for all a in X, we have a in X_i and a in X_i for some X_i, so a is related to a
#help-49
tired ferry
ok yeah that makes a lot more sense
Thats kinda what i was thinking, but " saying that a is related to b if there exists X_i with a,b both in X_i" felt like something i should need to prove
is there a reason we can assume it
radiant roost
we are asked to prove that "there is an equivalence relation on X"
we can define any relation we want, and show that it's an equivalence relation
tired ferry
ah
so we can define the relation as reflexive
which it would be to be an equivilance relation
radiant roost
not exactly
they defined it with the phrase "let two elements be equivalent if they are in the same partition"
i rephrased this into "a is related to b if there exists X_i with a,b both in X_i" since i believe that's what they mean
tired ferry
Oh, and then they prove reflexivity as an a would be in the same partition in itsself
yeah that makes a bit more sense
radiant roost
yes
tired ferry
tysm
@radiant roost actually sorry rq
is there more clarifiction of the motiviation behind this relation
is it just becasue we know that its true, or does it come from the definition of a partition
radiant roost
they defined it with the phrase "let two elements be equivalent if they are in t...
this one?
tired ferry
yes
radiant roost
you're asking why they chose to define it that way?
tired ferry
yes, it seams to be the only place they do that and im assuming it gives some assumtions
radiant roost
assumptions?
midnight plankBOT
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digital walrus
hi
lean dock
sin(pi/2) is 1 innit
midnight plankBOT
digital walrus
.reopen
midnight plankBOT
✅
digital walrus
can i pls have some help differentiating this
i used the chain rule but i got it wrong
lean dock
can u show ur working out
digital walrus
ok wait
digital walrus
can u show ur working out
actually im redoing it and im a bit confused now
because this one seems to have 2
lean dock
seems to have 2?
lean dock
but you do realise this is also chain rule right
digital walrus
oh
ok let me try again
im confused on how they got 2x+pi/4
in the brackets
@lean dock Hi soz r u still here?
lean dock
yeah im here
lean dock
lemme see
digital walrus
heres my working
lean dock
i think theyre the same thing
digital walrus
what how
digital walrus
oke
lean dock
yeah i got the same answer as you
digital walrus
ok thanks
flat spire
Bum chicken
flat spire
u just started it 😔
flat spire
ok i lied specialist is really hard
just wait till ur vectors amp up in y12
If ur having trouble wit it you’re gonna have even more
digital walrus
bruh
idc nothing is going to be as hard as the in class tests my teacher gave us
because theres 1 smart indian kid
luckily i got a new teacher this year!!!
flat spire
Haha I remember school!
digital walrus
wait i cant lie my formula sheet kinda scary
flat spire
Show
last arch
digital walrus
Those are all chill
rn im doing de moivres theorem
and i dont understand it at all
i just copy the formula 😭
flat spire
rn im doing de moivres theorem
Hahahaa
gg
it’s over
Drop spec and methods
Do the normal maths
midnight plankBOT
@digital walrus Has your question been resolved?
zinc leaf
solve dN/dt + (5/t)N =0, t>0
midnight plankBOT
solve dN/dt + (5/t)N =0, t>0
jolly nimbus
y' + a(x)y = 0
=> y = exp(-A(x))
where a is the derivative of A.
Here you have a(t) = 5/t
jolly nimbus
You can also solve by integrating :
dN/dt + (5/t)N = 0
<=> dN/dt = -(5/t)N
<=> dN/N = -(5/t)dt
You integrate on both sides and then you solve for N
zinc leaf
oh yeah but how would i do it witha n integrating factor
because i know how to do that method
but i think the focus is supposed to be with integrating factor
wait so is it 5/t...
jolly nimbus
I don't think you need an integrating factor for such an easy equation
It would be juste dumb to ask for one
fallen galleon
yes it is
zinc leaf
ok nvm then
what about for
y' + 4y/x = x^(-4)
is that also
without an integrating factor
fallen galleon
you have a 4/x multiplied with the y
jolly nimbus
The integrating factor is always exp(A(x)) if I'm not mistaken
zinc leaf
yeah so
what is it specifically
like
4y?
but x is like
on the bottom
so idk..
fallen galleon
the integrating factor is exp(integral of thing multiplied next to the y)
the thing multiplied next to y is 4/x
fallen galleon
so its exp(integral 4/x)
sullen jungle
what is it specifically
for every diff eq you gotta make sure to put the dy/dx as the first term with nothing in front, then you want the y term and everything else on the right hand side with no y.
whatever is in front of the y will be part of the integrating factor
$e^\int A(x) dx$
A(x) is the function in front of y
multiply the entire equation by the result of the e^integration and solve it
grand pondBOT
Ryse
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
tall rapids
$$
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\textbf{Problem:} \
Let $a$, $b$, and $c$ be positive integers such that:
[
a + b + c = 2025
]
and
[
\gcd(a, b) = \gcd(b, c) = \gcd(c, a).
