oh well that definitely changes things lol
#help-27
cinder bobcat
sterile pewter
so in this case if I move it up, d(n) won't be going to zero, instead zero so I'm not so sure
sterile pewter
oh well that definitely changes things lol
yh my bad sry😭😭
I figured out the answer for a>=1, just not for a<1
cinder bobcat
if α is less than 1 then you'll need to do something else. can you tell what happens if α is negative?
just by directly looking at the limit you started with
sterile pewter
I think if alpha is negative, then theta/yn^alpha would go to infinity? so 1- that will also go to infinity?
sterile pewter
so it's absolute will go to infinity
and depending on n being even or odd it'll be positive/negative infinity
cinder bobcat
exactly
so the limit doesnt exist in that case
hard to immediately see what happens when α is between 0 and 1 🤔
sterile pewter
well if it's between 0 and 1, won't the fraction be going to zero, thus making the entire thing going to infinity?
cinder bobcat
well the fraction goes to 0, but then you have a (1-0)^infinity type of situation
which is indeterminate
sterile pewter
ahhhh ok makes sense
cinder bobcat
not sure what your teacher wants exactly
sterile pewter
I'm also a bit confused, because the subsequent question requires this to have a limiting distribution, so if that's the case it won't be having a limiting distribution for a<1
thanks a lot for the help @cinder bobcat
cinder bobcat
yeah sure thing, I guess for α in (0,1) you could prove monotonicity and boundedness, then use logs to find the limit if it exists
devout snowBOT
devout snowBOT
Available help channel!
shell laurel
can someone explain this sequence stuff
devout snowBOT
can someone explain this sequence stuff
upper bolt
can someone explain this sequence stuff
Where exactly do you need the explaining to begin - Are you familiar with what a sequence is? Were you able to graph it out yet?
carmine veldt
can someone explain this sequence stuff
ru familiar w stirling's approximation?
shell laurel
ru familiar w stirling's approximation?
no im not
shell laurel
Where exactly do you need the explaining to begin - Are you familiar with what a...
im familiar with sequences but using limits to find convergence and divergence i dont understand that well
upper bolt
im familiar with sequences but using limits to find convergence and divergence i...
Right. Do you know what it means if a sequence is convergent or divergent?
shell laurel
if it converges thene the limit exists and if not then it is divergent?
woven radishBOT
Donkey
woven radishBOT
Donkey
upper bolt
if it converges thene the limit exists and if not then it is divergent?
That's correct so far. Now if you take any sequence, say, a sequence with limit 2 for n->∞, what would you expect the values of the sequence to look like for very high n?
shell laurel
2?
upper bolt
Not exactly. They would be very close to 2
If it has a limit of 2 vor n->∞ it means that the higher your n is, the closer the sequence gets to 2
In fact it gets infinitely close to 2
You with me so far?
shell laurel
yes
upper bolt
yes
Great! Now you know, given a limit, what the sequence should look like. Do you think you could reverse it now and, given a graph of a sequence, see what the limit might be?
upper bolt
Yup 
upper bolt
just by looking at a graph and guessing?
That's the first step. Then you have to prove your guess.
shell laurel
hwo do u prove
upper bolt
hwo do u prove
Let's go step by step, did you graph it out and have a guess now - and could you show that?
shell laurel
devout snowBOT
@shell laurel Has your question been resolved?
lunar harbor
their intention was probably for you to graph out enough terms and notice that the sequence approaches zero
you can also do this more rigorously without stirling's
consider the ratio of successive terms
proud sparrow
can someone explain this sequence stuff
is it (9n)! or is it 9(n!)
lunar harbor
is it (9n)! or is it 9(n!)
It says 9n!, which is 9(n!).
proud sparrow
then the product of odd numbers till 2n-1 is just (2n)!/(2^n*n!)
so An would just be( 1/9*2^n) ( 2nCn)
devout snowBOT
@shell laurel Has your question been resolved?
thick ledge
Is it even possible to do this question without stirlings approximation
vale raft
hello, i'm trying to do this summ but i cant really get what's the method
vale raft
i clearly see that it's about the newton binomial
but i cant get it to work
i tried changing the variable to j=k+1
to get x^j
sand dove
but i cant get it to work
Hint: remember your calculus
vale raft
but i cant work around the k
vale raft
Hint: remember your calculus
idk what's calculus we don't have that where i am
we have maths
that's all
sand dove
like, derivatives
vale raft
oh okay
sand dove
does kx^(k-1) look like anything
vale raft
ooh yeah okay
it's the derivative of x^k
but how do i make it appear like it's a summ i cant just do the derivative
can i???
sand dove
but how do i make it appear like it's a summ i cant just do the derivative
the derivative of a sum behaves pretty well doesn't it
vale raft
i swtg i've seen it in class but i dont remember anything sorry lmao
sand dove
i swtg i've seen it in class but i dont remember anything sorry lmao
derivative of sum = sum of derivatives
olive snow
And two things equal have their derivatives equal
vale raft
derivative of sum = sum of derivatives
but i'd have to derivate k choose n dont i ????
untold ravine
but i'd have to derivate k choose n dont i ????
