fell appart when copy pasting

[fathom jewel]

Oh ok

Dachte

So like A approx {1,2,...,n} means there is a bijection cause they have the same cardinality

[sturdy raven]

is that not standard notation ? i've been told this lecture has weird notation last time i asked for help here

[fathom jewel]

no it isnt

for me at least it was weird seind approx

[fathom jewel]

Anyway, can't you just say AuB={c_1,c_2, ... c_{a+b}} and then define f(i)=c_i

with index set {1,2,...,a+b}

[sturdy raven]

i think so ? the way i did it also workd tho, right ? i'm kind of focused on the second part of the hint from the problem right now

btw, since you speak german, is it correct that abbildung is called projection in english ?

[fathom jewel]

no we say map

[sturdy raven]

oh, i see

[safe radish]

@sturdy raven Has your question been resolved?

[sturdy raven]

i mean, technically the initial one has but when i'm done with the whole task i'd like to ask if i did it right (also i'm pretty sure another question on this will arise before i'm done)

[sturdy raven]

this is correct ? (other than that i accidentally wrote "mit" thrice instead of "mit" and "und")

[fathom jewel]

f : A U B -> {1,2, ... a+b} makes no sense it should be the other way

A U B approx A U B is also redundant

also the notation xalpha is weird, probably means alpha(x)

[solar hazel]

[fathom jewel]

I think your proof goes into the right direction but you have the same issue with g you did in f

Oh wait I think I am mistaken actullay bruh

[sturdy raven]

I've been told the same last time I asked something here but that's the way we did it in that lecture and the prof said this was standard notation but analysis uses the "strange" notation f(x)

[fathom jewel]

I went by the definition {1,2,...,n} -> A but the other way should still work by the assumptions

[sturdy raven]

I think I might have misread that part in 3.1b (screenshot from before) by accident

[solar hazel]

i can’t read german but why is the proof so long?

[fathom jewel]

It should work either way though

[sturdy raven]

The proof in English is on the right side of the screenshot

[solar hazel]

oh the right side is a translation?

still so long

[sturdy raven]

Because both sets have the same cardinality anyways or why ?

[fathom jewel]

Like if f : {1,2,...a} -> A is bijection then g : A ->{1,2,...a} as well (inverse)

and you stayed consistent, so I think your proof works

The steps are sound to me

[sturdy raven]

I think I'll switch it around to match the definition anyways just to be safe tho

[fathom jewel]

Sure

[solar hazel]

this is not right

[fathom jewel]

yeah typo

[sturdy raven]

It should be f:AuB->[im not sure here] ?

[fathom jewel]

I would omit it but it should be technically something like A U B -> {1,2, ..., l+m+n} or something

[sturdy raven]

you mean you would just cut out that whole line ?

[fathom jewel]

ja, because it only states the goal of what you want to prove

it's like a note for yourself but not part of the proof

[sturdy raven]

also, do i understand correctly that there are three variables l,n,b because the left part of the map has three summands (that sounds wrong but several translation dictionarries gave me that translation for the german "Summand")

[fathom jewel]

yes

The partioning follows from your disjunct case

Now that I think about it, maybe you dont even need to reconstruct g

From the first case we know |A+B|=|A|+|B| so you would apply that twice basically and be done

[sturdy raven]

what if we had something like |AuB|=|A|*|B| instead ? would we get AuB->{1,...,m*n} ?

[fathom jewel]

do you mean |AxB|?

[sturdy raven]

no but that case would also interest me

[fathom jewel]

this isnt always true tho

{1,2,3} U {1,2,3} = {1,2,3} => |{1,2,3}|=3
but |{1,2,3}| * |{1,2,3}| = 3 * 3 = 9

I think your real question is how the index set is supposed to look like

and the idea is to have as many elements as the set you map on

[sturdy raven]

i just noticed, it should be like this towards the end, right ?

[fathom jewel]

ja but i dont think you need this

[sturdy raven]

tbh, if the solution i have works i'm fine with that one even if something more elegant exists. we discuss different students' solutions a week after the due date anyways

[fathom jewel]

I mean yeah it's mechanical but my point is you could use what you had already else why consider the disjoint case at all

[sturdy raven]

i just don't want to rewrite it (plus i already get the logic of the way i wrote it down and kind of want to take a break for a bit before doing the next task)

thanks for helping tho

[fathom jewel]

[sturdy raven]

cool profile picture btw. i think i can close this thread now, or is there something more to be said ?

[fathom jewel]

i have nothing else to say other than thank you

your lidl pfp is also fire

[sturdy raven]

thanks

.close

[safe radish]

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[exotic gull]

Is it true that
cos(x) = cos(|x| mod 2π)

It makes sense and gives proper results, but my calculator says that "parameters are out of bounds"

[honest perch]

yes

the mod 2pi because cos is periodic

the |x| because cos is even

[exotic gull]

Aight, thanks.

Do you perchance know what could my calculator mean by that error?

