ok but how do we count that as bijection
Since in bijection we would be connecting every even subset to a " unique " odd subset and vice versa
#help-49
robust forge
round parcel
OK, now we just said that we could add or remove an element to get a bijection.
So, we'd need to decide which element to do that with.
robust forge
So, we'd need to decide which element to do that with.
yes
how are we going to decide that?
round parcel
Well, what's an idea for picking that element?
Let's examine it.
Even if it won't work.
Like what would you suggest as a first try?
robust forge
Well, what's an idea for picking that element?
adding a element that is smaller than the smallest element in the subset or removing the smallest element or we could do it with the biggest too
round parcel
OK, so let's try the smallest first.
Let's say we have a subset {5, 6, 7}.
How would we convert that to an even subset?
robust forge
OK, so let's try the smallest first.
the problem will be with the null set since the term smallest isn't applicable there
round parcel
Well, we'd have to add an element to that one.
robust forge
Let's say we have a subset {5, 6, 7}.
Removing 5 which is the smallest element
round parcel
OK, so that gives us {6, 7}.
Now, let's forget for a moment that {5, 6, 7} gave us {6, 7}.
If we started from the beginning, how would we handle {6, 7}?
robust forge
If we started from the beginning, how would we handle {6, 7}?
adding a element smaller than 6 which "could " be 5
robust forge
What would we do with that?
we can't remove 3 since {6,7} is already bijected to {5,6,7}
so does that mean our strategy of smaller elements doesn't work ?
round parcel
Not necessarily.
It just means that removing the smallest element in the set won't necessarily work.
robust forge
It just means that removing the smallest element in the set won't necessarily wo...
Hmm I see
By the same logic above we could argue that the greatest no strategy won't necessarily work either
let me think of something else
ehh nothing's coming to mind, any hints ?
round parcel
adding a element that is smaller than the smallest element in the subset or remo...
Well, this isn't a bad starting place.
Removing the smallest element in the subset wouldn't work, but the first part could work for part of it.
robust forge
Removing the smallest element in the subset wouldn't work, but the first part co...
yea I figured that out since two subsets can have diff smallest elements while the rest of subset remains same so they biject to the same even subset
robust forge
hmm so the only choice is of adding
adding a element smaller than the smallest element of the subset
round parcel
OK, so how would you determine which element to add?
Like let's say you have {5, 6, 7}.
robust forge
Like let's say you have {5, 6, 7}.
adding a element smaller than 5
round parcel
Right, but which element?
robust forge
Right, but which element?
One smaller than 5 which would be 4
but If the smallest element is 1 then it won't work
round parcel
Why not?
round parcel
OK, so it won't work for a set with 1 in it.
robust forge
yes
round parcel
So, let's talk about sets without 1 in them.
robust forge
So, let's talk about sets without 1 in them.
Well then our strategy of adding one less than smallest element works
round parcel
Well, we did {5, 6, 7} -> {4, 5, 6, 7}.
But if we forget that and work with {4, 5, 6, 7}, what do we get?
robust forge
{3,4,5,6,7}
round parcel
Right, so with {5, 6, 7}, the element we add can't be 4.
But it might be something else in line with adding a smaller element.
So, we have 1, 2, and 3.
robust forge
So, we have 1, 2, and 3.
ahh adding 1 would be it
round parcel
OK, so {5, 6, 7} -> {1, 5, 6, 7}.
Now if we forget that, what does {1, 5, 6, 7} go to?
robust forge
Now if we forget that, what does {1, 5, 6, 7} go to?
{1,5,6,7}
So the modification should be that we add to those which don't have 1 in them and remove 1 from those which have it in them ?
round parcel
OK, so let's look at that idea.
{5, 6, 7} -> {1, 5, 6, 7} and {1, 5, 6, 7} -> {5, 6, 7}, so that works.
Are there any ways it wouldn't work?
robust forge
Are there any ways it wouldn't work?
Doesn't seem like it
round parcel
Right, each odd set pairs with an even set, and the pairing is unique.
robust forge
Doesn't seem like it
Also we can do this with any element r <= n right?
adding r if the set doesn't contain it and remove if it does
robust forge
I see and this must be called xoring
round parcel
Yeah, XOR is, loosely, toggling whether something is there.
robust forge
But 1 is nice because it works for any starting n.
I see
robust forge
Yeah, XOR is, loosely, toggling whether something is there.
