#help-41
vague thorn
trim crypt
Personally i want diamond to make a good comeback, one sided win is no fun
humble topaz
result = 0
for x in k:
result += (1/0.5) * (x/k)
I am trying calculate this in one go instead of having to add up the parts one by one.
k is any number above 1
x is the part
k=3, x=3: (3/3)
k=3, x=2: (2/3)
k=3, x=1: (1/3)
leaden sonnet
Personally i want diamond to make a good comeback, one sided win is no fun
lmao idk if he can
scarlet trail
Personally i want diamond to make a good comeback, one sided win is no fun
come back from what
im correct
leaden sonnet
lmao
vague thorn
vague thorn
dmn you got the whole squad bawling
trim crypt
Imma respect standing on business
leaden sonnet
good thing hes blocked 😉
vague thorn
thing* right
leaden sonnet
dmn you got the whole squad bawling
damn * right
humble topaz
Did you guys understand what I am trying to do? 😂
vague thorn
no x2
humble topaz
Lets make it simpler then.
leaden sonnet
!original
amber waspBOT
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.
scarlet trail
Lets make it simpler then.
young padowan look at the answer i suggested and try it
humble topaz
Please show the original problem, **exactly as it was stated to you**, with the ...
No textbook problem.
leaden sonnet
young padowan look at the answer i suggested and try it
stop degrading people bc your life is sad, him not know how to do something doesnt make him any less than you
scarlet trail
stop degrading people bc your life is sad, him not know how to do something does...
how is that degrading?....
humble topaz
Can I simplify this without solving it?: (3/3) * (2/3) * (1/3)
scarlet trail
apparentley I am
leaden sonnet
apparentley I am
so can you stop reading then?
reef mesa
Can I simplify this without solving it?: `(3/3) * (2/3) * (1/3)`
is this the slots thing again?
leaden sonnet
is this the slots thing again?
yea i think
humble topaz
is this the slots thing again?
This is just a different part of the topic, but it is not helping to solve the slot problem..
reef mesa
so can you stop reading then?
bro dont give them the satisfaction of a response just ignore it
leaden sonnet
bro dont give them the satisfaction of a response just ignore it
im having fun
idrc lol
i needa stop tho before i get banned
reef mesa
This is just a different part of the topic, but it is not helping to solve the s...
i mean your approach is wrong so even if you try to simplify that expression you’re not going to get what you need
plus i think you’re only allowed to occupy one channel at a time
leaden sonnet
yea he is flagged for that
you can go close the other channel with .close
make a diff help channel then 🙂
humble topaz
We can't solve the other issue right now so I thought about continuing this a different issue for now.
Still following the other channel though.
scarlet trail
Still following the other channel though.
did you see what i suggested the answer to your problem could be
humble topaz
i mean your approach is wrong so even if you try to simplify that expression you...
And I am not sure which approach you are referring to. I have not claimed to have any working approach.
Yes this is correct.
n can be any positive number.
i starts at 0 and until i = 3 all results are summed up.
humble topaz
I wonder though if I can express it in an easy to compute way.
reef mesa
What does this even entail
fr
humble topaz
(k!) / (k^k)
leaden sonnet
yessss
humble topaz
.close
weak portal
Can someone make my brain understand if you have a probability of success of 2 independent things 0.7 and 0.8 where both success is $0.7\times0.8=0.56$, and if you want to get the probability of atleast one success, why is it $0.7+0.8-0.56$ ?
grizzled pagodaBOT
Totalani
weak portal
I hope I could explain it properly lol, but in essense why subtract the probability of both succeeding, to get the probability to get atleast one succeeeding?
hearty inlet
what's the probability of the first succeeding but not the second?
hearty inlet
call the events of both succeeding A and B respectively
weak portal
ok
hearty inlet
A and B are independent yes
0.7 is the probabilty of A = (A and B) or (A and B^c)
i want A and B^c
weak portal
ok so the probability of B failing is 0.2
hearty inlet
mhm
weak portal
a and B^c should be 0.14
weak portal
still dont really get it 😦
hearty inlet
what about B winning but not A
hearty inlet
still dont really get it 😦
we're getting there
weak portal
0.8*(1-0.7)
weak portal
0.56
hearty inlet
yes 0.7*0.8
so either A or B winning means any of these 3 cases
and these 3 cases are disjoint
ie they can't happen simulatneously
weak portal
disjoint was that they cant happen at the same time right?
hearty inlet
yes
hearty inlet
**Totalani**
yeah which is the same as this
weak portal
so why do you 0.7+0.8?
weak portal
ok
weak portal
P=p-pq+q-qp+pq=p+q+pq ?
hearty inlet
yh
p + q - pq which is what u had
u can also think of it as a venn diagram
you want the area of both circles
weak portal
am I misunderstanding independent?
hearty inlet
idts
weak portal
english isnt my first language but I thought that ment first prob wont affect the other
weak portal
right
hearty inlet
a probability is just a number
weak portal
fair
weak portal
Ok i think its starting to make sense, but I dont like probabilities man its confusing to me
weak portal
yea
hearty inlet
in our case A and B are indep so we replace P(AnB) with P(A)P(B) = pq
hearty inlet
happy to help
hearty inlet
k
weak portal
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amber waspBOT
hearty inlet
cool
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sour hornet
how do you do partial fractions of 2/(s^2+2)(s^2+1)?
amber waspBOT
how do you do partial fractions of 2/(s^2+2)(s^2+1)?
sour hornet
.close
amber waspBOT
kindred ocean
how do you do partial fractions of 2/(s^2+2)(s^2+1)?
