please ping if you are replying
#help-43
wintry knot
compact pewterBOT
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gilded lark
compact pewterBOT
kind viper
,rccw
gilded lark
,rccw
boreal girderBOT
cursive harbor
Factorise the second equation
gilded lark
With the constant?
kind viper
progress on 13?
cursive harbor
Right sorry Ann
gilded lark
I did but got nothing
kind viper
it'll be important that x and y are positive integers here
in particular you might find it useful to add 1 in the first equation to get
xy + x + y + 1 = 72
think about why i am suggesting this
gilded lark
Idk why but my image quality 🤌
gilded lark
in particular you might find it useful to _add 1_ in the first equation to get
...
I want this intuition
kind viper
yeah ok this is a no-go
cursive harbor
in particular you might find it useful to _add 1_ in the first equation to get
...
That is genius
kind viper
ig i can say it: xy+x+y+1 factors as (x+1)(y+1).
gilded lark
ig i can say it: xy+x+y+1 factors as (x+1)(y+1).
🤌
kind viper
you can then streamline your guess-and-check for factorizations of 72 into two factors by making a table with the columns x+1, y+1, x, y, x+y
and checking whether the product of the last three can possibly be equal to 880 (sometimes it will be divisible by primes it's not supposed to be)
gilded lark
and checking whether the product of the last three can possibly be equal to 880 ...
Long process must wait
kind viper
it isn't that long
gilded lark
Ohk
kind viper
do you want me to show you my thought process for how i did it just now
gilded lark
Pls
gilded lark
do you want me to show you my thought process for how i did it just now
Pls show
kind viper
gilded lark
Ty for this
Using colourful pens for me 🤌
kind viper
point is i dont really ever work out the product except in the one green case
instead i find a reason why it can't be what im looking for (880) and then move on knowing this case needs no more attention
gilded lark
146
kind viper
of course it helps to know the first few primes
gilded lark
of course it helps to know the first few primes
Fr
gilded lark
of course it helps to know the first few primes
How shall I develop more Intuition in maths🙏
🙌let's move to 14th
kind viper
does the polynomial x^2 - x + 1 remind you of anything
kind viper
How shall I develop more Intuition in maths🙏
no royal road to that unfortunately
gilded lark
does the polynomial x^2 - x + 1 remind you of anything
Complex roots
kind viper
you need to do questions that tickle your brain and make you spot patterns
kind viper
Complex roots
well sure but this one's a bit special
kind viper
consider: (x+1)(x^2-x+1) = x^3 + 1
this means alpha and beta are both also roots of x^3 + 1 (without being -1 themselves)
alpha^3 = beta^3 = -1
i will be right back but this is already a big step in the right direction
gilded lark
alpha^3 = beta^3 = -1
Yea I knew it we called it Omega^3=-1
kind viper
i don't think you used omega this way
i think you used omega to mean a cube root of unity ie omega^3=+1
gilded lark
But 3 and hundred are not pretty factors😭
kind viper
(also definitely omega ω and not Omega Ω)
gilded lark
(also definitely omega ω and not Omega Ω)
Yea
kind viper
But 3 and hundred are not pretty factors😭
doesn't matter
gilded lark
(also definitely omega ω and not Omega Ω)
Isn't the last one sign of ohm
kind viper
it is, but it is also just the uppercase letter omega
gilded lark
K=2?
kind viper
it is, but it is also just the uppercase letter omega
greek alphabet has upper and lower case just like Latin and Cyrillic both do
gilded lark
Yea answer is 146 how ‼️
gilded lark
in fact i know it's wrong
??
kind viper
Yea answer is 146 how ‼️
answer to which one
gilded lark
answer to which one
14th
kind viper
i thought we were done with 13 and now we are doing 14
the answer to 14 isn't 146 either
kind viper
alpha^3 = beta^3 = -1
see here
gilded lark
This is pretty mathemystic that idk
kind viper
alpha^100 = -alpha^97 = +alpha^94 = ...
