lofty monolith
thanks
#help-27
devout snowBOT
devout snowBOT
Available help channel!
sage socket
So, what does knowing that P(A) < P(B|A) give me exactly?
devout snowBOT
So, what does knowing that P(A) < P(B|A) give me exactly?
rotund crypt
If P(A) < P(B|A) the events can't be independent
sage socket
Wouldn't that be true if it were P(A|B) ?
rotund crypt
Oops, yeah
sage socket
Like P(A|B) = P(A intersection B) / P(B) = P(A) P(B) / ~~P(B) ~~ = P(A)
That tells me that knowing that B occured did not change the P(A)
rotund crypt
What is the question exactly?
You can of course use Bayes' theorem to relate P(B|A) and P(A|B)
sage socket
If I know that the P(A) is less than P(B|A), what does that tell me about P(A) or P(B|A) other than that one is smaller than the other?
sage socket
You can of course use Bayes' theorem to relate P(B|A) and P(A|B)
For this section, we aren't using Baye's theorem.
or how about this
what does the statement above tell me about P(B) or P(A|B) ?
rotund crypt
Without assuming anything about the events I don't think you can extract any information. If you do assume that the events are independent, for example, then P(B|A) = P(B)>P(A) is a conclusion
sage socket
Wait can you explain the rhs?
The P(B) > P(A) , so this is equal to P(B|A)
I need to think about that
rotund crypt
If the events are independent, then P(B|A) = P(B)
But since P(B|A)>P(A), P(B|A)>P(A)
untold pivot
Show that Axiom $3.11$ can be deduced using the preceding axiom of set theory and the power set axiom.
Axiom $3.11$ Let $X$ and $Y$ be sets. Then there exists a set denoted $Y^X$, which consists of all the functions from $X$ to $Y$
I saw some work on proofing the result using the product of sets but this has not been used at this point of the book.
I am not sure how to even start. I am guessing I have to use the axiom of replacement somewhere but I don’t see how having the power sets $2^X$ or $2^Y$ helps me at all.
devout snowBOT
Show that Axiom $3.11$ can be deduced using the preceding axiom of set theory an...
lusty sapphire
The set of all functions from X to Y
hmm
Well, do you know the set-theory definition of function? (Hint: in terms of relation)
untold pivot
i think so.
Let $X$ and $Y$ be sets. Let $P(x,y)$ be a property that relates $x \in X$ and $y \in Y$. st $P(x,y)$ is true for exactly one $y$
untold pivot
no actually
lusty sapphire
Oh
That would help a lot 
For sets X and Y, a relation is a set of ordered pairs (x, y) such that x is in X and y is in Y
untold pivot
Ok nvm, i just check the errata section online, they moved the exercise forward till after i learnt about cartesian products
oops.
lusty sapphire
More generally, a relation is a subset of the cartesian product of X and Y
untold pivot
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lusty sapphire
untold pivot
thanks for your time sorry about that
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devout snowBOT
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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
1
sullen island
alright then
last wasp
hii
sullen island
hii
last wasp
go on
sullen island
it's prolly a good idea to start with a drawing of the situation
last wasp
ye true
sullen island
last wasp
i like ur art skills
last wasp
the dimater
sullen island
yeah 13m
last wasp
yes
sullen island
alright so 13m
last wasp
correct
sullen island
now we want to put a pole somewhere on the circle
"A pole has to be erected ... on the boundary of a circular park"
last wasp
so on the circumfernce
sullen island
say there
last wasp
ye so go on
sullen island
"A pole has to be erected ... in such a way that the difference of its distances from the gates is 7m"
neon aspen
Are we done here?
last wasp
Are we done here?
nope
sullen island
:/
neon aspen
Just checking
neon aspen
Right
sullen island
"A pole has to be erected ... in such a way that the difference of its distances...
so yeah do you get what they mean here ?
neon aspen
Mind your tone.
last wasp
Mind your tone.
no idnt mean it that way 😭
neon aspen
Tell that to the moderators.
last wasp
i thought u busy checking by help channels if trhey r free to lose or nah
last wasp
Tell that to the moderators.
what why
neon aspen
You're walking on thin ice buddy.
last wasp
dont take it personally plsss i beg u
i didnt mean it the way it sounded
i just wanted sum help
neon aspen
Chill dude I was joking
last wasp
even i was
get ur ass out the chanell (JK PLS THIS WAS THE LAST TIME PLSPSLPSLSPSLSPSLSPSLSPSLSPSLSSSSSSSSSSSSSSSS)
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sullen island
.reopen
devout snowBOT
✅
sullen island
it's ok mate you'll survive
last wasp
@neon aspen 🙏
last wasp
it's ok mate you'll survive
nahnah
ill learn how to do it
u can close it for the others
trust me
sullen island
there's no limit of channels mate
last wasp
true
neon aspen
You don't need to close the channel because of me
last wasp
idk ill do it on my own issokk
last wasp
You don't need to close the channel because of me
i dont mind u at all
last wasp
i mean as you wish
thnx again
last wasp
I can't force you to stay here
yye npp
main gull
I can't force you to stay here
That's a lie, I'm here against my own will
sullen island
That's a lie, I'm here against my own will
I didn't mean you couldn't be forced by other ppl
do you mean I in particular force you to stay here ?
weak cove
aPlatypus forces me to be here
main gull
do you mean I in particular force you to stay here ?