1/(s²+1) -1/(s²+2)
how? solve $\frac{1}{(s^2+1)(s^2+2)} = \frac{\alpha}{s^2+1} + \frac{\beta}{s^2+2}$
grizzled pagodaBOT
Goëtia
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glass wraith
To calculate an inflection point I calculate when the 2nd derivative is 0, however, I heard that the 3rd derivative also needs to be non zero at that point, why is that/is this even true?
glass wraith
f''x = 6x
f'''x = 6?
grizzled plume
Second derivative is 72x^8
glass wraith
that doesnt add up
grizzled plume
Ya
glass wraith
10x^9 f'x and f''x = 90x^8?
grizzled plume
Inflection pt at 0
grizzled plume
10x^9 f'x and f''x = 90x^8?
Yes
grizzled plume
Then f''' is 0 at 0
grizzled plume
Statement is disprove
grizzled plume
is it always an inflection point?
Only if sign changes before and after 0
grizzled plume
Wdym
grizzled plume
like forever 0?
glass wraith
ye
glass wraith
can I somehow determine an inflection point without needing to do 3 calculations
as in 1 for the 0 point
1 before the 0 point and one afterwards
grizzled plume
can I somehow determine an inflection point without needing to do 3 calculations
I don't know tbh
Sry
grizzled plume
Np
glass wraith
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amber waspBOT
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narrow cloud
Hi, could someone explain to me how did they find an expression for the n-th term of this series?
patent raptor
Hi, could someone explain to me how did they find an expression for the n-th ter...
It looks like a telescoping series
Try to write down the first few terms
amber waspBOT
@narrow cloud Has your question been resolved?
narrow cloud
Try to write down the first few terms
Okay, I'll try that
amber waspBOT
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narrow cloud
Yea, it worked, how didn't I notice? 😭 Any tips for recognizing "recurring/special types" of series?
patent raptor
Yea, it worked, how didn't I notice? 😭 Any tips for recognizing "recurring/spec...
yea, try to analyze and observe the term inside the sum
it's a difference of a term and its consecutive
narrow cloud
i thought u still didnt get it
No, I took so long cause I was having breakfast
narrow cloud
it's a difference of a term and its consecutive
I still don't recognize the patterns "intuitively" I guess I have to practice with more exercises
grizzled pagodaBOT
𝔸dωn𝓲²s
narrow cloud
**𝔸dωn𝓲²s**
I'll keep this form in mind
patent raptor
I still don't recognize the patterns "intuitively" I guess I have to practice wi...
You can always write down a few terms and try to figure a pattern
If you can't see it right away
If you can't do great things, start by doing small things in a great way
narrow cloud
You can always write down a few terms and try to figure a pattern
Yes, I'll do that, I got intimidated at first cause the initial expression looked complicated, but the terms actually cancelled out nicely
Thanks for your help!!
patent raptor
Thank you too!
narrow cloud
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weak portal
A machine with 2 components A and B only works if both components works, the probability of A not working is 0.2 and 0.3 for B. Which means 0.8*0.7=0.56 for a machine to work.
weak portal
My question is why isnt it 0.2*0.3 for a machine not to work?
spiral zealot
that's both components failing
but just one failing is enough for it not to work
spiral zealot
only A fails: 0.2 * 0.7
only B fails: 0.8 * 0.3
both fail: 0.2 * 0.3
which added up will give 1 - 0.56 = 0.44
weak portal
right ok
but yea 0.2*0.3 is both failing
makes sense
Thank you, im gonna bang my head some more.
.close
amber waspBOT
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stoic locust
Solve this tendency:
sin(x)/x where x tends to 0.
amber waspBOT
Solve this tendency:
sin(x)/x where x tends to 0.
alpine flower
L'Hopital
stoic locust
Considering I do not know L' Hospital
indigo cloud
LH would be circular anyway, as that needs the derivative of sin and to get the derivative of sin you need to solve this specific limit in the first place
stoic locust
ikr
alpine flower
my bad
vast spade
Considering I do not know L' Hospital
sandwich theorem
amber waspBOT
@stoic locust Has your question been resolved?
stoic locust
considerin i don't know sandwich theorem
vast spade
considerin i don't know sandwich theorem
wdym considering Idk sandwhich theorem?
alpine flower
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compact lava
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compact lava
someone please explain why we didn't take the derivative of 2x and e^x+3
why did we take 3 seperately?
signal estuary
isnt that just the standard way to find the derivative of a sum?