]
What is the smallest possible value of $\max(a, b, c)$?
\end{document}
$$
midnight plankBOT
$$
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\textbf{Probl...
grand pondBOT
.
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
tall rapids
hlp
midnight plankBOT
@tall rapids Has your question been resolved?
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somber thorn
midnight plankBOT
white gate
Hint: Khoảng cách gần bờ biển nhất là khoảng cách giữa tiếp tuyến tại "đỉnh" của đường cong với đường thẳng x+2y+1=0
vale zephyr
is this vietnamese
white gate
hay tìm x y của điểm đó luôn cho nhanh anh
k cần đâu e
đoạn xanh kia = 9 r
góc xanh biển tính đc qua hệ số góc
mình có bờ biển là y = -1/2 x - 1/2 nên suy ra k = tan(a) = 1/2 với a là góc nhọn xanh biển
midnight plankBOT
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ionic magnet
At what value of the parameter does the quadratic level have one real root?
midnight plankBOT
At what value of the parameter does the quadratic level have one real root?
thorn sorrel
use the discriminant
wind oxide
At what value of the parameter does the quadratic level have one real root?
as he said, use the discriminant
can you tell me what the discriminant is?
ionic magnet
b^2- 4 ac
wind oxide
well, comparing $ax^2 + bx + c$ to your equation, can you tell what $a, b,$ and $c$ are?
grand pondBOT
rak³en
wind oxide
yeah sorry let me check
ionic magnet
thank you
wind oxide
well now tell me
how do you know a quadratic has real roots??
hint: u equate the determinant to a specific value, and u can prove that its that value using the quadratic formula
ionic magnet
how do you know a quadratic has real roots??
i just know that if one real root then m has to be = 0
wind oxide
i just know that if one real root then m has to be = 0
wrong u see
m cant be 0
discriminant has to be 0
last slate
yup
wind oxide
i just know that if one real root then m has to be = 0
if u put m=0, u no longer have a quadratic equation, u have a linear equation which always has 1 real root
ionic magnet
refer to this <@657491085659996172>
yes
last slate
$$D = b^2 - 4ac$$
we know that
$$a = m$$
$$b = -1 + 2m$$
$$c = m-2$$
just substitute this into D and set D to 0
$$(2m - 1)^2 - 4(m)(m - 2) = 0$$
solve for m
grand pondBOT
Edmund Cloudsley
ionic magnet
wind oxide
yep thats correct
midnight plankBOT
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Available help channel!
restive pivot
Can someone explain reducibility to me
midnight plankBOT
Can someone explain reducibility to me
restive pivot
Basically why is it that if A is reducible to B then
(1) If B is easy -> A is easy
(2) If A is hard -> B is hard
(3) If B is hard -> can't say anything about A
(4) If A is easy -> can't say anything about B
How can I understand this in a more intuitive way
midnight plankBOT
@restive pivot Has your question been resolved?
restive pivot
I guess just cuz we can transform a problem into a hard problem doesn't mean it itself is hard
It can be easy
So even if A is easy it can be transformed into a hard problem B
But a hard problem A cant be transformed into an easy problem B
So if B is easy so A has to be easy
And so if A is hard B has to be hard
I guess
lusty bough
question about trigonometry,
midnight plankBOT
question about trigonometry,
lusty bough
is it standard that you have to also do a pi-
like here
i get that the thing above results in 3x = sin (pi/2 -x)
and the k + 2pi i also get, because that's a x amount of 360 rotations
but the pi - on the right side isn't clear to me yet
dusty portal
is it standard that you have to also do a pi-
Yes.
$\sin(\pi-x)=\sin(x)$, prove this.
grand pondBOT
;(
lusty bough
thanks!
grand pondBOT
;(
orchid ibex
ans cos2pi = 1
and sin2pi = 0
lusty bough
thanks all
got my reexam tommorow 😭
had 50% previous time, so I need a little extra boost
orchid ibex
got my reexam tommorow 😭
best of luck buddy
lusty bough
🙏
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Available help channel!
main frigate
(super duper easy)easiest way of solving this with the subsititutemethod 1. 2x - y = 3 2. 2y - x = 9
midnight plankBOT
(super duper easy)easiest way of solving this with the subsititutemethod 1. 2x -...
sudden yacht
Get y from the first equation and replace it in the second
Or get x from the second equation and replace it in the first
lone whale
u could also multiply the first equation by two and add the equations to each other
sudden yacht
Yeah but I don't think that can be properly called substitution, I'd say it's sorta reduction
main frigate
u could also multiply the first equation by two and add the equations to each ot...
how does this work
sudden yacht
u could also multiply the first equation by two and add the equations to each ot...