Yes you
main gull
You're using the strong force
neon aspen
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@last wasp Has your question been resolved?
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viscid goblet
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viscid goblet
pseudo basin
any progress thus far?
viscid goblet
this question is from my younger sister's book
it tried solving it
but i feel there is something wrong with the question
pseudo basin
show what you've got so far.
viscid goblet
sure but i can't take an image tho
pseudo basin
why not
pseudo basin
how did you work these out
i can see triangle=0 fine, but what makes you conclude pentagon=0 as well?
pseudo basin
ah, right, that...
yeah, this looks like it was done out of order.
i think square-1-triangle-triangle is the product of 50 and pentagon-square, not square and pentagon-square.
viscid goblet
so
pseudo basin
PS
* 5S
-----
8P
S100
-----
S18P
this is how i think it should have been written "properly"
S = square, P = pentagon
viscid goblet
ok
yeah am not sure becuse this was exactly the question from the book
i just screen shot this image from the pdf
pseudo basin
i don't doubt that
since pentagon is the last digit obtained from multiplying together two numbers ending with the same digit, this limits what pentagon could be quite a lot
viscid goblet
yeah
pseudo basin
specifically pentagon can only be 0, 1, 4, 5, 6 or 9
0 is already taken by triangle
viscid goblet
yeah true
pseudo basin
can't be 1 since that would make the product of PS and 5S too small
try the other options
viscid goblet
alr
devout snowBOT
@viscid goblet Has your question been resolved?
jaunty musk
Hello ! I am searching help for something which, unfortunately, I couldn't find help with. Basically, I'm trying to prove (or disprove) that the function $$(x,y) \mapsto \max{ |x+3y|,|x-y|}$$ is a norm. The only one left which I found out to be the least obvious one is the triangular inequality.
It made me search for information about the max function, as well as potential proofs. Unfortunately, I couldn't even find the first part, so I could really give up on finding a proof.
The property I wanted to find a proof of (or finding a counterexample which I couldn't find) is the following :
$$\forall a,b,c,d \in \mathbb{R}^+,$$
$$max{a+b,c+d} = max{a,c}+max{b,d}$$
devout snowBOT
Hello ! I am searching help for something which, unfortunately, I couldn't find ...
woven radishBOT
_epsilia
jaunty musk
I already tried to prove it, but the problem came into how much hypothesis I would have needed to do in order to come to something, and it would have been quite laborious
So here's my synthetic question : Can someone help me into proving (or disproving) the triangular inequality for this function above ?
(Going to eat so I'll be right back if anyone offered some help, I'll probably ping you)
devout snowBOT
@jaunty musk Has your question been resolved?
supple knot
into how much hypothesis I would have needed to do in order to come to something
what does this even mean
show your math
jaunty musk
Ow hello, thanks for helping !
jaunty musk
> into how much hypothesis I would have needed to do in order to come to somethi...
Coming to calculations, I realized how when I start with $$\max{a+b,c+d}=...$$, I already find myself stuck and in order to un-stuck myself, I need to suppose what the max is in order to continue. That's a first assumption (or "hypothesis" if I quote myself).
When I look at the other side of the inequality I want to prove, I have $$\max{a,c}+\max{b,d}=...$$
Here I would need two assumptions in order to proceed further ... This way, I come into so much assumptions without even knowing how I will get to an end at the first place (I thought about how to but it didn't help, which is why I'm still stuck)
woven radishBOT
_epsilia
jaunty musk
If I showed you my notes, you would have had even less things written so I'm writing in detail what's going on in my head
It's probably the way I'm handling the max function which isn't helping ??
jaunty musk
I showed all of them
supple knot
oh
jaunty musk
Or, if you think there isn't
Then ... I never found any step at the first place 
Ow, I didn't detail in the case of contradiction, but I've tried out for different positive real numbers a,b,c,d with, for example, a large or a tiny difference between the four numbers, seeking the extremes in order to find a contradiction
And it seems to always work
That's when I started trying to prove it
And we come back to what I wrote before
winter torrent
could try proving it by cases, like what if a>c and b>d
jaunty musk
Okay, I thought a bit on my side and I think there is stuff we can assume, without loss of generality
We have addition, which is commutative
And max{a,b}=max{b,a}
So a, b, c and d play symmetric roles
It should simplify stuff out by a lot
jaunty musk
could try proving it by cases, like what if a>c and b>d
So now if I consider these different cases, do we cover all of them ?
winter torrent
I haven't done it but yeah that's possible
devout snowBOT
@jaunty musk Has your question been resolved?
devout snowBOT
@jaunty musk Has your question been resolved?
jaunty musk
I think I know where to search for ! Still thanks a lot for trying to guide me into a right path @supple knot @winter torrent, it really helped me to search into it more thoroughly 
.close
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sour meteor
!show
devout snowBOT
Show your work, and if possible, explain where you are stuck.
