ok c'est bon j'ai compris
#help-39
pearl pondBOT
stray canyon
.reopen
pearl pondBOT
Available help channel!
amber compass
is the answer I put in correct?
pearl pondBOT
is the answer I put in correct?
shut elm
Looks good to me
pearl pondBOT
pearl pondBOT
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north talon
Let $m,n,x$ be positive integers, prove that [\sum_{i=1}^n \min (\lfloor \frac{x}{i} \rfloor , m) = \sum_{i=1}^m \min (\lfloor \frac{x}{i} \rfloor , n) ]
jolly parrotBOT
Copter
north talon
i think the way to do this is by counting? but im not sure how
rotund ferry
i think the way to do this is by counting? but im not sure how
could you elaborate what you mean by "counting"?
pearl pondBOT
@north talon Has your question been resolved?
naive zinc
It’s clear if you use induction
On n
Prove that P(n,m) is true then P(n+1, m) is true. Since P(m,m) is true for any m you obtain all are true
$\sum_{i=1}^{m}(\text{min}([\frac{x}{i}], n+1)-\text{min}([\frac{x}{i}], n))=\text{min}([\frac{x}{n+1}], m)$ can be shown by direct calculation (discuss x/i value compared to n, n+1)
jolly parrotBOT
Cogwheels of the mind
pearl pondBOT
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worn mango
,, \frac{4}{x-1} \ge \frac{1}{2x-1}
jolly parrotBOT
Madhup
worn mango
in this why cant we just multiply both sides by 2x-1
rather than transposing 1/2x-1 to LHS
versed mica
because multiplying by a negative flips the sign
worn mango
because multiplying by a negative flips the sign
2x-1 isn't a negative?
versed mica
2x - 1 < 0 for x < 1/2
versed mica
2x-1 isn't a negative?
sure it can be
restive garnet
Because you don't know if 2x-1 is positive or negative
worn mango
sure it can be
Oh wait
worn mango
Because you don't know if 2x-1 is positive or negative
Oh shit
versed mica
it isn't valid for all values of x
versed mica
if you multiply buy 2x - 1 then your solution is only valid for x > 1/2
worn mango
oh
versed mica
and you miss the full solution
versed mica
you can consider these cases separately
worn mango
yes
versed mica
or
worn mango
thanks
worn mango
multiply both sides by (2x - 1)^2
That's genius too
versed mica
rather than doing case work
worn mango
I also had one other doubt
versed mica
ask away sir
worn mango
ask away sir
While solving inequalities, why do we first plot the roots
restive garnet
<@&268886789983436800>
versed mica
While solving inequalities, why do we first plot the roots
because they are the points at which the factors change sign
worn mango
because they are the points at which the factors change sign
I see
slow oak
<@&268886789983436800>
Tq :)
worn mango
And my teacher also taught me that the signsof alternative intervals are opposite
But why does that not work when degree of a factor is more than 1
versed mica
not sure what you mean by alternative intervals
worn mango
like -infinity, 0 and 0,infinity
versed mica
But why does that not work when degree of a factor is more than 1
factors with even degree don't change sign
versed mica
you're welcome sir
latent quail
versed mica
^
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worn mango
how to do this
pearl pondBOT
how to do this
worn mango
the critical points are going to be -2, 8, 2, 0, and 15 right?
proper nova
yeah
proper nova
how to do this
prove this?
worn mango
after that what do i have to do?
worn mango
prove this?
no find the solution set
worn mango
yes
proper nova
$(x + 2)^6$ and $x^{100}$ is already $\geq 0$
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
yes
proper nova
so probably ignore that
worn mango
but it'll still be a part of the critical points though right?
proper nova
we will only care about the signs of $(x - 8)^5$, $(x - 2)^3$ and $(x - 15)^7$
worn mango
aight
proper nova
but it'll still be a part of the critical points though right?
uh yeah
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
the critical points are going to be **-2, 8, 2, 0, and 15** right?
0 and 15 are unfortunately not the critical point
because then the denominator = 0
bronze heath
but we still gotta include them on the number line
Yes
worn mango
mhm
proper nova
**1 divided by 0 equals Infinity**
try to consider the sign of $(x - 2)^3$ first
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
if $x < 2$ then of course $(x - 2)^3$ is negative right?
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
but what happens to other terms when $x < 2$?
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
specifically $(x - 8)^5$ and $(x - 15)^7$?
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
x-8 is also negative
proper nova
yes
worn mango
and x-15 too
proper nova
cool
worn mango
mhm
proper nova
now since $(x - 8)^5$ is negative, $(x - 2)^3$ is negative, so the numerator is positive
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
and the denominator is negative
worn mango
agreed
proper nova
leading to the whole fraction $< 0$
worn mango
so the function is negative?
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
alr
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
mhm
proper nova
of couse i don't talk about x = 2
proper nova
now $x > 2$
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
mhm
proper nova
then we consider the signs of $(x - 8)^5$ now, what happens if $x < 8$
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
its negative
proper nova
and what happens to $(x - 2)^3$?
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
it can be positive
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
**1 divided by 0 equals Infinity**
so $(x - 2)^3$ is positive
worn mango
yea
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
so the numerator is negative
proper nova
what happens to $(x - 15)^7$ btw?
worn mango
denominator is also negative
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
cool
worn mango
so it becomes positive?
proper nova
so function > 0
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
yes
worn mango
**1 divided by 0 equals Infinity**
wait what
proper nova
now it's your turn to consider the signs of $(x - 15)^7$
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
**1 divided by 0 equals Infinity**
so what happens if $8 < x < 15$?