(3x²-1)' = 6x ? no?
compact lava
why isnt it this
cus in the previous one im getting 2xe^x + 2e^x, this one i'm getting 2e^x ( x + 1 ) + 6
signal estuary
show the previous one
compact lava
compact lava
its not though
signal estuary
and then adapt the product rule
compact lava
ok wait
signal estuary
but the exponential function has it's own derivative
perhaps that is confusing you?
signal estuary
if you have 3(3x + 2) then you do 9x + 6 not 3 * 3 + 3(x+2) do you?
im not saying that you can adapt the same rules for order of operations to derivatives but perhaps this will help your reasoning for this situation
amber waspBOT
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mystic lotus
someone please explain why we didn't take the derivative of 2x and e^x+3
this is the order of operations
multiplication happens before addition
you can think of 2x e^x + 3 as being the same thing as (2x e^x) + 3
it is not 2x (e^x + 3)
so (2x e^x) + 3 is a sum of two things, which means you need to first break it up according to the sum
d/dx (2x e^x) + d/dx (3)
after that you can use the product rule on 2x e^x
does that make sense @compact lava
compact lava
if you have 3(3x + 2) then you do 9x + 6 not 3 * 3 + 3(x+2) do you?
ohr yeah
mystic lotus
no worries lol
compact lava
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amber waspBOT
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untold rivet
amber waspBOT
untold rivet
no one has helped me, and the chat i was in was empty so. can some one help me solve this, i dont know how to solve it
tawdry lintel
Start by forming an equation
You know the input and output must be the same
I’ll give you the first half: 5(x-9)=…
untold rivet
okay and then is the second -3(x+11)
is that correct?
why does everyone begin to help me then leave?!
slow atlas
okay and then is the second -3(x+11)
No in the second case, first -3 is multiplied to the input. Then 11 is added
So it will be -3x + 11
slow atlas
Yes
untold rivet
okay, and thats 5x-45=-3x+11
slow atlas
Yes
untold rivet
okay lemme solve it rq
woeful dirge
untold rivet
oh
slow atlas
It's 5x - 45 = -3x + 11
-3 x
untold rivet
so x=7
slow atlas
thank you sm for your help!
Welcome
untold rivet
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mellow hornet
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mellow hornet
Could I have some help with Part (1)? Not really sure how to approach this problem
gonna move to #groups-rings-fields actually
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amber waspBOT
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kindred grail
amber waspBOT
kindred grail
I am not sure if i am correct I got the answer 5.2 seconds
honest galleon
it's not correct. can you show me your process?
kindred grail
ouch
okay
so
using the formula
s=ut+1/2at^2
then i sub in values
to get
19.6=29.4t+-4.8t^2
then i make it a quadratic
so its
4.8t^2-29.4t+19.6=0
then i sub it into quadratic formula
and get t to be
5.23
i thought i was right ngl thats humbling
honest galleon
i think you should just have 0=ut+1/2at^2, because once the ball comes down and reaches it's initial height while going down, it's velocity is -29.4, so any time after, it's greater
since you could think of s as $\Delta{x}$, or change in position, and the net change in height is 0
grizzled pagodaBOT
fish
latent nest
no its right
latent nest
displacement is 19
latent nest
yeah your answer is correct
honest galleon
no, it's not
kindred grail
oh sweet
honest galleon
your intitial height is 19.6, you have a velocity of 29.4m/s, when you reach 19.6 as a height again, you have a velocity of -29.4m/s
latent nest
oh yeah the velocity should be negative
honest galleon
so any time after t such that 0=vt-1/2gt^2, the maginitude of velocity is greater than 29.4m/s
kindred grail
so i should just sub in my values nd solve for t essentially
latent nest
i think the question is asking for how long it has a velocity greater than 29, not at what time it reaches a velocity greater than 29
honest galleon
still, it's not 5.2, since it reaches height 19.6 at t=6s, and reaches the ground at t=6.6s
latent nest
yes the original was wrong because the velocity should have been negative
but it should give the right answer if he just makes it negative and solves it the same way
honest galleon
also s should be -19.6, because the final height is less than initial
kindred grail
lemme make velocity a negative
and solve it
i got a negative value
wtf am i doing wrong
honest galleon
make s=-19.6
honest galleon
what equation do you have?
kindred grail
-19.6=-29.4t-4.8t^2
honest galleon
that has a root at 0.6
honest galleon
so, that's the answer
kindred grail
so, that's the answer
okay lemme try it rq
fuck my life i wrote 0.6 instead of 0.61 and got it wrong
honest galleon
i was about to tell you to change it, but i was too slow
i realized it should've probably been rounded
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Available help channel!
errant drum
I keep trying it but I can't prove it true nor false
amber waspBOT
I keep trying it but I can't prove it true nor false
@errant drum Has your question been resolved?
amber waspBOT
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errant drum
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amber waspBOT
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young seal
What do I do next
amber waspBOT
What do I do next
young seal
What is BD
solid mortar
bd is the length from point b to d
solid mortar
just compute whatever equation you have for your length bd
so just calculate 22.9 * sin26* = BD
young seal
do I put that in the calculator
solid mortar
type in 22.9
then the multiply button
then find the sin button and then press that and then type 26 in
solid mortar
nah nah im in canada 😭
young seal
then find the sin button and then press that and then type 26 in
bc = 17.5?