(hint: its a constant)
olive snow
What is k choose n here if its a polynomial expression
vale raft
(hint: its a constant)
how is it a constant if it's in the summ and changing when we go one up??
sand dove
but i'd have to derivate k choose n dont i ????
if the variable is x
then what's the derivative of k choose n with respect to x
vale raft
then what's the derivative of k choose n with respect to x
ooh yeah okay
so it's a constant regarding the derivative of the f function
okay
but how do i put it into the summ then??
like it's not just the summ of the x^k is it????
vale raft
oh no it's k choose n
- x^k
fck
i wanted to do a *
(n k) * x^k
and it's the summ from 1 to n
vale raft
then i just have to multiply by 1^n-k
vale raft
you can add the k = 0 term
and yeah i did that too when i was searching but it didnt work as i couldnt do it
olive snow
You don't have to, its already here
vale raft
You don't have to, its already here
wdym it's already there??
i mean to be rigorous i have to write it down dont i??
olive snow
You already have 1^(n-k) in the expression
vale raft
i think my teacher wants us to write the 1
sand dove
i think my teacher wants us to write the 1
fair enough
olive snow
i mean to be rigorous i have to write it down dont i??
Sure but you don't multiply by 1^n-k
sand dove
why
just write a = a * 1^(n-k) if you want to
but you're not multiplying by 1^(n-k)
you're replacing every implicit "1" by 1^(n-k)
olive snow
^
vale raft
yeah
i get you
but thanks
i'll have to ask for other things later lmao i have a test tomorrow and i understood nothing about summs and products
integrals are a bit easier that i get it
but once i calculated the summ, i have to do the primitive of what it gives me dont i??
and does this just gives (-2)^k ??
as i can just make the 1^n-k appear
and say it's (x-y)^n which is (-1-1)^n
hardy trail
and does this just gives (-2)^k ??
Not quite
hardy trail
as i can just make the 1^n-k appear
(-1 - 1)^n doesn't give this term :)
hardy trail
but once i calculated the summ, i have to do the primitive of what it gives me d...
Wdym by this
If you mean like taking the derivative and then integrating it to find a desired result, then yeah sure.
vale raft
Wdym by this
the one before, as i calculated it by doing the derivative
i should do the primitive
hardy trail
No, because you can get the desired result just by taking the derivative.
Like, the whole point is just to differentiate both sides; recognising that the derivative of x^k is kx^(k - 1).
vale raft
yeah but like
if i call this sum f(x)
i have f'(x) = sum from 0 to n of x^k * 1^(n-k) (n c k)
olive snow
Here its pure application of newtons binome
vale raft
i have f'(x) = sum from 0 to n of x^k * 1^(n-k) (n c k)
that i can calculate it
but like
im looking for f(x) not f'(x)
hardy trail
im looking for f(x) not f'(x)
Well...
vale raft
??
inland carbon
and say it's (x-y)^n which is (-1-1)^n
write it again on a piece of paper
hardy trail
The whole point of your question was to find: [ \sum_{k=0}^{n}k\binom{n}{k}x^{k-1} ] When you were doing that question, you were considering: [ f\left(x\right)=\sum_{k=0}^{n}\binom{n}{k}x^{k}=\left(1+x\right)^{n} ] The point is that when you differentiate each of the functions, you get:
[ f'(x) = \sum_{k=0}^{n}k\binom{n}{k}x^{k-1}=n\left(1+x\right)^{n-1} ]
This is already the desired result via taking the derivative. So there is no need to actually find any primitives afterwards.
vale raft
oh my bad i wrote -1 where it's +1
woven radishBOT
Redfern Station
vale raft
**Redfern Station**
ooooh yeah okay i did it the wrong way
i did the derivative while it was the derivative
vale raft
write it again on a piece of paper
and that it gives me 0 while i dont think it does 0??
inland carbon
why
vale raft
because it's -1+1
and i find it weird that an example gives 0
but if i had to write something i'd say 0
but it seems weird
inland carbon
did you try on small examples
vale raft
nope
but i'd say it's 0 as n c k is symetric
but it's not as if i try for n=1 it doesnt gives 0
inland carbon
n=0 is the only exception
vale raft
what about n=1
inland carbon
then 1 - 1 = 0
vale raft
it gives 0-n
inland carbon
are you sure
vale raft
then 1 - 1 = 0
0^0 is equal to 1 or undefined??
inland carbon
in combinatorics it's easier when 0^0 = 1
vale raft
i see i see
but i'd have to say that for n=0, it gives out 1 then
and then say it would always be 0
thanks then
now products yay
i cant bear it anymore lmao
btw i cant do alternate series anymore idk why
i dont remember how to do them
i simplified it to
(-1)^k * 2^k
and put in factor the 1/2^n
as idc much about that
inland carbon
it's just a geometric sum
vale raft
i dont have to do the odd and even thing??
inland carbon
no
vale raft
so it's the form of 1-q^n / 1-q
wait i think i just dont remember my power rules
is (-1)^k * 2^k just 1^k
inland carbon
no
inland carbon
yes
vale raft
sorry im slow
it's getting late and all but i need to know the basics lmao
i cant have less than 10 to my test
which will be difficult given how far i am from the level they're waiting me to have
but then idk how to do that
i know for the first one i could start at 2
but can i just divide by 2 after that??
inland carbon
you need to "parametrize" each term of the sum
for example, if k is even you can say k=2m for m in [[1; n]]
inland carbon
after the you just need to calculate
vale raft
so i just like
divide by 2 the n
the 2n i mean
and say k = 2k ?? or something written a bit better
inland carbon
write it then we'll see
vale raft
on my copybook or here ???
ig on my copybook
oh i get it
is it not just 2 times the sum from 1 to n of k??
inland carbon
yes
vale raft
okay yeah
i wrote it down already
i should demonstrate it like step by step if i do it tomorrow
like writing out each number
and for k odd now
i'm at 2n-1
inland carbon
it depends on what your prof. wants
vale raft
it depends on what your prof. wants
he told us it's changes nothing if we write out the terms
we can
it's allowed even for the concours
inland carbon
general rule is that at the start of the exam, you should give a little more details but at the end you can go fast
do it if you want then
vale raft
and for k odd now
1+3+5+7...