loud dagger
For all of these, you would take each pair of functions and add them, subtract them, multiply them ,etc. So for 1, (f+g)(x) = 3x+4 + 2x-3 and so on like that. Let us know if you're stuck somewhere specific
sour meteor
soma, subtração, multiplicação, divisão das funções
patent finch
oh ya
jaunty musk
Pretty sure that wasn't spanish but seems to have worked, right ?? 
sour meteor
XD
patent finch
Pretty sure that wasn't spanish but seems to have worked, right ?? <:opencry:586...
potugis i think
near stone
spanish plis 😭
use a translation
patent finch
good point
tepid shadow
I am looking at more mathematical induction, and this is the solution they provide in the slides
devout snowBOT
I am looking at more mathematical induction, and this is the solution they provi...
tepid shadow
I did the question myself without looking at the solution in the slides and got this instead:
tepid shadow
what's the question
Sorry, I was just getting a screenshot of my work
keen oxide
I understood the question from your solution
So generally the expectation is to prove s(k+1) given s(1), s(2), s(3), ... s(k) is true
you are not wrong but the general procedure is what I have mentioned above
tepid shadow
hmm, nuts. Okay thank you. Imma try it again using the k+1 instead
looks like that one went alright. Thank you @keen oxide
.close
normal forum
If we say that $\lim_{x\rightarrow 2}\frac{x^2+x-6}{x-3}$ is continues at x=2. Does that mean that it is continuous from that point? So can we say that $\lim_{x\rightarrow 3}\frac{x^3-2x^2}{x-2}$ is continuous from x=3? If not, how do we express that in words? Do we say $\lim_{x\rightarrow 3}\frac{x^3-2x^2}{x-2}$ is discontinuous at x=3??
devout snowBOT
If we say that $\lim_{x\rightarrow 2}\frac{x^2+x-6}{x-3}$ is continues at x=2. D...
woven radishBOT
Bennxy
supple knot
continuous refers to points in the domain of the function
although it's confusing since all your limits points are defined
maybe you have some typos
normal forum
I don't know, I just tried to implement something with x=3 with $\frac{x^3-2x^2}{x-2}$. I did see my book say that $\lim_{x\rightarrow 2}\frac{x^2+x-6}{x-3}$ is continuous at x=2 because this function doesn't have holes right? So I was thinking how we should write it with a discontinuous function.
woven radishBOT
Bennxy
supple knot
you can just plug in x=3 to (x^3 - 2x^2)/(x-2)
normal forum
Yeah we then get a real point that exist.
supple knot
then what's the issue
normal forum
But the thing is, how should we write it down? Do we say it is continuous or discontinuous?
supple knot
write what down?
maybe it'd help to do some more examples
https://tutorial.math.lamar.edu/classes/calci/TheLimit.aspx
normal forum
Do we say \frac{x^3-2x^2}{x-2} is continuous at x=3 or \frac{x^3-2x^2}{x-2} is discontinuous at x=3?
supple knot
why would it be discontinuous at x=3?
supple knot
you can just plug in x=3 to (x^3 - 2x^2)/(x-2)
.
normal forum
maybe it'd help to do some more examples
https://tutorial.math.lamar.edu/classes...
Thanks I will check it out
normal forum
why would it be discontinuous at x=3?
I dunno I'm a bit confused because at x=2 it is undefined
So it doesn't effect other point? So we can just say \frac{x^3-2x^2}{x-2} is continuous at x=3?
supple knot
yes you can
normal forum
Ooohh
supple knot
,calc (3^3 - 2(3)^2) / (3-2)
woven radishBOT
Result:
9
Aha
Ok
But when we refer to the function we need to say it is a discontinuous function right?
Or am I wrong?
normal forum
Hmm
Okok
So I should say
The function \frac{x^3-2x^2}{x-2} has a point of discontinuity
I'm damn confused
woven radishBOT
supple knot
that function is perfectly continuous everywhere it's defined
normal forum
But I get undefined for x=2 in desmos
supple knot
that function is perfectly continuous _everywhere it's defined_
did you read this
normal forum
The function \frac{x^3-2x^2}{x-2} has a point of discontinuity
But this would be accurate right when a correct function is used
devout snowBOT
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Available help channel!
bitter dove
is this a good way to show that Any common divisor of m - n and n is also a common divisor of their sum, m
If I have
na - nb = nc
rearrange the equation i get
na = nb + nc
which still shows that n is a common divisor
devout snowBOT
is this a good way to show that `Any common divisor of m - n and n is also a co...
@bitter dove Has your question been resolved?
mild comet
is this a good way to show that `Any common divisor of m - n and n is also a co...
what are a, b and c?
mild comet
is this a good way to show that `Any common divisor of m - n and n is also a co...
can you write out your full proof so we can check it
winter torrent
where's m? I thought there was an m in this story
bitter dove
there is i think i should use a different variable other than n
xa - xb = x(a - b)
x(a - b) + xb = xa
can this be used to show that the above statement is true?
i want to know if my understanding of the above statement is correct or not
winter torrent
I don't understand what that's supposed to show. You should use the variables defined in the problem statement
so, let m and n be integers, and k be a divisor of (m-n) and of n
you don't have to use k if you don't want to
worn cloud
the problem with this proof is that x dividing (m-n) doesn't necessarily mean that m-n factors as x(a-b) for the b such that n=xb
youre using what you want to prove
bitter dove
so if k is a divisor of (m - n) and n. i need to show that k is also a divisor of m
winter torrent
exactly
bitter dove
m = 6 n = -4
a divisor of m and n is also the divisor of it's sum and difference
is that not enough?