[honest perch]

what did you type in ur calc

[exotic gull]

Hiper calc on google play

[honest perch]

idk

like, we got desmos

which is way more modern

[exotic gull]

I assumed it is i who made a mistake
But maybe the calc just isnt equipped to handle things like that idk

Ill try on desmos

So in desmos i got scattered vertical lines for every x=π in both direction

And a warning that the plot may not have fine details

But ill presume the identity to be true, it works good enough for me.

!solved

!solved

.close

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[fringe dock]

part b

[fringe dock]

i'm not sure how to go about this

[cunning pasture]

Well, theta is assumed to be small.

[hidden gyro]

Small angle approximations

[stoic torrent]

do you know about the taylors expansions?

[cunning pasture]

Only when theta is small does the equation turn into a quadratic equation, so that assumption must be maintained; extraneous solutions are exiled from the solution set.

[safe radish]

@fringe dock Has your question been resolved?

[safe radish]

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[upper forge]

<@&268886789983436800>

[safe radish]

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[vast pilot]

hi

[cunning pasture]

!da2a

[safe radish]

Asking the actual question right away is more likely to get responses.

Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.

[vast pilot]

f(a) = b
f(b) = a

Determine the linear form f(x) = mx + b

I don’t rlly understand how to get it

[cunning pasture]

What is meant by 'determine the linear form f(x) = mx + b'?

[vast pilot]

I mean determine the form

Srry, my English isn’t great

[normal moss]

hmm f(x) = mx + b

Is the b intentional

Because b is already used

[vast pilot]

Oh the b just means y intercept

[normal moss]

Right

But I assume it's not the same b as the one in f(a) = b and f(b) = a

[vast pilot]

Nvm i solved the thing by pointing it in point slope form

.close

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[normal moss]

[safe radish]

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[wise lotus]

revisiting some linear algebra and I'm having some trouble with how to read this. Specifically what "G x G -> G defined on G" means.

[split kayak]

AxB refers to the cartesian product of two sets, in this case, the same set.

The easiest way to showcase how it works is to do a basic example:
A = {1,2}
B = {3,4}
AxB = { (1,3) , (1,4) , (2,3) , (2,4) }

You take one element of the first set and one element of the second and write them as an ordered pair.

For instance, IR^2 (the set of two-tuples of real numbers, usual vectors) is in reality IR x IR

Now, if some function maps from
AxA -> A

We say that its a binary operation over A.

For example:

• : IN x IN -> IN
  Is the "addition" function over natural numbers.

Take 2 and 4, as (2,4); and as we all know, 2+4 = 6.

So + maps (2,4) -> 6

[wise lotus]

Oh okay, so ⊗ is a binary operation defining the cartesian product of two sets?

[split kayak]

No, it doesnt define it; it maps from the cartesian product of a set, to that original set.

The cartesian product is just a construction of two sets.

[wise lotus]

Okay so a binary operation that maps from the cartesian product of a set G to the original set

That's not so bad

[split kayak]

Its just the formalization of what addition,multiplication,division,exponentiation,etc are.

If youve ever had a calculus 2/3 course, you technically already know how they work

f(x,y), f : IR^2 -> IR

is technically a binary operation.

[wise lotus]

Thanks man I appreciate the help

[split kayak]

np.

[wise lotus]

.close

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[barren fjord]

Im stuck on bi

[safe radish]

@barren fjord Has your question been resolved?

[icy halo]

I'm curious about it actually

I found 10+dx/dt (2+t) - d^2 x/dt^2
all evaluated in t=0 as the maclaurin series

how can I deal with the derivatives?

[barren fjord]

I quite literally dk what to do

I tried taking f(t)=5(t+1) cosht

But that got me 5 and not 6

Hello!

[severe pond]

hello

[severe pond]

not quite

[barren fjord]

Can someone help pls

[severe pond]

you have 5(t + 1)cosh(t) - 0.5x + dx/dt = 0.5 d^2x/dt^2 and we know f is a solution to this. we want f(0) + f’(0)t. we know from before that f(0) = 6 so we just need to find f’(0). to find f’(0) you can just plug in t = 0 and x = f(t) to the differential equation

[barren fjord]

Hmm

[barren fjord]

Im a bit confused

Which part of the equation is f(t)

What does it mean by particular sol

[severe pond]

they say x = f(t) is a solution

[severe pond]

it solves the differential equation with the proper initial conditions

f(0) = 6

[barren fjord]

Ohhhh

I was getting mixed up

[severe pond]

did you solve a ii

[barren fjord]

Particular integral is this right ?

[severe pond]

youll need it here

[severe pond]

never heard of "particular integral"

[barren fjord]

Oh ok

[barren fjord]

I thought this might like plugging x into the solution

[severe pond]

yes wherever you see x write f(t)

$5(t + 1)\cosh t - 0.5f(t) + \dv{f}{t} = 0.5\frac{d^2f}{dt^2}$

[flat frigate]

knief

[severe pond]

so our goal is to find f’(0)

or df/dt evaluated at 0

[barren fjord]

Ahhh

[severe pond]

so let’s just put t = 0 in and figure it out

since this equation must hold if f is a solution

[barren fjord]

Ok

[severe pond]

you know the definition of cosh t right

[barren fjord]

yes

As in exponential form?