inland carbon
i'm at 2n-1
use the properties
vale raft
use the properties
i mean i'd like to
but i cant really divide by 2
but i think we just wrote n-1
when doing it in class
inland carbon
split the sum into sum (2k) - sum (1)
vale raft
nah it's wrong
i thought about that and gave up on it lmao
i feel so dumb
and now just one or two product things ig
thanks tho
.close
devout snowBOT
vale raft
what's ur problem??
restive river
If it’s a fracture when graphing what number do I go up by
devout snowBOT
If it’s a fracture when graphing what number do I go up by
brisk raptor
If it’s a fracture when graphing what number do I go up by
a fraction?
restive river
what fracture, what are you trying to graph?
1/4x
winter patrol
if you want to work with integers,
then for this, use multiples of 4 for your x values
or w/e gives you something nice enough to plot on your grid
next saddle
if you want to work with integers,
then for this, use multiples of 4 for your x ...
do you need mults of 4? wouldnt that be for something like 4/x
i think any integers work, no?
restive river
So like do I got up four then over one
next saddle
well i would make a table of values
winter patrol
other way around
next saddle
for x=1, f(x)=_, for x=2, f(x)=_, etc
winter patrol
comparing: \ \
$\frac 14$ to $\frac{\text{rise}}{\text{run}}$ \
you'd want to go up 1 and the 4 to the right
woven radishBOT
ραμOmeganato5
winter patrol
(assuming you meant $y = \frac14 x$)
woven radishBOT
ραμOmeganato5
woven radishBOT
Mathemusician
restive river
ah, wait, is this $\frac{1}{4}x $ or $\frac{1}{4x}$?
First one
real sail
Hello, I need help with part 3 of this quadratic regression problem please!
real sail
winter patrol
how did you get from the left to the right
real sail
I went to a new area of the page to solve that part
winter patrol
yes, but i'm asking how you got the equation on the right
what rule/property were you applying
why did t just vanish
real sail
I wanted to get rid of the t, so I used the substitution property of equality to solve for t
to isolate the expression inside the parenthesis
winter patrol
what were you subbing for the value of t there?
sub property of equality requires you to have a known value for t or equivalent expression
which you didn't have yet
what i see there is that you're conflating \textbf{zero product property}
$$ab = 0 \implies a=0 \vee b=0$$
\red{but this doesn't apply for $ab = k$}
woven radishBOT
ραμOmeganato5
winter patrol
you have a quadratic equation, with values that aren't nice
so the best approach would probably be to arrange it into general form
then apply the quadratic formula
winter patrol
that property works since 0 * any number = 0
real sail
so applying the quadratic function, I get t = -88.46866465, t = -97.6619476
I feel like that's wrong because it's asking how long it will take the object to reach 18 meters, and I doubt it can be a negative time
(by x I meant t)
oh I entered it on the calculator incorrectly
the correct solutions are t = -1.515008824, t = 7.67827413
I will ignore the negative result
hmm it was still marked incorrect
winter patrol
why is your c 57
you didn't rearrange your equation to general form
you need to ensure that whenever you're applying
rules/identites/etc
where one side of the equation is 0
that your equation actually has that 0
here you determined when the height is 0
and not 18
real sail
how do I convert h = -4.9t^2+30.2t+57 to standard form? I thought it already was
winter patrol
first replace h with the desired height (in this case 18)
which you did in your first step
then you want to rearrange that to
stuff = 0
real sail
ahhhh
that's right
ok, now the solutions are t = -1.096362498, t = 7.259627804
I will ignore the negative and enter 7.260
that was correct, thank you for your help!
amber sluice
devout snowBOT
toxic grove
Why do you think that?
amber sluice
an cant be0 oor something
amber sluice
Why do you think that?
limit 5/n / 1 + 2 /n
0 / 1 + 0
but im thinking of something else i forgot which one
toxic grove
where it cant be 0?
It never equals zero (set it equal to zero and solve for n to see why) but that doesn't stop it from converging to 0
toxic grove
limit 5/n / 1 + 2 /n
Yep that's one way of looking at it
But it is a bit circular
Because 5/n tends to zero for exactly the same reason as 5/(n + 2)
So you have to find a different way of proving it
toxic grove
Not unrelated but it doesn't show what you want
toxic grove
What is the question?
toxic grove
Okay so this question is asking about sequences
toxic grove
But this chart is about series
Bit of a difference there
But series aren't very relevant for this question
amber sluice
But this chart is about series
ah okay
yeah i gotcha
i was going back to 9.1 to 9.6
last tihng was root and ratio series test
so im reviewing
so this one would be divergent
cause its infinithe
n+ 1 N! / n! cancel n! then n+1 , infinity + 1 - divergebt
think i got it thanks
.close
devout snowBOT
toxic grove
np
amber sluice
.close
devout snowBOT
Available help channel!
short hare
two players, lets call a and b, are playing a game where they have 12 chips on a table. player a marks one of the chips puts it back on the table (player b wasn't looking) player b marks a different chip with wasabi sauce and puts it back on the table (player a wasn't looking)
If they took rounds of eating 1 chip at a time (assuming player a wont choose the chip he marked, same for b)
what is the chance that they keep eating unmarked chips (unknowningly) until they're left with the last two marked chips
devout snowBOT
two players, lets call a and b, are playing a game where they have 12 chips on a...