you step the exponent down by 3 until it's too small to go further, where do you stop
gilded lark
I'll stop at 1
Fr
Also +alpha^4 is equal to it
kind viper
why probably
could it also be 42069 steps instead or some other wacky number
there is no probability or uncertainty here
only your self doubt. yes it's 33 steps.
thus alpha^100 = (-1)^33 alpha^1 = -alpha
and same for beta
gilded lark
It's just I did that easy calculations in mind only
kind viper
It's just I did that easy calculations in mind only
what have i told you about doing calculations in mind.
gilded lark
what have i told you about doing calculations in mind.
😭
gilded lark
thus alpha^100 = (-1)^33 alpha^1 = -alpha
?
kind viper
what's unclear?
gilded lark
Alpha^100= -alpha^1 ? How
Isn't it +
Bcoz Alpha^100=-alpha^4
kind viper
no you fucked up the signs
kind viper
thinking somewhat in the wrong direction here
gilded lark
?
kind viper
thing is, every time you step down by 3, you pick up a minus
100 to 1 takes 33 steps so you pick up 33 minuses ie (-1)^33
the parity thing is a side effect and not a cause
kind viper
Define alpha^1=?
i have no idea wtf you actually meant here
but alpha^1 is just alpha
gilded lark
And what is alpha?
gilded lark
And how can I can answer by doing -alpha=alpha^100
random mica
And how can I can answer by doing -alpha=alpha^100
meaning how you got there?
look very simply alpha^100 = alpha * (alpha^3)^33 = alpha * (-1)^33 = -alpha
gilded lark
And how can I can answer by doing -alpha=alpha^100
Ik how did I get it but..... How shall I define it for the answer
random mica
well first show that alpha is a solution to x^3 + 1 = 0 so alpha^3 = -1
then alpha^100 = alpha^100 = alpha * (alpha^3)^33 = alpha * (-1)^33 = -alpha
so alpha^100 + beta^100 = -(alpha + beta)
do you know how to get alpha + beta?
gilded lark
well first show that alpha is a solution to x^3 + 1 = 0 so alpha^3 = -1
Why the hell alpha can't be -1 then too imaginary 😭
kind viper
And how can I can answer by doing -alpha=alpha^100
this and same for beta gives -K = -alpha - beta.
gilded lark
do you know how to get alpha + beta?
Yes
random mica
so whats the answer?
gilded lark
1
random mica
correct k=1
gilded lark
Ty 🙇♂️
random mica
next question?
gilded lark
Done ty
I'll be back after lunch if I need help
random mica
kk
gilded lark
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compact pewterBOT
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cold sail
!status
compact pewterBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
which part are u stuck at? and what have you tried? any relevant information will be appreciate
foggy cobalt
which part are u stuck at? and what have you tried? any relevant information wil...
question1,2,3,6
cold sail
aight, start with q1 shall we
you jst need to use sum of angles in a triangle=180 degrees
can you do that (we need to find A1 a.k.a BAC)
foggy cobalt
how should I write it
if its 180 deg and we have to find A1 should we calculate every side in the angle
cold sail
no
foggy cobalt
dame
cold sail
the other r given
foggy cobalt
sorry I am bad in maths
foggy cobalt
can I add you and do it there I dont like to talk on servers
then I can understand better
cold sail
i prefer do the bussiness here
foggy cobalt
nevermind thx
compact pewterBOT
cold sail
u have done solving it i assume
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strange bolt
I was learning about the inverse radon transform bc of tomography but I don't understand the last step cause I thought the intergration of the fourier transform was on dw but there is a function of w and s at the end. Am I just tripping?
compact pewterBOT
I was learning about the inverse radon transform bc of tomography but I don't un...