[severe pond]

yes

[barren fjord]

Ye

[severe pond]

e^x + e^-x / 2

[barren fjord]

Im allowed a calculator tho

[severe pond]

i see

[barren fjord]

I see where it comes from now!

[severe pond]

you’re welcome

what a long question

[barren fjord]

😭😭😭

[barren fjord]

Also for ci

When I subbed in 0.25 I got 4.57

Why do you subtract that from 6?

[severe pond]

are you reading a solution key or something?

oh

[barren fjord]

Noo I was able to do ci without doing b

I just noticed that 6-4.57 =1.43

[severe pond]

i see

yes because f gives the displacement from O

it starts 6 meters away from O and starts traveling towards O

if after a quarter of a second it is now 4.57 meters away then it has traveled 6 - 4.57 =1.43 meters towards O

maybe draw a picture if you have to

if we are 6 feet apart from each other then after 0.25 seconds we are 4.57 feet apart

[barren fjord]

Oh i assumed it started at the origin

[severe pond]

how far did i move in that time?

[barren fjord]

1.43feet

[severe pond]

yep

[severe pond]

yea there’s a lot to keep track of in this question

[barren fjord]

Yeah☹️

Im terrified for my exam now

[severe pond]

when is it

[barren fjord]

Thursday

[severe pond]

is this ap calc?

[barren fjord]

Im not American

[severe pond]

oh

[barren fjord]

A levels

[severe pond]

i see

what do they say over there

something about gold stars

[barren fjord]

😅not sure what ur referring to

[severe pond]

i thought that’s how they scored those

some sort of star system

[barren fjord]

Ohhh

Yeah we have letter grades

[severe pond]

maybe i’m thinking of A*

[barren fjord]

From an E to A*

[severe pond]

yea

well good luck

[barren fjord]

Thanks

[barren fjord]

Do you have any last minute tips for questions like these

[barren fjord]

I tend to just get overwhelmed and confused I wasn’t able to get past thinking that f(t)=5(t+1)cosht

[severe pond]

i don’t know that there are any tricks but a major issue for most people is picking up on the little things they say in these paragraphs and then translating it into whatever math they gave you beforehand

like you have to be able to read something then connect it to what they just told you

here the big thing was knowing to use x = f(t) in the differential equation

and knowing that since it’s a solution we can just plug everything in

[severe pond]

yes this can be hard

just read it slow

read every sentence a few times over if you have to

if they say something it’s almost always important

they rarely give irrelevant information

[barren fjord]

That’s true

[barren fjord]

How did you know to do that

[severe pond]

experience

[barren fjord]

😭

[severe pond]

yea i mean unfortunately there is no shortcut

you have to familiarize yourself with these patterns of thinking

until it’s just natural

[barren fjord]

Guess I left it too late

[severe pond]

you have time

a lot of these questions lack variety

[barren fjord]

A day

[severe pond]

they are pretty much all the same format

[barren fjord]

I’ve done almost all of the past papers by now and I still got stuck

[severe pond]

hmm

are you checking solutions too soon?

[barren fjord]

Probably

[severe pond]

everyone i’ve helped with this subject has this habit

[barren fjord]

But I only started actually revising recently so I feel like I don’t have enough time to actually attempt the question

[severe pond]

they don’t immediately know what to do and don’t bother spending more than a minute on a problem grappling with the words

[barren fjord]

Bc my exam is so close

[severe pond]

yea this is unfortunate

being in the panic mindset of needing to get everything done quick is counterproductive

learning math requires a lot of focus

[barren fjord]

Yeah

I guess

[severe pond]

i would say spend more time on one problem

[barren fjord]

Im just hoping in the exam im able to see what i need to do

[severe pond]

don’t think you need to get through every paper in such a short time frame

make sure you feel comfortable with a few after a lot of investment

[barren fjord]

But to be honest

Theres a couple topics that I spent ages learning

Like polar coords and de movires theorem

And I’m still not able to solve them even after like 30 mins

[severe pond]

what did you use to study

[barren fjord]

So even with practice the maths doesn’t come naturally to me

My notes from class and the textbook

[severe pond]

i can’t really comment much on your "studying technique" but i’m sure there is something there that is the problem

even if you don’t know it

math is hard to learn though

i can’t imagine trying to learn math if i wasn’t so interested in it

[barren fjord]

I used to enjoy it a lot

But it’s not fun when it becomes hard and I just struggle with the paper

[severe pond]

yea i know what you mean

the exam stress is tough too

[barren fjord]

Yeah if I don’t get an A I won’t get into uni

[severe pond]

well

i don’t know about that

maybe you won’t get into the cambridge math department

but i’m sure you can still get into uni

[barren fjord]

Yeah true haha

[severe pond]

what do you want to do

[barren fjord]

Engineering

I don’t have it in me to do a maths degree 😭

[severe pond]

yea so youll definitely need strong math

are you doing better in physics

chemistry

[barren fjord]

Yh

[severe pond]

so it’s not so bad then

[barren fjord]

Yeah true

I have four papers tho and this one I’m the most prepared for

[severe pond]

you’re most confident in math?

[barren fjord]

I have four maths papers

[severe pond]

or you just mean you’ve spent the most time on it

oh wow

what for?

