lol
#help-28
rare dock
that’s the only one
sterile torrent
are you sure that table is correct
Chat gpt but smn from my team already checked using python
stone meteor
and they got that?
sterile torrent
They did that for every n between 1 and 9
And it's all correct
We even checked manually until n=6
stone meteor
this is not making sense
did you mean A+B = (0,1,2,4,8,...,2^n)
your definition asks for pairs of sets that would add up to {0,1,2,3} for n =3
sterile torrent
no
i just rephrased the question to make the context more clear
sterile torrent
In case you want to check the original question its problem 4
A B
0 1 2 0 1
0 1 0 1 2
0 2 0 1
0 1 0 2
sterile torrent
Yes
stone meteor
and I have no idea how it corresponds to what you asked
sterile torrent
multiplying 2^m x 2^n is like adding m and n
stone meteor
maybe if you could translate the question in english I could help
fierce cedar
I landedddd
stone meteor
{0,1,2,3,4}
{0,1,2,3}, {0,1}
{0,1}, {0,1,2,3}
{0, 1, 2}, {0, 1, 2}
{0,1,3}, {0,1}
{0,1}, {0,1,3}
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@sterile torrent Has your question been resolved?
cold sail
been trying 2b for ages now
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been trying 2b for ages now
cold sail
im just lost
rustic frigate
What have u tried
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@cold sail Has your question been resolved?
cold sail
What have u tried
bunch of things which led nowhere
rustic frigate
It’s a bit obvious once you see it
cold sail
i think i realilsed something tho
ye i wanna tyr solving it myself
i just realoised that R is a subset of R- tho lol
im gonna try showing E\M is in R- then prolly
cold sail
It’s a bit obvious once you see it
ye ik ive been humbled hard by this module
🙏
wait
cold sail
im gonna try showing E\M is in R- then prolly
nvm this wont give me what i want
cold sail
It’s a bit obvious once you see it
ye i need a hint
so far (E\M) U (N\M) and i know N\M is null
i dont think this route goes anywhere tho
cz i cant show E\M is in R ( i think)
rustic frigate
cz i cant show E\M is in R ( i think)
Yes not possible, think through a counter example
Might be helpful
cold sail
Yes not possible, think through a counter example
uhhh
maybe i should sleep on it idk cz ive been trying since like before 2am
my brain is tired
lol
rustic frigate
No
cold sail
Do u think I should just try tmrow lowkey I wanna solve it tho
zenith kernel
(E cup N)-M = (E-(E cap M)) cup (N-(N cap M)). Clearly N-(N cap M) is null so you are trying to prove E-(E cap M) is in R?
rustic frigate
E\M must not lie in R
rustic frigate
Do u think I should just try tmrow lowkey I wanna solve it tho
Yeah
Good night rest is important
zenith kernel
Equivalent to E cap M being in R
cold sail
But I kinda have to I wanna ace these exams
zenith kernel
How though… E cap M in R… this null set definition, can it somehow make us write E cap M is countable union or intersection?
rustic frigate
Intersection
zenith kernel
E-E cap M is in R iff its complement , which is E^c cup (E cap M) is in R
cold sail
This is the first time I struggled so much on a module
zenith kernel
E^c is in R so we need E cap M is in R right?
rustic frigate
Take e.g. Lebesgue measure on [0,1]
Then pick your favourite non measurable null set
Let E = [0,1]
Notice that E\M is non measurable
And so is E cap M
So it does not at all need to be in R
zenith kernel
I can’t think of a null set this case. You have any?
rustic frigate
It’s non constructive
rustic frigate
What does lebesgue measure on [0, 1] mean
Measure is interval length
rustic frigate
For lebesgue measure that’s the definition
cold sail
Oh y we were given a defn with symmetric differences
zenith kernel
Lebesgue case it’s done by defining outer measure μ first then measurable sets are A satisfying μ(X cap A)+μ(X-A)=μ(X) for any out measurable X
zenith kernel
But yours is a general one not having outer measure things…
cold sail
Ye
rustic frigate
That’s a condition for a set to be measureable given any outer measure…
rustic frigate
But yours is a general one not having outer measure things…
??
cold sail
Ok so E\M
What do I do with this
I was thinking of stuff like sunsets but
*subsets
rustic frigate
All im saying so far is that you can’t show this is in R
cold sail
But I’m not sure how taking a subset will help me
Ye I get that
Doesn’t follow from sigma algebra axioms
Ok so
zenith kernel
I was suggesting trying to show that E cap M, which is null and a subset of a member of R, is in R. Aslan thinks it’s wrong but I haven’t come up with a counterexample following his idea.
cold sail
I know M is a null set which means there exists F in R with 0 measure and M is a subset of F
rustic frigate
I was suggesting trying to show that E cap M, which is null and a subset of a me...