But yeah this works perfectly as well
lone whale
the three ways work equally as well
main frigate
But yeah this works perfectly as well
I have other questions where i can use whatever method
sudden yacht
Whatever between which?
sudden yacht
I don't get you 😢
main frigate
Get y from the first equation and replace it in the second
i dont think this works first 1
buoyant yoke
i dont think this works first 1
it does
main frigate
i will get decimals
sudden yacht
i will get decimals
Well, you should work with fractions in maths, not decimals
buoyant yoke
recheck your calculations
sudden yacht
But even if you get decimals/fractions, where's the issue?
main frigate
okay so when i add -y = 3 - 2x to 2y - x = 9 what does it become
buoyant yoke
okay so when i add -y = 3 - 2x to 2y - x = 9 what does it become
why would you write it like this?
why not write it as y=2x-3
main frigate
cuz the y is negative
buoyant yoke
wdym?
main frigate
2x (-) y = 3 --> -2x to isolate y
sudden yacht
By the way, how do you know y is negative if you still haven't found its value?
main frigate
cuz its there
buoyant yoke
\begin{align*}2x-y&=3\+y&+y\\hline 2x&=3+y\-3&-3\\hline 2x-3&=y\end{align*}
main frigate
why does it become positive
sudden yacht
Because you have to isolate it
main frigate
what*?
grand pondBOT
Bonk
main frigate
oh right i didnt see my bad
but why do you add positives instead of just solving it normally
buoyant yoke
what?
buoyant yoke
i just isolate the y in the first equation
and then you plug that into the second equation
main frigate
Why do you have to make y positive to isolate it is what i meant
becuase it would be easier to just subtract 2x
buoyant yoke
i can also write $-y=-2x+3$, but thats not useful
grand pondBOT
Bonk
buoyant yoke
because we need to fill in positive y into the other equation
main frigate
why do you need to do that
buoyant yoke
the most useful form actually owuld be $2y=4x-6$
grand pondBOT
Bonk
buoyant yoke
look at the other equation $2y-x=9$
grand pondBOT
Bonk
buoyant yoke
there is a 2y
and we know 2y=4x-6
thus, we can substitute
$\left.\begin{aligned}2y-x&=9\2y&=4x-6\end{aligned}\right}\implies (4x-6)-x=9$
grand pondBOT
Bonk
buoyant yoke
make sense?
main frigate
what are we even doing can we quick restart
buoyant yoke
we are substituting
main frigate
right
main frigate
No I do know I just want to restart so I can see it from the begining
it looks too messy
HELLOOO
buoyant yoke
it looks too messy
how is it messy?
main frigate
I dont know where you started solving for 2y = 4x -6
main frigate
right
buoyant yoke
(write it as y=...)
buoyant yoke
yes
main frigate
so now we add it to the second equation right
buoyant yoke
now, fill that into 2y-x=9
main frigate
wait i need to write this down its a question from my mathbook
buoyant yoke
you werent writing along?
buoyant yoke
what did you get
main frigate
i got x= 5 y = 7
buoyant yoke
thats correct
main frigate
yee
thanks
can u show me the other method
you know where you multiply the first equation by 2
main frigate
???
buoyant yoke
its just changing in which step you do the *2
i genuinely have no clue how youre confused
main frigate
wasnt it some other method where you would combine both the equestions and make it to 1 solo equastion
main frigate
whyd he say it was diffret
main frigate
u could also multiply the first equation by two and add the equations to each ot...
i meant this
it wasnt u bruh
either way to you get it?
buoyant yoke
why didnt you say so then
main frigate
i dont understand
lone whale
que
buoyant yoke
i thoguht you were talking about mine
main frigate
i thought it was u
buoyant yoke
but its adding the equations
sudden yacht
Ohh that's completely another method, called reduction
buoyant yoke
you explicitly said with substitution
sudden yacht
The one of Bonk is called substitution
main frigate
yeah but it seemed intresting
sudden yacht
you explicitly said with substitution
Indeed 👍
sudden yacht
yeah but it seemed intresting
It is
lone whale
do u want me to explain it or r u covered
buoyant yoke
personally, i wouldnt ever solve this with substitution
main frigate
just ur method
main frigate
do u want me to explain it or r u covered
ur method
lone whale
ok so have you added equations before?
main frigate
personally, i wouldnt ever solve this with substitution
we only learnt the subsitution and with a graf
main frigate
ok so have you added equations before?
no i dont think so
is it like where you make everything = to y and combine them
sudden yacht
No no
lone whale
do u wanna explain?
sudden yacht
is it like where you make everything = to y and combine them
That's called comparison (at least in my country)
buoyant yoke
do u wanna explain?