hmm
winter torrent
a divisor of m and n is also the divisor of it's sum and difference
but we don't know that k is a divisor of m yet
bitter dove
hmm
we only know it's a divisor of n and (m - n). if k divides n and k divides (m - n). when we say k divides n that means there an integer multiplied by k which gets us n
i don't know how to put what i know together!!! i know what if two numbers have a common divisor, if we add those numbers up or subtract them, the result will also have the common divisor
worn cloud
we only know it's a divisor of n and (m - n). if k divides n and k divides (m - ...
Lets be a bit more precise. From this statement, we can write that there exists some $a \in \mathbb{Z}$ such that $n=ka$, that is what you said right?
woven radishBOT
KB0
worn cloud
(If you havent seen it before, the funny Z means the integers)
and the symbol next to it means 'in'
bitter dove
yes
worn cloud
can we make a similar statement for $(m-n)$?
woven radishBOT
KB0
bitter dove
yes we can
i want to make a similar statement, but idk if i need to replace n with ka
worn cloud
I would hold off on that!
m-n is an integer in its own right, and we know it has k as a factor, so (m-n)=?
bitter dove
(m - n) = kb
worn cloud
sure, so lets hold onto that information on the one hand. on the other hand, there is another relation we're interested in here
from the information of n and (m-n), can we make m?
worn cloud
sure you could do it that way, but i think youve made a sign error
worn cloud
i always make these small errors.... it's frustrating
youre definitely not alone on that front
worn cloud
oops m = kb + ka
from here, i will say were one step from showing that m is divisible by k, can you spot it?
bitter dove
do we just divide k on both sides?
worn cloud
not quite, we need to show that m can be written in the form $m=kc$ for some integer $c$
woven radishBOT
KB0
bitter dove
oh
m = k(b - a)
if i want to show something divisible k i need to show that it's a multiple of k!
restive river
Butene (C4H8) molecular geometry are tetrahedral and trigonal planar right?
main gull
Butene (C4H8) molecular geometry are tetrahedral and trigonal planar right?
You know that this is a math server, right? You'll get limited Chem help here
restive river
Geometry = math
orchid glade
ok no
main gull
Geometry = math
One could argue that English = math because math has words and so does English
orchid glade
ok done
topaz beacon
Have you drawn what it would look like
orchid glade
its not relevant
restive river
Have you drawn what it would look like
Yes
orchid glade
i think it would have that >-< planar thing to one side
and the tetrahedral part to other
topaz beacon
No judgment here
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Available help channel!
terse cloak
would anyone be able to help? I dont know why Im stuck on this
devout snowBOT
would anyone be able to help? I dont know why Im stuck on this
terse cloak
sorry forgot the times
devout snowBOT
can anyone help me with math
Please read #❓how-to-get-help
@terse cloak Has your question been resolved?
terse cloak
.close
viral edge
To find the solutions of the statement function P(x): x>3→2|x, which is defined on the set M={1,2,3,…,9,10}.
winter torrent
that would help, particularly the p -> q column
do you understand what x>3 -> 2|x means?
viral edge
If x is higher than 3 and the second part is if x can be divided by 2
or?
@winter torrent
winter torrent
that's correct, so that would be read as "x is greater than 3 implies x is divisible by 2"
so you just see which of the numbers 1..10 satisfy that
viral edge
yeah but my question was if the number for the first part is true and for the second part is not true, lets say number 5 then we dont count that number right?
winter torrent
look at the truth table
for P(5) you'd have 5 > 3 -> 2|5, which reduces to T -> F
viral edge
okay so I wont add number 5 in this example
thanks that was the answer that I was looking for
.close
lyric night
Change the integral bounds by identifying the regions where the following integrals are taken.
devout snowBOT
Change the integral bounds by identifying the regions where the following integr...
scarlet sequoia
start from the sketch
lyric night
then?
devout snowBOT
Available help channel!
lucid oracle
Simple group theory proof that i cant figure out (lol): Let G be a group with an element x, if for all y in G: xyx = y^3, then x^2 = e
devout snowBOT
Simple group theory proof that i cant figure out (lol): Let G be a group with an...
lucid oracle
I've tried a few stuff but i cant seem to find anything usefull for a proof
For example the identity x^2 = y^3x^-1y^-1 *x
wooden veldt
pick a nice y
lucid oracle
hmmm
wooden veldt
pick the nicest y you can think of
lucid oracle
Haha im trying a few that come to mind rn
so far for y = x ^-1 i got x^2=x^-2 and from there x^4 = e
hmmmmmm lemme think
wooden veldt
an even simpler y
lucid oracle
OHH
Wow my laptop died in the middle of typing my reasoning and now i forgot
I think it worked out if i picked y = x
OH
y = e
LMAO
DHIKJFLJLMG
im mad.
wooden veldt
lmao
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Available help channel!
left robin
hey
restive river
hey
left robin
$|A-\int_0^1 e^{-x^2}dx|<\frac{1}{100}$
woven radishBOT
martin3125
left robin
I have to find such an A out of Q
the answer is 3/4
at least that works
the hint was that the leibniz criteria might help
which is for alternating series
any idea where that might help?