Ohh I see
Thanks a ton for the help 🙏 ❤️
Have a good day ahead 🎈
round parcel
You're welcome. You too.
robust forge
.close
midnight plankBOT
midnight plankBOT
Available help channel!
elfin fox
What should my opening line be to prove this by contradiction?
midnight plankBOT
What should my opening line be to prove this by contradiction?
solid iris
What should my opening line be to prove this by contradiction?
doesnt look like you need contradiction
the claim is $W^\perp=\brc{x\in V:\lbr{x,w_j}=0 \text{ for } 1\le j\le k}$
grand pondBOT
𝓡𝓞𝓚𝓔𝓣𝓣𝓞─୨ৎ─❥ ♡ <𝟹❤
solid iris
try a good ol double inclusion
midnight plankBOT
@elfin fox Has your question been resolved?
gritty hatch
Ok.
gritty hatch
This video explains how to use an algorithm that let's you determine the greatest common denominator.
buoyant yoke
yoooo smth new
buoyant yoke
so you want to find the greatest common divisor, right?
grim vector
Thats a cool thing to do in cs if interested
buoyant yoke
cs?
grim vector
Coding
dawn dagger
my recursive sportsfreunde
buoyant yoke
so you want to find the greatest common divisor, right?
@gritty hatch
grim vector
my recursive sportsfreunde
Everybody's gangsta till you don't have to draw the memory state at line 43
buoyant yoke
dont abandon me so soon techno
gritty hatch
I'm back
buoyant yoke
aight
grim vector
Yeah don't let us alone techno
gritty hatch
I has abandoned you!!! Your korps is defunded! Rahhh!!! Hahahaha all your base are belong to me.
Alr anyways
gritty hatch
so you want to find the greatest common divisor, right?
._. Maybe,
buoyant yoke
so, greatest common divisor (gcd) between two numbers right?
gritty hatch
So they said if I understood that
buoyant yoke
you can do smth rly simple
gritty hatch
rdapp.net/attachments/903463354654404658/1341121680579494001/1340453306207572073...
That i would understand this
buoyant yoke
ill explain it to you
buoyant yoke
rdapp.net/attachments/903463354654404658/1341121680579494001/1340453306207572073...
idk what this is
gritty hatch
buoyant yoke
i dont wanna open random link
gritty hatch
This
grim vector
https://media.discordapp.net/attachments/903463354654404658/1341121680579494001/...
Isleep
buoyant yoke
i can explain it super simple
gritty hatch
Alr
gritty hatch
ight
buoyant yoke
we take the smaller number
and subtract it from the larger number
16 is the smalle rnumber
so we subtract it from 40
so we get our new numbers, 40-16 and 16
which are 24 and 16
yes?
gritty hatch
Alr
buoyant yoke
now, we repeat this process until we get the same number
so we have 24 and 16
16 is smaller so we subtract it from the larger one
to get 24-16 and 16
thus 8 and 16
now 8 is smaller than 16
we subtract it from 16
8 and 16-8
so we get 8 and 8
now, this is the same number twice
thus the greatest common divisor of 40 and 16 is 8
make sense?
buoyant yoke
yes
gritty hatch
Alr
buoyant yoke
because the greatest common divisor of 40 and 16 is the same as 16 and 40
ill give you an example
and see if you can find the GCD yourself
alright?
grim vector
The result (gcd) is the last non zero number in the algorithm
gritty hatch
Yes
buoyant yoke
lets use the numbers 133 and 21
follow the algorithm
(write out all your steps)
(you can write out one step at a time, no need for doing all at once)
gritty hatch
Alrrrr
buoyant yoke
continue
gritty hatch
48 27
buoyant yoke
typo?
gritty hatch
Where
buoyant yoke
IM SORRYYY 😭 😭 😭 😭
grim vector
Kekw
buoyant yoke
yes!
grim vector
Gut
buoyant yoke
so, now what is the gcd?
buoyant yoke
yup
gritty hatch
Alrrr
grim vector
Euclide is really quiet since techno is using the algorithm
buoyant yoke
70-21 49 -21 = 28 = 7 21-7 = 14
do you see how you subtracted 21, 6 times?
instead of doing it 6 times, you can subtract 21*6=126 in one go
gritty hatch
Euclide is really quiet since techno is using the algorithm
I will use the algorithm on euclid to make him non-euclidean.
gritty hatch
Wow
buoyant yoke
yeah, magic isnt it
grim vector
That would result in an euclidian division with the |_ symbol yknow
gritty hatch
Wait you said i subtracted 21 6 times but near the end it was 7 and no longer 21
gritty hatch
Oh
buoyant yoke
There is also some proprety to gcd calculuation
dont make it too complex
grim vector
Ok
grim vector
Imma just sit and watch
gritty hatch
I think so so do you multiply by the end of the usage of 21s? So like after it goes below 21 do you not count the 7-21 as would that not make 6
buoyant yoke
I think so so do you multiply by the end of the usage of 21s? So like after it g...
yeah
you go until you make a negativ enumber
if you did 7-21 you get -14
which we dont wanna go to
gritty hatch
So 7-7 would also apply to the 21?