I have no idea
I am so confused
what do I do after
<@&286206848099549185>
young seal
22.9 * sin(26°) = 10 cm
why does it say 17.46 in my calculator 😭
solid mortar
why does it say 17.46 in my calculator 😭
is your calculator in radians?
young seal
is your calculator in radians?
yep 😦
solid mortar
make sure its in degrees
amber waspBOT
@young seal Has your question been resolved?
young seal
yo
amber waspBOT
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.
young seal
Please only use the <@&286206848099549185> ping **once** if your question has no...
I am
wdym
pallid canopy
<@&268886789983436800> copy pastagpt
young seal
why do we use the reverse, like how did u get that
split sail
I get it from online and
young seal
I get it from online and
bruh chatgpt..
severe scroll
can you stop using gpt in the help channels please
young seal
@severe scroll @pallid canopy Can u guys help pls 🙂
severe scroll
i cannot, i am a bit busy.
split sail
can you stop using gpt in the help channels please
okok sry
young seal
i cannot, i am a bit busy.
okie 🙂
young seal
like after tanf = 8.1/9.6
young seal
I will be back
bruh
<@&286206848099549185>
<@&286206848099549185>
😭
<@&286206848099549185>
it's been half and hour 
drowsy lava
hi
drowsy lava
i suggest u leave intermediate working in 3 dp but yes thats right
young seal
did u find eg yet
yes
drowsy lava
use the tangent inverse function?
young seal
use the tangent inverse function?
like how?
drowsy lava
do u know what tan inverse does
$tan(f)=\frac{8.1}{9.6}\$
$tan^{-1}(tan(f))=tan^{-1}(\frac{8.1}{9.6})$
young seal
do u know what tan inverse does
nope haven't learned, or maybe I forgot
grizzled pagodaBOT
Kai The Cat
young seal
kinda know it now
drowsy lava
yes ok great
young seal
since I haven't then it yet
drowsy lava
theres a shift function right
usually that lets u swap to tan inverse instead of tan
young seal
usually that lets u swap to tan inverse instead of tan
I see, but I am using the google one rn 
grizzled pagodaBOT
young seal
oh, ty
drowsy lava
np
amber waspBOT
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@young seal Has your question been resolved?
verbal lark
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amber waspBOT
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quick ridge
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quick ridge
$L(x) = 7 + 10(x-2)$
grizzled pagodaBOT
knief
quick ridge
did you get that
verbal lark
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grizzled pagodaBOT
Kenzo
quick ridge
L(2.1) = 7 + 1 = 8
verbal lark
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quick ridge
f(2.1)
verbal lark
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quick ridge
,calc (2.1)^3 - 2(2.1) + 3
grizzled pagodaBOT
Result:
8.061
nope
verbal lark
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quick ridge
x
verbal lark
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quick ridge
a + 0.1 = 2 + 0.1 = 2.1
quick ridge
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what
verbal lark
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quick ridge
,calc (2.1)^3 - 2(2.1) + 3
x^3-2x+3
verbal lark
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quick ridge
rookie mistake
verbal lark
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verbal lark
rookie mistake
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quick ridge
yes
verbal lark
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quick ridge
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that’s your error
verbal lark
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quick ridge
we really only care about its magnitude
verbal lark
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quick ridge
what
verbal lark
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quick ridge
10^?
verbal lark
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quick ridge
-1
10^-1 =0.1 > 0.061
you’d never write that on the ap exam btw
you’d write the actual error
0.061
verbal lark
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amber waspBOT
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proven vapor
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proven vapor
why can we let n go from 0 to m?
is it because the forward difference operator reduces the degree of a polynomial by 1 each time
okay yeah that's it
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loud edge
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near flower
how do you do the last part of this?
amber waspBOT
how do you do the last part of this?
near flower
I have no idea how to start
normal dove
how do you do the last part of this?
third inequality looks awful like the (a+b+c)^2= a^2+b^2+c^2+2(ab+bc+ca) type shit maybe u can manipulate it
graph would be of a cubic with real roots all. so it cuts it at three different places
amber waspBOT
@near flower Has your question been resolved?
languid lichen
can someone confirm the height of the hill is 1072 ft?
amber waspBOT
can someone confirm the height of the hill is 1072 ft?
fierce edge
,w tan(24 deg) = (50 + h)/x and tan(23 deg) = h/x
grizzled pagodaBOT
fierce edge
how 1072?
vestal hatch
probably adding 50 by mistake lol
fierce edge
ohh
languid lichen
probably adding 50 by mistake lol
OH
fierce edge
yeah that’s probably it
amber waspBOT
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vernal warren
I feel like these tasks have similar ways to solve them
amber waspBOT
I feel like these tasks have similar ways to solve them
vernal warren
similar logic. How do I solve these kind of tasks where I have find an equation?