[barren fjord]

I have 2 are core further maths like de movires, differential equations, hyperbolics etc

1 stats paper and one further pure paper with has number theory and group theory which I hate

[severe pond]

oh wow

they teach that in high school?

[barren fjord]

I also have three standard maths paper pure, mech and stats

[severe pond]

secondary school

[barren fjord]

But those should be easy enough

[severe pond]

american education is terrible lmao

[barren fjord]

Sixth form

[severe pond]

😭

[barren fjord]

16-18

[severe pond]

yea so like 10th-12th grade for americans

[barren fjord]

Yeh

[normal moss]

||-2||

[severe pond]

where i live the only requirement was to finish algebra 2

but they actually set that back further i think

so you don’t need to finish algebra 2

you can just take some remedial algebra along with a financial math class to finish your math in high school

most kids took algebra 2

probably about a third took precalculus

and like a fifth took calculus

that was it though

and the curriculums for each of those courses covered far less than you do in the uk

i went to public school though

[barren fjord]

So do I

[severe pond]

there are private schools that can be pretty intense

[barren fjord]

To be honest I just chose really hard courses

It’s only necessary to do maths up to a gcse level

[severe pond]

yea you want to go to uni though so you pretty much have to do a levels

[barren fjord]

Yh

But I do further maths which is only needed to top unis

You get can get away with just doing standard maths for most unis

[severe pond]

so you want to go to oxford or cambridge or something

[barren fjord]

Well I did but I was rejected

[severe pond]

damn

already?

you didn’t even take the test

[barren fjord]

Yeah

We apply with predicted grades

[severe pond]

i see

[barren fjord]

Which is really stupid because they’re often inflated and based on ur teacher

[severe pond]

yea that’s dumb

imperial it is then

[barren fjord]

🤣

[severe pond]

did they reject you too?

😭

[barren fjord]

Yep

It’s tough out here

[severe pond]

damn

might have to go to scotland

[barren fjord]

Im having a crisis abt whether i should even do engineering 😭😭😭

[severe pond]

what makes you want to do engineering

did you google careers with the highest paying salaries or do you genuinely like stem

[barren fjord]

Engineering isn’t even well paid in the Uk

[severe pond]

really?

what does pay well then

[barren fjord]

It’s better than average but nothing like American salaries

[barren fjord]

Ib and medicine

[severe pond]

ib pays well everywhere i’m sure

[barren fjord]

Yh true

[barren fjord]

Honestly I didn’t know what to pick

[severe pond]

no lockheed martin huh

[severe pond]

fair

most people don’t know what they want to do at 17-18

[barren fjord]

Yeah I had to pick my a levels at 16😭😭😭 I basically spun a wheel

[severe pond]

😬

[barren fjord]

😅

[severe pond]

become a vet

[barren fjord]

That would be kinda fun

Yk I regret not doing bio bc I feel like doing med would be cool

[severe pond]

maybe

opens a lot of doors for you

[barren fjord]

Yep

[severe pond]

there’s a lot you can pivot to afterwards

[barren fjord]

My mum said that abt engineering

Like u can basically work anywhere after

But with med ur stuck in the nhs

[severe pond]

yea this is true

you can work in a lot of different industries

[severe pond]

maybe this isn’t such a bad thing

[barren fjord]

idk

I mean it’s better than the American system

Anyways ty for ur help it’s getting kinda late so byebye

[severe pond]

no worries

have a good one

[safe radish]

@barren fjord Has your question been resolved?

[safe radish]

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[tranquil geyser]

\textbf{Definition - Class Functional:} A \textit{class functional} is an L-formula $\varphi(x,y)$ with 2 free variables (might mention other sets) such that $\forall x \exists^{\leq 1}y \varphi(x, y)$.
\\
Fix a class functional $\varphi(x,y)$ possibly mention sets $b_1, b_2$.
\\
Slang: For any set $A$, the imagine of $\varphi$ restricted to $A$ is a set. i.e ${y : (\exists x \in A)\varphi(x,y)}$ is a set.
\\
Example ${{n}: n \in \omega}$ is a set.
\\
More formally, for all L-formulas, $\varphi(x,y,z_1, ..., z_k)$ we have an axiom.
\\
$\forall z_1, .., \forall z_k$ [If $\forall x \exists^{\leq 1} y \varphi(x,y,z_1,...,z_k)$ then $\forall A\exists B \forall w ( w\in b \iff (\exists u \in A)\varphi(u, w, z_1, ..., z_k)$]

[flat frigate]

toast

[tranquil geyser]

hi i just want to review, can someone explain what a class functional exactly is with examples, and possibly explain what the rigorous definition of replacement is doing?

well

in my opinion replacement scheme kinda just lets me make sets like

{x in X : f(x)}

a little different from separation scheme since a set generated from separation is awlays a subset of the original set itself

or maybe more accurartely its. {f(x) : x in X}

[safe radish]

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[tranquil geyser]

\textbf{Cardinal Exponentiation} \
Suppose $X, Y$ are sets, ${}^{X}Y = {f: X \to Y }$ (f function). So ${}^{\lambda}\kappa = {$ all functions $f: \lambda \to \kappa }$.
\\
\textbf{Definition:} $\kappa^\lambda = |{}^\lambda \kappa|$.