Another thing i noticed is that:
$\int_0^1 e^{-x^2}dx=\sum_{k=0}^{\infty}\frac{(-1)^k}{(1+2k)k!}$
woven radishBOT
martin3125
left robin
which is an alternating series
restive river
i think you should wait for someone who knows this arrives
left robin
i could bruteforce
to get the first few terms of the series
but then, i cant compare it to the integral
because i dont have the value of the integral
since i cant quote wolframalpha haha
final laurel
are you allowed to use any numerical methods other than WA to estimate $\int_{0}^{1} e^{-x^{2}}dx$?
woven radishBOT
keto11
left robin
yeah, i dont think im meant to use any numerical stuff
final laurel
yeah I'm not really sure either
maybe using something like the alternating series estimation theorem will yield results
but that's just my guess
left robin
i will look into that
supple knot
but then, i cant compare it to the integral
you don't need to know the exact value of the integral.
Here, you correctly found L by integrating term by term. now you just have to find the upper limit in the partial sum
so that the next term is less than 1/100
devout snowBOT
@left robin Has your question been resolved?
left robin
hmm ok
so what i would do is, set
1/((2k+3)(k+1)!)=1/100
and then round up to the next whole k
then k=3
which gives a working approximation
thx^^
.close
pine gulch
Quick question, if you have 10 chances (total) to complete multiple challenges which' percentage is as follows
- 90% (to pass)
- 80%
- 70%
- 60%
- 50%
would i be right in the answer being: 0,9x0,8x0,7x0,6x0,5 = 15.12%*10 (CDF) = 80.59%??
(for clarification if you fail lets say the 80% challenge you can try again at 80% if you pass you move on to the 70% and try there but you only have 10 tries total, not per challenge)
devout snowBOT
Quick question, if you have 10 chances (total) to complete multiple challenges w...
pseudo basin
so essentially you have 5 levels and 10 attempts total, and if you win a level you move on to the next but if you lose you get sent back to level 1?
i think this wont be as simple as a product like you've written
(also what's 10(CDF)??)
devout snowBOT
@pine gulch Has your question been resolved?
pine gulch
i think this wont be as simple as a product like you've written
if you lose you dont get sent back to level 1, you can try again until you beat the level (though you have a maximum of 10 tries total, so if you fail 3 times in a row then pass you would only have 6 tries left total (minus and tries before if any)) CDF is Cumulative Distribution Function i think its necessary for the calculation but im not sure
pseudo basin
i think you are misusing the term CDF then
anyway i think you need to account for all possible paths of getting to level 5 and passing
pine gulch
anyway i think you need to account for all possible paths of getting to level 5 ...
oh... i see... 😭
there isn't some easy way to go about that is there?
pseudo basin
not that i can think of.
grim ginkgo
i know this may look easy but i forgot how to do it
devout snowBOT
i know this may look easy but i forgot how to do it
torn vessel
use the formulas $P(A\cup B) = P(A) + P(B) - P(A\cap B)$ and $P(\overline A) = 1 - P(A)$
woven radishBOT
Zybikron
torn vessel
yes
torn vessel
oh, you can also use the total probability formula $P(B) = P(A\cap B) + P(\overline A \cap B)$
woven radishBOT
Zybikron
grim ginkgo
for what?
devout snowBOT
@grim ginkgo Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
naive steeple
devout snowBOT
@naive steeple Has your question been resolved?
devout snowBOT
@naive steeple Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
dry rain
Question is on the right, im not sure if im setting this up right or where to go from here
devout snowBOT
Question is on the right, im not sure if im setting this up right or where to go...
neon aspen
how much sugar does 1 batch require?
dry rain
3/4 or .75
dry rain
would that be 2/4 cups? .5 cups
dry rain
so half of 3/5 and .75
neon aspen
right so how many cups?
dry rain
i dont know how to find that, do i do .75/2
dry rain
oh 3/8
neon aspen
yep
neon aspen
nah
dry rain
ok so half a batch requires 3/8 and 1 full batch requires 3/4 that means 1 and a half batch is 3/8 + 3/4? i feel like im messing up
neon aspen
yes that's correct
dry rain
so that wouldbe 6/12 or 1/2, the answer being 1 1/2
neon aspen
that's now how u add fractions
dry rain
do i have to find common bottom number like with subtracting
neon aspen
lcm ye
dry rain
ok so its 1 1/8
neon aspen
ru sure
dry rain
i get 9/8 so that would be 1 1/8
neon aspen
correct
dry rain
nice thank you i was struggling
neon aspen
wl
dry rain
.close
blazing summit
for what values of a does the series 1/x^a converge?
devout snowBOT
for what values of a does the series 1/x^a converge?
pseudo basin
you mean $\sum_{n=1}^{\infty} \frac{1}{n^a}$?
neon aspen
a > 1
woven radishBOT
Ann (glomed)
blazing summit
you mean $\sum_{n=1}^{\infty} \frac{1}{n^a}$?