I gave one
zenith kernel
That’s the thing. I can’t come up with a null set.
cold sail
And that (E\F) is a subset of (E\M)
zenith kernel
A set contained in any measure 0 set, doesn’t exist right? Only empty
rustic frigate
I said pick M to be any non measurable null set
This can only be done non-constructively
Using e.g. axiom of choice
zenith kernel
This null set definition, X is a null set iff X is contained in any measure zero set, like {r}. So X is empty in Lebesgue case
rustic frigate
Nope
cold sail
Fine I’ll go sleep lol
rustic frigate
But that doesn’t seem bad
zenith kernel
What’s your idea again? I also want to figure it out.
rustic frigate
I can give it away but I don’t think you’ll find it satisfying
cold sail
True
rustic frigate
It’s obvious once you see it
cold sail
I’m currently straddling this lane of
zenith kernel
A sketch maybe?
rustic frigate
And is not really illuminating
cold sail
I wanna solve t but I also can’t afford to waste too much time as I have other modules to revise for
rustic frigate
Sleep is also very important
cold sail
True
rustic frigate
Should be top priority
cold sail
- acing exams 🥶
My notes teaches measure theory so terribly
I only know lebesgue measure through the outer measure definition
It’s also cz this isn’t even a measure theory course it’s a functional analysis course with some measure theory in it
cold sail
50x better than me 😭
rustic frigate
No?
cold sail
My functional analysis course is kinda light too
rustic frigate
You’re doing well
By getting a well night sleep you’ll feel and perform alot better
cold sail
Prob I don’t think I’ll still be able to solve it tho
But I’ll try
I need to catch up on complex analysis I’m so behind on that as well
Luckily for me complex analysis is kinda trivial
Compared to functional analysis anyways
rustic frigate
Yes I’m struggling with functional analysis
cold sail
It’s just like
How do u just think of a Cauchy sequence of functions that does not converge in the sup norm
😭
rustic frigate
Once we started with Hahn-Banach and duals, weak-topologies I started getting overwhelmed
zenith kernel
How about I find a null set M’ containing E cap M, such that E-M’ is measurable?
By definition we have this M’, just that F in the definition “for some F”, right?
zenith kernel
This is the union:
E- (E cap F)
union
(E cap F) - (E cap M)
union
N-E-M
Measurable null null
null cup null is still null by a)
rustic frigate
And that (E\F) is a subset of (E\M)
This is the right approach btw
Anyways good night!
zenith kernel
Thanks. Didn’t think of that F.
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@cold sail Has your question been resolved?
solemn fable
Hello?
drifting summit
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drifting summit
can someone fully work out the algebra for this 😭
zenith kernel
Which part you don’t understand?
It seems very clear, on one hand the residue equals the coefficient a_ -1 of the Laurent series, on the other hand by Residue theorem it equals the integral in the bottom
.
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@drifting summit Has your question been resolved?
zenith kernel
Or your issue is not understanding why we come up the other integral?
We let w=1/z, f(w)dw=f(1/z)d(1/z)=-(1/z^2)f(1/z)dz so the original integral over C clockwise become (1/z^2)f(1/z) integral over C’=circle of radius 1/3 anti clockwise. That’s why we got that new integral
C’ is of radius 1/3 and it has only one singular point 0 inside it
drifting summit
@zenith kernel getting 1/z^2f(1/z) = 1/z(-3/2 + a1z + ...)
zenith kernel
Go on, what is the issue
(5 singularities in C, that’s why we change into integral on C’ where only one singularity 0 is inside C’ by the way, by w=1/z substitute)
If you prefer I can calculate directly for you without this substitute, and obtain the same result
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@drifting summit Has your question been resolved?
zenith kernel
Anyway it will be calculating the sum of value of f(z)(z-a) plugging in z=a, like -4/3+Σ_i (s_i ^3-3/2)/(s_i+1)Π(s_i-s_j) for j !=i, where s_1,…,s_4 are four roots of z^4+1/2=0, then multiply by 2πi. i am sure it will be the same result, but ton of calculations involved
$\frac{4}{3}+\sum_{i=1}^{4}\frac{s_{i}^{3}-\frac{3}{2}}{(1+s_{i})\prod_{j \neq i}(s_{i}-s_{j})}$
glossy valveBOT
Cogwheels of the mind
dawn owl
how did they find A^(-1) ? i mean ik how to find it by RREF with identity matrix, but is there a shortcut of doing so?
zenith kernel
You know A^-1=(1/det(A))(the adjoint matrix of A)?
2 by 2 case, A=(a,b;c,d). The adjoint of A is simply (d,-b; -c,a)
dawn owl
You know A^-1=(1/det(A))(the adjoint matrix of A)?
nah i didnt know that, all ik is that det(A^-1)=1/(det(A))
smth like that
zenith kernel
Then you need to know adjoint matrix. Laplace expansion in the case of one row/one column gives you A(adjoint of A)=det(A)I
dawn owl
Then you need to know adjoint matrix. Laplace expansion in the case of one row/o...
yep youre right thx
but i mean i could have done this the long way ig
but youre right anyways, i should learn that
zenith kernel
If you just want to solve your problem, compute directly that (a,b;c,d)(d,-b;-c,a)=(ad-bc,0;0,ad-bc)
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low violet
help
low violet
heres my question:
Mass if Box A: Mass of box B = 4:7
The mass of box B is 2.4kg more than the mass of box A. Calculate the mass of box A and the mass of box B.
and
chrome dew
**Mass if Box A: Mass of box B = 4:7
The mass of box B is 2.4kg more than the ma...