To make 6
Wdym 7-21
It was 21-7
Unsure actually
You know how at one point you reach 7-7?
gritty hatch
You know how at one point you reach 7-7?
Does this count towards the 6 you mentioned
buoyant yoke
no
the 6 refers to the times that you can subtract 21 from 133
if you do 133-21, 6 times you get 7
if you do 133-21 7 times then you get -14
which is one too many
gritty hatch
https://media.discordapp.net/attachments/903463354654404658/1341121680579494001/...
How does that apply to this
Bc they said it did somehow
buoyant yoke
okay
ignore the x for now
p=q*g+r
yes?
now
p=133
q=21
g=6
r=7
p is the number we are subtracting from (the bigger number)
q is the number we are subtracting (the smaller number)
g is the amount of times we subtract the smaller number
r is the remainder after subtracing q, g times
gritty hatch
One second
buoyant yoke
that your time
gritty hatch
im back
ashen yarrow
hi guys
ashen yarrow
does this discord do patten maths puzzles ?
buoyant yoke
does this discord do patten maths puzzles ?
sometimes
but read #❓how-to-get-help first
gritty hatch
you asked the question tho
grim vector
you gotta give me a better question techno
How
gritty hatch
How
you put a question mark after yes implying you wanted to know if p=q*g+r and i said i didnt know why
buoyant yoke
that we ignore the (x) stuff in the equation
you see here p(x)=q(x)g(x)+r(x)?
we just remove the (x) stuff and turn it into p=qg+r
gritty hatch
how?
grim vector
Just put p(x), g(x), q(x), r(x) as constants functions and this is the same mindset to the algorithm
gritty hatch
q is the number we are subtracting (the smaller number)
wdym wasnt q the number after the division
gritty hatch
?
buoyant yoke
p is the larger number
q is the smaller number
g is the amount of times you can subtact q from p
r is the amount leftover after your subtraction
so, from our 133 and 21 example
gritty hatch
buoyant yoke
!nogpt
midnight plankBOT
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
buoyant yoke
techno pls
gritty hatch
!nogpt
i doub that was only gpt
bc it said that bc before i interpreted that
as what someone else said
gritty hatch
that was not gpt ._.
that was gpts interpretation of my interpretation of what someone else said
buoyant yoke
whatever
gritty hatch
just talk to me instead
but... what if you are secretly a gpt?
buoyant yoke
perhaps i am
buoyant yoke
anyyway
gritty hatch
alr
gritty hatch
i believe so you subtract until you get to 7 or whatever value you put
to get the number
buoyant yoke
that for this example, yes
gritty hatch
to subtract the big number by
buoyant yoke
wanna do another example?
gritty hatch
alr
dawn dagger
you can also try to find gcd(-1,0)
dawn dagger
hmm?
gritty hatch
55-9 = 46
46-9 = 37
37-9 = 28
28-9 = 19
19-9 = 10
10-9 = 1
9x6 = 54
54-55 = 1
buoyant yoke
continue
gritty hatch
it was 6 right?
gritty hatch
it
well i divided 6 times so i thought it was 6 or 1 and after i subtracted i had 1
so i guess 1?
gritty hatch
p=55
q=6
g=9
r=1
buoyant yoke
okay, do you still have a question?
dusk pier
@gritty hatch What's up man
gritty hatch
<@563179988191281222> What's up man
buoyant yoke
any more questions?
buoyant yoke
🤣
gritty hatch
buoyant yoke
yoiu you wanna go
dusk pier
This is the incorrect one
dusk pier
Lord why is this your first problem
gritty hatch
ok so
8, 4 / 3, 5
dusk pier
Nevermind
gritty hatch
8/3 8/5
4/3 4/5
uhm........
wait a scond
SOB:
is it all the top numbers and all the bottem numbers individually
or is it the fractions themselves
humble wolf
it says "synthetic division", and by inspection, you can easily spot out one root
then you can carry out synthetic division to reduce it to a cubic polynomial
buoyant yoke
and how do they get this?
gritty hatch
and how do they get this?
simple
they divide the first and last numbers
to get a list
of available options
buoyant yoke
ah i see
gritty hatch
15 on top is 1 3 5 and 15 and 16 is 1 2 and 4
buoyant yoke
its the factors
round parcel
I think for 16, it should be 1, 2, 4, 8, 16.
gritty hatch
alr
round parcel
There are like 40 possible rational roots.
gritty hatch
15 + 1 - 52 + 20 + 16
gritty hatch
There are like 40 possible rational roots.