strange bolt
This is the link to the video: https://www.youtube.com/watch?v=f0sxjhGHRPo
compact pewterBOT
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sudden carbon
how do i do this
compact pewterBOT
how do i do this
weak cobalt
A first step would be to graph the function
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sick gale
compact pewterBOT
sick gale
Need a double check
compact pewterBOT
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peak turret
Maximum area of rectangle that can be inscribed in an equilateral triangle of side length 'a'?
compact pewterBOT
Maximum area of rectangle that can be inscribed in an equilateral triangle of si...
peak turret
Ping when replying
runic creek
Okay
Chill bro, I'mma make a diagram fisrt
But the main idea here is that, you should construct a function
Hold up
Okay
You want to find y in terms of x
Then set up :
f(x) = x • y
Graph it and find the maximum
@peak turret
peak turret
y= (a/2- x) tan60°?
peak turret
Why graph?
runic creek
So u can easily spot the max
peak turret
a/2-x=x
Wouldn't this just give the maxima?
peak turret
Derivatives?
runic creek
Derivatives?
Tiring
But ik how to do them
Actually not that bad in this case, but graphing is just my fav
peak turret
Thanks anyways
peak turret
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compact pewterBOT
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sullen gorge
To differentiate $\cot^{-1}\left(\frac{1-x}{1+x}\right)$ with respect to x
compact pewterBOT
To differentiate $\cot^{-1}\left(\frac{1-x}{1+x}\right)$ with respect to x
boreal girderBOT
Normie
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
sullen gorge
My problem is what to do after substituting the value of u which is 1-x/(1+x) into the cot inverse derivative
bleak dock
My problem is what to do after substituting the value of u which is 1-x/(1+x) in...
you need the chain rule
so you need to multiply by the derivative of (1-x)/(1+x) afterwards
sullen gorge
The derivative I am getting is
bleak dock
,w derivative of arccot(x)
boreal girderBOT
sullen gorge
-2/(x²+2x+1) for 1-x/(1+x) which is not going to give me the answer I am sure
,w derivative of 1-x/(1+x)
boreal girderBOT
sullen gorge
,w derivative of (1-x)/(1+x)
boreal girderBOT
sullen gorge
Which is basically what I got but it seems like there is a way here
They haven't opened the bracket
bleak dock
**Normie**
Compile Error! Click the <:errors:504266556054962196> reaction for mo...
- sub in (1 - x)/(1 + x) into x for the derivative of arccot(x)
- multiply this by -2/(1 + x)^2
sullen gorge
That multiplication is so complex omg
Never done such before
Is this what you mean? @bleak dock
Forget the tan theta part
sullen gorge
No I am not using tan theta
bleak dock
instead part 1) is $-\frac{1}{\left(\frac{1-x}{1+x} \right)^2 + 1}$
boreal girderBOT
south
boreal girderBOT
Normie
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
bleak dock
it's better not to expand the denominator
why? cause in total you have $-\frac{1}{\left(\frac{1-x}{1+x} \right)^2 + 1} \cdot \frac{-2}{(1 + x)^2}$
sullen gorge
Oh
boreal girderBOT
south
sullen gorge
This still doesn't give the correct answer
sullen gorge
I did what you said to do and continued from there
My conclusion isn't correct though
As you said I solved the cot-1 u by expanding only the numerator
From there I added 1 to it to get a single fraction after dividing by 1
Simplified and got it incorrect
boreal girderBOT
south
sullen gorge
Because it's not possible
bleak dock
bruh
sullen gorge
I will appreciate if you solve it yourself and show me
sullen gorge
Oh true
Oh false
Oh right I should have expanded the bracket first
Looking back at it there is no bracket to expand
I did everything step by step first expanding bracket then dividing then multiplying then simplifying
<@&286206848099549185>
,w 2x × x²
boreal girderBOT
sullen gorge
Chain rule method is too niche for this
Can someone explain the following
This is useful I think
,rotate
boreal girderBOT
bleak dock
Chain rule method is too niche for this
they literally used the chain rule
sullen gorge
Okay so we had two functions
F'(u) and g'(x)
We then chained them together with multiplication
Lemme do this myself
I think I made a mistake differentiating u
bleak dock
oh, it's a different order
sullen gorge
Nvm
sullen gorge
Na they did v u' - u v'
Which is a funny way to arrange it
I guess that cot-1 u derivative is hard for me to divide
Maybe I should stop expanding the brackets entirely
Oh yeah I think that's what happened
I was getting the answer in its expanded form
Shoots
Thank you though 👨🦳💀
.close
tender robin
how do u do b)?