Try forming two equations and solving them simultaneously
low violet
Q9: a group of 12 adults and 9 children travel on a bus. The cost of an adult ticket is n$. The cost of the child ticket is $(n-10). The total cost of the ticket is $277.50. Find the cost of one adult ticket.
chrome dew
**Q9: a group of 12 adults and 9 children travel on a bus. The cost of an adult ...
Try and form an equation out of this as well
low violet
how?
chrome dew
Which one do you want to start with
onyx glen
!1q
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low violet
4/11 x b+2.4
fallen salmon
**Q9: a group of 12 adults and 9 children travel on a bus. The cost of an adult ...
this is the stuff that makes me think a lot 🧠
low violet
4/11 x b+2.4
do i do it like this?
chrome dew
Not quite 4/11
chrome dew
Think about it like this: if Box A is 4kg then Box B must be 7kg right
So whatever equation you come up with that relates the two, plugging in these numbers should work
You can also rewrite the ratios as fractions if that helps
low violet
ok
chrome dew
Did you manage to get the equations yet?
chrome dew
You know how the ratio is 4:7
Like 4 parts to 7 parts
Here x represents the value of one 'part'
This is another way to do the question but not the one I was describing earlier
low violet
11 is total ratio
low violet
This is another way to do the question but not the one I was describing earlier
could u tell em how to do it?
chrome dew
You want me to ask copilot to do it??
low violet
?
chrome dew
could u tell em how to do it?
Did you mean "me" instead of "em"?
chrome dew
Oh ok ok
low violet
i have an exam tmrw
chrome dew
Let's say m_a and m_b represent the mass of box A and B respectively
If we rewrite the ratio as a fraction then we get m_a/m_b = 4/7
low violet
than what?
chrome dew
Then the second piece of information can be written as m_b = 2.4+m_a
low violet
execution method?
chrome dew
Did you mean elimination?
chrome dew
Yeah you could do that but I think substitution method is easier here
low violet
could u make the simultanious any easier fo rme to learn cuz that is so f'd up for me
chrome dew
wdym
low violet
ok nvm'
low violet
**Q9: a group of 12 adults and 9 children travel on a bus. The cost of an adult ...
wb this one
chrome dew
Ok let's try break it down
If the cost of an adult ticket is $n then how much does it cost for 12 adults?
low violet
12n
chrome dew
Yes good
If the cost of a child ticket is $(n-10) then how much does it cost for 9 children?
low violet
9(n-10)
chrome dew
Good good
low violet
wat...
chrome dew
So then what's the total cost in terms of n
low violet
ok ok
low violet
So then what's the total cost in terms of n
12n + 9(n-10)
chrome dew
Type ".close"
low violet
.close
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low violet
no u guys have to do it
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low violet
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dawn owl
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dawn owl
any hint for solving it will be cool
brittle sun
Well you could just substitute the two vectors into AX = b
And then sub the third and use information from the first two subs
dawn owl
Well you could just substitute the two vectors into AX = b
but can i just make A and b as i want to ?
subtitue only gives A*(5,0,1)=b for example
fast peak
can you find a solution to Ax=0 ?
dawn owl
can you find a solution to Ax=0 ?
the only thing ik about A (m*n) is that n=3
and perhaps m>3
dawn owl
what can you do with those
Ax=Ay
zenith kernel
{x: Ax=b} is a linear space of {x: Ax=0} plus some vector. So Ax_1=b=Ax_2, you can make x_3-x_1 and x_2-x_1 in that same null space. And this null space you already know a one dimensional subspace of it so…
dawn owl
{x: Ax=b} is a linear space of {x: Ax=0} plus some vector. So Ax_1=b=Ax_2, you c...
@fast peak linear space isnt included in the upcoming test, but i think that those 3 sols which gives b are from the same plane
fast peak
yes they are but I deliberately did not use those words
Ax=Ay
so A(x-y)=0
what can you say about A(x+lambda(x-y)) ?
zenith kernel
x_3-x_1 is in that space, the natural way is to let it in F(x_2-x_1) I meant. But anyway I don’t know how to phrase it without mentioning linear space
dawn owl
what can you say about A(x+lambda(x-y)) ?
they dont teach us how lambda is related to linear algebra
dawn owl
what can you say about A(x+lambda(x-y)) ?
ik that if v1 and v2 are vectors so v1 + t(v2-v1) is a line
if thats the direction of what youre sayin
fast peak
I just want you to multiply it out
dawn owl
i aint sure
dawn owl
I just want you to multiply it out
i mean its A(18x-17y)
but im trying to understand your stance
ive found how to get the answer , thx
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torn jolt
What in the wee man?! HELP PLS😰
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What in the wee man?! HELP PLS😰
torn jolt
trigonometry is my weak point too
It’s fun when you get it😃🥲 but when you don’t👹
torn jolt
on the line of 2tan/1-cos^2delta is where I messed up after
Cause idk what to do next and I assumed a bunch of stuff then it made it look all funky
burnt grotto
check line 5
torn jolt
check line 5
Yeahh what’s it suppose to be?
sudden chasm
What in the wee man?! HELP PLS😰
after step 4
1-cos² = sin²
foggy vapor
What in the wee man?! HELP PLS😰
Here you said $\cos^2(\theta) = 1 - \sin(\theta)$ seemingly?
glossy valveBOT
torn jolt
Here you said $\cos^2(\theta) = 1 - \sin(\theta)$ seemingly?