[flat frigate]

toast

[tranquil geyser]

I am having a tough time "understanding" this definition

[crude bear]

Why?

[tranquil geyser]

hmm

just like

why do we care about functions

[primal bone]

We can talk about function spaces as sets as well

Elements of sets don't just have to be numbers (or similar items)

[tranquil geyser]

makes sense

[primal bone]

We can talk about sets whose elements are entire functions

[tranquil geyser]

ye

[primal bone]

This is the notation we need for such a thing, at its most base case

[tranquil geyser]

oh

is this why

the caridnality of functions from R to R is |R|^|R|

[primal bone]

Well, the simplest way to consider a set of functions would be the set of all functions with the same input space and the same output space

e.g. the set of all functions f with domain X and codomain Y

i.e. the ^X Y definition you've just sent

Note this is a set

Again, notation-wise, we're denoting here Y^X to mean the cardinality of the set ^X Y

[primal bone]

We essentially need some notation to discuss things like this, yes

else this shit gets unwieldy fast

[tranquil geyser]

hmm

everything makes sense here

it just seems like we consider elements of functions without providing motivation for it

or is just

really hard to define it otherwise

[primal bone]

It's more that we can consider functions themselves as elements of a set

That's the sole motivation for now

Lemma to prove btw - check that such a "set" {f:X -> Y} is in fact a set

[tranquil geyser]

ye but why functions and not like relations or something

[tranquil geyser]

hmm

[primal bone]

Relations are rather simple in comparison that this is almost trivial

[tranquil geyser]

functions are subsets of relations right

[primal bone]

Any relation R comparing to elements X and Y results in a set some subset of the product X x Y

[tranquil geyser]

ye

and a function is a special subset of a relation right

[primal bone]

ye

Because given x in X, there can be at most one y in Y such that f(x) = y, else it's not a function

But since functions are more useful in that regard, it might be more useful to jump straight to defining this sort of set instead

Incidentally, this sort of thing is called a https://en.wikipedia.org/wiki/Function_space

[tranquil geyser]

P(X x Y) by powerset axiom, then we can just use separation to create a set {f : X -> Y}

[tranquil geyser]

ah

ah okay i think here iis motivation

since we really cared about ordinals

a function like

f: 2 -> 3 would be like "combinations"

and ig a function is a good way

[primal bone]

Also, I might be wrong, but I believe the restriction with relations is that the codomain IS the domain

You're talking about how two elements in the same set relate

But functions don't necessarily have that restriction per se

[tranquil geyser]

ah

[primal bone]

x |-> x+1000 with input space [0,1] has a range far away from the domain, for instance

So I could let X = [0,1] and Y = [1000, 1001] to describe f

And then I could consider the set of functions with those two sets X, Y

[tranquil geyser]

hmm

[primal bone]

That said, there is apparently something called heterogenous relations, where xRy requires x in X, y in Y, where X != Y

[tranquil geyser]

kk

i think this more or less makes sense

especially when considering ordinals

.solved

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[unreal mango]

How do I do these

[outer pollen]

<@&268886789983436800> there's one more here.

[ionic blaze]

is it normally purple
do PEMDAS

or rather GEMDAS

whichever ones you guys have

simplify the inside of the groupings, then do exponents, then multiplication/division then addition/subtraction

oh wait it's naming properties

[unreal mango]

nvm it’s been almost an hour and no one has answered the same question

I just used AI atp

[ionic blaze]

ai bad

[unreal mango]

I know

No one in here was answered me for 10 straight minutes and some other dude was bringing me in 5 different directions at once

.close

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[waxen prawn]

i need help with 7 and 8

[summer coral]

if you take just any point (x, y) on the plane and reflect it over the line y=x, what happens to that point?

in other words, what are the new coordinates?

[waxen prawn]

(x,y)->(y,x)

i did that already

to the two points

[summer coral]

yes so if the points swap, then you can directly just swap x and y in the equation and solve for y

this is the same as finding the inverse, if you have covered that in class yet

[waxen prawn]

what does that mean yo

[summer coral]

so you have y = 2x + 5

[waxen prawn]

yes

[summer coral]

set x = y and y = x

x = 2y + 5, and then solve for y

[waxen prawn]

ohh i see

no i dont

i dont get it

[summer coral]

your points are being "inverted," right?

or "swapped"

that's all we're doing, just swapping x and y

[waxen prawn]

yeah

so the. what

then

[summer coral]

^

that's all you need to do, and you have your equation y = something

[waxen prawn]

y=5?

that doesnt freaking make sense yo 😭

[summer coral]

no, say you have some equation 5H + 2J = 3K and i ask you to solve for H in terms of J and K

you would just isolate H by itself on one side so you get H = something, right?

[waxen prawn]

that doesnt even make sense

oh yaah

yeah

[summer coral]

that's all we're doing here

x = 2y + 5 is your answer, we just want it in the form y = something

[waxen prawn]

is it

y=-1/2x+5

bruh thats not even right

[summer coral]

close

[waxen prawn]

its supposed to be 2.5 but wtf do i get 2.5 from

[summer coral]

show your work step by step

[summer coral]

instead of dividing by 2 first try to get 2y by itself on one side to start.