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
here is when we consider the case of $x > 8$
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
so function is -ve
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
yes
proper nova
now $x = 15$ is not satisfying the function
worn mango
yeah
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
so $x > 15$?
worn mango
cause it would lead to indefinite form
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
**1 divided by 0 equals Infinity**
function is +ve
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
yeah
proper nova
so to sum up, what's the solution set?
worn mango
wait let me write the solution set
(- infinity, 0) group (0,8] group (15, infinity)
this?
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
lemme check the answer in my book
it says that {-2} is also a solution
singleton set
proper nova
oh yeah we forgot $(x + 2)^6$
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
my bad
worn mango
how is -2 lso a solution?
proper nova
what's $(x + 2)^6$?
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
**1 divided by 0 equals Infinity**
.
proper nova
**1 divided by 0 equals Infinity**
$(x + 2)^6 = 0$ when $x = -2$
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
yes
proper nova
and $(x + 2)^6$ is in the numerator
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
yea
worn mango
thanks man
proper nova
so ${-2} \cap [2, 8] \cap (15, +\infty)$
jolly parrotBOT
1 divided by 0 equals Infinity
worn mango
wrong set symbol bro
proper nova
also i accepted your friend request
worn mango
but yeah
worn mango
also i accepted your friend request
thanks
proper nova
im not used to this ranges thing
worn mango
im not used to this ranges thing
i had to ask something related to how inequalities work
proper nova
i had to ask something related to how inequalities work
lol
pearl pondBOT
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Available help channel!
elfin geyser
I do not get part iii
pearl pondBOT
I do not get part iii
elfin geyser
like I cannot get an answer
pearl pondBOT
@elfin geyser Has your question been resolved?
sharp smelt
3(c)
sharp smelt
Forward direction first
I was thinking $\sum_{n=0}^{\infty} \left( \sum_{k=0}^{n} a_kb_{n-k} \right) x^n=1$
jolly parrotBOT
wai
sharp smelt
now the cofficients of all terms but the constant term should be 0
so $\sum_{n=0}^{0} \sum_{k=0}^{n} a_k b_{n-k} x^n=1$?
jolly parrotBOT
wai
jolly parrotBOT
wai
pearl pondBOT
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Available help channel!
dense blade
yeah im here
when we divide by 0 the answer is infinite 0
I don't know why people say smt else like
iron basin
.close
pearl pondBOT
sharp smelt
when we divide by 0 the answer is infinite 0
I don't know why people say smt el...
it isn't
iron basin
bruv
warped violet
Infinite 0?
sharp smelt
You can explain why using field axioms ( the reals form a field)
maybe that's overkill , but it works
sharp smelt
division is just multiplying by an elements' multipliactive inverse
it can be shown 0 has NO multiplicative inverse in R
warped violet
It is bruh if you understand division it is
It is not
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Available help channel!
fading nexus
For number 5 am I right to say every thing cancels out? And if so what would be the product
merry carbon
For number 5 am I right to say every thing cancels out? And if so what would be ...
They do cancel out, sure 
and do you have an idea of what the product would become? (if I gave you a fraction such as a/a, do you know what that becomes?)
fading nexus
They do cancel out, sure <:SCgoodjob2:750436894915166321> and do you have an ide...
1?
Kk ty!
How do I close the channel I forgot the command
merry carbon
(sorry I had to deal with the spambot 
)
merry carbon
How do I close the channel I forgot the command
You just type .close 
fading nexus
Welp I got another question just went to check if this one is correct since I am not confident. And I don’t want to do the rest of the problems the wrong way and have to redo them alll
merry carbon
,rccw
jolly parrotBOT
cedar sentinel
hint: find the hypotenuse first
then use the ratio definition of each function
you don't need to know what theta is
merry carbon
Welp I got another question just went to check if this one is correct since I am...
(What they're asking you to do for that part of the question is to find all of these, you don't need to (and shouldn't) find the angle theta itself)
merry carbon
you don't need to know what theta is
Sniped 
cedar sentinel
fading nexus
hint: find the hypotenuse first
Wait I thought we where finding the angles
I’m confused 😭
I just don’t understand the wording of the text book
cedar sentinel
you want the values of the six functions
not theta
you were never asked to find theta
fading nexus
Wdym by the values
cedar sentinel
,rccw
jolly parrotBOT
fading nexus
Wdym by missing the theta?
cedar sentinel
remember, never give a pedantic examiner the chance to dock your marks
fading nexus
What’s that?
cedar sentinel
wdym what's that
north pulsar
basically, assume that your examiners are going to find every fault with you
merry carbon
What’s that?
(Someone who's a pain in the ass, who gets what you mean, but will still say "nah" because they can)
fading nexus
I don’t know what it is or at least don’t know the word for it
oh wait
I reread the message nvm
I am also confused on what I should be even doing here
cedar sentinel
,rccw
jolly parrotBOT
cedar sentinel
same concept as q5
draw your triangles, then use the ratio definitions of the trig functions to map the numerator and denom to two of the three sides of the triangle
then do your thing
once you get skilled enough, you can skip the drawing and just note that sin = opp/hyp, for instance, and then assign the lengths as such
fading nexus
draw your triangles, then use the ratio definitions of the trig functions to map...