Ahhh ok will do that merci🙏
ancient zinc
This is the error that makes the rest wrong
torn jolt
Am I on the right track now?
burnt grotto
why’d you undo the thing
you get 2/sinxcosx
that’s exactly what they ask for
one of the sinx cancels
torn jolt
😅I confused
burnt grotto
from 2sinx/cosx whole divided by sin^2x
the sinx in the numerator and denominator cancels
and the cosx goes into the denominator
so you get 2/sinxcosx
torn jolt
No burp I need to equate the denominator to have a sin at the bottom
burnt grotto
No burp I need to equate the denominator to have a sin at the bottom
it has sinx times sinx
torn jolt
No it’s squared
burnt grotto
sin^2x = sinx times sinx?
queen flame
No it’s squared
a^2 = a*a
glossy valveBOT
BKB
burnt grotto
$tan(x)=\frac{sin(x)}{cos(x)}$ ok?
glossy valveBOT
BKB
torn jolt
you with me?
Yes
burnt grotto
$2\frac{\frac{sin(x)}{cos(x)}}{sin^2(x)}$
torn jolt
Is sin over cos equals to 1? But my teacher didn’t teach us that for AS
burnt grotto
Is sin over cos equals to 1? But my teacher didn’t teach us that for AS
sim over cos is tan
glossy valveBOT
BKB
slate violet
and that's in AS
burnt grotto
so now we split $sin^2(x) = sin(x)\cdot sin(x)$
burnt grotto
The bottom can be 1-cos squared delta
that’s gonna ruin the purpose
torn jolt
Ok…
glossy valveBOT
BKB
glossy valveBOT
BKB
burnt grotto
the answer is right in front of you now
torn jolt
Sin x means cos times tan?
burnt grotto
yeah but that identity doesn’t help herr
burnt grotto
**BKB**
@torn jolt did you understand this part?
if you did then congratulations, you are one step away from the answer
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austere cove
From Dummit and Foote:
Let $N \trianglelefteq G$. If $\overline{H} \leq G / N$ is a subgroup of $G / N$, the complete preimage of $\overline{H}$ in $G$ is the preimage of $\overline{H}$ under the natural projection homomorpishm.
As an exercise, to understand the definition, I attempted to construct a very simple example.
Let $G$ be $Z_8$, and let $N$ be $Z_2$, then $G/N = {{0, 1}, {2, 3}, {4, 5}, {6, 7}} \cong Z_4$. If we have $\overline{H} = Z_2 \leq Z_4$ then $H = {{0,1}, {4,5}}$ so its preimage in $G$ would be ${0,1,4,5}$.
Is this accurate? This would seem to imply that $\overline{H}$ is not a subgroup in $G$. Is this expected? What is the complete preimage used for? (Why is it complete?) All of the references to this concept online seem connected to Dummit and Foote, is this concept known by a different name in other texts?
glossy valveBOT
OmnipotentEntity
brittle sun
The way you've written G/N feels wrong
Like I get what you're trying to say but why is G/N = 0,1,.., 7
brittle sun
Oh lmao
fast peak
but something is wrong with those cosets
brittle sun
There is
fast peak
so N={0,2,4,6} ?
brittle sun
You can't just take Z_2
fast peak
or what do we mean by Z_2
brittle sun
You have to take a subset of G that's isomorphic to Z_2
fast peak
or do you want N={0,4} ?
fast peak
in the first case you would have G/N = {{0,2,4,6}, {1,3,5,7}}
in the other you would have G/N = {{0,4}, {1,5},...}
austere cove
oh my bad
brittle sun
From Dummit and Foote:
Let $N \trianglelefteq G$. If $\overline{H} \leq G / N$ ...
So the problem in what you've done is
You don't mod just by groups
You mod by subgroups isomorphic to those groups
austere cove
right
So I would have $G/N = { {0,4 }, {1, 5}, {2, 6}, {3, 7 } }$, thus $\overline{H} = { {0,4 }, {2, 6} }$, which is a subgroup of $G$, so that's cleared up.
glossy valveBOT
OmnipotentEntity
austere cove
well, it's a subgroup of G when unboxed.
ancient solar
btw, it's nice to know that the preimage of a subgroup under a homomorphism is always a subgroup
austere cove
That's something I figured that should be true, which is why I was confused.
brittle sun
well, it's a subgroup of G when unboxed.
When you take the preimage yes
austere cove
so, what's the motivation behind defining a "complete" preimage, and why is it considered complete? And does it have any other name? I grepped my entire digital textbook collection, and thumbed through about half a dozen abstract algebra books that I own, and I can't find another reference to it.
uncut cedar
When you take the preimage yes
off topic but pre-image is the domain of a function no
austere cove
it's a subset of the domain of a function.
brittle sun
so, what's the motivation behind defining a "complete" preimage, and why is it c...
I have never heard of this concept
uncut cedar
oh I see
brittle sun
The fact that it's in DnF feels weird as fuck
austere cove
lol
brittle sun
Complete seems to mean maximal in this context
"This is the set of elements that would map to H under the quotient"
Which, sure, but I don't see why we need a separate name for this cuz it doesn't seem like it'll ever show up again
ancient solar
yeah, I have also never heard of this term, I feel like there's no need to have a dedicated term for this
austere cove
ok, so the concept is uncommon enough that it only has one name that one author gave it, and it never really caught on because it isn't generally useful.
Thanks so much! 😄
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red rampart
Can someone give me some concrete directions on how to determine whether this expression is always greater than or equal to 0 for π/4<x<π/2? I’ve tried substituting t=tan(x/2) and also using derivatives, but I still can’t find a solution to this problem
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Can someone give me some concrete directions on how to determine whether this ex...