you go ahead
main frigate
That's called comparison (at least in my country)
its better than the substituion method, my teacher does not let me use it
lone whale
right so in order to add equations together you need to choose a variable that will cancel out
sudden yacht
its better than the substituion method, my teacher does not let me use it
Very weird, but ok let's stick to them
lone whale
yall like
the side convo here
continuing, you have 2x-y =3 and 2y-x = 9
to get a variable to cross out, you need the coefficients to match but their signs to be opposite (ex 2y and -2y)
we already have 2y in the second equation, but not in the first
so in order to change -y to -2y, we multiply the entire equation by 2
2x-y=3 turns into 4x-2y=6
now we can put the equations on top of each other to add them together
lemme write it out so its easier to see
now we add each variable in columns, add the ys together, the x together, and the other side of the equation
the +2y and -2y cancel out, giving you 0
2x + (-x) = x
and 3+9 = 12
main frigate
wait wait wait what did you do with 4x - 2y = 6
lone whale
holy im tweaking
one sec
mb
(u were supposed to use the altered equation lol)
ok now u have 3x = 15, which solves to be x = 5
then you can plug x=5 into either equation to solve for y
main frigate
that was way easier
lone whale
ye i like it better too lol
main frigate
how does gpa work
lone whale
gpa is different depending on the school you go to
lone whale
only if trump was as half smart as u
lmao
main frigate
only if trump was as half smart as u
thanks
lone whale
in my school, gpa goes to a scale of 4.4 - A+ = 4.4, A= 4.0, A- = 3.7... etc
main frigate
when someones says weighted and unweighted what does that mean
lone whale
each letter grade in the courses you take corresponds with a number on your schools scale
weighted GPA means your gpa is affected by the rigour of your classes
if you take all honors and APs, you get more points by getting a good grade
unweighted is only based on your grades, not on which courses you take
and A in low math and A in high math would amount to the same
main frigate
so having a 4.4 weighted is best
lone whale
for my school, 4.4 is unweighted and corresponds with A+
yeah its for college mainly
your school might be different tho, youll wanna check out how your school does gpa
finite karma
so having a 4.4 weighted is best
where do u live
main frigate
how do you solve for your number
main frigate
where do u live
sweden
finite karma
how do you solve for your number
most countries goes for /20 or /100 scores idk why they use gpa on america
main frigate
I think its good
lone whale
how do you solve for your number
i only know unweighted cuz thats what my school does, you multiply the grade you got on GPA scale to the credits you earned, and add them all up/out of total possible credits earned
main frigate
and also why do you guys have so many extra clubs and things
main frigate
i only know unweighted cuz thats what my school does, you multiply the grade you...
📲
lone whale
cuz its not enough to get into college with just grades now lol
main frigate
ye ye 100%
lone whale
sadly
main frigate
and also why do you guys have so many extra clubs and things
this one
how long is a school day for ur grade
actually nvm
thanks for the help guys
.close
midnight plankBOT
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radiant roost
midnight plankBOT
radiant roost
i dont get this
dawn dagger
so their p = p1 and phi = mi1 so i guess you wan show that p o phi for example equals some element of p0, p1, ... m1, ...
radiant roost
it seems like rho is working differently in these two cases
yeah i can say rho * phi = rho1 * mu1 = mu3
radiant roost
but it sounds like i'm supposed to draw a picture?
radiant roost
i relabeled the vertices according to the permutation
but like rho is supposed to correspond to a rotation? but in the first case it's clockwise and in the second case it's counter-clockwise
is it supposed to be like that?
dawn dagger
i think you applied it wrong
applying p, which is p1 means 1 -> 2, 2 -> 3 and 3 -> 1 which would give you the rotation
like it should be
3
1 2
radiant roost
radiant roost
yeah
dawn dagger
now applying rho
would rotate it ccw
once
1 -> 2 and 2 -> 3 so 1 -> 3
i think writing it down phi o rho would help
dawn dagger
you have phi and apply rho so phi o rho
1 -> 2 and 2 -> 3 so 1 -> 3
2 -> 3 and 3 - > 2 so 2 -> 2
3 -> 1 and 1 -> 1 so 3 -> 1
wait this basically just swaps 1 and 3
radiant roost
rho o phi should swap 1 and 2 i think
dawn dagger
1 -> 1 and 1 -> 2 so 1 -> 2
2 -> 3 and 3 -> 1 so 2 -> 1
yea
radiant roost
dawn dagger
radiant roost
i'm confused because this is a group action but we haven't defined the action
dawn dagger
you mean what u do with rho and phi?