amber waspBOT
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manic veldt
hi, i'll be doing a final exam of the course. My teacher made a check list for everything i should know. Can anyone ask me qustions and tell me if i know it or not?
outer hull
no
amber waspBOT
Please post images (such as PNGs or JPGs) of the question rather than other filetypes such as PDFs which have to be downloaded. Non-image downloads can potentially contain viruses or other security risks.
outer hull
the help channels are for asking specific problems or explaining specific concepts, they are not your quizlet
outer hull
pardon? i am respectful
proven vapor
ill ping mods next time
please do
outer hull
ask a direct question and get a direct response 
manic veldt
ask a direct question and get a direct response <:shrug:1159570387953258616>
Okay
Can anyone help me to know what I know in maths?
outer hull
no
the help channels are for asking specific problems or explaining specific concepts, they are not your quizlet
outer hull
a website that is pretty much what you are asking for
amber locust
You’re being disrespectful now @outer hull
outer hull
who are you?
manic veldt
a website that is pretty much what you are asking for
How to use it?
outer hull
idk, do it yourself
amber locust
….
manic veldt
...
neat wind
..
proven vapor
.
outer hull
.
ah damn beat me to it
@manic veldt anyhow, you can only ask a specific question in here
that is "can someone explain why 2+2=4" something like that
you cannot ask something like "help me revise calculus"
proven vapor
yeah if you send a bunch of text and expect us to parse through it and tell you exactly whatever you want to know then you're setting yourself up for failure
manic veldt
my thing is
THIS IS A CHECLIST
it asks me what i CAN DO.
I'd wanna know if I KNOW that
proven vapor
then go through the checklist and see if you know it
outer hull
I'd wanna know if I KNOW that
no one knows you better than you
sonic fiber
Math
proven vapor
HOW AM I SUPPOSED TO KNOW IF I DID THAT RIGHT?
you still haven't sent a single question here other than saying "ask me questions"
sonic fiber
Fight him
manic veldt
**Professional ** Mathematics server!
sonic fiber
**Professional ** Mathematics server!
What u need help with
manic veldt
read this smh
outer hull
brother thats not in english
proven vapor
LOL
outer hull
amber locust
then learn it
Wow baddie comeback
manic veldt
fucking first grade
outer hull
!vol
amber waspBOT
Helpers are just people volunteering their time to help you. Be polite and patient.
sonic fiber
Wait so you are 8?
manic veldt
r u-
sonic fiber
fucking first grade
Isn’t that too young to be swearing
manic veldt
ronaldo is hotter
sonic fiber
ronaldo is hotter
How dare you
outer hull
stay on topic yall
@manic veldt if you dont have any specific question, do close this channel please
sonic fiber
stay on topic yall
Mb mb
outer hull
this server is not for tutoring
proven vapor
yeah if you want someone who combs through your syllabus/checksheet and ask you hand-crafted questions to check your knowledge get a personal tutor
sonic fiber
yeah if you want someone who combs through your syllabus/checksheet and ask you ...
Or go to school?
proven vapor
or that
sonic fiber
Cheaper imo
manic veldt
school is free wym?
sonic fiber
school is free wym?
That’s the idea
amber waspBOT
@manic veldt Has your question been resolved?
mint nacelle
school is free wym?
absolutely, teachers are an underrated resource
unless you know for sure that they are not helping you in that class
vagrant acorn
i got the same calculations as the answer sheet but how does this equal 2 and not 0 ?
vagrant acorn
used this rule for the problem
mint nacelle
i got the same calculations as the answer sheet but how does this equal 2 and no...
$\frac{1}{x - y} (2x - 2y)$
grizzled pagodaBOT
south
vagrant acorn
i dont see it, if we set (x,y) -> (0,0) doesnt everything go to 0?
vagrant acorn
$\frac{1}{x - y} (2x - 2y)$
^
mint nacelle
i dont see it, if we set (x,y) -> (0,0) doesnt everything go to 0?
no, you have a 0/0 form
vagrant acorn
.close
amber waspBOT
amber waspBOT
Available help channel!
swift flume
Position was 10sin-10cos, I got 10(sin-cos) for acceleration which is wrong and I can’t figure out why, the image is the answer key
swift flume
My velocity came out to be 10(cos+sin)
amber waspBOT
@swift flume Has your question been resolved?
mint nacelle
Position was 10sin-10cos, I got 10(sin-cos) for acceleration which is wrong and ...
if you differentiate 10(cos + sin) you get 10(-sin + cos)
cause the derivative of 10(sin - cos) is 10(cos - (-sin))
swift flume
cause the derivative of 10(sin - cos) is 10(cos - (-sin))
Ohhh, I see why now, but tbh idk what I was thinking when doing the first derivative, somehow added extra negative or smth
Thanks
.close
amber waspBOT
amber waspBOT
Available help channel!
livid bolt
proof that multiplication on Z is associative
amber waspBOT
proof that multiplication on Z is associative
livid bolt
did i skip some steps or is it correct
in the step where i multiply e^i(theta1 + theta2) * e^i(theta3), am i not using the associative property ?
it looks circular reasoning to me
grizzled pagodaBOT
Wild123
ionic terrace
Instead of moving from first form to 2nd, try expanding them both separately? I suppose. Hope it makes sense.