yes
devout snowBOT
neon aspen
wl
devout snowBOT
Available help channel!
devout snowBOT
devout snowBOT
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.
obtuse aurora
How can I start this? (I dont want a full solution, but maybe a hint on how to go about it...):
Consider the set of N distinct vectors {a1, a2 , a3 , · · · , aN } to be clustered in K non-empty
indistinguishable clusters. Derive the total number of such possible clusters.
devout snowBOT
How can I start this? (I dont want a full solution, but maybe a hint on how to g...
wooden veldt
you have no other information about the a_i?
obtuse aurora
No, I think it doesnt matter. what the vectors are I think it has something to do with Stirling Numbers
wooden veldt
well then what is a cluster? and what does it mean for two clusters to be indistinguishable?
obtuse aurora
This is the whole instruction
I got "3" as the solution for Part 1:
{a1,a2}, {a3}
{a1,a3}, {a2}
{a2,a3}, {a1}
wooden veldt
right what this really is is partitioning a set of size N into K nonempty subsets
obtuse aurora
I guess you are right
wooden veldt
just translating it into more mathematical language for people
it does seem to involve stirling numbers but as for the derivation i am not familiar
obtuse aurora
For K = 2, I get 2^{N-1} - 1
this is just by testing out though and I guess there is no way to use that for generalization :
devout snowBOT
@obtuse aurora Has your question been resolved?
devout snowBOT
@obtuse aurora Has your question been resolved?
devout snowBOT
@obtuse aurora Has your question been resolved?
supple knot
this is just by testing out though and I guess there is no way to use that for g...
Now that you have some initial values, try finding a recurrence relation. Use some notation like S(n, k) to be the number you're looking to solve for
supple knot
right what this really is is partitioning a set of size N into K nonempty subse...
Or S( N, K )
obtuse aurora
Now that you have some initial values, try finding a recurrence relation. Use so...
I kind of found the formula for Sterlin Numbers but that is not deriving it then 🤔
supple knot
I kind of found the formula for Sterlin Numbers but that is not deriving it then...
What does kind of mean
supple knot
Now that you have some initial values, try finding a recurrence relation. Use so...
If you did this then you derived them
obtuse aurora
I mean I found the formula online without deriving it myself.
devout snowBOT
@obtuse aurora Has your question been resolved?
restive river
is this error in the book, or am i missing something.
I highlighted two expressions.
Shouldn't the next step be:
pi/4 - x/2?
Seems like they didn't divide x by 2.
devout snowBOT
is this error in the book, or am i missing something.
I highlighted two expressi...
@restive river Has your question been resolved?
supple knot
is this error in the book, or am i missing something.
I highlighted two expressi...
Try plotting both curves in desmos
restive river
I did and it seems that there is indeed a mistake.
It should be x/2
ionic gazelle
yeah looks like the book goofed up
restive river
I bought this workbook few months ago, but new version came out recently 😮💨.. they probably fixed it there. I can't really spend more money on the same book..
Anyway, I have another question
Let's say we have this equation
sinx = - 1/2
One of the solutions is:
- pi/6 + 2k*pi
(where k elem of Z set - whole numbers)
ok, but this solution can also be written as
11pi/6 +2k*pi
So,the question is, how do I convert one solution into the other, in general?
Oh.. i think I got it while writing this..
I just add 2pi or subtract 2pi.
-pi/6 + 2pi = 11pi/6
11pi/6 - 2pi = - pi/6
ionic gazelle
yeah they're the same thing
but generally speaking the integers (Z) are not the same thing as whole numbers
restive river
hm.. Z is a set of numbers such as - 5,-3, 0, 1, 2... those are whole numbers @ionic gazelle
ionic gazelle
whole numbers start from 0, and go up by 1
integers include the negative numbers you listed out
restive river
oh.. i see..
English is not my first language..
in my first language the word for integers has the literal translation for "whole".. bit confusing.
I will have this in mind... until I forget 😅
.close
jovial blaze
Is this correct?
graceful cosmos
No
digital fossil
@jovial blaze you should use u substitution
graceful cosmos
Straight substitutions don't work like that. See "u-substitution"
digital fossil
let u = e^t
then top becomes du
bottom becomes
1-u sqaured
sqrt
become a sin inverse
then make u back into t
also becareful of bounds
digital fossil
Straight substitutions don't work like that. See "u-substitution"
hi kaynes
i needed help with one of my questions
can you check my channel plz thanks
digital fossil
is the question starting bound 2 and 1?
jovial blaze
1 to 2
digital fossil
ok
bounds become e^2
and e
since you use u sub
after integrate
just use bounds to find answer
jovial blaze
But I can just sub for u
and then put those bounds instead of changing
That works too no?
digital fossil
yea yea
no need to change back to t
canse you get answer
since this is a bound question
jovial blaze
Is this correct?
Then this was right?
digital fossil
Then this was right?
yea should be
jovial blaze
Alright
jovial blaze
I just did a u sub in my head
digital fossil
ok
jovial blaze
Alright
devout snowBOT
@jovial blaze Has your question been resolved?
reef pilot
how can i go about proving this?
devout snowBOT
how can i go about proving this?
reef pilot
The result is obvious but i dont know what arguement i should use
pseudo basin
consider: f(x) = x is the same as f(x) - x = 0
edgy thicket
Okayy
edgy thicket
So how does one calculate the trigonometric ratios in terms of just theta(an acute angle) for for the angle (90+θ)
wooden wraith
Do you mean you're given tan(90+theta), and you want to find tan(theta)?