short hare
the way i approached this...
mighty thicket
idk how to help u without giving the straight up awnser lmao, lemme try to come up with a text for ya
short hare
the player that's gonna start is going to have 11/12 chance of eating a non-marked chip no?
self doubting myself
mighty thicket
a non marked
short hare
oh yes osrry
mighty thicket
do you know combinatorial analysis?
short hare
absolutely not
the last piece of combinatorics i learned was like
2 years ago
im a highschool senior
short hare
what's that?
mighty thicket
oh were here then
okay i think i know how to explain it to you
if you have 1 banana and 1 apple
in how many orders can you eat them?
short hare
2
short hare
yes
short hare
in AC the OR sign means +
what
mighty thicket
1 of them has a banana and an apple
and the other one has a potato and a pineaple
in how many ways can you eat them?
short hare
are there rules for hte baskets
mighty thicket
(were gonna get to the chip problem im just explaining to you the basics of arranging then well get into probability into no time)
short hare
i mean 4! if not
mighty thicket
nice
now
what if
you gotta finish basket 1 before basket 2
so 2 independent sets
mighty thicket
i mean 4! if not
(thats a permutation if u didnt know what it was btw)
short hare
2 * 2?
mighty thicket
yes
nice nice
cause ur gonna eat the basket 1 AND basket 2
in this case it wouldnt vary, but if u would eat basket 1 OR basket 2 it be 2 + 2
im still thinking on how i get to combinations
never taught this to any1 before im sorry lmao
mighty thicket
ah you know AC
mighty thicket
okay so in your game do you want to consider player 1 eating the chip that he himself marked?
short hare
i was wondering where we stop
short hare
okay so in your game do you want to consider player 1 eating the chip that he hi...
nope!
neither will
mighty thicket
so u start at 10/11
short hare
they know where their chips are
mighty thicket
because ur disregarding that one chip
short hare
mhm
short hare
ahaaaa
so
this was a self asked question
from a yt short
where they played that game
mighty thicket
yeah i thought about that when i read the question lmao
short hare
and someone sent this
and when i calculated it
i got 11!/12! (mistake)
and i was wondering
how the hell did they even get that
so i assumed they're right and came here
mighty thicket
close enough lmao
mighty thicket
close enough lmao
was talking about ur first calculation lmao
short hare
(thats a permutation if u didnt know what it was btw)
i did learn permutations and combinations
just in arabic
so i didnt recognize the name
hence why i asked "what's that" inmstead of "no i didnt"
devout snowBOT
devout snowBOT
Available help channel!
rustic lynx
Qn 20
pseudo basin
no one has right hand neighbor same
... what does this even mean lol
devout snowBOT
@rustic lynx Has your question been resolved?
rustic lynx
devout snowBOT
rustic lynx
Qn 20
rustic lynx
> no one has right hand neighbor same
... what does this even mean lol
Like if their houses are in order a1 a2 a3 …. Then a2 can’t be right of a1
a2 a1 a3 is a acceptable sequence
eager cobalt
invictus?
eager cobalt
im a spotlight guy
rustic lynx
Ic
pseudo basin
Like if their houses are in order a1 a2 a3 …. Then a2 can’t be right of a1
so we're arranging the numbers 1 through n in a line and the constraint is k+1 cannot be immediately right of k for any k,
correct?
rustic lynx
so we're arranging the numbers 1 through n in a line and the constraint is _k+1 ...
Yes
devout snowBOT
@rustic lynx Has your question been resolved?
violet wind
You can generate most configurations for n+1 houses uniquely by taking every configuration for n houses and adding in the (n+1)th house anywhere except after the nth house
I'll let you calculate how many possibilities this accounts for exactly
The remaining cases are where removing the n+1th house would cause an invalid state, by smooshing a k and a k+1 together
A hint for counting these cases: ||remove the n+1th house, and merge the k and k+1th house into a single house||
This should give you a recursive definition for t_n, which it's hopefully easy to prove that D_n+1 + D_n satisfies
If someone finds a bijection I'll be impressed
rustic lynx
This should give you a recursive definition for t_n, which it's hopefully easy t...
I got the recursion but can’t see how to prove it equal to Dn + Dn+1
violet wind
Hmm, do you have a recursion for D_n?
rustic lynx
I have tn+1 = ntn + (n-1)tn-1
violet wind
Ok, that's t_n
rustic lynx
Hmm, do you have a recursion for D_n?
Dn = (n-1)[ Dn-1 + Dn-2]
violet wind
I feel like the inductive proof would follow pretty naturally
But I haven't checked it
Write t_n, rewrite using the t_n recurrence, rewrite that using the inductive hypothesis, and then I would expect the answer to rewrite neatly back to D_n+D_n-1 using the D_n recurrence
rustic lynx
Alr tysm
violet wind
Np
devout snowBOT
@rustic lynx Has your question been resolved?
keen turret
What's the mesh equation for the second loop?
devout snowBOT
What's the mesh equation for the second loop?
@keen turret Has your question been resolved?
dark talon
What's the mesh equation for the second loop?