compact pewterBOT
how do u do b)?
tender robin
am i meant to use vector resolute or something
tender robin
i have no idea
kind viper
kind viper
this is actually just a matter of adding the unit vectors of a and b
and normalizing the result
tired rock
sounds like "vector resolute" is another way of saying "vector projection"?
tender robin
this is actually just a matter of adding the unit vectors of a and b
why?
kind viper
normalize = divide a vector by its own length
tender robin
ohh okok
kind viper
why?
gonna need me to wait about 20 min to get back home before i can show you a diagram
tender robin
okay
for adding the a hat and b hat is it meant to be just
expanding
and collecting
strange ermine
wit idk how to normalise
lavish gull
yes. a vector's length is its magnitude
bleak dock
this is actually just a matter of adding the unit vectors of a and b
diagram
strange ermine
yeah but idk what the length is
When Ann told that, you could have asked her the meaning of length
bleak dock
you have two congruent triangles with sides v, w and v + w
hence two of the angles make up the big angle between v, w
bleak dock
yes and those two angles are the same (by SSS congruence)
tender robin
and what exactly is a hat? the magnitude?
bleak dock
and what exactly is a hat? the magnitude?
it's a symbol to say something is a unit vector
$\hat{a}, \hat{b}$
boreal girderBOT
south
tender robin
isnt a bolded letter just a vector
strange ermine
and what exactly is a hat? the magnitude?
It's the notation for unit vectors
bleak dock
yeah this is one of these things that is ambiguous
bleak dock
It's the notation for _unit_ vectors
but this convention is more common
and it's the one we're all using
here
strange ermine
isnt a bolded letter just a vector
In textbooks, usually yes
But when you handwrite you can't use bold face font, so we write them in multiple possible ways: $\vec{v}$ or $\overline{v}$ or $\underline{v}$
boreal girderBOT
Alberto Z.
strange ermine
but this convention is more common
Mmh I'm not that sure actually
tender robin
my teacher got me writing a squiggly line on top of the letter
and then we gotta add hats or something
strange ermine
my teacher got me writing a squiggly line on top of the letter
Yeah that's another possible notation (though way more rare)
bleak dock
in handwriting you can easily confuse v-hat with (v + squiggly line on top)
so to be unambiguous you should write the squiggle on the bottom
but yeah notation is one big clusterf*ck
strange ermine
Yeah that's another possible notation (though way more rare)
I had a teacher for continuum mechanics that wrote vectors with a tilde (squiggly line) but below it, not above
tender robin
Yeah that's another possible notation (though way more rare)
oh
tender robin
so to be unambiguous you should write the squiggle on the bottom
ohh ok
strange ermine
But in contexts where it's understood that things are vectors (such as in linear algebra) you might as well see no notation at all, neither bold font
strange ermine
Sure, we know
tender robin
could you teach me the process?
strange ermine
I'm sorry, I don't know what resolution means
compact pewterBOT
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runic pagoda
hellooo i have a question
compact pewterBOT
hellooo i have a question
runic pagoda
so i have to create an equation
chilly basalt
Yes 🙂
mild sky
Take a circle and scale it?
runic pagoda
uhhh
its like a scenario
i have to make a mug or cup looking thing
and the volume of that has to be 400-600
chilly basalt
How about a rectangle?
mild sky
400-600 i.e between those values?