A
wait are those rational roots correct/usable?
round parcel
If the roots are going to be rational numbers, they each have to be one of those forty.
So, they're not necessarily the answers, but they're possibly some of the answers.
gritty hatch
are there multiple values (more than two) that would make the X's equal to zero?
round parcel
This is a fourth-degree polynomial, so it'll have four roots.
round parcel
OK 🙂
gritty hatch
.close
midnight plankBOT
lapis slate
Maniacs
midnight plankBOT
Available help channel!
round parcel
is it all the top numbers and all the bottem numbers individually
That's incorrect. The constant term's divisors go on top, and the leading coefficient's divisors go on bottom.
midnight plankBOT
That's incorrect. The constant term's divisors go on top, and the leading coeffi...
round parcel
(\pm \frac{1, 2, 4, 8, 16}{1, 3, 5, 15})
grand pondBOT
Chai T. Rex
round parcel
Plus, once you use synthetic division to get rid of one of the roots, you can work with the divisors of the new constant term and leading coefficient .
.close
midnight plankBOT
dusk pier
https://cdn.discordapp.com/attachments/1018703621841494076/1341534878986014810/i...
Techno
There is a very nice trick here if you understand it
We can define $P(x)=\sum_{k=0}^n a_k x^k$ where $a_k$ are the coefficients of the polynomial
grand pondBOT
mathisfun
dusk pier
$P(1)=\sum_{k=0}^n a_k=a_0+a_1+a_2...a_n$, so if $(x-1)$ is a factor of the polynomial then the sum of the coefficients must be zero since $P(1)=\sum_{k=0}^n a_k=a_0+a_1+a_2...a_n=0$
grand pondBOT
mathisfun
midnight plankBOT
Available help channel!
gritty hatch
There is a very nice trick here if you understand it
1?!??!1
buoyant yoke
here we go....
dawn dagger
rings...
buoyant yoke
@dusk pier go on, explain
gritty hatch
i was about to compile them
buoyant yoke
lol
gritty hatch
https://www.youtube.com/watch?v=AM-iv51sX1k
also this is a very good song at nvm it was 2:35
dawn dagger
it really starts off good at 1:05:33
gritty hatch
real
Rational roots test```Solving for this polynomial via RRtest and synthetic division.```
```lua
'Solving for'
[15 + 1 + 52 + 20 + 16]
3 5 15 1
over
8 5 2 1
Context to the Euclidean Greatest Common Factor method and its related variables to Synthetic Division.```
Mathisfun describing a more advanced but effective technique.```@gritty hatch Has your question been resolved?
humble wolf
```lua
Mathisfun describing a more advanced but effective technique.```
i also spotted that out earlier
humble wolf
explain further
math is fun's solution in your screen cap alr does the job
humble wolf
and I am curious as to what that is
you know the factor theorem?
gritty hatch
describe it
humble wolf
describe it
humble wolf
i think so, yes, its just the one saying if x - 1 yields zero then x = 1
actually P(1) equals to the sum of coefficient
say it this way:
when x = 1, P(x) = 0
or even simpler "P(1) = 0"
"x - 1 yields zero" doesn't make sense to me
then you can carry out synthetic division
to simply it to a cubic polynomial
dawn dagger
[professional](https://media.discordapp.net/stickers/1280966947198074952.png?siz...
dangerous man
gritty hatch
dangerous man
gritty hatch
actually *P*(1) equals to the sum of coefficient
say it this way:
> when *x* = 1...
- If the goal is to solve for X and x = 1 and if P is the polynomial pre division then why is P(x) zero? Is this attempting to describe how when you solve the goal is to define a set of operations thatd generate zero?
humble wolf
1. If the goal is to solve for X and x = 1 and if P is the polynomial pre divisi...
sorry i dunno what you're asking ("pre division"?)
the goal is to find the roots of P, and we can start with a simple root, then use synthetic division to make the problem easier to solve
gritty hatch
how come there are multiple possible roots?
urban widget
There just is
dawn dagger
fundamental theorem of algebra
midnight plankBOT
@gritty hatch Has your question been resolved?
molten bay
midnight plankBOT
midnight plankBOT
Available help channel!
sharp wave
tried taking e^sqrtalpha common from the denominator but how do i proceed?
midnight plankBOT
tried taking e^sqrtalpha common from the denominator but how do i proceed?
fallow scarab
tried taking e^sqrtalpha common from the denominator but how do i proceed?