dawn dagger
i think the action is clearly applying one to the other, like rho o phi
you apply phi onto rho
radiant roost
this is what they say
dawn dagger
but it should be reversed actually phi o rho
radiant roost
the binary operation is composition
dawn dagger
yea
radiant roost
they said different authors use different orders for that
radiant roost
yeah it says WARNING: some texts compute a product sigma mu of permutations in left-to-right order, so that (sigma mu)(a) = mu(sigma(a))
dawn dagger
bruh
ok but then you still should end up showing it correctly
if you decide for one
oh wait so your rho * phi means phi(rho)
dawn dagger
ok i see what u mean lol
dawn dagger
i relabeled the vertices according to the permutation
but like rho is supposed t...
i think that's like it
i think this only works if you apply it with itself
that it resembles rotation
cobalt nest
whats the question? @radiant roost
radiant roost
like how do i answer this?
were any of the images i posted correct?
dawn dagger
like rho tells u to swap 1 with the 2, the 2 with the 3 and 3 with the 1
cobalt nest
eeeh, looks tedious
midnight plankBOT
dawn dagger
it said it here too
dawn dagger
yes
midnight plankBOT
Available help channel!
obtuse totem
hellooo
cobalt roost
hiii
cobalt roost
ok
buoyant yoke
!da2a
midnight plankBOT
No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/
obtuse totem
honestly I keep getting distracted
buoyant yoke
ideally you open a channel with a quesiton
buoyant yoke
so whyd you open a channel?
obtuse totem
omg hiii @tribal temple
buoyant yoke
you can open a channel when you have a question
obtuse totem
hii @dawn dagger
@tribal temple I did it lol
remember how I was rlly sad cause no one wanted to hire me
i got an internship position!!!
at a bank!!!
as my first internhsip
tribal temple
Awwww well done 
that’s amazing, it’s always a matter of time with these things, hope it goes well 
obtuse totem
I knowww
for the longest time I thought my major wasnt right for me
ngl I still have imposter syndrome
blissful trench
<#268882317391429632>
no access
blissful trench
yeah I have perma-studying for 5 months lol
buoyant yoke
whats the difference between perma-studying and studying
blissful trench
perma-studying is requested by the mods and studying is just self-role
buoyant yoke
you cant remove it?
blissful trench
you can get rid of it at anytime
yeah I cant lol
it will automatically gonna get removed
after 5 months
buoyant yoke
yeah thats what i meant
blissful trench
ok this is getting off-topic I should stop now 
dawn dagger
i'll prob have questions
whats 14 = x mod 7
buoyant yoke
i'll prob have questions
but then open a channel the moment you have a channel
obtuse totem
0
buoyant yoke
youre not supposed to claim a channel when you have no question
obtuse totem
ok here it talks about using one model
I guess they are considering 4 diff models, Im not sure whats happening tho, so eacg model has one feature?
i think so?
ok so just adding a new feature to create new model
and you select best one before adding each feature
midnight plankBOT
@obtuse totem Has your question been resolved?
midnight plankBOT
@obtuse totem Has your question been resolved?
obtuse totem
.close
midnight plankBOT
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fresh abyss
Can someone explain this change in indices?
midnight plankBOT
Can someone explain this change in indices?
fresh abyss
Also, I'm very terrible in general with these indices changes, does anyone have tips/tricks?
nova yoke
this equality is not true
try it with n=2
left hand side is 6, right hand side is 3
it would be true if it were $$n \sum_{k=0}^{n-1}\frac{1}{n-k} = n\sum_{k=1}^n \frac{1}{k}$$
grand pondBOT
Bungo
nova yoke
that's just summing the same terms in forward order vs reverse order
midnight plankBOT
@fresh abyss Has your question been resolved?
midnight plankBOT
@fresh abyss Has your question been resolved?
gaunt zinc
(Linear Algebra) Hate to be a bother, but could someone help me with (e), I'm not sure why it's being so confused.
Gimme a second to explain.
midnight plankBOT
(Linear Algebra) Hate to be a bother, but could someone help me with (e), I'm no...
gaunt zinc
Finding the dot-product. If we where to consider the vector y and z, they are of different dimensions, with y = 4 x 1 and z = 3 x 1, however, aren't we allowed to tranpose a vector to get the columns of vector y to be the same as the rows of vector z? such that y = 4 x 1 and z^T= 1 x 3?