[waxen prawn]

ohh i see

so i just cant do simple math

anyway

[summer coral]

if you wanted to divide by 2 first you can, you just have to divide everything by 2

[waxen prawn]

this is js an example but what would happen if i had to relfect jt over a y=2 line

[summer coral]

reflecting over a horizontal line is a bit different from this, because you would keep the same x-values but different y-values

a rule you can use is that the new y-value would be 2k - (the original y-value)

where "k" is the number in y = k, in this case 2

[waxen prawn]

okay i understand that

i bet i could figure it out

how to do #8

just kidding

[summer coral]

well, what does a 180 degree rotation do intuitively

[waxen prawn]

i think i can figure ut out

its just like rotating a regular point

thanks for ur help

[summer coral]

yes

[waxen prawn]

how do i quit

[summer coral]

ofc!

!done

[safe radish]

If you are done with this channel, please mark your problem as solved by typing .close

[waxen prawn]

thank u

.close

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[tranquil geyser]

What is the cofinality of $\omega + \omega + \omega$?

[flat frigate]

toast

[tranquil geyser]

I thought it would just be $\aleph_0$

[flat frigate]

toast

[tranquil geyser]

Well specifically the question is what is the cofinality of $\omega_1 + 5 + \omega \cdot 3$

[flat frigate]

toast

[safe radish]

@tranquil geyser Has your question been resolved?

[tranquil geyser]

<@&286206848099549185>

.sovled

.solved

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[median sand]

Would it just be aleph 0?

[tranquil geyser]

i got that

yes

[median sand]

Or just omega.3.?

[tranquil geyser]

i think aleph 0

well the cofinality is in terms of cardinals

[median sand]

Make sense there.

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[hoary seal]

What’s going on with the yellow?

[median vigil]

solving the system of equations above

first step subtracting equation 3 from equation 1 and then using that to solve for lambda 2

then substituting that value for lambda 2 into equation 1 to find lambda 1

since equation 2 was not used at all in that process, you still have to double check lambda 1 and lambda 2 values you found satisfy it

[hoary seal]

Could we use substitution method

I’m reality confident with that

Really *

[median vigil]

sure

it doesn't really matter how you solve the system of equations

[hoary seal]

I see

My second question

Is the span of two vectors same as a plane

[median vigil]

if they are linearly independent, yes

[hoary seal]

Linearly independent?

[median vigil]

well for sets of two vectors they're linearly independent if one can't be expressed as a scalar multiple of the other

[safe radish]

@hoary seal Has your question been resolved?

[hoary seal]

I see

Thanks

.close

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[daring sonnet]

Part c

[daring sonnet]

Could someone please spot where I went wrong

Only consider the working above the red line

[tardy mango]

This is wrong for one

sin(pi) is 0

Also what happened to distributing the pi to the integral on the line above

etc

[daring sonnet]

Hmm I forgot to do that…

[daring sonnet]

How many marks should I award myself?

[tardy mango]

3 ig?

[daring sonnet]

I see, and which marks did I get

[tardy mango]

The first three

[daring sonnet]

Thank you so much!

Btw

[daring sonnet]

Is it ok if I plug in the limits here to avoid writing a long expression in terms of trig functions?

[tardy mango]

[tardy mango]

If you do it correctly, then sure

[daring sonnet]

Noted, thanks again!

[tardy mango]

[daring sonnet]

.close

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[daring sonnet]

.reopen

[safe radish]

✅ Original question: #help-23 message

[daring sonnet]

Different question but what’s the logic for part b

[daring sonnet]

,pin

[frank widget]

look d^y/ dx^2

=0

then d(dy/dx)/dx =0

now use that

and u will get the 2 points where maximum and minimum occurs

of dy/dx

[safe radish]

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[safe radish]

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[hoary seal]

What’s this slide saying?

[worldly lantern]

basically asking whether a multiplicative identity exists in the sense of vector dot product

can you find a vector x such that dot product of a and x is a

[hoary seal]

What’s multiplicative identity

[worldly lantern]

if x multiplied by a gives a , x is a multiplicative identity

like 1 multiplied by a gives a , in the sense of multiplying in real numbers

[hoary seal]

I see

[hoary seal]

Yeah the identity vector of <1, 1>?

[worldly lantern]

can you observe that , a definite “multiplicative identity” does not exist in the vector space

[hoary seal]

Why not

I mean we can’t multiply a vector by a scalar of 1

[quiet plume]

The main difference here is that a conventional product takes two objects and produces another object (all of the same type).
Notice that the dot product takes two vectors, but it’s doesn’t produce a vector: it produces a scalar. Then as they hint, it doesn’t really make sense to speak of some multiplicative identity.

[hoary seal]

But we can multiply a vector by an identity vector

[worldly lantern]

thats scalar multiplication, not dot product

[hoary seal]

I mean compute dot product

Not multiplication

It should give us back the vector we started

[worldly lantern]

notice how $\vec{A} \cdot \vec{X} = A$ only has 2 trivial solutions for X , which will depend on what $\vec{A}$ is, hence there will not be any special vector that will always satisfy $\vec{A} \cdot \vec{X} = A$

[hoary seal]

Why?