Show
sharp wave
Ignore the part below that’s for another question
fallow scarab
Yea I rarely suggest L'Hopital, but that seems useful for this problem
sharp wave
wont that be crazy
fallow scarab
No
sharp wave
sin and then quotient rule
sharp wave
Oh yes
let me try
it isnt leading anywhere
i thought of one thing
cant we let e^t -1 be something like K
so that t = ln(k+1)
and the limit converts to ln(k+1) /(sqrt alpha +lnK)
where K tends to 0
sharp wave
it isnt leading anywhere
no nvm it does i tried applying it twice
thanks riemann
.close
midnight plankBOT
midnight plankBOT
Available help channel!
final shale
midnight plankBOT
sharp coral
how did you get your answer?
final shale
how did you get your answer?
In my statistics class we have a chart to work on
sharp coral
and how did you use the chart?
final shale
and how did you use the chart?
I found 2.45 in the z scores column, and then found the area between the mean and z and plugged that as my answer
sharp coral
and what in the problem indicated to you that it required the area between the mean and z?
final shale
and what in the problem indicated to you that it required the area between the m...
I'm not sure, the question was so foreign to me that I just had to put something in.
static arrow
I'm not sure, the question was so foreign to me that I just had to put something...
you don’t need the chart here, do you know the definition for z-score?
final shale
you don’t need the chart here, do you know the definition for z-score?
I know z-score = standard score
static arrow
yes but what information does the z-score/standard score actually tell us
final shale
yes but what information does the z-score/standard score actually tell us
How many SD units a score is from the mean
lyric charm
note btw this question does not involve any probabilities directly at all
fallen galleon
static arrow
How many SD units a score is from the mean
can you represent that relationship as an equation
final shale
can you represent that relationship as an equation
x minus the mean divided by the standard deviation
static arrow
x minus the mean divided by the standard deviation
yes and now can you figure out what to do with the numbers given in the question?
final shale
yes and now can you figure out what to do with the numbers given in the question...
I know it's x-88/26
static arrow
I know it's x-88/26
you’re missing the other half of your equation
final shale
What do I do with the z score
static arrow
what does (x-88)/26 equal
final shale
I don't know what x is.
static arrow
but what you meant is z = (x - μ)/σ
you mean to use this formula right
final shale
Yes, I am using that
static arrow
you can solve for x but you have to use the whole equation
both sides
what does the z in the equation represent
final shale
-2.45
static arrow
yes
final shale
What now?
static arrow
plug in z = -2.45 into the equation
final shale
So z will equal x?
static arrow
no because your equation is not z = x it’s z = (x - μ)/σ
final shale
no because your equation is not z = x it’s z = (x - μ)/σ
Okay so how do I find x then?
static arrow
i need to know that we are on the same page of plugging the numbers into the equation lol
what does the equation become when you plug in all the numbers you have
final shale
what does the equation become when you plug in all the numbers you have
It shows us the z score
final shale
x-88/26
final shale
x-88/26 = -2.45
final shale
so that means we need to isolate it
I'm guessing we could add 88 to both sides.
static arrow
so be careful with your order of operations there
final shale
We divide first?
static arrow
so be careful with your order of operations there
it should actually be (x-88)/26 = -2.45 if we want to make that notation less ambiguous
not divide in this case, but we don’t wanna deal with the 88 first, we wanna get rid of the 26 first and leave the brackets for last
final shale
not divide in this case, but we don’t wanna deal with the 88 first, we wanna get...
Which leaves us x-88 = -2.45
static arrow
when we solve equations we have to do the same thing to both sides
final shale
x-88 = -28.45
static arrow
no i’m not sure where you got the number 28.45 from
static arrow
no we can’t subtract on both sides because subtracting is the opposite of adding
how would we get rid of 26 if the 26 is the denominator
final shale
how would we get rid of 26 if the 26 is the denominator
We divide by 26 (on both sides)
static arrow
no we are already dividing by 26 in the equation so we have to do the opposite of dividing
final shale
How do we multiply then?
static arrow
the opposite of dividing by 26 is multiplying by 26
final shale
Do we only multiply the numerator?
static arrow
when you multiply (x-88)/26 by 26 the 26s will cancel each other out and you will have x-88
final shale
when you multiply (x-88)/26 by 26 the 26s will cancel each other out and you wil...
So then we just have x-88 = -63.7
final shale
now we just need one more step almost there 🤠
We add 63.7 on both sides?
final shale
So we add 88 instead
static arrow
yes
final shale
Which means x=24.3
static arrow
yes
i would recommend that you do some practice questions and maybe watch some videos to just review the actual algebra and solving equations, you will want to be pretty solid with these skills to make it through a stats class 🫡
final shale
i would recommend that you do some practice questions and maybe watch some video...