Is that a legal thing we can do, and then we can find the dot-product with that, even when the dimensions are not the same?
rain wasp
the fact that the two vectors are of different dimension already means addition is not defined, let alone dot product
gaunt zinc
Yeah, I just realized that. I'm not sure why I got confused so much. Much love big man
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worthy vector
how do you simplify (1-x^2)(1/2 square root x)
midnight plankBOT
how do you simplify (1-x^2)(1/2 square root x)
tidal turret
rationalize, what I mean is, multiply sqrt(x) in the numerator and denominator of 1/(2sqrt(x)) to bring the square root to the numerator, then distribute
worthy vector
like this?
tidal turret
you added an extra 2sqrt(x) in the numerator next to the 1-x^2
should be (1-x^2)(2sqrt{x})/((2sqrt{x} • 2sqrt{x})
worthy vector
i dont understand how it is just times 2sqrt(x) because don't you multiply it by 1/2sqrt(x), or is it because it already has the same deminator so you only need to multiply the 1-x^2
tidal turret
this is multiplication is not addition. so for example
a/b • c/d = (a×c)/(b×d) even if the denominator is not a common multiple, we multiply whats in the numerator with what it is in the numerator and we multiply what it is in the denominator with what its in the denominator
tidal turret
i dont understand how it is just times 2sqrt(x) because don't you multiply it by...
also, use parenthesis when multiplying with (1-x^2) otherwise it seems like you are only multiplying x^2 with sqrt{x}
worthy vector
this is multiplication is not addition. so for example
a/b • c/d = (a×c)/(b×d) e...
ohh i understand it now, thank you so much!
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arctic crescent
Hey, unfortunately two transformations in the proof above are unclear to me. I suspect it's because the textbook I'm working with leaves out many steps and presents them in a compressed manner.
-
When estimating row 1 to 2, I don't understand why the first row member is multiplied by the other row members. Shouldn't it be added up?
-
In the 3rd line, the conversion to b^-m is a bit too quick for my understanding.
Maybe someone can give me a slightly more detailed explanation. Thank you very much
midnight plankBOT
Hey, unfortunately two transformations in the proof above are unclear to me. I s...
lyric charm
,rccw
grand pondBOT
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@arctic crescent Has your question been resolved?
arctic crescent
No
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@arctic crescent Has your question been resolved?
zinc leaf
solve xy''+y'=x
midnight plankBOT
solve xy''+y'=x
lyric charm
zinc leaf
lyric charm
ok first off the prime symbol is supposed to face the other way. you've got it more like a backtick
anyway u := y' is good
but then afterwards trying to integrate the thing as $\int \paren{-\frac{u}{x} + 1} \dd{x}$ will lead you nowhere
grand pondBOT
ann.in.a.teacup
lyric charm
actually you can notice that $xu' + u = [xu]'$
grand pondBOT
ann.in.a.teacup
zinc leaf
oh yeah.
lyric charm
so you get $[xu]' = x$
grand pondBOT
ann.in.a.teacup
lyric charm
thus $xu = \frac12 x^2 + C_1$
grand pondBOT
ann.in.a.teacup
zinc leaf
ohhhhhhhh
lyric charm
calling it C1 because you will also have a C2 from integrating to get back to y
(but divide by x first)
zinc leaf
mhm
finite karma
u can slo just transform it to u'+(1/x)u=1 for x!=0 which u can solve with the general formula
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zinc leaf
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zinc leaf
yes
finite karma
then take for x=0 u will get y'=0
which mean y=Constant
since u solved earlier for y in geenral form
if he asked u for a solution in the whole domain
u just stretch the function
meaning u take the limit for x=0 and see where y goes
if its not infinity
then its stretchable
since the condition for the general solution was x!=0
finite karma
yw
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zinc leaf
solve y'' + a^2/y^3 =0
midnight plankBOT
solve y'' + a^2/y^3 =0
dusty portal
solve y'' + a^2/y^3 =0
What have you tried?
zinc leaf
ohh no its fine dw dw
soz
wait sorry also for this question
you find the orthogal trajectories of the family of curves y^2 = 4cx
why do you have to replace dy/dx by -dx/dy
dusty portal
Since m=-1/m_1 if m_1 is a perpendicular slope
dusty portal
And vice versa
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wintry tinsel
I'm trying to figure out the oblique asymptote for x to -inf, which is equal to y=-2x-2, as confirmed by geogebra. However, I get y=-2x. I do it as follows: for a, I just take the limit for f(x)/x = 2|x|/x, meaning a=-2. For b though, I take the limit x->-inf of f(x)-x= f(x)+2x . if I calculate the limit of f(x), I end up at 2|x|. Plugging that in for b I get 2|x|-2x=0 , which would mean y=-2x for the asymptote.
midnight plankBOT
I'm trying to figure out the oblique asymptote for x to -inf, which is equal to ...
wintry tinsel
What am I missing?
wintry tinsel
f(x)-mx
I forgot the m in my explanation, you're right, but I did include it in my calculations (a)
cedar coral
show them
wintry tinsel
you mean limx->-inf f(x)? for x -> -inf, constants can be ignored so you get -(4x^3)/(-x) in the sqrt, simplifying to 4x². Taking the sqrt gives 2|x|
so for f(x)/x it would simply be = -2, which gives m in y=mx+b. for f(x)-mx that would give 2|x|+2x= -2x+2x= 0 for x=-inf?