You’re better off looking for help in the electrical engineering server as this isn’t exactly a math question
keen turret
<#406129911992942593>
ooh thanks
dark talon
You’re welcome
keen turret
didn't notice that
subtle ermine
devout snowBOT
sinful pier
what you wrote is equivalent to N
subtle ermine
i meant to write {{},1,2,3,4.....}
sinful pier
i meant to write {{},1,2,3,4.....}
union of {0} and {} is {0}
subtle ermine
{{},n:n is any natural number}
subtle ermine
union of {0} and {} is {0}
0 isnt a natural number?
sinful pier
0 isnt a natural number?
in the general case $X\cup\emptyset=X$
woven radishBOT
Estelle
subtle ermine
**Estelle**
yeah my bad i didnt write what i meant to
subtle ermine
i meant to write {{},1,2,3,4.....}
i meant this
is that the answer?
because the set of subsets of any set also contains the empty set
subtle ermine
the question
subtle ermine
technically what i wrote is correct, i checked N is the answer
sinful pier
you don't have sets of subsets
subtle ermine
but like why?
subtle ermine
you don't have sets of subsets
X was just any member of the set P(N)
sinful pier
what you wrote is technically correct because $\mathbb{N}\cup\emptyset=\mathbb{N}$
woven radishBOT
Estelle
sinful pier
X was just any member of the set P(N)
yk the definition of P(N)
subtle ermine
i think i get it
subtle ermine
yk the definition of P(N)
yeah
sinful pier
you can verify that $X\subseteq\mathbb{N}$
woven radishBOT
Estelle
subtle ermine
cuz like we do end up doing N U {} but thats just N
sinful pier
so $x\in X\Rightarrow x\in\mathbb{N}$, not $x\in X\Rightarrow x\subseteq\mathbb{N}$
woven radishBOT
Estelle
subtle ermine
**Estelle**
yeah that i know
subtle ermine
isnt it just because N U {} = N?
because {} is a member of that power set
but it dosent matter
sinful pier
have fun w the proof
subtle ermine
i started the book yesterday
i have no idea how to prove it but i get why the answer is N
.close
mighty pendant
E1 = After waking up from sleep, you find yourself on Pluto. (Impossible)
E2 = After waking up from sleep, you have a headache.
E3 = After waking up from sleep, you have a headache on Pluto.
Is P(E2|E1) = P(E3)? If so, what does it equal?
devout snowBOT
E1 = After waking up from sleep, you find yourself on Pluto. (Impossible)
E2 = A...
spice token
E1 = After waking up from sleep, you find yourself on Pluto. (Impossible)
E2 = A...
E1 is impossible
so how come P(E2|E1) exists?
mighty pendant
So you'd say that P(E2|E1) is undefined right?
spice token
yes
mighty pendant
But what about P(E3)?
Linguistically, both say the same thing
Mathematically, one is undefined and the other is zero (if it even is, im not sure)
how does that make any sense?
subtle ermine
first of all, the answer set cannot possibly contain any sets, because we are taking the union of sets that only contain numbers
the book of proof by richard hammach
its totally free online
subtle ermine
first of all, the answer set cannot possibly contain any sets, because we are ta...
and P(N) is a set which contains other sets
the answer is just N, the set of natural numbers
subtle ermine
first of all, the answer set cannot possibly contain any sets, because we are ta...
u get it right, @arctic oriole ?
spice token
E1 = After waking up from sleep, you find yourself on Pluto. (Impossible)
E2 = A...
if you want to make it more sense
probably P(E1∩E2)=P(E3) thats want you're asking for
spice token
if you want to make it more sense
probably P(E1∩E2)=P(E3) thats want you're aski...
but the condition P(A|B) still would not exist and makes 0 sense because P(A∩B)/P(B) is undefined
and so just make it not, this is a false proposition
mighty pendant
Hmm..
I'd argue that E3 is impossible because you can't be on pluto in the first place to have a headache there
And then I'd add that E2|E1 is also impossible by that very logic -- not undefined
But its undefined according to maths. 🥲
lyric hornet
P(E2|E1)=P(E1 and E2)/P(E1), but P(E1 and E2)=P(E3) so P(E2|E1)=P(E3)/P(E1)=0/0
but in any case, if you have wake up on Pluto whether you have a headache or not doesn't matter for the implication
vacuous truth, ex falso quodlibet, etc etc
mighty pendant
I'd argue that E3 is impossible because you can't be on pluto in the first place...
I dont see why we need to shrug and avoid concretely defining it as impossible
for example letting 1/0 be defined poses a number of different contradictions to proven facts, so it 100% makes sense to keep that undefined
Why the uncertainty here? What could possibly go wrong if we call it "impossible" as a special case where the denominator is 0?
the former
lyric hornet
it's a vacuous truth, it doesn't make sense to ask anything given you wake up on pluto
P(2+2=5|I'm the pope)=1 could be true
asking in terms of probability is weird, but logically the implication could be true yes, if P is false then P==>Q is true
irregardless of the truth value of Q
hence ex falso quodlibet -- "from falsehood, anything"
[
\begin{array}{c|c|c}
P & Q & P \implies Q \
\hline
T & T & T \
T & F & F \
F & T & T \
F & F & T \
\end{array}
]
woven radishBOT
ΠαϳαμαΜαμαΛλαμα
mighty pendant
it's a vacuous truth, it doesn't make sense to ask anything given you wake up on...
If a child came up to you and asked "Can I, one day, fly planes on pluto?" — both aware of the fact that going there is impossible — would you say "Hmm.. You couldn't go there in the first place so no. Its impossible." or would you say "I can't answer that. Your question makes no sense."
I'd expect the majority of people put in this situation to go with the former reply
lyric hornet
idk I know it's called Aristotelian logic, so maybe that's one of them ¯_(ツ)_/¯
lyric hornet
If a child came up to you and asked "Can I, one day, fly planes on pluto?" — bot...