@red rampart Has your question been resolved?
red rampart
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lucid cairn
hello why is this wrong
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hello why is this wrong
lucid cairn
i looked at this
robust slate
$(a^b)^c=a^{bc}$, not $(a^b)^{ac}$
glossy valveBOT
Civil Service Pigeon
robust slate
idk what happened at the end tbh
lucid cairn
idk what happened at the end tbh
i did 2 squared which is 4
so 2 to the power of 2+6=8
robust slate
**Civil Service Pigeon**
even if 4^6 were right, the same exponent rule still applies ^
lucid cairn
even if 4^6 were right, the same exponent rule still applies ^
idk what is goin on
blissful sail
i did 2 squared which is 4
You don't need to do 2 squared
delicate torrent
hello why is this wrong
you mean by doing $4^6 = (2^2)^6 = 2^{2+6} = 2^8$?
glossy valveBOT
1 divided by 0 equals Infinity
blissful sail
Just multiply the outer most power with the one inside
lucid cairn
**1 divided by 0 equals Infinity**
yes yes
delicate torrent
hello why is this wrong
$(2^3)^2 = 2^6$, not $4^6$
glossy valveBOT
1 divided by 0 equals Infinity
delicate torrent
@lucid cairn
bitter star
i did 2 squared which is 4
$(2^3)^2=(2^2)^3$ so it should become $4^3$
glossy valveBOT
Yash_AR
delicate torrent
**1 divided by 0 equals Infinity**
and even if until $4^6$ is correct, this is still wrong because you didn't apply the exponent rules here
glossy valveBOT
1 divided by 0 equals Infinity
lucid cairn
why dont i power the 2 aswell
bitter star
**Yash\_AR**
^this, cause you already evalutated 2^2 which is 4 you dont need to put it again
blissful sail
in here they power the 2 aswell
They are two different cases
lucid cairn
wdm
blissful sail
In the problem you originally posted, there was (2^3)^2
delicate torrent
in here they power the 2 aswell
$(ab)^c = a^c \cdot b^c$
glossy valveBOT
1 divided by 0 equals Infinity
blissful sail
Yeah this
lucid cairn
In the problem you originally posted, there was (2^3)^2
yh
delicate torrent
hello why is this wrong
And this one is $(a^b)^c = a^{bc}$
glossy valveBOT
1 divided by 0 equals Infinity
delicate torrent
These are 2 different things
delicate torrent
**1 divided by 0 equals Infinity**
This one is $b$ in the product
glossy valveBOT
1 divided by 0 equals Infinity
delicate torrent
**1 divided by 0 equals Infinity**
And this one is $b$ in the exponent
glossy valveBOT
1 divided by 0 equals Infinity
blissful sail
yh
But in the second case there were two terms multiplied (2 and x^2) and the whole thing was raised to the power 3, so the power gets distributed to the multiplied terms
lucid cairn
ok
so whats the rule here
if its 2 stuff inside the bracket then u power both
but 1 stuff inside the brackter you power it once?
like this?
lucid cairn
yh
glossy valveBOT
1 divided by 0 equals Infinity
delicate torrent
And according to the definition, what's $(a^b)^c$?
glossy valveBOT
1 divided by 0 equals Infinity
lucid cairn
a to the power bc
glossy valveBOT
1 divided by 0 equals Infinity
lucid cairn
yh
delicate torrent
And because $a^b$ is just $a$ multiplied $b$ times
glossy valveBOT
1 divided by 0 equals Infinity
delicate torrent
How many $a$s are multipled together in $(a^b)^c$?
glossy valveBOT
1 divided by 0 equals Infinity
lucid cairn
a^b multiplied c times
delicate torrent
a^b multiplied c times
$a$s, not $a^b$s
glossy valveBOT
1 divided by 0 equals Infinity
lucid cairn
a is beinb multiplied b times
delicate torrent
a is beinb multiplied b times
And that thing is multiplied $c$ times
glossy valveBOT
1 divided by 0 equals Infinity
lucid cairn
yh
delicate torrent
So it's just $a$ multiplied $bc$ times
glossy valveBOT
1 divided by 0 equals Infinity
lucid cairn
yh ok
so in this case
2 is being mutiplied 3 times which is being multiplied 2 times
yh?
delicate torrent
Which case?
lucid cairn
Which case?
delicate torrent
Yup
lucid cairn
is there anything else i need to understand
for this
so how about this
x is being multiplied 3 times
and 2 is being powerd by 3 times?
blissful sail
x is being multiplied 3 times
Inside the cube, x is first multipled 2 times (cuz squared) and two was multiplied 1 time
But when you cube the whole thing, the cube gets distributed to both 2 and x^2
So you get $2^3 \times (x^2)^3$
glossy valveBOT
Ishmam
lucid cairn
aaa yes
lucid cairn
yh thanks
blissful sail
And as you've seen, for x, we multiply the powers
lucid cairn
got it
blissful sail
Nice
lucid cairn
.close
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boreal flare
Une fonction (f) est majorée sur un intervalle (I) s’il existe un réel (M) tel que pour tout (x) dans (I), (f(x)\le M). Une fonction (f) est minoreée sur un intervalle (I) s’il existe un réel (m) tel que pour tout (x) dans (I), (f(x)\ge m). Une fonction est bornée sur un intervalle (I) si elle est à la fois majorée et minorée sur (I)
glossy valveBOT
StellarPhoton
onyx glen
ok well ig here is a translation
let f : I -> R be a function. we say that:
- f is bounded above on the interval I if there exists a real number M such that for all x ∈ I we have f(x) ≤ M
- f is bounded below on the interval I if there exists a real number m such that for all x ∈ I we have f(x) ≥ m
if your function is bounded above AND bouned below, then we just say it is bounded
boreal flare
ty, can u give me an example
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fallen jasper
Really hard
noble pivot
this just looks like tedious work, it doesnt look like any real math skills are being put to use
you just need to brute force it
narrow path
this just looks like tedious work, it doesnt look like any real math skills are ...