Although I think 1) you wrote Z in initial post, but actually mean C (complex numbers), and 2) using this form does indeed seem circular, moving the problem of associativity from the e^{} numbers to the exponents, which are also complex. Maybe instead go for the good old a+bi form?
Maybe I misunderstood the exercise. Just giving my 2 cents.
livid bolt
Maybe I misunderstood the exercise. Just giving my 2 cents.
does this make more sense now?
livid bolt
Although I think 1) you wrote Z in initial post, but actually mean C (complex nu...
yea C
ionic terrace
I think it's good, although factoring i was unnecessary, and uses another property which you might have not proven already.
livid bolt
I think it's good, although factoring i was unnecessary, and uses another proper...
yea to use that i should prove distributivity too
ionic terrace
instead, could just add the exponents, even though you'd write a bit more
grizzled pagodaBOT
Wild123
livid bolt
instead, could just add the exponents, even though you'd write a bit more
okay thanks! these proofs on associativity, distributivity, etc always confuse me because i always risk doing circular reasonong
.close
keen pawn
1.2.8. If impossible, do I show it's impossible?
amber waspBOT
1.2.8. If impossible, do I show it's impossible?
void canyon
It is possible
keen pawn
my example for (a) is y= x^2, for (c) it would be y=x
void canyon
Sorry, for some reasons I thought you were asking only for 1.2.8.c
void canyon
Your example for (c) is not good. If you take the identity function, i.e., f(n) = n, it won't be onto
keen pawn
Isn't it onto
void canyon
Also I recommend using the letter n instead of x for the variable when describing functions that start from N
vast spade
literally any function thats range is N will work as an example of (c)
keen pawn
range is z you mean
void canyon
And not necessarily any function
vast spade
range is z you mean
ah my bad
keen pawn
okay, so I want f(1)=1, f(3)=2 \dots
f(2)=-1, f(4)=-2 \dots
Now to find a formula for this
f(2n) = -n
f(2n+1)=n
keen pawn
It's fine lol
void canyon
f(2n) = -n
f(2n+1)=n
Yes
keen pawn
cunning birch
then f(?) = 0?
vast spade
0 is a natural number
cunning birch
oh f(1)
void canyon
You would also need f(1) = 0
void canyon
0 is a natural number
Is it?
cunning birch
0 is a natural number
shush they will burn us for this
vast spade
yeah 
keen pawn
Now you would need to include 0 in **N** for me to say yes <:opencry:58607861486...
Not really?
f(1)=f(3)
void canyon
What is f(2)?
keen pawn
severe scroll
keen pawn
Hi!
severe scroll
which one are we working on
void canyon
Do you want wifi?
keen pawn
Alternatively f(n) = n-1 for n \geq 2, f(1)=1
keen pawn
which one are we working on
(b)
keen pawn
[
f(n) =
\begin{cases}
1 & \text{if } n = 1, \
n - 1 & \text{if } n \geq 2.
\end{cases}
]
Possible
severe scroll
cool
grizzled pagodaBOT
ƒ(Why am. I here)=I don't Know
severe scroll
does this book define N to start at 1
very sad
whatever though
this looks right
amber waspBOT
amber waspBOT
Available help channel!
orchid summit
How do we solve this queestion
amber waspBOT
How do we solve this queestion
orchid summit
Chat GPT failed to solve
spiral zealot
what if i rephrased the question as
for what values of k will y=k intersect the curve exactly twce
orchid summit
what if i rephrased the question as
for what values of k will `y=k` intersect th...
I didn't understand
spiral zealot
draw a horizontal line on the graph, and tell me how many times it intersects the curve
orchid summit
depends
somewhere it does 3
somewhere it does 2
somewhere it does 1
somewhere it doesn't
0-3
hello
spiral zealot
when will it be 2
orchid summit
spiral zealot
yes
spiral zealot
yes
orchid summit
.close
amber waspBOT
amber waspBOT
Available help channel!
stuck kindle
every open subset of R can be represented as a countable union of disjunct open intervals
foggy kayak
what's the question?
stuck kindle
how can I get started proving it?
undone merlin
consider the following motivational sets on R
---(--)---(---][--)---(--)(--)---
--[-----)----...
stuck kindle
how would I do the second one?
undone merlin
To formalise your question a little more, do you mean
"every set of disjunct open subsets on R is isomorphic to a single open subset on R"?
stuck kindle
no I have an open set $A \subseteq \mathbb{R}$
grizzled pagodaBOT
hanno
stuck kindle
and I want to write it as $A = \bigcup_{n=1}^{\infty} I_n$
grizzled pagodaBOT
hanno
stuck kindle
and $I_n$ is some $(a_n, b_n)$
grizzled pagodaBOT
hanno
undone merlin
Ok, you need to show the two subset inclusions.