Well the question you've posed "Can I, one day, fly planes on pluto?" isn't an implication, it's a statement rephrased as a question
For example "if the sun is shining, then I go for a walk"
then if the sun is shining and I am not walking, the implication could still be true
because it's a necessary but not sufficient condition
because to go for a walk I also may need to be not tired
mighty pendant
im not sure i understand :/
Are you saying that this hypothetical child already assumes that planes can be flown on pluto and his question is ill-formed due to it being based on that assumption?
lyric hornet
hmm yeah perhaps I could've worded that better...
mighty pendant
Are you saying that this hypothetical child already assumes that planes can be f...
probably not cause that doesn't make sense either 😅
the child knows that no one can ever go to pluto
Now that I think about it.........
I may just need sleep.
lyric hornet
Apparently I do too, because I think my sunshine and walking example is wrong 
but you're asking treating a theoretical with a nonsensical premise and either false or meaningless?
mighty pendant
Let me go again
A child, unaware that no one can go to pluto, asks you "Can I, one day, fly planes on pluto?"
Do you say "You can't go there in the first place so no. Its impossible." or "You can't go there in the first place. Your question makes no sense."
lyric hornet
causally, the former; logically, the latter
"Can I, one day, fly planes on pluto?" presupposes them being on pluto as a premise. A premise that we know is impossible.
mighty pendant
Oh thats excellent
Thats exactly what I was looking for
Tyvm man
ill head to bed now 😭
lyric hornet
Tyvm man
ill head to bed now 😭
gn 
mighty pendant
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fluid wharf
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fluid wharf
help me
eager cobalt
has to b proved right?
wind mason
Given […] prove that […]
fluid wharf
Given […] prove that […]
yessss
eager cobalt
Given […] prove that […]
fine man try it out now
wind mason
i actually have no idea how to approach this
I would consult my textbook but I don’t have it with me right now
I’ll think though, see if anything pops up
eager cobalt
bc im in grade 8
im in 12th
fluid wharf
im in 12th
help me
fluid wharf
ty
wind mason
I thought of a calculus method but I assume you haven’t learned calc 💀
eager cobalt
I thought of a calculus method but I assume you haven’t learned calc 💀
which method
mighty thicket
multiply ur first equation by all the denominators and call what you get X
same for ur second and call it Y
then u factor Y in terms of X
and then proof is implied
wind mason
which method
differentiating w.r.t. each variable
i think multiplying by (a-b)+(b-c)+(c-a) might work
mighty thicket
without + right
eager cobalt
without + right
yes
mighty thicket
yw
eager cobalt
np
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river lotus
I’m confused about the algebraic reasoning in the solution to the q - if we consider those inequalities and consider them together, its saying and e could be equal to each other when they clearly cant ? Ik the mark scheme lists abc and then cde separately but is it not still saying a could be e? If a=b=c and then c=d=e then a=e is possible - am I reading too much into it
devout snowBOT
I’m confused about the algebraic reasoning in the solution to the q - if we cons...
fossil locust
it's saying that a = b and c = d = e
river lotus
But it also says b <= c
river lotus
it's saying that a = b and c = d = e
Sorry i meant in the bit where the solution lists the inequalities
Not after it works stuff out using them
So the line before the (e-20) + …. Bit
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river lotus
<@&286206848099549185> 😭🙏🏻
turbid bane
I’m confused about the algebraic reasoning in the solution to the q - if we cons...
That's a big question
lunar harbor
I’m confused about the algebraic reasoning in the solution to the q - if we cons...
the idea is that it doesn't matter what order the numbers are in since neither mean nor range depends on the order that the numbers are in
so we can let them be in non-decreasing order without loss of generality
(you may be familiar with this if you ever had to find something like median by hand, where you'd first list the numbers in non-increasing order)
so by definition of a non-increasing sequence, $a \leq b \leq c \leq d \leq e$
woven radishBOT
Civil Service Pigeon
river lotus
Wdym by non-increasing function
lunar harbor
the numbers can't go down (decreasing)
river lotus
Yea
lunar harbor
More formally, a sequence $(a_n)$ is non-decreasing if $a_1 \leq a_2 \leq \dots$
woven radishBOT
Civil Service Pigeon
river lotus
I meant the solutions oops loll
lunar harbor
since the range depends on the largest and smallest numbers so we want to "fix" those
river lotus
Yup
river lotus
**Civil Service Pigeon**
But if we use this inequality and say that all the numbers are equal
Then aren’t we saying a could be equal to e
When thats not rly possible since e-a=20
lunar harbor
reposting b/c I don't want to scroll
lunar harbor
When thats not rly possible since e-a=20
mhm
wait so do you have any other questions
river lotus
Yea i do but i wanted to do this one first so i sent it before those
river lotus
mhm
Do u get what i mean
Like why is it ok to define it like that and then go ahead and solve it
lunar harbor
Like why is it ok to define it like that and then go ahead and solve it
the idea is that it doesn't matter what order the numbers are in since neither mean nor range depends on the order that the numbers are in
so we can let them be in non-decreasing order without loss of generality
the idea being the argument would be the same as if, say, $a \leq b \leq c \leq e \leq d$ instead
river lotus
What’s loss of generality
woven radishBOT
Civil Service Pigeon
lunar harbor
the letters you write would be different
but the argument would be the same after swapping the positions of the respective letters
so you can impose an order for convenience
lunar harbor
What’s loss of generality
"without loss of generality" is usually used to indicate that you're narrowing your analysis to a particular case (often chosen arbitrarily), but the proof still holds in general (ex. because all the other possibilities are just "permutations" of the case you consider, like here)
lunar harbor
this is just having good observation skills imo
consider what happens if you send $x \mapsto -x$
woven radishBOT
Civil Service Pigeon
river lotus
I got it right but i didn’t understand the way the mark scheme went about this
Uh
What happens
The way i went ab it was just rearranging so it was x cubed= the quadratic
And then sketching a graph and seeing
But the mark scheme does that but a bit more algebraically
lunar harbor
oh called it lol
river lotus
I just wanted to understand their way
lunar harbor
What does that mean/ do - ik they multiplied it by -1 but why
uhhhh
ok honestly I wouldn't do this but it does have teaching value
the idea is to notice that the coefficients of the odd powered terms are negated
and in general, $(-x)^n=-x^n$ for odd $n$
woven radishBOT
Civil Service Pigeon
lunar harbor
whereas the coefficients of the even powered terms (including $x^0$) are the same
woven radishBOT
Civil Service Pigeon
river lotus
ok honestly I wouldn't do this but it does have teaching value
Would u do my way then ? Or is there some elite better way
lunar harbor
and in general, $x^n=(-x)^n$ for odd $n$
woven radishBOT
Civil Service Pigeon
lunar harbor
so it just makes the negating thing more obvious
lunar harbor
Would u do my way then ? Or is there some elite better way
ngl this was my solution lol
in my head it was roughly
"send x to -x -> two of the negated roots are negative and one negated root is positive -> two positive and one negative"
probably not very helpful ik
lunar harbor
What’s negated
multiply by -1
river lotus
.. making me feel like idk English
lunar harbor
you should've just asked what "negated" meant bruh
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lunar harbor
Thanks for the help dude!!