Why do we have two solutions for x

[worldly lantern]

ok wait

mb

two trivial solutions

[hoary seal]

If there is 2 solutions then shoudnt a dot x = a be satisfied?

[flat frigate]

Mr. Smith

[hoary seal]

Wdym?

[worldly lantern]

okay wait

so what do you understand by the word “multiplicative identity”

[hoary seal]

I don’t know tbh

What is it

Oh ur multiplying by 1

Which leaves the original object unchanged

[safe radish]

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[scarlet mantle]

I am doing C.A.T Math comp, and my questions are like this one, could someone please explain to me how I do it?

[buoyant shadow]

ignore the thing before the picture

there's no information there

no wait

it lists windows that don't count

still you can ignore it

at the beginning

[scarlet mantle]

So is the answer 4?

Since one window already got replace, and the others are next to 4.

[buoyant shadow]

it should be 2 at most

24 25 have 4 neighbors

ok i guess there's 2 interpretations, ignore the door or not doesn't matter

[scarlet mantle]

Wait, so how do I work this out?

[buoyant shadow]

guessing

idk

[scarlet mantle]

k

[buoyant shadow]

it's unconscious

[scarlet mantle]

I think I got it

[buoyant shadow]

there are no steps to determining when a window has 4 neighbour

the answer should be 1

it's 24 and 25 but 25 is already listed

[scarlet mantle]

Yep, I got it now.

.close

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[forest osprey]

Help

[nimble wyvern]

!status

[safe radish]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[forest osprey]

This what I did but it’s wrong because I remember school saying it was 6

Question is to find d btw

[nimble wyvern]

there is an easier way

instead of manual counting

[forest osprey]

What is it?

[nimble wyvern]

do you remember the formula to find the sum of n natural numbers?

[forest osprey]

Yea 1/2(n)(n+1)

[nimble wyvern]

can we figure out a way to apply that formula for the summation on the right hand side?

[forest osprey]

I’m not really sure

T would be 2 summations

And u take away the one with n 11?

[nimble wyvern]

can we break 11 to 14 into 1 to 14 minus 1 to 10?

[forest osprey]

Uhh wait 1s

[nimble wyvern]

no

the upper limit must be 10

[forest osprey]

Is it 10 instead of 11

Ok

To include 11!,

So the two summations of 1 to 10 cancel

[nimble wyvern]

yes exactly

wait

one question

[nimble wyvern]

why did u start with a-d here

on the left hand side

[forest osprey]

Because r-1

[nimble wyvern]

but we are starting with r=1

[forest osprey]

Ah

[nimble wyvern]

not r=0

[forest osprey]

Wait it doesn’t cancel tho

[nimble wyvern]

?

[forest osprey]

It would be 2 of the summation on the left

[nimble wyvern]

yes

we did that so we could just calculate faster

we can directly apply the summation formula n(n-1)/2 now

[forest osprey]

Ok

I have to go to class rn

Cya and thank u

.close

[safe radish]

Channel closed

Closed by @forest osprey

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[sullen orchid]

Why’s there only one set of coordinates when x can = 0. AI claims it’s a mistake. Is this true?

[sullen orchid]

Also is it wrong to calculate instead of simplifying like there

[opaque lintel]

there only one point where x = 0, because only (x, 2 + (0)^2e^-4(0)) is on the graph of the function?

or do you mean something else

[sullen orchid]

I mean in the mark scheme, the set of coordinates doesn’t include that point

Is there a reason

[opaque lintel]

the question is asking for stationary points

so dy/dx = 0

[sullen orchid]

Right

X=1/2 or x=0

[opaque lintel]

oh right okay sorry I misunderstood

[sullen orchid]

Do u get now

[opaque lintel]

no yeah that's just a mistake

[sullen orchid]

Only one set of coordinates

Ahh okay, and in the exam how do u write out the two set of coordinates?

Is there a plus between them

[opaque lintel]

You can never go wrong writing it out: "The stationary points are (a, b) and (c, d)"

[sullen orchid]

Alr thank u and one last question

[opaque lintel]

mhm?

[sullen orchid]

Is this fine

[opaque lintel]

It might want an exact value, because that's what the marking key has

[sullen orchid]

In the worked answer they just gave the simplified terms no calculation

was left in the form of e

Some simplification

[opaque lintel]

Substitution in values and evaluating usually isn't what they are testing for so it should be fine to do that

[sullen orchid]

I see so either is fine

If not rounded

[opaque lintel]

and that's as an exact value, because e is irrational so you can't really simplify it further without doing some rounding. Both can

[sullen orchid]

Oh okay I didn’t know

[opaque lintel]

I would check with my teacher if I were you. I don't know what your marking keys would look like

[sullen orchid]

Alr tysm for helping

[opaque lintel]

np!