Yeah I mean I already technically passed this quiz with an 80% but I have an extra retake so I still want to try to get a higher grade
Statistics has not been super difficult but every single point counts for me and I'd like to be far from failing as possible
static arrow
yeah i would say just practice solving and rearranging equations in general, it doesn’t even have to be stats specific questions but you can just google like “solving linear equations worksheet” and then the actual algebra will become easier which will save you time and stop you from making arithmetic errors
final shale
yeah i would say just practice solving and rearranging equations in general, it ...
I actually sort of enjoy statistics since it's an easier form of math
static arrow
lol idk if i would put it like that, it will get pretty hard at a certain point 😅
final shale
lol idk if i would put it like that, it will get pretty hard at a certain point ...
Well this is only an introductory class I'm taking since i'm a psychology major
young spire
Sorry it's on tilt I'm stuck on q12
midnight plankBOT
Sorry it's on tilt I'm stuck on q12
young spire
I tried subbing in -3 but the polynomial qx is confusing me
lyric charm
,rccw
grand pondBOT
lyric charm
lyric charm
I tried subbing in -3 but the polynomial qx is confusing me
so what do you actually get once you sub in x=-3?
like, if you are confused about something then just leave it as is, but show me what you get anyway
young spire
I got -3(a+b+q(x))
lyric charm
that sounds (i) wrong and (ii) like you didnt actually substitute properly
or maybe you severely screwed up the simplification
$P(-3) = (-3-1)(-3+3)Q(-3) + a \cdot (-3) + b$
grand pondBOT
ann.in.a.teacup
lyric charm
**ann.in.a.teacup**
this is what you get when you sub in x=-3.
do you understand?
im deliberately leaving shit unsimplified.
young spire
Ok
lyric charm
do you understand what i've done so far?
young spire
Ye
lyric charm
ok now look at this -3 + 3. what is -3 + 3 equal to?
young spire
0
lyric charm
yes, the first product just ends up as 0 so you're left with a * (-3) + b.
i would have liked for you to verbalize the "anything times 0 is 0" thing explicitly.
but w/e
midnight plankBOT
midnight plankBOT
Available help channel!
dusk raven
- Combination + No replacement
- Permutation + No replacement
- Permutation + Replacement
midnight plankBOT
1) Combination + No replacement
2) Permutation + No replacement
3) Permutation +...
nimble copper
1
nimble copper
thorn burrow
we have total 44 balls (including bonus balls)
dusk raven
I have the answer provided by the teacher but I dont understand
thorn burrow
do you know how nCr and nPr work?
thorn burrow
nCr = number of ways to select r objects from n distinc objects
dusk raven
nCr = number of ways to select r objects from n distinc objects
ik the basic knowledge
but Idk how to apply in this question
this question get really complicated
midnight plankBOT
@dusk raven Has your question been resolved?
dusk raven
<@&286206848099549185>
midnight plankBOT
@dusk raven Has your question been resolved?
dusk raven
<@&286206848099549185>
midnight plankBOT
@dusk raven Has your question been resolved?
dusk raven
.close
marsh hawk
can someone explain 1c to me
midnight plankBOT
can someone explain 1c to me
sudden yacht
What's cos(30°)?
midnight plankBOT
@marsh hawk Has your question been resolved?
runic pawn
Des francais
midnight plankBOT
midnight plankBOT
Available help channel!
gusty falcon
hi so my my question is more about why we can't do it rather than how to do it (because i know of a valid way to do it)
midnight plankBOT
hi so my my question is more about why we can't do it rather than how to do it (...
gusty falcon
The question is: compute the remainder when 9^1 + 9^2 + 9^3 + 9^4 + 9^5 + 9^6 + 9^7 + 9^8 + 9^9 is divided by 6
buoyant yoke
what is the remainder when 9 is divided by 6?
gusty falcon
,, \frac{\sum_{k = 1}^9 9^k}{6} = \frac{9( \sum_{k = 0}^8 9^k)}{6}
grand pondBOT
mmmm7
gusty falcon
so far so good right
nowww uh
why can't i simplify the 9 and 6
$\frac{3( \sum_{k = 0}^8 9^k)}{2}$
grand pondBOT
mmmm7
buoyant yoke
its much simpler than this
gusty falcon
no 😭 it's not about how to solve it
yeah i see that it's cyclic
9 = 3 mod 6
and it's like 331 331 331 and so on
buoyant yoke
okay, yes
buoyant yoke
7*3?