runic pagoda
so i have a random sin equation
ye
so i have this crazy sin equation, and i left the scalar as a constant
using the volume formula
like the v= pi integration of x^2 dy
can i put in the x^2 and make it equal to 500
and solve for a
?
lol does that make sense?
mild sky
No
mild sky
You should have started with that...
runic pagoda
AHAHA SORRY
runic pagoda
right
compact pewterBOT
@runic pagoda Has your question been resolved?
proper dawn
Hey does anyone here know actuarial mathematics?
compact pewterBOT
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bleak garnet
when they ask for interval I, do they mean the domain the independent variable can cover without making the dependent variable invalid?
compact pewterBOT
when they ask for interval I, do they mean the domain the independent variable c...
bleak garnet
for example y = 1/x its invalid when the independent var x = 0
lime juniper
when they ask for interval I, do they mean the domain the independent variable c...
yes i think so
compact pewterBOT
lime juniper
.
bleak garnet
😭
old mica
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strange pendant
compact pewterBOT
strange pendant
winged lion
strange pendant
can someone check my shit?
winged lion
your shit looks good
dim raven
i gave this exam today and there are a lot of question i got stuck in its my first time doing those type of questions with some knowledge to them
the questions i got stuck in
14
16
13
10
compact pewterBOT
i gave this exam today and there are a lot of question i got stuck in its my fir...
@dim raven Has your question been resolved?
remote bane
hey guys
remote bane
chilly basalt
remote bane
i can tell easily that B and D are answers
but how'd is it also A and not C
I ended up saying all of them were answers but I got C being an answer wrong
kind viper
if $b = xv_1 + yv_2$ then you can also say $b = (x-y)v_1 + yv_1 + yv_2 = (x-y)v_1 + y(v_1+v_2)$
boreal girderBOT
Ann
kind viper
however with C note that span{v1+v2, -v1-v2} = span{v1+v2} and this consists only of vectors whose v1-coordinate and v2-coordinate are equal (which those of a vector in span {v1, v2} need not be!)
remote bane
hol on let me try to understand this
sorry i don't really understand your method
what i try to do was use arbitrary vectors
b vector = the 0 vector
oooooh I get it kinda
since v1 and v2 are the same vector in 3, it can't be in the span?
upbeat gorge
b vector = the 0 vector
this is the opposite of arbitrary, actually ._. if you want an arbitrary vector, you have to work with v_1 or v_2 as the symbols
remote bane
oooh wait I can't use the zero vector cuz it's in the span of everything
upbeat gorge
oooh wait I can't use the zero vector cuz it's in the span of everything
yes exactly
upbeat gorge
since v1 and v2 are the same vector in 3, it can't be in the span?
v1 and v2 aren't the same vector or scalar multiples, but (v1 + v2) and (-v1 + v2) are scalar multiples
so C is the span of one vector - precisely, it's what ann described
there's another way to think about this: if you can get to $v_1$ and $v_2$ using the vectors in the spanning set, then $\spn{v_1, v_2}$ is going to be inside it
boreal girderBOT
haseeb
upbeat gorge
like for A: $v_2 = (v_1 + v_2) + (-1)(v_1)$ is a linear combination of $(v_1 + v_2)$ and $v_1$
boreal girderBOT
haseeb
upbeat gorge
and of course $v_1$ is in the spanning set
boreal girderBOT
haseeb
remote bane
ur method is very high intelect for me
i just picked b to be 0,1
v = 1,0
v2 = 0,1
and i got an inconsistent matrix when tryna solve for C
yall helped me understand span damn. thanks yall
.close
compact pewterBOT
upbeat gorge
i just picked b to be 0,1
v = 1,0
v2 = 0,1
im glad it worked out, but picking a vector is not always going to work out
remote bane
.reopen
compact pewterBOT
✅
remote bane
i mean
im a fast solver i don't mind picking 3 different vectors and trying them all
i understand ur concern tho
upbeat gorge
let's say you picked v1 = (1,1) and v2 = (-1, -1), then you would have gotten something consistent, even though it doesnt work for all vector
upbeat gorge
im a fast solver i don't mind picking 3 different vectors and trying them all
can you try infinitely many though?