<@&286206848099549185> sorry for bothering you all, I can't figure out my mistake
wind oxide
is log gamma of exponential type
midnight plankBOT
is log gamma of exponential type
wind oxide
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wind oxide
<@&286206848099549185>
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carmine sigil
By Stirling's approximation we have Γ(1+z) ≈ √(2πz) (z/e)^z
Taking log of both sides we get
ln(Γ(1+z)) ≈ ln(√(2πz)) + ln(z^z) - ln(e^z), which is asymptomatically log-linear in z. @wind oxide
wind oxide
By Stirling's approximation we have Γ(1+z) ≈ √(2πz) (z/e)^z
Taking log of both ...
meaning it isnt of exponential type right
carmine sigil
Only because it has poles. Otherwise it would be.
wind oxide
Only because it has poles. Otherwise it would be.
it doesnt have poles on re z > 0
so why do they matter
carmine sigil
Because |z| -> inf, not z -> inf
wind oxide
oh
wind oxide
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verbal pumice
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verbal pumice
For the first one I suppose I can just do something like:
$$A_1 = {a_{1,1}, a_{1,2}, a_{1,3}, \cdots }$$
$$A_2 = {a_{2, 1}, a_{2, 2}, a_{2, 3}, \cdots }$$
$$ \vdots$$
$$A_n = {a_{n, 1}, a_{n,2}, a_{n,3}, \cdots }$$
So then, define the set $A = A_1 \cup A_2 \cup A_3 \cup \cdots \cup A_n$:
$$A = {a_{1,1}, a_{2, 1}, a_{3, 1}, \cdots a_{n, 1}, a_{1, 2}, a_{2, 2}, a_{3, 2}, \cdots, a_{n, 2}, \cdots , a_{n,n} }$$
lyric charm
that ending is a bit troublesome
grand pondBOT
dyxn
lyric charm
it feels as if you are stopping at just the first n elements of the n'th set
also in general this notation is kind of treacherous
verbal pumice
Hmm
lyric charm
you should instead work explicitly with bijections
verbal pumice
How would I do that?
lyric charm
also you should definitely say that the A_i can be assumed disjoint, and argue why it can be assumed so
ok first
does your N start from 0 or from 1
verbal pumice
1
lyric charm
damn ok thats gonna be mildly annoying
whatever
ok so suppose all the $A_i$ are disjoint (we will figure out how to ensure this later on).
let the bijections $f_i : \bN \to A_i$ be given for each $i \in 1:n$, and also let $A := \bigcup_{i=1}^n A_i$.
grand pondBOT
ann.in.a.teacup
lyric charm
our goal will be to construct an explicit bijection f: N -> A
verbal pumice
Oh
lyric charm
your idea of taking the first elements from each set followed by the second elements from each set and so on (where e.g. the element we reckon as first in A_i is f_i(1), the second is f_i(2) etc.) is fruitful
verbal pumice
Interlacing is always powerful
lyric charm
so what you want is that 1, n+1, 2n+1, ... map to elements of A_1
2, n+2, 2n+2, ... map to elements of A_2
etc
verbal pumice
Wait isn't this literally Hilbert's hotel thing
lyric charm
it kind of is yes
it's how to accommodate a finite number of countably-infinite buses of guests
(taking the people already at the hotel as a bus in their own right and not otherwise giving them any special treatment)
verbal pumice
so what you want is that 1, n+1, 2n+1, ... map to elements of A_1
I can just say this right?
lyric charm
i do not know the level of formality expected of you here
verbal pumice
Fair
I guess I'll just say something like
For the $k$th element in $A_i$, let $f$ map $f_i(k)$ to $in + k$
grand pondBOT
dyxn
verbal pumice
works?
lyric charm
i... think you've mixed a few things up
verbal pumice
does it matter if its a bijection
lyric charm
of course it does...
simple field
you can have X injecting into N to show countability
verbal pumice
Assume $A_i$ for $i \in {1, 2, 3, \dots, n}$ are disjoint countably infinite sets. Thus, there exist bijects $f_i : \mathbb{N} \to A_i$ for each $i$. Define $A = \bigcup_{i=1}^n A_i$. Our goal is to show that $A$ is also countably infinite. \
Consider the function $f$ that maps the positive integer $k(n-1) + i$ to $f_i(k)$.