So like this ?
merry carbon
,rccw
jolly parrotBOT
cedar sentinel
allie please rotate your images hahah
my neck is about to become a right triangle at this rate
yes that's the idea
cedar sentinel
thank you for your contribution mr oppenheimer
merry carbon
(a tiny comment, some of your handwriting here is a little bit hard to read, so it may be worth being a bit careful to make sure it's clear to everyone, lest you get that one mean person grading you 
)
bronze heath
(a tiny comment, some of your handwriting here is a little bit hard to read, so ...
handwritng on this server...
fading nexus
My phones on its last legs and it lagging a ton 😭
cedar sentinel
if that was a typo I'm kinda curious what you meant to type lol
fading nexus
if that was a typo I'm kinda curious what you meant to type lol
Sorry
I think maybe it is because of the r and t being next to eachother
The keyboard has also been messed up since the new iOS 26
fading nexus
Is there anyway to break down the square root of 130? For example of you have the square root of fifty you can do the square root of 25 • 2 ( this could be wrong )
uneven cypress
Try writing the prime factors of 130 @fading nexus
fading nexus
I don’t know the prime factors by heart 😭
fading nexus
are prime numbers the ones that can be divided by the same number?
Or multiplied
Like 3•3 = 9
shut vortex
are prime numbers the ones that can be divided by the same number?
No, they’re numbers who can only be divided by 1 and themselves
uneven cypress
Like 3•3 = 9
so the prime factors of 9 are 3 x 3, for 10, it would 2 x 5, and so on
viscid shale
are prime numbers the ones that can be divided by the same number?
Prime Numbers are those which can only be evenly divided by themselves and one.
Composite Numbers are those which are formed by the multiplication of prime numbers (which we usually call the prime factorization)
fading nexus
so the prime factors of 9 are 3 x 3, for 10, it would 2 x 5, and so on
Ohhh ok
For this one I think I did the process correct but I do not know if I was allowed to use the given angle. Or if I should of found the other one that is touching both the hyp and opp side
cedar sentinel
nothing wrong here, except what is that symbol in the first line used in the argument of sine
fading nexus
nothing wrong here, except what is that symbol in the first line used in the arg...
a badly written theta 💀 I just wrote that for me to remeber what sin looks like
pearl pondBOT
@fading nexus Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
rustic ice
In the attached image you will see the problem and attempts to solve. I am lost how to continue. Really do not get the bare concept of ‚variation of constants‘. My main question is how to use it and what it does. I have basic problems in understanding.
pastel umbra
What's the problem here though?
Because, I mean, you've already answered the question in the image on the first line:
You might have forgotten to attach the question-part (b)
pearl pondBOT
@rustic ice Has your question been resolved?
kind herald
Can someone help me figure out what formula I need to solve this problem?
I want to find the "opportunity cost" of 3 variables
Each variable can take a step, when they do, they increase
A increases by 10
B by 5
C by 2
I want to assign a "cost" value to each variable so I can find out, given 3 values, how many "steps" were taken.
Example:
A 100
B 25
C 8
steps:
A took 10
B 5
C 4
total 19
In this case I can just use the value of each step for this "opportunity cost", but if I wanted to add a $ cost of taking each step for each variable, I couldn't do that
kind herald
sorry if this is confusing, I'm very confused too
pearl pondBOT
@kind herald Has your question been resolved?
somber adder
Can someone help me figure out what formula I need to solve this problem?
I wa...
Can you state this more precisely?
digital summit
Does this residual plot not match well with the data because it has a curvature?
digital summit
does this mean that the data should be x^2 or higher in the equation?
shut elm
The residual plot matches well because about half the residuals are above 0, and about half the residuals are below 0
When analyzing residuals you want to look for even distribution and not too much clustering
river hazel
Does this residual plot not match well with the data because it has a curvature?
it kind of shows non-linearity, but the dataset is too small to really judge correctly
digital summit
When analyzing residuals you want to look for even distribution and not too much...
ohhh I get it! Thank you :)
digital summit
it kind of shows non-linearity, but the dataset is too small to really judge cor...
thanks for letting me know, i'll take note of both of your responses :)
digital summit
The residual plot matches well because about half the residuals are above 0, and...
btw how close to zero is considered generally accurate and whats too far?
shut elm
Definitely like c2b7 said you'll notice if the points continued that way they would start going below 0 and never come back (non linearity), or if they oscillate around (also non linearity)
shut elm
btw how close to zero is considered generally accurate and whats too far?
haha it's statistics so it's all up to looks
I actually am not sure what they taught us in our class
Maybe below 0.1ish
digital summit
haha it's statistics so it's all up to looks
xD true
maybe 🤷♂️
im going ahead in my class so I can learn geometry
thank you both so much!
shut elm
im going ahead in my class so I can learn geometry
Good luck! any questions you can always come here 👍
!done
pearl pondBOT
If you are done with this channel, please mark your problem as solved by typing .close
digital summit
.close
sharp smelt
pearl pondBOT
sharp smelt
I don't really get what $\mathcal{K}_{i}$ is
jolly parrotBOT
wai
toxic lichen
it's the same thing as in part a but now you are considering all conjugacy classes at once
sharp smelt
it's the same thing as in part a but now you are considering all conjugacy class...