remote bane
those are the same vectors though i wouldn't do that
upbeat gorge
i suppose you can pick vectors to disprove certain options in this case, but working generically is better
remote bane
from what i understand and correct me, if the two vectors are the same vector or colinear, then the dimension can't line up correctly
so it won't be in the span
upbeat gorge
that is a really good explanation :)
remote bane
that is a really good explanation :)
i only thought about it cuz of u bro thanks
❤️
and thanks @kind viper
.close
compact pewterBOT
upbeat gorge
. uh np, although i think that was all you :)
also, you could probably make a situation where picking linearly independent vectors will give you the wrong answer - particularly past 2 dimensions
remote bane
so it's either R^2 or R^1 cuz we're given 2 vectors?
upbeat gorge
. or R^0, a singular point
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floral kelp
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winged lion
guys, how do i make that into (m-20)^3 = -4000 form?
you can expand (x-a)^3 and then match coefficients
floral kelp
you can expand (x-a)^3 and then match coefficients
ohhh okayy thanks
winged lion
i dont get the -4000
can you show your work?
in order to factor you need on both sides -8000, so the right side would become -4000
boreal girderBOT
floral kelp
floral kelp
in order to factor you need on both sides -8000, so the right side would become ...
why minus -8000 on both sides?
winged lion
You got right now
winged lion
you cheated the -8000
winged lion
You didn't have a -8000 before, but you need it, in order to factor
so you add on both sides -8000 and so the right side becomes 4000-8000=-4000
floral kelp
so you add on both sides -8000 and so the right side becomes 4000-8000=-4000
okayy thanku so muchh!
will close the channel
.close
compact pewterBOT
sick kiln
if an equation of y, that can be represented in terms of x, has a stationary point, does it follow that y^2 has a stationary point at the same value?
winged lion
if an equation of y, that can be represented in terms of x, has a stationary poi...
This channel closed, you can grab an own #❓how-to-get-help
sick kiln
oops
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eager abyss
someone help i need to know how to do all this by tommorow and i know nothing except the first one
compact pewterBOT
someone help i need to know how to do all this by tommorow and i know nothing ex...
tired rock
it's better to show specific questions in help channels so helpers can walk you through
eager abyss
uhmmm
compact pewterBOT
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compact pewterBOT
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pure hornet
why is -9-(-8) 17
compact pewterBOT
why is -9-(-8) 17
weak cobalt
If you're asking why -9-(-8) = 17
It isn't
weak cobalt
You sure you "lagged"? 'cause -(-9)-(-8) = 17
kind viper
for addition purposes, -9 means go nine steps left and +8 means go eight steps right
pure hornet
th
hollow oasis
-- means turn left twice, you end up going right
fresh breach
i have two sets which are not subset of each other but they share common elements
compact pewterBOT
i have two sets which are not subset of each other but they share common element...
fresh breach
can we talk about complement of one set with respect to another?
because in definitions i found everywhere there is explicitly said that one set needs to be subset of another
kind viper
can we talk about complement of one set with respect to another?
just say set difference tbh
A \ B
fresh breach
lets say A={1,2,3,5} B={3,5,7,10} what is compleement of A with respect to B
weak cobalt
To talk about a complement, you do need one to be a subset
kind viper
lets say A={1,2,3,5} B={3,5,7,10} what is compleement of A with respect to B
that would be B \ A
but also,
!xy
compact pewterBOT
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.
weak cobalt
But you can talk about the set difference in general
fresh breach
lets say A={1,2,3,5} B={3,5,7,10} what is compleement of A with respect to B
this
@kind viper
kind viper
that would be B \ A
weak cobalt
Actually I'm probably wrong; the relative complement doesn't need subsets, it's just synonymous with set difference
fresh breach
> that would be B \ A
but they are not subsets of each othe
fresh breach
do you consider B\A to be undefined unless A ⊆ B
hm dunno we didnt specify
so were talking about relative complement or absolute one?
kind viper
"absolute" complement is only ever "complement of <set>" and implies some universal set that we fix and work inside of.