there is a diophantine equation hidden in the amount of ways you can partition it
but that seems like about it to me
noble pivot
i'd do this with code, trial and error would take literal hours, but if you can only trial and error then i'd split the binary question into smaller parts
narrow path
there is quite literally 4,450,838 possible guesses
but the search space could be lowered heuristically
if you create standards for what could be a potential string of words
are you allowed to use code @fallen jasper
cause if you arent thats just screwed
torn sand
Can you paste the binary string here
fallen jasper
are you allowed to use code <@730786238105256056>
nope
noble pivot
the way i'd do this manually is to just to find every possible letter for the first 5 characters
then i move one step and find all possible letts for the next 5 characters
its very tedious but its the only way i can think of
narrow path
then i move one step and find all possible letts for the next 5 characters
That has to be worse than exponential im not even able to conceive of how long that would take but maybe its possible since its obvious it wont be an x at the start
raw kiln
the best unc meme
noble pivot
That has to be worse than exponential im not even able to conceive of how long t...
it'd probably take several hours
you dont have to count duplicates within the string of 5 as well since you shift by 1 character at a time, which helps i guess
either way, chatgpt says that the answer is: "questionwhatisoneplusone"
just lie and say you bruteforced it
narrow path
either way, chatgpt says that the answer is: "questionwhatisoneplusone"
!nogpt
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either way, chatgpt says that the answer is: "questionwhatisoneplusone"
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
narrow path
but can we get a fact check on this
noble pivot
this is barely math tho
narrow path
this is barely math tho
This is in the domain of math where else would it be lmao
fuh naw open your own channel bruh
noble pivot
This is in the domain of math where else would it be lmao
i guess but its like saying that counting to 10000 manually is maths
torn sand
i would suggest enumerating a wordlist would be more feasible than enumerating all possible letter combinations
but code isn't allowed, so... 🤷
reef gale
Hi
torn sand
either way, chatgpt says that the answer is: "questionwhatisoneplusone"
q doesn't even start the binary string...
noble pivot
q doesn't even start the binary string...
the question was asked at the end according to it
torn sand
actually?
noble pivot
either way, your best bet would to take the first 5 binaries, write down all the possible characters it can be (from left to right, no duplicates), then shift by 1 binary and repeat about 300 times
torn sand
"qu" strat might work lol
noble pivot
actually?
sthisismurydsvineltpufverythoughtitwouldhelpstalnnlyymplequestionwhatisoneplusone
is what chatgpt got after 6 minutes of thinking lol
torn sand
using the fact that u always appears after q in english
noble pivot
using the fact that u always appears after q in english
oo didnt know that
narrow path
if you just spam rules like that it could help atleast
but the english language is notoriously stupid
torn sand
anyways i tried searching for common words like 'and' and 'the' but came up with nothing
noble pivot
if the teacher assigned you this i'd suggest just asking them to solve it
they seem to know something we dont
narrow path
Its funny cause even computing the amount of solutions is stupidly difficult cause its like the coefficient of some weird generating function (unless you can think of an easier way)
narrow path
please close ticket <@730786238105256056>
<@&268886789983436800> this guy should just be banned from help channels
LETS GOO TROPOSPHERE
torn sand
how do you think you could extend this concept to the other letters?
on second thought, i dont think you can
marsh vault
please close ticket <@730786238105256056>
Apparently the timeout you just came back from was not long enough. Fixing that now.
noble pivot
on second thought, i dont think you can
o and p only
torn sand
o and p only
no because o could be ag
noble pivot
no because o could be ag
iforgotabouttdatdoe
narrow path
iforgotabouttdatdoe
omg bro solved it
noble pivot
this seems like a better problem for coding than for math
torn sand
another idea i have: questions usually use 5W1H -- who what when why where, and how.
you could try searching for those
narrow path
I thought of a computational approach which will probably work but its maybe a bit unsatisfactory due to it's involvement with code you just throw each possible solution into an NLP and perplexity score them and then just manually scan the top ranked ones
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reef crystal
Is A1 correct?
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Is A1 correct?
reef crystal
forest trail
Hi! In case 2, how did you know that a^(-k) * a^(k + r) was equal to a^r?
forest trail
I think there is a way to prove this by induction without all of those cases. You should just prove that a^m * a^n = a^(m + n) by induction on n (keeping m fixed arbitrarily, you don't want to say something like m = -k because that overcomplicates things). Then repeat the same thing but prove that a^m * a^(-n) = a^(m - n).