@edgy thicket
You can probably use the sum formula for tan
or whatever trig function you need
edgy thicket
Oh okay lemme explain
@wooden wraith
This is a table that shows the relations between sins, cosine and tans across multiple angles in many quadrants such as (90+ theta), (180+/- theta) etc. to the angles in the first quadrant the; the original angle theta in the first quadrant
What doesnt make sense to me is the relation between sin(90+theta) = cosine theta
When we plug the values/draw the sides in a graph
sine (90+ theta)= -x/r
Which is supposed to be - cosine
But they used + cosine instead
Why!!
@wooden wraith
devout snowBOT
@edgy thicket Has your question been resolved?
craggy escarp
sin(x+y) = sin(x).cos(y) + sin(y).cos(x)
edgy thicket
Yea…
craggy escarp
You see?
craggy escarp
Even geometrically it's x/r
not -x/r
you should not mind sign!
it's not physics
edgy thicket
Umm
craggy escarp
Even geometrically it's x/r
not -x/r
you should not mind sign!
it's not physics
edgy thicket
Even geometrically it's x/r
not -x/r
you should not mind sign!
it's not physics
We have to mind the sign in every other quadrant wdym????
craggy escarp
it's not physics, it's not some vector or something
after all, it's a triangle that results right?
and when you compute the sine of some angle in a triangle
you just do opposite/hypo
length of opposite side / length of hypo
LENGTH!
edgy thicket
Okayy since we hate the signs
hahhh
Why this
Cos(90+ theta) should be positive sine
Butt
The boek says
edgy thicket
<@490649542077841408>
This is a table that shows the relations between sins, co...
Its Negative
craggy escarp
I give you another way of thinking of it
Wait, I draw
good old paint
Look at the first angle, x
its cosine is high, and its sine is low
Then look at the second angle, x+90°
its cosine is low and negative
its sine is high
cos(90°+x) = -sin(x)
sin(90°+x) = cos(x)
edgy thicket
Then look at the second angle, x+90°
Second angle as in.. the angle between the center line(the black one) and the red line right??
craggy escarp
edgy thicket
Thiss???
craggy escarp
We always start measuring from the positive x-axis, remember that!
edgy thicket
Aooah
edgy thicket
We always start measuring from the positive x-axis, remember that!
Ofc im not a doughnut
craggy escarp
😝
copy paste
"Look at the first angle, x
its cosine is high, and its sine is low
Then look at the second angle, x+90°
its cosine is low and negative
its sine is high
cos(90°+x) = -sin(x)
sin(90°+x) = cos(x)"
edgy thicket
Question timeee
craggy escarp
first, 90+x extends all the way from positive x-axis to the 2nd red line
edgy thicket
edgy thicket
first, 90+x extends all the way from positive x-axis to the 2nd red line
Yea sorry ab that
craggy escarp
How to get its cosine?
edgy thicket
Yess
edgy thicket
PROJECT IT?
edgy thicket
On this???
craggy escarp
edgy thicket
Im sorry im just confused rn
craggy escarp
There you go, more details
When you drop a vertical line from the angle to the x-axis, you get the angle's cosine
Horizontal line from angle to y-axis, you get angle's sine
You see x, its cosine and its sine
craggy escarp
And 90+x, its cosine and its sine
edgy thicket
This is just.. no. Like ive seen people explain this and i always go.. “BUT WHY Big angle have the cosine value in the y axis should it not be sine”
😓
Maybe im just cursed or something
craggy escarp
Haha
What I drew is called the unit circle
it's well known
x-axis is cosine
y-axis is sine
craggy escarp
idk how to summon tex
edgy thicket
, tex. Unitcircle
woven radishBOT
maybeinactive
winter torrent
,tex .unit circle
woven radishBOT
Hayley
winter torrent
aha
edgy thicket
,tex \.unit circle
Youre a wizard harry
craggy escarp
Holy
How
I literally wrote that
Thank you @winter torrent
Why does it not work for everyone
winter torrent
gotta set it up with https://github.com/riemann-discord/math-discord
craggy escarp
,tex .unit circle
edgy thicket
@craggy escarp so explain why we use cosine for y axis in a unit circle?
winter torrent
cosine is x and it's because that's where the Adjacent side is (remember sohcahtoa)
craggy escarp
No lol
in the unit circle, you always project down to the x-axis
and read the cosine of the angle
project to the y-axis to read the sine of the angle
no matter which quadrant
or how big your angle is
craggy escarp
in physics, you may sometimes have the y-component of some force
be related to cosine and not sine
but that's not here
There are many videos on youtube that explain the unit circle
Watch some!