$$A \subseteq \bigcup I_n \implies \forall; x \in A .; \exists; I_n .; x \in I_n$$
$$A \supseteq \bigcup I_n \implies \forall; I_n .; x \in I_n \implies x \in A$$
grizzled pagodaBOT
Shuba
stuck kindle
but how should I first get started defining these intervals? I only have A
undone merlin
There's the trivial option; I_0 = A and I_n = \emptyset for all n > 0. 
stuck kindle
but if A is not an interval?
undone merlin
If A is not disjoint, then you can just let {I_n} be the non-disjoint subsets of A.
stuck kindle
what does it mean for A to be disjoint?
undone merlin
You won't be able to split up an interval (a,c) because you'd either have (a,b)u[b,c) (up to symmetry) or (a,b)u{b}u(b,c) because {b} and [... are both not open.
undone merlin
what does it mean for A to be disjoint?
two subsets are disjoint if their intersection is the emptyset.
stuck kindle
if A itself is an interval that's fine, I can just define I_0 to be A
undone merlin
that's the ONLY thing you can do if a is not disjoint
stuck kindle
two subsets are disjoint if their intersection is the emptyset.
but you said A is disjoint
undone merlin
oops
I meant not disjoint
my bad
sorry
Basically;
- If
Ais not disjoint, then letI_0 = Aand all otherI_n = 0 - Otherwise, let
{I_n}be the set of all open non-disjoint subsets of A
(1.) is the trivial case for (2.)
undone merlin
I only know about two sets can be disjoint
$$A = \left{ 0 < x < 2 : x \ne 1 \right}$$
grizzled pagodaBOT
Shuba
undone merlin
----(---)(---)----
0 1 2
stuck kindle
so if A is not that interval it is disjoint, is that how it's defined?
undone merlin
are you familiar with (or starting to learn) topology?
stuck kindle
I'm starting
undone merlin
thought maybe so XD
So, consider the usual topology on the real numbers (R,T) where T is the set of all open subsets on R.
A is open in R because, for all points x in A there exists a small region epsilon such that x - epsilon and x + epsilon are also in A.
A is disjoint because it is the union of the open subsets (0,1) and (1,2) whose intersection is the empty set.
I'm trying not to simply say "a doesn't contain 1" because that requires the topology to induce a metric, which is not always the case. It just happens to be so for the real numbers, which are an incredibly rich topology.
stuck kindle
oh not going that deep into topology, just learning about some basic properties of R and R^n
I know a set is open if you can fit a small open ball around any point
but I haven't heard before how a set by itself can be disjoint
undone merlin
You could define I = { I_n \subseteq A : I_n is open in R }.
by the axiom of choice
stuck kindle
the set of all open intervals in A?
stuck kindle
but then they are not disjoint no?
undone merlin
For example, if A = (0,1) U (2,3) U (5,6), then I = { {}, (0,1), (2,3), (5,6), (0,1)U(2,3), (0,1)U(5,6), (2,3)U(5,6), (0,1)U(2,3)U(5,6) }
stuck kindle
but shouldn't it also contain (0, 0.5)?
undone merlin
ah yes, very good point
so it's there's f****n loads of open subsets in I XD
That's probably a good way to define I.
Then, you want to prove A = U I_n
Again, you'd need to show A ⊆ U I_n and A ⊇ U I_n.
what would it mean for an element x ∈ A and x ∈ U I_n?
stuck kindle
but still I don't see how these I_n are disjoint like this?
undone merlin
actually, this is a great exercise to learn topology with
undone merlin
but still I don't see how these I_n are disjoint like this?
Take, for example, the I_n = (0, 0.5) U (2.1, 2.9)
is it a subset of A?
is it open?
is it disjoint?
undone merlin
this one is all of the three
yep
undone merlin
but still I don't see how these I_n are disjoint like this?
i see the confusion. sorry. I've mislead you.
The term "disjoint" is meaningless for a set on its own. It has meaning for two subsets. (i.e. two subsets are disjoint if their intersection is empty).
Still, for your question, I would define I to be the set of all open subsets of A. And then prove A is the union of all elements in I.
stuck kindle
but I would contain A itself so it would be trivial no?
amber waspBOT
@stuck kindle Has your question been resolved?
undone merlin
but I would contain A itself so it would be trivial no?
you could say proper open subsets. i.e. subsets that aren't the set itself.
stuck kindle
and how is it guaranteed that they are disjoint?
undone merlin
why do you want disjointedness?
For any open set of the real numbers, you can construct two disjoint open subsets.
stuck kindle
oh and they have to be intervals
I want to represent A as a disjoint union of open intervals
countably many actually
undone merlin
oh and they have to be intervals
with that requirement, you wouldn't need that the set itself necesserily not be in I (because A could be irrepresentable as an interval).
For example, A = (0,1) U (5,6) clearly isn't an interval, and yet we can construct I as the set of all open intervals on A.
stuck kindle
yeah but then I will also contain stuff like (0, 0.5), (0.2, 0.8) and it won't be disjoint
undone merlin
if you're after disjoint intervals, that's impossible.
by which I mean, if you want I to be made of intervals I_n who are the union of two disjoint open subsets.
stuck kindle
but clearly I_1 = (0, 1) and I_2 = (5, 6) in this case right?
undone merlin
And (0,0.5) etc...