simple field
For the $k$th element in $A_i$, let $f$ map $f_i(k)$ to $in + k$
so something like this good with a couple tweaks
grand pondBOT
dyxn
verbal pumice
spacing 😵💫
Wait I'm having trouble convinving myself that this is a bijection
Okay wait nvm I was thinking of this wrongly
@lyric charm for the second part will it do to just let n --> infinity
Define a sequence of sets A_i
lyric charm
no
you do not "let n -> ∞"
you're not doing limits
that is not a thing you can do here
you need an actually radically different construction
verbal pumice
lyric charm
for this, i recommend first thinking about a bijection between N^2 and N.
simple field
the issue with the idea of letting n->inf is that you would never reach k > 0, i.e. you would be stuck only choosing the first elements of each set
verbal pumice
But n is still finite?
simple field
well as ann said you need a totally different appraoch
i would suspect you have seen the idea you need to use at some point in your lectures / textbook or however youre learning
if not its a bit cruel of a question but not impossible
so i would say go back over your notes see if anything clicks
simple field
From when I was proving the rationals are countable
ah missed this, this is where you should have seen the idea
verbal pumice
This isn't a regular class so there's actually no lecture textbook/notes
simple field
so go and have a look at that and see if you can adapt it to this
verbal pumice
It's complicated 😭
verbal pumice
so go and have a look at that and see if you can adapt it to this
But the problem with that was that
You have a finite number of elements in an infinite number of sets
This is infinite elements, infinite sets
simple field
You have a finite number of elements in an infinite number of sets
im not sure i follow ?
grand pondBOT
dyxn
verbal pumice
Which is essentially the diagonal elements
Each of these sets had finitely many elements
So you could just count them
It was the number of sets which was infinite which didn't really matter
But for this case it's the number of the elements in the set itself that's infinite 
verbal pumice
yeah I'm aware
simple field
sure nws
simple field
got it?
verbal pumice
I can regard each column as a set
simple field
yep !
verbal pumice
And traverse in the same way
simple field
perfect
verbal pumice
I actually am interlacing but in a different way
simple field
sometimes you just need to look at a picture haha
simple field
nice
verbal pumice
I only have one more question, for which I'll open a new channel
Thanks a lot!
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alpine cedar
Hello I am having a bit of a hard time proving this lemme
\textbf{Lemma 1: }If $\gcd(a,b) = 1$ then $\gcd(a,b^n) = 1$
midnight plankBOT
Hello I am having a bit of a hard time proving this lemme
\textbf{Lemma 1: }If...
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alpine cedar
can someone push me on to the right track
so far i tried contradcition but i feel like it doesnt work well
alpine cedar
what are you planning on studying @mortal sleet
mortal sleet
Albania 💔
alpine cedar
thats fair, its expensive to study abroad
mortal sleet
My long term plan is either europe or usa
alpine cedar
like western europe?
mortal sleet
germany switzerland and so
mortal sleet
Which country
alpine cedar
he worked in kosovo so he knows a tad bit of albanian
mortal sleet
Yooo actually?
alpine cedar
ion really remember, but i think he visited greece, macadeonia, and some other countries
yea, they speak albanian in kosovo no
alpine cedar
no
mortal sleet
Oh 😅
alpine cedar
its not high level
its an entry level proofs course
like discrete math, math for cs
my prof describes discrete math as the art of translating one thing into another thing
alpine cedar
its just saying the greater common denominator of a and b
mortal sleet
Oh
mortal sleet
Its seems simple to prove but lmao i cant imagine how to
alpine cedar
and you prove different things with it
alpine cedar
Its seems simple to prove but lmao i cant imagine how to
thats kinda the point of the course
mortal sleet
Yh
alpine cedar
even some of the simplier things have a pretty rigourous proof if you look at it
alpine quiver
so far i tried contradcition but i feel like it doesnt work well
what do you have so far
alpine cedar
what do you have so far
well i tried looking into proof by contradiction
and just direct proof
i have been playing with bezouts for a bit, i havent gotten anywhere tho
alpine quiver
ah idk sorry
alpine cedar
ah idk sorry
so you dont know how to help
alpine quiver
so you dont know how to help
no I was just seeing if I had an idea
mortal sleet
If gcd of 2 numbers is 1 that means they are prime between each other?
alpine cedar
If gcd of 2 numbers is 1 that means they are prime between each other?
yes they are co prime
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mortal sleet
@alpine cedar
If a,b are coprime
Would that also mean that a,b^n are also coprime
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last slate
Hi
polar star
hello
dawn dagger
greetings
last slate
Are these answers correct? Also I have no idea how to go about the last question
polar star
(-m)^2 ≠ -m^2
for part a
4*4 ≠ 8
your idea for part b and c is correct but fix part a to get the right answers
polar star
m^2-16m
for the last part, use the vertex formula
x = -b/2a
you will get the x value at which max height is achieved. then plug it into the quadratic to get the max height
last slate
you will get the x value at which max height is achieved. then plug it into the...
So is this right so far?
polar star
why do you have n in your equation