edgy thicket
I got no clue what any of the witchcraft you have been saying lol
craggy escarp
😂
edgy thicket
https://www.youtube.com/watch?v=1m9p9iubMLU
Haha ive watched that vid like 20 times before
craggy escarp
How the hell do you know sin(x+y) = sin(x).cos(y) + sin(y).cos(x)
And not this unit circle :P
gaunt helm
have you seen this before
edgy thicket
How the hell do you know sin(x+y) = sin(x).cos(y) + sin(y).cos(x)
No i dont know that shit
edgy thicket
have you seen this before
Who hasn’t?
craggy escarp
Watch the video again
edgy thicket
Watch the video again
Fine
craggy escarp
Today's the day you understand this
edgy thicket
Today's the day you understand this
Ahh.. ive been at it since yesterday lol
edgy thicket
He is simplifying it as much as possible
Sal khan?
craggy escarp
yes
edgy thicket
Okai il will waitch it again
Ugh just dont go anywhere
I have been abandoned by two people already
gaunt helm
When we plug the values/draw the sides in a graph
sine (90+ theta)= -x/r
it's better to ignore the signs at first, then apply the negatives because you're working with magnitudes
for example, your r is a length, but you're applying direction to your x and y
gaunt helm
Who hasn’t?
also no need to be so aggressive 
edgy thicket
also no need to be so aggressive <:Hehe:380943705776979969>
Sorryy just a lil vexed rn
gaunt helm
have you seen this before
based on this, your related angle will be 180 - (90 + x)
which is 90-x
edgy thicket
it's better to ignore the signs at first, then apply the negatives because you'r...
We tried that strategy. Doesnt work for the sine ratio cause sine has to equal to the negative of cosine and im like waaaaaaa?
gaunt helm
it's also much easier to visualise using the graphs of cos and sin
transformation of graphs
edgy thicket
based on this, your related angle will be 180 - (90 + x)
which is 90-x
Um.. yea but im looking for (90+ theta) which isn’t the same as 180-(90+x)
gaunt helm
it's 180 - (90 + x)
gaunt helm
basically, sin(90+theta)
is the same as sin(180-(90+theta))
and it retains positive because 2nd quadrant
and thus it'll simplify to sin(90-theta)
which is your cos(theta)
edgy thicket
@gaunt helm lemem draw one sec
edgy thicket
basically, sin(90+theta)
is the same as sin(180-(90+theta))
and it retains posit...
Umm
Again i dont have a problem with 90-theta
Were solving 90+ theta
@craggy escarp oi
craggy escarp
Hi
edgy thicket
Watched itt
And it was just basic trig functiona
He didn’t say anything about big angles or how the cosine is measured in the unit circle
craggy escarp
He said...
gaunt helm
Again i dont have a problem with 90-theta
it's precisely because you don't have a problem with 90-theta, that's why I simplified 90+theta to 90-theta
edgy thicket
it's precisely because you don't have a problem with 90-theta, that's why I simp...
Ightt
So we have 180-(90+theta)
craggy escarp
He didn’t say anything about big angles or how the cosine is measured in the uni...
The cosine in the unit circle is always the x-axis
gaunt helm
(using the related angle and quadrants)
edgy thicket
The cosine in the unit circle is always the x-axis
This is too frustrating
Ahm.. are we allowed to vc?
edgy thicket
Voice.. chat?
craggy escarp
No lol
Look, for any angle
Wherever it is
Whichever quadrant
its cosine is just the x-component
and its sine is just the y-component
edgy thicket
its cosine is just the x-component
Yes ik that
craggy escarp
so just look at
craggy escarp
edgy thicket
For the angle 90+theta
craggy escarp
yep
edgy thicket
We say the cosine is in the y axis
craggy escarp
No, its cosine is the one below it
edgy thicket
The green line that says cos
edgy thicket
For the (90+ theta) waat did you says the sins and cosines were?
craggy escarp
For x, look, its cosine is high, looks like 0.95
And its sine is low, looks like 0.35
For 90°+x, its sine is high, looks like 0.95
and its cosine is low, looks like 0.35, but also negative
sine of 90+x is just the cosine of x
cosine of 90+x is just - the sine of x
by just observing, you can deduce
😃
edgy thicket
What do you mean by sine of 90+x
The entire thingh??? The big angle thats over 90 degrees?
How is anyone able to determine the sine or the cosine value of it??
craggy escarp
The yellow is the sine
I told you, you just follow the horizontal line (green) to the y-axis
the intersection is the sine
my english is not 100%, i cannot convey what I want to say
im not american
edgy thicket
Me neither lol
craggy escarp
maybe others can explain better
edgy thicket
my english is not 100%, i cannot convey what I want to say
Okay.. i want you to tell me about the angle.. 90+x
I understand that the yellow line on the vertical direction represents sine
craggy escarp
Since now you can see the cosine, sine of x
And the cosine, sine of 90+x
Observe that sine of 90+x is same as cosine of x
Look, they both look like 0.9
devout snowBOT
@edgy thicket Has your question been resolved?
edgy thicket
@craggy escarp so we’re gonna go with this explanation?
They both look the same qed there both the same?
edgy thicket
Hmm…
craggy escarp
cos(90+x) sin(90+x) cos(90-x) sin(90-x)
cos(180+x) sin(180+x)
cos(180-x) sin(180-x)
cos(270-x) sin(270+x)
All of them the same way
you draw the 2 angles and you see
edgy thicket
Wiatt