But aren't conjugacy classes defined wrt a certain element
toxic lichen
wdym
sharp smelt
toxic lichen
conjugacy in G is an equivalence relation. you were able to deal with one single such class in part a. now all you're doing is looking at all of them partitioning G
you just aren't fixated on any specific representative for any class
sharp smelt
Okay , so we just are looking at the set of conjugation calsses
toxic lichen
well yes, the collection of all conjugacy classes is {mathcal K_i : i = 1,...,r}
sharp smelt
I see
naive zinc
Technically, sum of (1_R times element of K_i)
sharp smelt
Technically, sum of (1_R times element of K_i)
I mean they are the same thing, no
by distrbutivity of rings
naive zinc
Your (sum of element of K_i) by definition is already my sum of (1_R times element of K_i), multiply by 1_R outside isn’t necessary
naive zinc
Sure
You mean (b)?
(Σx: x in K_i itself is only an abbreviation, it actually is Σ 1_R x: x in K_i, the latter is well defined)
pearl pondBOT
pearl pondBOT
Available help channel!
sharp smelt
pearl pondBOT
bitter herald
what is your question
sharp smelt
I want my proof checked
bitter herald
ok go for it
waxen agate
Go for it
sharp smelt
Chose $N$ such that $n,m≥N \implies \abs {a_n}< \varepsilon; \abs{a_m}< \varepsilon$.
\Then $\abs{s_m-s_n} = \abs{a_{n+1}-a_{n+2}+ \dots -a_{m}}≤ \abs{a_{n+1}+a_{n+3}+ \dots +a_{m-1}}+\abs{a_{n}+ \dots + a_{m}}< 2m \varepsilon$
jolly parrotBOT
wai
spare lark
Is it not a problem that it depends on m ?
tropic saddle
you havent used that the sequence is monotone
so your proof cannot be correct
(exercise: find an alternating series which diverges because the sequence is not monotone)
sharp smelt
you havent used that the sequence is monotone
so basically a_n converging isn't enough
tropic saddle
yes
sharp smelt
(because that's what this proof implictly assumes)
I'll continue working on this
thanks
.close
crimson nebula
Let m,n € N and m>n .
Show that there does not exist an injection from Nm into Nn.
pearl pondBOT
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
Okay, what did you assume for your contradiction?
crimson nebula
Let there exists an injection from Nm into Nn
There was another theroem which might me helpful in proving this
If S is a finite set , and A is a proper subset of S . Then there is no bijection from S onto A
shut elm
Let there exists an injection from Nm into Nn
Where m >= n sure
shut elm
If S is a finite set , and A is a proper subset of S . Then there is no bijectio...
Thats a nice theorem, but no bijection doesn't necessarily mean no injection
Let's think about this from the definition and try to gain an intuition
What's the definition of injection?
crimson nebula
Where m >= n sure
m>n
shut elm
m>n yes* sorry
crimson nebula
Injection means every element in domain has a unique image in codomain
lilac ocean
Let m,n € N and m>n .
Show that there does not exist an injection from Nm into...
i love the usage of the euro symbol for element of 😭
shut elm
That means that for every element of the domain, how many elements of the codomain must there be?
crimson nebula
One exactly
random ermine
no
lilac ocean
not in an occupied help channel
shut elm
One exactly
Yes, good
random ermine
Let m,n € N and m>n .
Show that there does not exist an injection from Nm into...
is Nm N^m
shut elm
So if there's one element in the codomain for every element in the domain
And we started with m elements in the domain
crimson nebula
Nm ={1,2,3,.....,m}
Subset of N
shut elm
How many must be in the codomain
shut elm
Yes
crimson nebula
One common element in codomain for all..?
shut elm
The element has to be unique since its injective, remember?
midnight haven
Wrong channel, sorry
shut elm
So each one in the domain has a unique one in the codomain
There are m in the domain, there must be _ in the codomain
crimson nebula
Yes so minimum no of elements in codomain should be equal to no of elements in domain
crimson nebula
Sir , i respect your help
But actually these statements won't constitute a rigorous proof
shut elm
Right, now we have to translate our intuition into a proof
shut elm
We understand what we're doing and how we want to do it, so thats the last step
Let's just change all of our statements to math
crimson nebula
Sure
shut elm
Let there exists an injection from Nm into Nn
You have the first part
Let there be an injection f from Nm to Nn, where m > n (feel free to write this as you please)
shut elm
Injection means every element in domain has a unique image in codomain
Now because f is an injection, for every a in Nm, f(a) in Nn is unique
Do you follow so far
crimson nebula
Yes
shut elm
Good
shut elm
Yes so minimum no of elements in codomain should be equal to no of elements in d...
Therefore since there are m unique elements in Nm, there are at least m unique elements in Nn, because f(a) in Nn is unique for every a in Nm.
(There are many ways to write this, if you are familiar with image notation you could use that as well)
shut elm
So can we have m > n?
But m elements in Nn contradicts n < m, so f cannot exist 
shut elm
Therefore since there are m unique elements in Nm, there are at least m unique e...
Other ways to write this would be
f(Nm) is a subset of Nn
f(Nm) has m unique elements, so n >= m (because its a subset) - which gives us our contradiction
shut elm
Great!