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cold haven
<@&268886789983436800> scam
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restive river
@cold haven
hello, is your DMS open?
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hello, is your DMS open?
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@restive river Has your question been resolved?
cold haven
@restive river you might wanna close this, unless you plan on asking in here as well?
restive river
Ohhh how do I close it
cold haven
.close
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keen sundial
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keen sundial
time stamp 8:10
how did he find the intervals
3 pi/4, pi, 5pi/4
I know how he derived the first and last interval
keen sundial
3 pi/4, pi, 5pi/4
but not the ones in the niddle
dense jay
its just splitting the pi interval into 4
so halfway is pi/2 from the end points - ergo pi is your middle
the other two are half way between the start, pi and end
dense jay
we found the new period to be pi, correct?
and the 'start' has been shifted right to pi/2
when we look at a sin(x), just from 0 to 2pi, we can split it into 4 sections, 0->pi/2->pi->3pi/2->pi 2pi, these are the clear points
over this new pi period we're splitting it into 4 again, so each gap is now pi/4 instead of pi/2
so, pi/2 -> pi/2 + pi/4 = 3pi/4 -> 3pi/4+pi/4=pi -> pi+pi/4 =5pi/4 -> 5pi/4+pi/4=3pi/2
Thats one way to look at it, the way they did it was by finding the midpoint of pi/2 and 3pi/2, which is pi, so thats the middle of your cycle
and then finding the midpoints of (pi/2 and pi) and (pi and 3pi/2), these are our stationary points
keen sundial
"we found the new period to be pi, correct?"
I thought its 3pi/2
we applied trnsofmration
dense jay
3pi/2 is just the end of one of the cycles
the actual period is 2pi/2=pi
keen sundial
we found the new period to be pi, correct?
and the 'start' has been shifted rig...
which way is better?
dense jay
this is what they are graphing at the moment you showed
dense jay
which way is better?
I wouldn't say either is better, just do whichever makes the most sense to you
the first is probably more widely applicable
the second works fine when you have easy numbers to work with
keen sundial
this is what they are graphing at the moment you showed
does the period start at pi/2? or its just a cycle that has been aplied transofmration
dense jay
you can consider it to be that I have just transformed the graph of sin(x) for 0<=x<=2pi
then thats where that cycle ends up
the x=pi/2 is where x=0 would have been before the transformations
dense jay
ah, yes my bad
keen sundial
"so, pi/2 -> pi/2 + pi/4 = 3pi/4 -> 3pi/4+pi/4=pi -> pi+pi/4 =5pi/4 -> 5pi/4+pi/4=3pi/2"
don't we need to add it by pi/2 since thats start of
the phase shift
dense jay
i started at pi/2
keen sundial
your referring to this example right?
keen sundial
so we just add pi/2 to the intervals of normal sin increments (0, pi/2, 3pi/2, 2 pi --> 0+pi/2, 3pi/2+pi/2, 2pi +pi/2)
keen sundial
Starting at pi/2 I added pi/4 4 times
why did u add pi/4 4 times? I think thats where Im confused at
dense jay
if it was just sin(x-pi/2), then yes
you said you were alright with the endpoints being pi/2 and 3pi/2
we found 3pi/2 because pi/2+pi=3pi/2
if we want to split that region into 4, then we divide pi by 4, so pi/4
and then by adding that to pi/2 we will get the bounds of each part
If this is confusing then we can do it like this:
Midpoint of pi/2 and 3pi/2: [pi/2+3pi/2]/2=2pi/2=pi
Then do it again:
Midpoint of pi/2 and pi: [pi/2+pi]/2=[3pi/2]/2=3pi/4
then the same for pi and 3pi/2
its the same thing
keen sundial
yes I understand his method and found the points
u specefically said it works better or only for easy numbers
what if thats not the case?
dense jay
Yeah I may have been hasty there, when I said that I was thinking about just looking at a graph rather than doing simple midpoint calculations
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keen sundial
Hey also wondering how did u make it so you can view the critical points
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velvet karma
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velvet karma
hpw do i start
small nest
Checking...