gusty falcon
each group adds up to 7
buoyant yoke
ohhh, nvm
gusty falcon
**mmmm7**
i can now simplify (i don't see why i can't; in fact, that's my question cuz u get the wrong answer now)
now we're looking at the remainder the numerator leaves mod 2
which can technically only be 0 or 1
it's obvious that this is the issue maker in the question
so yeah i wanted to know what's the reasoning behind it
buoyant yoke
happens
buoyant yoke
continue
buoyant yoke
oh, i thought you were gonna complete the pattern
buoyant yoke
okay im a bit confused what youre doing
gusty falcon
and that gets you the answer immediately
,, \frac{\sum_{k = 1}^9 9^k}{6} = \frac{9( \sum_{k = 0}^8 9^k)}{6}
grand pondBOT
mmmm7
buoyant yoke
ah yes, i see that its true now
lyric charm
,, \frac{\sum_{k = 1}^9 9^k}{6} = \frac{9( \sum_{k = 0}^8 9^k)}{6}
you aren't dividing the sum by 6 tho you want its remainder modulo 6
gusty falcon
you aren't dividing the sum by 6 tho you want its remainder **modulo** 6
yep that's fair but uhhh
the algebra checks out tho
😭
lyric charm
so such simplifications are sussy
gusty falcon
i see that modulo 2 now causes us to only have remainders {0,1} so that was the issue but tbh it didn't make much sense
lyric charm
10 % 4 and 5 % 2 don't have a lot to do w each other
gusty falcon
10 % 4 and 5 % 2 don't have a lot to do w each other
oh
is there something (theorem of what not)
talking about this?
that i can read
like the reasoning i figured is basically what u said yeah;
we were initially dealing with modulo 6, in which we had 6 integers possible as remainders
when we transformed to modulo 2 with "simplification"
that reduces our choices to only 2 possible integers
which is fishy yep, but is there something about equivalence classes or whatever that doesn't permit this?
midnight plankBOT
@gusty falcon Has your question been resolved?
crimson stone
midnight plankBOT
lethal path
Do I use the point-line formula?
you need r.n = d
lethal path
yep so a normal vector of plane Pi_1 is (0, 2, 2)
but to apply the formula correctly, you need the unit normal vector
which would be n = (0, 1/sqrt2, 1/sqrt2)
crimson stone
What is unit normal vector?
lethal path
and then r is just the position vector (a, a, a - 7)
lethal path
What is unit normal vector?
the normal vector which has magnitude (length) equal to 1
lethal path
and then r is just the position vector (a, a, a - 7)
(a, a, a - 7) dot (0, 1/sqrt2, 1/sqrt2) = sqrt2
lethal path
Why do you sqrt2?
sqrt2 is the distance
midnight plankBOT
midnight plankBOT
Available help channel!
atomic swallow
here, the top statement has been proven (context is A = LU, u_j denotes the j'th row of U)
do these two logically follow? if so, what lets you just "norm upon thee" something like this?
midnight plankBOT
here, the top statement has been proven (context is A = LU, u_j denotes the j'th...
atomic swallow
.....?
atomic swallow
I've been writing here for 5 minutes.
oh, i didn't see anyone 💀 my b
runic hamlet
in general you cant just apply norms to each individual term
atomic swallow
is there some way to reduce a norm around everything to what is written up there though?
hmm
brisk iris
is there some way to reduce a norm around everything to what is written up there...
I don’t think it works in a general case
Can you share the original problem. We might be able to help
atomic swallow
here, the top statement has been proven (context is A = LU, u_j denotes the j'th...
i have the LU factorization of some matrix A, and have proven the first statement here
i now need to prove this
oh wait
atomic swallow
here, the top statement has been proven (context is A = LU, u_j denotes the j'th...
oop
thats what i'm given to use, at least
for some rows u_i a_i of that LU factorization
runic hamlet
well ok you only want <= so its probably enough if you just apply the triangle inequality
do you have a bound for L_ij ?
atomic swallow
i'm assuming row pivoting so yeah, absval of any element <=1
runic hamlet
ok then just induction
atomic swallow
oh?
runic hamlet
start with u_1, u_2, u_3 and see how those expressions actually look like
midnight plankBOT
@atomic swallow Has your question been resolved?
atomic swallow
ok got it
cool question
ive also just realized that the inifnite norm of one of the vectors u_j^T is actually just the maximum value oops
uhhh
i'll add a note :)
ty for the help!