In general this is called the pigeonhole principle
I cannot fit m unique elements (pigeons) into n slots (holes) when n < m
!done
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shut elm
Too many pigeons for holes
pearl pondBOT
pearl pondBOT
Available help channel!
tepid pebble
.close
pearl pondBOT
Available help channel!
bronze heath
claim
bronze heath
shut elm
I would think so
bronze heath
why
shut elm
Actually now I wouldn't think so
Maybe a simpson's paradox but in any case even though C always happens with B there may be (a majority of) cases of C without B, such that the inequality is reversed in those other cases
proper nova
Actually now I wouldn't think so
add me
shut elm
added
bronze heath
Maybe a simpson's paradox but in any case even though C always happens with B th...
hmm..
makes sense
shut elm
$P(B | A) = \frac{P(B \cap A)}{P(A)} > P(B | A^c) = \frac{P(B \cap A^c)}{P(A^c)}$
$P(C | B) = 1$
$P(C | A) = \frac{P(C \cap A)}{P(A)} = \frac{P(C \cap A)}{P(B \cap A)}\frac{P(B \cap A)}{P(A)} > \frac{P(C \cap A)}{P(B \cap A)}\frac{P(B \cap A^c)}{P(A^c)}$
jolly parrotBOT
Coolempire93
shut elm
Inconclusive?
shut elm
Applying the identity in the first line
bronze heath
hmm..
but we have counter
wait
but we had to compare cintersectiona/anot
idon't think if its the same
@shut elm
shut elm
but we had to compare cintersectiona/anot
Right and that's the thing
shut elm
Yeah
pearl pondBOT
shut elm
Np
shut elm
Prove that it's not necessarily true?
By contradiction naturally
We can easily produce such a dataset
pearl pondBOT
Available help channel!
solid torrent
In the below question how do I plot the points and get the intervals on the number line after getting the critical points?
solid torrent
<@&286206848099549185>
pearl pondBOT
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.
uneven cypress
,w plot \sin 3x - \cos 3x where 0<x<\pi
jolly parrotBOT
uneven cypress
@solid torrent do you see a pattern here, particularly what is different is the increasing and decreasing regions
solid torrent
This is how I worked on it till now..
solid torrent
<@1399760013068599369> do you see a pattern here, particularly what is differen...
I do see a pattern and now the positive and negative interval makes sense.. but how am I supposed to solve this without plotting the sin3x-cos3x graph cuz i dont know how to plot it
uneven cypress
Well the slope is positive for increasing intervals. Does this make sense? @solid torrent
solid torrent
Well the slope is positive for increasing intervals. Does this make sense? <@139...
Yes
solid torrent
It says strictly increasing and decreasing so the points are required where the slope changes
So I won't be able to solve this wo plotting this graph?
uneven cypress
So I won't be able to solve this wo plotting this graph?
You can find the derivative
solid torrent
You can find the derivative
Yes alr
uneven cypress
Observe the slope at the points where it is stagnant, ie peaks and troughs
These points lie in neither interval
I think you can attempt it now
uneven cypress
Yes they r on the x axis
No wait
Along the x axis the function is still decreasing
Or increasing
uneven cypress
Observe the slope at the points where it is stagnant, ie peaks and troughs
@solid torrent read the question again and retry this part
Again, look at the graph and observe the slopes
tranquil ridge
ts too advanced for me
solid torrent
uneven cypress
Yes
unique viper
Hint: ||How can one write an interval without its endpoints?||
solid torrent
Again, look at the graph and observe the slopes
When I see the graph I can see the increasing and decreasing interval.. but i have to do this question in exam and I wouldn't have any resource to check the graph for sin3x - cos3x.. so how do I do it then?
solid torrent
U want me to calculate the slope?
frozen bluff
no need to plot just do a derivative sign chart
uneven cypress
no need to plot just do a derivative sign chart
Yeah that's what I am explaining to OP
solid torrent
2.5?
solid torrent
no need to plot just do a derivative sign chart
😭
uneven cypress
Think
solid torrent
Shouldn't it be 0?
Wait how
uneven cypress
The change in y is 0 for a flatline
solid torrent
y2 -y1
solid torrent
So 1.5--1
solid torrent
The red lines you drew
Ohhh tht
uneven cypress
Yeah tangent
solid torrent
Yeah the slope is zero there
uneven cypress
So whenever the derivative is 0, it lies in neither interval
solid torrent
So whenever the derivative is 0, it lies in neither interval
Understood tht
uneven cypress
When derivative is positive, is it increasing/decreasing
uneven cypress
When derivative is positive, is it increasing/decreasing
Answer this
uneven cypress
Yes
solid torrent
Decreasing
uneven cypress
So is that positive or negative?
solid torrent
Positive
solid torrent
YEAH OK
uneven cypress
Note that the derivative is defined as the rate of change so this makes sense
If velocity is positive for example, then the displacement must grow. Ignore if you don't get this, it's a physics example
I think you got it now
do the opposite for negative because yes
solid torrent
no need to plot just do a derivative sign chart
Could u please help
uneven cypress
Well what I explained works without the graph
It's the same as a derivative sign chart
The Principle above like slope is 0 when it is neither is true for all functions @solid torrent
I gave the graph so you can understand better bc its your first time
solid torrent
Ty for that... I appreciate it.. but now I need something easy and fast for the answer
They ask me interval
And I need to provide intervals in the answer
frozen bluff
when we said slope we meant the derivative
frozen bluff
Could u please help
compute the derivative find where it is zero and then check whether the derivative is positive or negative in each interval
pearl pondBOT
@solid torrent Has your question been resolved?