I think the problem you have is you're trying to define A as the union of a set of intervals, then construct a set of intervals I whose union is A. this is circular reasoning.
We instead want any arbitrary A, and construct some set of open intervals I whose union is A.
But even that is a little circular. Depending on how you define the Is it may or may not span A.
There might be a third option for constructing I, but the two we've come up so far is
- trivially
I_0 = Aand all otherI_n = 0 - let
Ibe the set of all open subsets ofA.
stuck kindle
And `(0,0.5)` etc...
but I don't need to include those in my definition of I_n
undone merlin
but I don't need to include those in my definition of I_n
yes, you don't need to. but the proof that A is the union of all Is is circular if disclude them. How do you disclude them without looking at A?
stuck kindle
but why do I have to include them all?
my task is just to show there is some way to define I_n that works
undone merlin
but why do I have to include them all?
How would you construct I indepent of A otherwise?
Otherwise, it's trivial.
If $A$ is a single interval, then let $I_0 = A$ and all other $I_n = \emptyset$. Otherwise $$\qq{let} A = \bigcup_n J_n$$. Then you trivially have $$A = \bigcup_n J_n = \bigcup_n I_n \implies J_n = I_n$$
grizzled pagodaBOT
Shuba
undone merlin
Yes, you don't need ALL the open subsets. But you can't define I otherwise without defining A as a union of open intervals.
stuck kindle
but I'm not allowed to define A myself, I have to represent it as a union of open intervals instead
undone merlin
So how do you define I without knowing what the maximal open intervals on A are?
stuck kindle
I don't know, but that's what I'm trying to figure out
undone merlin
You cant.
You can construct a set of maximal open intervals FROM the set of all open intervals.
But you can't just construct the set of maximal open intervals because how do you know each interval is indeed maximal without defining A as a set of maximal open intervals.
Anyway, what's so bad about having some overlap XD
stuck kindle
that's just what the exercise wants
undone merlin
Can you post the full text of the exercise?
I think I'm not understanding what it wants.
stuck kindle
Show that every open set $A \subseteq \mathbb{R}$ is a countable and disjoint union of open intervals.
grizzled pagodaBOT
hanno
undone merlin
Let $O$ be the set of all open intervals of $A$. We define $\sim$ for all $X,Y \in O$ as $$X \sim Y \iff X \cap Y \ne \emptyset$$
Then $\sim$ forms an equivalence relation on $O$. Let $[I]$ be the equivalence class of $I$. Then $\left{ \cup [I] : I \in O \right}$ is a decomposition of $A$ into pairwise disjoint open intervals.
\
Now you just need to show \begin{enumerate}
\item $x \in A \implies \exists I \in O .; x \in [I]$ (i.e. $A \subseteq \left{ \cup [I] : I \in O \right}$
\item $\exists I \in O .; x \in [I] \implies x \in A$ (i.e. $\left{ \cup [I] : I \in O \right} \subseteq A$
\end{enumerate}
grizzled pagodaBOT
Shuba
undone merlin
Perhaps a better way...
Let $A$ be a non-empty open subset of $\mathbb{R}$. for all $x,y \in U$ we define the relation $$x \sim y \iff \left[\mbox{min}(x,y), \mbox{max}(x,y)\right] \in A$$. Then $\sim$ forms an equivalence relation on $A$ whose equivalence classes are pairwise disjoint open intervals in $\mathbb{R}$. Let $I = \left{ \cup [x] : x \in \mathbb{R} \right}$.
grizzled pagodaBOT
Shuba
undone merlin
What I'm trying to construct is, given the set of all open intervals of A, we construct the elements of I as the union of all overlapping open intervals.
I need to go. I'll be back in an hour or so if you want me to continue this thought.
amber waspBOT
@stuck kindle Has your question been resolved?
stuck kindle
that might work, it would only be left to show these are indeed open intervals and there are countably many
hmm if y is in [x], then y is also in A. Then there exists an open ball around y with radius epsilon in A, let's call it B. Then if z is in B, we have of course that [min(y, z), max(y, z)] is in B and thus in A too, so y~z and z is in [x] too, which means that B is a subset of [x], so [x] is open
[x] is also an interval, because for any two points y < z in [x], we have that y ~ z and so [y, z] is a subset of [x]
oh we also haven't shown that it's even an equivalence relation
I think reflexivity and symmetry should be trivial
for transitivity, let's say x ~ y and y ~ z. If x < y and y < z, then we have that [x, y] and [y, z] are subsets of A, thus their union is a subset of A too, which is precisely [x, z]. If y < x and y < z, we have that [y, x] and [y, z] are subsets of A. Then [x, y] is a subset of [y, z] and thus a subset of A. The remaining cases are similar
not sure how to show they are countable yet
amber waspBOT
@stuck kindle Has your question been resolved?
undone merlin
right, I'm back 🙂
amber waspBOT
@stuck kindle Has your question been resolved?
amber waspBOT
@stuck kindle Has your question been resolved?
amber waspBOT