Unless that was literally how the question was structured, in that case 🤷
prime pier
depending on where you start your induction, you may need to add n=0, etc., but that will blow it clean out the water and allow you to prove other exponent laws easier too
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verbal sky
not sure how to do b
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not sure how to do b
verbal sky
dont understand how this is true
lime ether
dont understand how this is true
remainder theorem
wide sundial
What does it mean if f(x) divided by (x+p) has remainder 9
sacred yarrow
@verbal sky Agree with what frosst mentioned, you should try to find the equation describing
f(x) divided by (x+p) has remainder 9
Here’s a simpler example
When 5 is divided by 3, the quotient is 1 and the remainder is 2
Try to find the relationship between these four numbers
verbal sky
Try to find the relationship between these four numbers
2 = 5 - 3
lime ether
5(1)
wide sundial
What does it mean if f(x) divided by (x+p) has remainder 9
Now apply the same thing to this statement
verbal sky
9 = f(x) - (x+p) ?
sacred yarrow
why is this important
Because you need to include the quotient
The quotient is not always 1
It makes things much clearer in a word
verbal sky
.close
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sacred yarrow
Has your doubt been resolved? @verbal sky
verbal sky
Has your doubt been resolved? <@1417933317537398824>
yeah i asked my teach
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zenith haven
hello
zenith haven
I am having an electrostatic quiz which I already solved but I want to check my steps. May anyone help me ?
neon basin
May we see the steps
zenith haven
May we see the steps
ofc one sec
neon basin
Thanks!
zenith haven
These are my steps I'd really appreciate your time to help
I think there is a mistake in region 1
that is my main concern actually
if u check here it says the outer shell is grounded.. should we have Electric field and Electric Potential in that first region as zero because of that ??
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@zenith haven Has your question been resolved?
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@zenith haven Has your question been resolved?
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reef crystal
Is A1 correct?
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@reef crystal Has your question been resolved?
rough tundra
Is A1 correct?
Sure, look like the standard proof and I don't see any glaring algebra mistakes
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candid hazel
not a command
mb
onyx glen
anyway, progress?
candid hazel
not a command
hello ann long time no see
onyx glen
hello. i don't remember you.
candid hazel
hello. i don't remember you.
onyx glen
glossy valveBOT
candid hazel
i tried to change it to this
but idk it's validity
or what to do after this
<@&286206848099549185>
blissful sail
Is that an s?
wise tartan
this is kind of ambiguous
did you mean $\sum_{s=0}^n {n \choose r} \left(\sum_{r=0}^s {n \choose s}\right)$
glossy valveBOT
CherryMan
wise tartan
i mean yeah thats the same thing
wise tartan
i tried to change it to this
so what have you tried after this?
queen flame
<@&286206848099549185>
u ain't tough (jk)
candid hazel
Is that an s?
candid hazel
this is the actual question
candid hazel
and i have turned it to this but idk if it is correct
wise tartan
its correct
candid hazel
hmm
wise tartan
but what did you try after
blissful sail
<a:intenseNod:1201182468666048512>
Just a suggestion: please don't write s like that otherwise you might mistake it for an 8 when you are in a hurry
candid hazel
Just a suggestion: please don't write s like that otherwise you might mistake it...
ok
wise tartan
wait wait
candid hazel
but what did you try after
well i take out the nCs from the first summation
wise tartan
it says r<s here
wise tartan
yeah the upper limit is s-1
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gilded forum
Help? I can’t multiply by nothing…
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Help? I can’t multiply by nothing…
sacred yarrow
@tall onyx
fast peak
wdym you cant multiply by nothing
scarlet pike
Perhaps can't solve by elimination
gilded forum
wdym you cant multiply by nothing
How do you get -y to -3y
scarlet pike
So just use substitution
scarlet pike
It does
fast peak
multiply by 3?
scarlet pike
Given the system has a solution of some kind
gilded forum
Something like this
cerulean trellis
Something like this
i think you should multiply the second equation by 3? so that y cancels out and you have the value of x
scarlet pike
Yeah elimination is much easier in this case
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gilded forum
,w 1/3 * 3/2
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,w 1/3 * 3/2
glossy valveBOT
onyx glen
,w 1/3 * 3/2
gilded forum
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royal dew
find the bounded area
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find the bounded area
royal dew
,w integrate 2-1/(sqrt(1-x^2)) from x = -sqrt(3)/2 to x = sqrt(3)/2
glossy valveBOT
glossy valveBOT
royal dew
IS THIS CORRECT GUYS
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royal dew
<@&286206848099549185>
queen flame
yeh looks good
royal dew
.close
daring patrol
If sin(θ+Φ)=2sin(θ-Φ)
prove that tanθ=3tanΦ
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If sin(θ+Φ)=2sin(θ-Φ)
prove that tanθ=3tanΦ
daring patrol
I am assuming expand and solve it but I am not sure ...
burnt turret
I am assuming expand and solve it but I am not sure ...
Did you try?
glacial stream
If sin(θ+Φ)=2sin(θ-Φ)
prove that tanθ=3tanΦ
!status
daring patrol
Did you try?
I did it rn but didn't prove it correctly
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If sin(θ+Φ)=2sin(θ-Φ)
prove that tanθ=3tanΦ
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
!showyourwork
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Show your work, and if possible, explain where you are stuck.
daring patrol
Oh wait , just solved it correctly
Just a small mistake in calculation
Ty!
.close
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left moon
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left moon
Could someone please check my work? There is no answer for me to check with in the back of the textbook
rare dock
,w partial fraction 3/(x^4+x)
glossy valveBOT
rare dock
yes it looks right
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digital arch
How do I do the equal volume part
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How do I do the equal volume part
hoary ember
I already did ts
integrate from 0 to b, and find b so the value is half of what you got here
digital arch
is it 512^1/5
hoary ember
yeah
digital void
Hello. so for δ, i had to express a sentence using "implies" into a sentence using only "or" or "not". is this the right way to do it? i could definitely do the entire board thing too and simplify it but our prof asked only for different expression, not simplified ones
digital void
,rccw
glossy valveBOT
foggy vapor
Its you 
foggy vapor
Hello. so for δ, i had to express a sentence using "implies" into a sentence usi...