sharp smelt
how are the fibres additive cosets
pearl pondBOT
how are the fibres additive cosets
sharp smelt
I feel like I'm missing some intution
shut elm
Ah this should have been one of the first things they showed
tropic saddle
this is something you should know from group theory
waxen agate
Same lol
shut elm
Think along the lines of $\phi(r + I) = \phi(r) + \phi(I) = \phi(r) + e_G = \phi(r)$ so the fiber of phi(r) is r+I
sharp smelt
this is something you should know from group theory
lemme see if I can recall it
jolly parrotBOT
Coolempire93
shut elm
Put spoiler on that image
tropic saddle
frankly you should even know it from linear algebra
sharp smelt
I think the use of fibres is making me take time to process this
tropic saddle
its like the first thing you prove about any homomorphism of any kind
sharp smelt
Okay, yea, makes sense
shut elm
I think the use of fibres is making me take time to process this
What's your intuition of fibers
Or knowledge/definition I guess
That may be what holds you up
sharp smelt
What's your intuition of fibers
First time I'm using them extensively is today, so none
shut elm
It's funny our book never even introduced them
They just introduced the inverse image and called it a day
sharp smelt
Or knowledge/definition I guess
the set of pre-images of a certain image
shut elm
It's funny our book never even introduced them
I did a control f for fiber and there was nothing 😂
shut elm
the set of pre-images of a certain image
Exactly yeah
In non math terms
The set of elements in $S$ such that $\phi(\text{element}) = \text{whatever}$
jolly parrotBOT
Coolempire93
shut elm
Is the fiber of whatever
tropic saddle
the idea that phi(something + (anything from kernel)) = phi(something) is fundamental
which is really just this fact in other words
shut elm
(we can even reword the definition of kernel to be "fiber of the identity element")
The set of elements that map to identity
The nullspace
etc.
jolly parrotBOT
Coolempire93
sharp smelt
the idea that phi(something + (anything from kernel)) = phi(something) is fundam...
the use of fibres tripped me 😭
shut elm
Actually if you want my book does have a few find the fiber problems if you want to try them
sharp smelt
I'll stick with D&F for now if that's fine
shut elm
I got too excited 😆
I just remembered the professor put one on the test like find $\phi^-1(4)$ for $\phi(x) = x$, $\phi \colon \bZ \to \bZ_{20}$ and it annoyed me
jolly parrotBOT
Coolempire93
pearl pondBOT
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royal harness
how do I do partial fraction decomposition in this
pearl pondBOT
how do I do partial fraction decomposition in this
versed mica
how do I do partial fraction decomposition in this
you have a quadratic and a linear do you know how to deal with the quadratic?
royal harness
you have a quadratic and a linear do you know how to deal with the quadratic?
no we haven't done integrals yet
versed mica
go to paul’s online math notes then if you’ve never seen the topic
royal harness
go to paul’s online math notes then if you’ve never seen the topic
oh thanks this is such an interesting website imma check it out !
.close
pearl pondBOT
cursive wraith
😭
i'm a minor dw
Sorry but after looking at your bio I have to ask you to change it to something appropriate. Even if you're a minor and even if it's a joke, it's not a cool one
royal harness
Sorry but after looking at your bio I have to ask you to change it to something ...
oh yeah sure ig i didn't think it through
jolly parrotBOT
cursive wraith
oh yeah sure ig i didn't think it through
thank you
royal harness
IT'S A COOL PFP
versed mica
ain’t no party like a diddy party
royal harness
it' showing occupied
royal harness
takes some time
ohh alr
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Available help channel!
rocky crow
if ther are 11 grades and to get minimum of 4,75 as average, how many grades can you get for 5 in a 1-6 grade system?
uneven cypress
!xy would help @rocky crow
pearl pondBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
versed mica
if ther are 11 grades and to get minimum of 4,75 as average, how many grades can...
assume there are n grades that get a 5 and assume the worst, that the remaining grades are a 1
so 11 - n get a 1
should be clear how to continue
versed mica
😭
uneven cypress
just calculate for me
!noans
pearl pondBOT
just calculate for me
The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.
versed mica
last moth
if ther are 11 grades and to get minimum of 4,75 as average, how many grades can...
Do you have a screenshot of the problem as originally worded? It's kinda hard to tell what you mean from this
Oh nvm I understand what you're asking here
shut elm
if ther are 11 grades and to get minimum of 4,75 as average, how many grades can...
Interesting
last moth
So as knief said, let's call the number of 5's n
and let's assume all the other grades are 1
so you'd get grades like 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1 for example
how would you calculate the average in that scenario?
pearl pondBOT
@rocky crow Has your question been resolved?
rocky crow
ya
graceful sigil
j
k
k
j
j
j
jj
j
j
j
j
j
j
j
j
jj
j
j
j
jj
j
j
j
sharp vigil
.close
clear shore
pearl pondBOT
pearl pondBOT
@clear shore Has your question been resolved?
mortal imp
a hint would be awesome, thanks
try pairing in summation
smallest of a series with largest of b
clear shore
try pairing in summation
smallest of a series with largest of b
That would give the largest sum which is 1012^2, on the other hand the smallest sum is 1012. But I'm stuck here
mortal imp
i think you misunderstood me
when i said paring i meant pairing a1 with b1 a2 with b2 and so on
see it like this , take an element k in a set then we can have 2025-k in other one in that way the summation can be written as
k-(2025-k) which sinplifies to 2k-2025
now see to it that there is a mod
so instead og going for 1012 terms
we go for 2024
and then again we convert it into 1012 sum and multiply it by 2
clear shore
i think you misunderstood me
when i said paring i meant pairing a1 with b1 a2 wi...