[sullen orchid]

I’ll write it In that form

.close

[safe radish]

Channel closed

Closed by @sullen orchid

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[bright bough]

im confused why both 8v and 8x are -

shouldnt they be opposite

since 8v is a resistive force to 8x

[gentle edge]

they both should be -ve here

since your finding max speed

[bright bough]

can u elaborate pls?

[gentle edge]

your partical would go ahead of point P

so your force would be in -ve dirn

and air resistance is in dirn opposite to dirn of motion

well actually the max speed would happen when partical is at point P

the partical is doing dampened occilatory motion

but max speed would still occur at P

oh wait your asking abt the intial time

mb i mesunderstood

[bright bough]

im mainly asking about why theyre both -ve

[gentle edge]

point P is at x=0 and your partical is at point x=a

[bright bough]

mb im struggling to process ur wxplanation

i dont do physics so all of this is so confusing 😭

[gentle edge]

i was explaining something different

[bright bough]

ohh

[gentle edge]

so your force would be in -ve x direction

like how a pendulum moves

its attracted to the middle position (lowest position)

your partical is at the far right end

[bright bough]

oh so its always working in the direction to x=0

even tho its moving away

?

[gentle edge]

yeah

[bright bough]

but still shouldnt the resistive force be opposite to it regardless

bcs its resistive

[gentle edge]

yeah it should be in the +ve x direction

wait lemme reread the question

[bright bough]

i think the equation written is the correct one tho

[gentle edge]

elaborate

[bright bough]

as in -8x-8v=2a is the answer

well

it gives u the correct DE to then find the answer

[split kayak]

A bit late to the discussion here tbh

But, x is the position over the line, v is the velocity at any point, formally known as dx/dt
And the force translated to some acceleration, which we know is d²x/dt²

[gentle edge]

this is pretty much whats going on

the black ball thingy is the object

its in +ve x direction

at rest initially

[split kayak]

Notably, consider that for us, the force doesnt immediatly translate to acceleration, since the mass ≠ 1

[bright bough]

wait uhh

so is it more like

v is negative

[gentle edge]

yeah

since its at rest initially and the force is applyed in -ve direction

so it gains an accel in -ve diretion which causes a velocity in -ve direction

which makes reistance force(air) be in +ve direction

[bright bough]

right

ok

so -8x+8v where v is speed

but since the velocity is negative its like -8x-8v

wait

[gentle edge]

v is speed

it doesnt have a direction

[bright bough]

yeah

brooo

im so cooked

[gentle edge]

its always positive

so to decide its direction you look at which direction the ball is going

and put it in opposide direction

[gentle edge]

tbh dwag all u can do now is just focus on makes the digrames correctly

and hoping you can do the math involved

[bright bough]

yh which is why when its speed the resistive force is opposite

but then cause we know its direction we can change to -??

[gentle edge]

which direction is the air resistance in this diagram

[bright bough]

the other way

but the answer is -8x-8v

like theyre the same

-ve

[gentle edge]

@split kayak did i do something wrong here

[split kayak]

Gimme a sec im literally in the gym rn

[gentle edge]

o

good luck on em lifts

[gentle edge]

is that the final answer or how they assumed the forces?

[split kayak]

This is gonna be the shittiest diagram ive ever done

[gentle edge]

lol

i useing a mouse n paint

[bright bough]

it gives u the correct DE u need to get the answer so ye

the final answer

[gentle edge]

wuts DE

[bright bough]

Differential equation

[gentle edge]

i think they did it wrong then

caus the force and resistance force would be opposite to each other unless they partical goes past point P

and the max speed would happen when Fnet=0 towards the right of P

[safe radish]

@bright bough Has your question been resolved?

[bright bough]

hmm ok

thanks

[split kayak]

Mb i took so fucking long 🥀

Assume the particle P at some point is at rest, at the left of O

We measure its position as a negative, or, equivalently, a vector pointing to the left.

Now, since we know the force pulls it to the origin, then we know that the force (F) and the position (x) have opposite direction

Which we denote as the force being -kx

In our case, its F = -8x

Now, if we start considering movement in itself, we know that air friction acts as a force that opposes the movement.

If you consider the situation after some time (where the force has managed to accelerate the particle) Then a velocity vector appears, and with it, its air friction force (μ)

[bright bough]

hmm pkay

i think its making sense so far

are you saying force us -ve and resistance is +ve?

[split kayak]

They come up from different magnitudes

F = -8x

μ = -8v

It happens to be that v points opposite to x in this case.

[split kayak]

If we define v = dx/dt

[split kayak]

So, yes, they are both negative, but its because one of them "resists the particle getting away" and the other "resists the particle going fast."

As in, one relates to the position, the other one to the current velocity. If the particle is at "O", then theres no force of attraction since its already at the point of the attractor

If its velocity turns 0, then there will be no air friction

@bright bough u with me so far?

[bright bough]

hold on im reading

and thinking about it

[split kayak]

In this part of the diagram you can see that F goes opposite to x and μ goes opposite to v.

We ignore the fact that these are vectors cause they all are alligned to this imaginary line

[bright bough]

i thinkkk its making sense

[split kayak]

,tex Well, now, from here, we have that $\Sigma F = ma \implies -8(x+\frac{dx}{dt}) = (2)a$