This seems fine to me for what is being asked, tbh
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maiden vapor
You’re welcome fijo
foggy vapor
You’re welcome fijo
Blud thinks he did something
nimble crane
maiden vapor
Blud thinks he did something
My presence alone was enough
foggy vapor
My presence alone was enough
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snow mural
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queen flame
<@&286206848099549185>
!15m
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Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
forest trail
Hi! How is part b) related to part a)?
hidden harbor
its a product of 4 things, and result is positive or 0
forest trail
In other words, can you translate the inequality into english?
hidden harbor
so either one of them is 0, or an even number of them is negative
forest trail
Well,
snow mural
I just want to solve b
next cedar
you have the graph, and also, remember the keyword hence.
queen flame
when is the graph above x axis and crosses it?
next cedar
if you see that keyword, consider using whatever you have proven in the subquestions prior
snow mural
Ohhh so I need to check graph
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signal solar
i feel like all are true so im confused
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i feel like all are true so im confused
rapid rain
i feel like all are true so im confused
Do you want a hint?
Hint: ||what does 6/x^3 integrate to||
fiery cedar
@rapid rain <@&286206848099549185>
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<@246787399991033858> <@&286206848099549185>
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
fiery cedar
Thanks
rapid rain
<@246787399991033858> <@&286206848099549185>
Also:
- Don't ping helpers until after 15min of your problem not being resolved
- Don't ping specific helpers unless they authorized you to
fiery cedar
Also:
- Don't ping helpers until after 15min of your problem not being resolved
...
I'll create new one
signal solar
Hint: ||what does 6/x^3 integrate to||
-3/x^2 +C?
rapid rain
-3/x^2 +C?
missing a sign
signal solar
yep
signal solar
ok but that hint i still not sure
fiery cedar
Help me too guy's😭 tomorrow is my exam i am cooked
rapid rain
ok but that hint i still not sure
what happens when we try to compute integral of 6/x^3 between 0 and 1 then
modest obsidian
p series is only for 1 to inf
signal solar
ah
modest obsidian
but you should realise
modest obsidian
what happens to p series from 0 to 1
queen flame
Help me too guy's😭 tomorrow is my exam i am cooked
post in #help-27
signal solar
what happens to p series from 0 to 1
im not sure
rapid rain
im not sure
ok, so the problem bound is 0 right
imagine, instead of integrating between 0 and 1
we integrate between a and 1, where a > 0 gets smaller and smaller
that's the definition of an improper integral
so
$\int_a^1 \frac{6}{x^3}dx$
glossy valveBOT
Raphaelisius Maximus MMIII
rapid rain
-3/x^2 +C?
(remember what you found)
signal solar
so what does FTC tell you?
ftc?
modest obsidian
fundamental yada yada
signal solar
evaluate antiderivate from 1-a?
glossy valveBOT
parabolicinsanity
modest obsidian
the point is that we can then look at the case where a approaches 0
signal solar
-3/1^2 - -3/a^2
signal solar
i see
rapid rain
let em think
signal solar
well since claim 1 is true
then i assume by comparison test then it would be divergent
rapid rain
then i assume by comparison test then it would be divergent
so, if you're smaller than a divergent integral, you're divergent?
signal solar
so do we have to make a new claim
rapid rain
yes, and before we do that, a quick point on "p tests"
You've probably seen how $\int_1^\infty \frac{1}{x^p}dx$ converges if and only if... what?
glossy valveBOT
Raphaelisius Maximus MMIII
signal solar
p>2 or something like that
rapid rain
that's not exactly correct
signal solar
p>1
rapid rain
yes
you remember why? It's pretty much what we did here
when $p > 1$, $\frac{1}{x^p}$ integrates into $-\frac{p-1}{x^{p-1}}$
glossy valveBOT
Raphaelisius Maximus MMIII
signal solar
right
rapid rain
**Raphaelisius Maximus MMIII**
when p < 1, this integration is still valid, but p-1 is negative
so it's actually "constant/0" when x = infinity
and diverges
signal solar
Yup
signal solar
Right
rapid rain
so that's the p-test around infinity
when we're around 0
the integrations we make are still valid
but we're worried about "x=0", not "x= infinity"
rapid rain
**Raphaelisius Maximus MMIII**
so when p > 1, and we plug x = 0, it's suddenly constant/0
which diverges
and p < 1 with x = 0 gives us constant/infinity
converges
(p = 1 still gives us log, and log(0) also diverges)
any questions about that or does that make sense?
signal solar
It makes sense
glossy valveBOT
Raphaelisius Maximus MMIII
rapid rain
so, if we want to use comparison test around 0
In our case
What should we keep in the denominator
signal solar
x^1/3
rapid rain
What should we keep in the denominator
So we write that this is smaller than 6/x^(1/3)
now, the integral linked to this function is convergent (p test around 0)
so using comparison test, the original integral is convergent
If you want to take a bonus problem: is $\int_0^\infty \frac{5-\sin x}{x^{1/3}+x^3}dx$ convergent?