I have hard time digest this, wdym by "in other one"
mortal imp
I have hard time digest this, wdym by "in other one"
take a series which consists a1 a2 a3 and so on and similarly there is a b series
now i picked a number k from a series
clear shore
and?
mortal imp
now for the question i pick number 2025-k
clear shore
okay?
mortal imp
so summation of |ai-bi|
becomes |2k-2025| where k goes from 1 to 1012
but since there is a mod and a sign change occurs
i wrote it as 2*summation of -(2k-2025) where k is from 1 to 1012
clear shore
so summation of |ai-bi|
becomes |2k-2025| where k goes from 1 to 1012
but since ...
You can't ensure that k and 2025-k paired together
bronze heath
You can't ensure that k and 2025-k paired together
you can
clear shore
How?
bronze heath
previous conditions
clear shore
How can we ensure that ai=k and bi=2025-k
clear shore
previous conditions
Can you elaborate
bronze heath
they say the sequences are a permutation of S so we can say they a1....a1012,b1...b1012 contains each element of S once
clear shore
yeah ik
bronze heath
then the previous condition says ascending and descending order, for any combination of 1012 elements there would be only one ascending and 1 descending order
clear shore
yes
clear shore
I'm not even sure if k and 2025-k would be in different sequence
bronze heath
they would be for 1 specific
pearl pondBOT
@clear shore Has your question been resolved?
naive zinc
I think if you can show that switching two elements won’t change the result it’s solved, any such partition works, 2024C1012
I mean 2024 length sequence of 0, 1. j-th element is 0 when j is in A, is 1 when j is in B. I think we should try to prove that switching any adjacent 0 1 won’t change that sum of | |
I tried for smaller number, like 4 instead of 2024, it seems to be correct
clear shore
I think if you can show that switching two elements won’t change the result it’s...
like, switching ai with bj or just switching order of a pair in the sum by that I meant ai and bi
naive zinc
It involves some other elements too, so I am not sure how to proceed
clear shore
Hmm, I'll try
They gave us 15min to do this problem btw, and it has taken 2 hours of my life so far
naive zinc
Oh
order doesn’t change this way
Adjacent 0 1 if this 0 is the p-th 0 from left, 1 is the q-th 1 from right
Swap this two 1 0, 0 is still the p-th from left and 1 is still the q-th from right
So simply |k-j|+|k+1-i|=|k-i|+|k+1-j| for any i, j that don’t equal k or k+1
|k-(i-1)|+|k-j|=|k-i|+|k-(j+1)| since (i, j+1) is just (i-1,j) shift right
(This 0 1 appear at k, k+1 . p-th 1 from right is at j, q-th 0 from left is at i)
For any 2n (your case 2024), the sum thus remains n^2, so done, binom(2n, n)
naive zinc
So simply |k-j|+|k+1-i|=|k-i|+|k+1-j| for any i, j that don’t equal k or k+1
As long as k k+1 are inside [i, j] though, this part I haven’t guaranteed
Oh I have completed, say p<=q, then the position is of this relation: k k+1 i j
(j-k)+(i-k-1)=i+j-2k-1=(i-k-1)-(j-k)
clear shore
As long as k k+1 are inside [i, j] though, this part I haven’t guaranteed
Ig well we could ,cuz |k-j|=|j-k|, pair it with |j+1-k| if that's not the case
Oh wait
No
naive zinc
Ig well we could ,cuz |k-j|=|j-k|, pair it with |j+1-k| if that's not the case
Turns out it’s always outside but I showed it’s still invariant
0 1 at k k+1, this 0 is p-th 0 from left, this 1 is the q-th 1 from right
q-th 0 from left is at i, p-th 1 from right is at j
Then
position relation when p<q:
k<k+1<i,j
position relation when p>q:
i,j<k<k+1
position relation when p=q:
j=k, i=k+1
clear shore
Hmm, this's some kind of "invariable" problem isn't it
naive zinc
p>q: (j-k)+(i-k-1)=(i-k)+(j-k-1)
p<q: (k+1-i)+(k-j)=(k+1-j)+(k-i)
p=q: obviously 1=1 unchanged
naive zinc
Hmm, this's some kind of "invariable" problem isn't it
Yeah. I guess we just need to find out the fact that it is invariant as soon as possible in a test
clear shore
0 1 at k k+1, this 0 is p-th 0 from left, this 1 is the q-th 1 from right
q-th 0...
That's just a different way we use to choose k,i,j so that i<k<k+1<j doesn't happen isn't it
naive zinc
No. Case by case there isn’t such case
naive zinc
0 1 at k k+1, this 0 is p-th 0 from left, this 1 is the q-th 1 from right
q-th 0...
Only these
clear shore
Sorry I was trying to do the problem my own after knowing that this's invariant kind of problem
clear shore
Only these
I mean using this apporach that case can't happen right?
I'm going through your sol after tried it my own
naive zinc
I mean using this apporach that case can't happen right?
Yeah. Can’t happen
clear shore
Gimme a min to digest all of them
Okay cool
One question
How am I supposed to spot this's invariant problem?
clear shore
Hmm okay yeah, I keep forget to try for small problem first
naive zinc
I just happened to notice some extreme cases 010101 , 000111, 111000, 101010 happen to have the same value, so guessed maybe it’s invariant
clear shore
This's like the second time a invariant problem appeared on the test btw
and I have taken many, should take some time to learn it? I mean some technique
naive zinc
I don’t know. I have never trained myself for competitions. I guess, stay sharp? And see more examples
clear shore
I don’t know. I have never trained myself for competitions. I guess, stay sharp?...