.clowse
.close
last slate
midnight plankBOT
last slate
hello
last slate
so here for T :R^3 -> R^3 to be a linear transformation,
T(u1,u2) = T(u1)+T(u2) and
T(a u_1)=aT(u_1)
dumb question but there are 3 over here
im stuck
there is u1,u2,u3
also what would t(ui) be ?
i =1,2,3,....
atomic swallow
so here for T :R^3 -> R^3 to be a linear transformation,
T(u1,u2) = T(u1)+T(u2) ...
i think your first one should be T(u1+u2) = T(u1) + T(u2)
it might help to think of these multivalued transformation as linear if ALL of the slots follow this rule
atomic swallow
you could just do x = (x1, x2, x3) as well yeah
y = (y1,y2,y3)
x+y = (x1+y1, x2+y2, x3+y3)
etc.
last slate
i think your first one should be T(u1+u2) = T(u1) + T(u2)
but this transformation only takes 3 entries right ?
atomic swallow
but this transformation only takes 3 entries right ?
the example you gave took 1
this one takes 3 yes
but like
atomic swallow
you could just do x = (x1, x2, x3) as well yeah
y = (y1,y2,y3)
x+y = (x1+y1, x2+...
dawn dagger
you could just do x = (x1, x2, x3) as well yeah
y = (y1,y2,y3)
x+y = (x1+y1, x2+...
exactly that
atomic swallow
yep
last slate
midnight plankBOT
last slate
Find if the transformation is Linear or not
whats C'(a,b) and C(a,b) ?
just some random set?
twilit field
Usually C^1 the set of once differentiable and differential is continuous functions
midnight plankBOT
midnight plankBOT
Available help channel!
twilit field
midnight plankBOT
twilit field
$a_{n}-a_{n-1} =K a_{n-1}(N-a_{n-1})$ is my best bet
Is that right
or is it $a_n-a_{n-1} = a_n(N-a_n)$
grand pondBOT
What a wonderful world it is !
visual tiger
both are "okay", but one would be a forward euler scheme while the other would be a backwards euler scheme
twilit field
what
visual tiger
eh don't worry too much about it
visual tiger
$a_{n}-a_{n-1} =K a_{n-1}(N-a_{n-1})$ is my best bet
keep this one
but add a constant in front
grand pondBOT
What a wonderful world it is !
visual tiger
as the word "proportional" in the sentence "the rate of diffusion is proportional to..." is important
twilit field
This now
What do they mean by discuss the model here?
Find conditions for equilibrium and what would a positive and negative k mean I suppose
Equilibrium would require k=0 or M=a_n or a_n=m
A negative k would mean a declining population
a positive k would mean an increasing population
sharp coral
also discuss what the end behavior would be depending on initial conditions
midnight plankBOT
@twilit field Has your question been resolved?
lament fox
I have a vector $\vec{\omega}$ and its time derivative $\frac{d\vec{\omega}}{dt}=\vec{\alpha}$ such that $\vec{\alpha}$ is perpendicular to $\vec{\omega}$ at all times.
midnight plankBOT
I have a vector $\vec{\omega}$ and its time derivative $\frac{d\vec{\omega}}{dt}...
grand pondBOT
The د
lament fox
how long does it take for $\vec{\omega}$ to make one revolution and return to its original state?
grand pondBOT
The د
lament fox
so $r = \frac{\omega^2}{\alpha}$?
grand pondBOT
The د
keen saddle
we are able to do this because "alpha is perp to omega always"
grand pondBOT
The د
keen saddle
i think so
lament fox
ok thanks!
keen saddle
!close
lament fox
.close
midnight plankBOT
keen saddle
midnight plankBOT
Available help channel!
atomic grail
midnight plankBOT
atomic grail
Can someone help me with this equation? I wrote x as the sum of [x] and {x} an did the calculus and it gave me smth wrong
runic hamlet
show your work
runic hamlet
also no clue what happens after that
runic hamlet
take just the two left expressions of the fourth line and cross multiply
write [x]+{x} as x again
grand pondBOT
atomic grail
What s a size argument
runic hamlet
as an example, if x was around 100, then how big would the lhs and rhs be
runic hamlet
play this idea through with different numbers
atomic grail
How will it help me
runic hamlet
because you will only be left with very few options of what [x] can be
and then you can just solve the resulting expression for {x}
atomic grail
Tell examples
runic hamlet
how about you do things yourself
atomic grail
But i don t see it
I mean if I write integer part
As the ratio of other two
And fractional part is denominator
runic hamlet
as an example, if x was around 100, then how big would the lhs and rhs be
do this but with other numbers instead of 100
could 50 work?
runic hamlet
abs(x) approx 50
atomic grail
What s abs
runic hamlet
well in the original equation we cant divide by 0 so {x} cant be zero
runic hamlet
whats wrong with that
atomic grail
Well 0<{x}<1