tribal crypt
u can easily caculate angle DOB
small nest
Yup
small nest
Tangents of a circle from a external point are congruent respecto to radiis (radiuses, IDK) coming from the circle center
velvet karma
Tangents of a circle from a external point are congruent respecto to radiis (rad...
i got 9
small nest
velvet karma
ye
tribal crypt
I dont think we can use that
subtle ermine
tribal crypt
We can use 90 degree
velvet karma
is it 9
tribal crypt
is it 9
What angle
I think the answer is 19
BCD=1/2BOD=1/2(360-ADO-BAD-ABO)=1/2(360-119-75-90)=1/2.38=19
velvet karma
BCD=1/2BOD=1/2(360-ADO-BAD-ABO)=1/2(360-119-75-90)=1/2.38=19
its wrng
velvet karma
wait,go ahead
rain summit
to ensure that our eyes aren't harmed, can you draw segment OC, OB and BD?
velvet karma
to ensure that our eyes aren't harmed, can you draw segment OC, OB and BD?
i got it lol
rain summit
mark the other point of the diameter as D
then BCD is an isosceles triangle
that's what im thinking
velvet karma
its 1/1
rain summit
AC/BC is just tan of B
or cot of A
so if you either find B or A
then the problem is solved
velvet karma
so if you either find B or A
how
rain summit
im trying to find how
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dusky belfry
how do I use walfram alpha to check this: ,calc 2x^3 - x^2 - 13x - 6 = 0
dusky belfry
because I'm going something I think it's the algebra
no it's 100% the algebra, but I'd like to know where
,calc 2x^3 - x^2 - 13x - 6 = 0
woven radishBOT
The following error occured while calculating:
Error: Invalid left hand side of assignment operator = (char 22)
pseudo basin
,w 2x^3 - x^2 - 13x - 6 = 0
woven radishBOT
pseudo basin
like this. the command is ,wolf or ,w.
or you could just go to http://wolframalpha.com/ and ask there directly.
dusky belfry
yup, it's completely wrong, good to know the command
pseudo basin
(without any ,w or other prefix, just type your question)
dusky belfry
(without any `,w` or other prefix, just type your question)
nah, I prefer chat
pseudo basin
ok you're glued to discord then got it
dusky belfry
anywho, it's wrong, first though $$0 = x(2x^2 - x - 13) - 6$$
woven radishBOT
ℕ∈ℝD ALERT: Gonçalo Gonçalves
pseudo basin
nope nope nope nope nope nope
pseudo basin
0 = x(2x^2 - x - 13) - 6
this is a pseudofactorization
it's correct algebraically but also completely useless
dusky belfry
how so
proud mural
when factorising polynomial equations which are equal to 0. you know that the equation will be solved if any factor if 0 and the factors are easier to solve than the non factorised version
acoustic leaf
when you factor something as like (x + 3) (x - 7) = 0 for example, that is taking advantage of the fact that if two things multiply to zero, then one of them must be zero individually. there is no analogous property for like (x - 2)(x + 5) = 13 for example
proud mural
if you set x = 0 in the above factorisation you get 0 = -6. you won't get zero because you aren't just multiplying you are also subtracting something after
pseudo basin
how so
the whole premise of factorization is to use the zero product law, which is ab = 0 <=> (a=0 OR b=0)
this DOES NOT WORK if you replace the zero with any other number
it HAS TO BE ZERO or else it doesnt work at all
two numbers can multiply to 4 without either one being 4.
two numbers can multiply to 20 without either one being 20.
two numbers can multiply to 69 without either one being 69.
and so on.
dusky belfry
the zero product law doesn't work, so what should I do?
pseudo basin
it's not that the zero product law doesn't work
it's that what you have isnt a factorization at all and so zero product law is INAPPLICABLE to it.
big difference there.
dusky belfry
ok, how do I factor it out?
pseudo basin
honestly with a cubic i would go for the rational root theorem to try and guess a root, then factor (x-r) out
dusky belfry
no easy way to do it huh? (rational root theorem sounds complicated, I might've heard of it, but it's in my language, so idk)
pseudo basin
no there's no easy way
rational root theorem is not really a method in itself
it's more of a shortlist for "nice roots"
pseudo basin
it can and does happen that none of the roots RRT gives are nice.
then you are cooked.
dusky belfry
the exercise does say "show that 2 is a root of the polinomial", but I already did the exercise, I just substituted x by 2 and it gave 0, but still I though, because I couldn't solve it with factorization I was doing something wrong (which I was) so i came here. That's kind of the guist of it
harsh sierra
Factorizarion is a really hard thing to compute
And factoring to linear terms is equivalent to finding the roots
pseudo basin
the exercise does say "show that 2 is a root of the polinomial", but I already d...
that changes things a lot
now you're actually given a root for free
this means you know (x-2) is a factor of your polynomial
so you can factorize it into (x-2)(quadratic)
dusky belfry
that changes things a lot
I know, but I though that I could solve it factorizing, to find the 2, and I wasn't able to, so I came here
turns out
it's only possible by guessing
that's the guist of it
harsh sierra
RRT can give you this 2
dusky belfry
right, but at the end of the day it's just a short list
useful when I'm stuck
when I don't know the right way to solve an exercise
but that isn't the goal, the goal is to never brute force
know what I mean?
anyways, that about clears it
can I close this
?
.close
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✅ Original question: #help-27 message
dusky belfry
there we go...
sorry
I just realized, on the next page, there is a continuation to the question
represent P(x) as P(x) = a(x-b)(x-c)(x-d)
oh yea I have no idea
pseudo basin
should i repeat myself now