Thanks
for the help also 💙
.close
naive zinc
Np
pearl pondBOT
pearl pondBOT
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bronze heath
claim
bronze heath
Four of the eight vertices of a regular octagon are chosen at random. Then probability that these four vertices forms a square, equals
lilac ocean
claim
woah I love this problem…
anyway idk the factoid thingies, how far along are you
lilac ocean
perhaps, think of when a square is actually formed
bronze heath
anyway idk the factoid thingies, how far along are you
btw #helpers-info
bronze heath
perhaps, think of when a square is actually formed
idk
lilac ocean
btw <#1164751389034041374>
it’s ok I don’t rly care about it
bronze heath
hmm.. i found 2
bronze heath
no
lilac ocean
right
another way to rephrase this question is, what proportion of combinations of 4 vertices form a square
so perhaps, you can finish it off?
bronze heath
2/8C4
lilac ocean
:)
lilac ocean
supposedly in the same way, since you have the general idea, the problem becomes whether or not finding a square is possible and if so how
bronze heath
hmm.. gif [n/4]
lilac ocean
I’m assuming you mean 4|n?
bronze heath
greatest integer function
lilac ocean
I’m not sure what your notation means
oh I see
well, is it always possible?
how would you find a square for a pentagon
lilac ocean
I thought you were asking solely about squares, this is your question 😂
I can’t tell you what you mean
bronze heath
I’m assuming you mean 4|n?
btw what is this?
lilac ocean
np, but you should know that the answer is different even in the case of an octagon in this case
lilac ocean
btw what is this?
4 divides n; n is divisible by 4
bronze heath
np, but you should know that the answer is different even in the case of an octa...
ok
.close
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hollow sonnet
interesting question
Four of the eight vertices of a regular octagon are chosen at random. Find the probability that these four vertices forms a square.
hollow sonnet
seems like binomial expansion can just be done here with the probabilities
(same + not same)
(1/8 + 7/8)^4
to get (7/8)^4 chance it forms a square
1 out of 4096 cases is a point
28 out of 2096 cases is a line
294 is a triangle
nvm forgot to add the uhh
should be 7^4 + 4×7^3 cases?
vagrant trout
P = number of successful outcomes of forming a square / total number of possible outcomes
find total number of possible outcomes with combinatorics
hollow sonnet
3773/4096
bronze heath
eh guys appreciate but it is solved
vagrant trout
and for number of successful outcomes think logically, if you have an octagon, and you connected 4 vertices, in how many ways can you do it
hollow sonnet
and for number of successful outcomes think logically, if you have an octagon, a...
just need to subtract all the cases where it isn't a square
vagrant trout
just need to subtract all the cases where it isn't a square
but there's only 2 possible cases when it is a square lol and 70 total choosing any 4
why would you go the hard way
hollow sonnet
oh wait its a rectangle
for that
LMAO calculated rectangle
mb thats embarrassing
is the word quadrilateral
hollow sonnet
.close
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abstract pagoda
In this property or theorem or something
We dont say that the restrictions of the binary operation * on G in H
Is equal to *?
abstract pagoda
In the end why they said * is binary operation on H
Its like they change the whole domain and codomain of * (as function definition of binary operation)
unborn abyss
In the end why they said * is binary operation on H
where did they say that?
brave helm
that sentence just means "if two elements are in H implies the result from applying * is als in H, then H is closed under *"
ie the sentence after ie
it is just a verbal vs mathematical way of ohrasing it
frozen bluff
strictly speaking the binary operation on $H$ is the restriction $!\mid_{H \times H}$ but by standard abuse of notation we denote it again by ∗
abstract pagoda
where did they say that?
- is automatically associative (or commutative) on H
Is similar for me saying *is binary operation on H
jolly parrotBOT
anflo
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
unborn abyss
* is automatically associative (or commutative) on H
Is similar for me saying *i...
Hmm. If I define + to be an operation on R, (hint: it's the one you know and love),
then + is associative and commutative on Z, but is it a binary operation on Z?
abstract pagoda
Yeah
Its
Then if thats true
The restrictions of a binary operation * on G in H its equal to *
abstract pagoda
Hmm. If I define `+` to be an operation on R, (hint: it's the one you know and l...
Then there RxR=ZxZ
And also R=Z
unborn abyss
yes, we get a little bit lazy and instead of talking about $+\Z$ or $+\Q$ we just say $+$ and let context do the work for us
jolly parrotBOT
boughs of hayley 🪷
abstract pagoda
yes, we get a little bit lazy and instead of talking about $+_\Z$ or $+_\Q$ we j...
So this is an idiomatic expression?
unborn abyss
i suppose in some sense yes, but it's normally unambiguous
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sand swallow
can anyone offer help by solving this lim using MVT?
pearl pondBOT
can anyone offer help by solving this lim using MVT?
oak quiver
mean value theorem?
cinder flower
do we need to use mvt somehow?
waxen agate
Cauchy’s MVT?
sand swallow
mean value theorem?
yes
sand swallow
do we need to use mvt somehow?
i mean i just rather use it lol
lilac ocean
can you go WITHOUT using it?