"complement of <set> wrt <other set>" can only be read as, in your terms, relative complement
or as it's more commonly known, set difference.
weak cobalt
because in definitions i found everywhere there is explicitly said that one set ...
Can you show one of those definitions you found? Including context
fresh breach
hm so its homework for my little brother but i can translate the definition their teacher gave them
kind viper
send a photo of the thing
weak cobalt
I'd rather you just take a picture
kind viper
like the entire worksheet
for future reference this kind of context should not be coming out this late into the help channel convo
but rather state it upfront next time
fresh breach
hm but its in slovak
fresh breach
for future reference this kind of context should not be coming out _this late_ i...
Yes sorry i thought it would be more straight forward
kind viper
hm but its in slovak
post it anyway
fresh breach
boreal girderBOT
Ann
fresh breach
alright then ill tell my brother to ask his teacher about their definition
weak cobalt
This looks like the work you (or your brother) have done, not the definition
You should be able to find the definition in the textbook
fresh breach
Doplnok množiny, nech B je podmnožina množiny A doplnok množiny B v množine A, je množina všetkých prvokv množiny A, ktoré nepatria do množiny B.
thats transcribed word to word
weak cobalt
The complement of a set, let B be a subset of set A, the complement of set B in set A, is the set of all elements of set A that do not belong to set B.
Sure
weak cobalt
But in general B is not required to be a subset of A
fresh breach
are you telling me THIS is `k`?
xd yes
fresh breach
xdd thats pretty standard handwiring in our area
but thanks guys
ill tell my brother to ask for clear clarification from his teacher
.close
compact pewterBOT
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Available help channel!
strange pendant
compact pewterBOT
fathom crest
You probably need to give the translation.
strange pendant
Exercise 12. Whenever Laly and Leo sleep at their grandmother's house, they return home with a package of cookies. If one day we saw them arrive with the package of cookies, we might be inclined to conclude that they slept at their grandmother's house. Can you identify the error in the reasoning above? Hint: It is known as the fallacy of affirming the consequent.
native shard
they gave you the answer
strange pendant
can we think it of terms of propositional logic?
native shard
sleep at grandmas house —> come home with cookies
just because you come home with cookies doesn’t mean they slept at grandmas
false —> true is still true
they may have some other reason for coming home with cookies
if you want it in precise logic
"p —> q
q
therefore p"
is invalid
strange pendant
,, p \to q \equiv \lnot p \lor q
native shard
\lnot
boreal girderBOT
Renato
native shard
$\lnot$
boreal girderBOT
knief
strange pendant
,align q \land (p \to q) &\equiv q \land (\neg p \lor q) \ &\equiv (q \land \neg p) \lor (q \lor q) \ &\equiv (q \land \neg p) \lor q
native shard
and not p or q is true
could be that not p is also true
boreal girderBOT
Renato
mint flower
the truth of the consequent does not guarantee the truth of the antecedent
native shard
my guy spent 5 minutes for that
mint flower
who
native shard
happens
compact pewterBOT
@strange pendant Has your question been resolved?
upbeat gorge
could be that not p is also true
did something not make sense in this explanation?
strange pendant
the truth of the consequent does not guarantee the truth of the antecedent
ye
strange pendant
did something not make sense in this explanation?
is it not possible to use propositional logic
upbeat gorge
it probably is, but showing that q and not(p) can be true at the same time is enough
a lot of logic proofs like this rely on assuming certain truth values (i.e. starting with q = true), and it tends to be faster
upbeat gorge
**Renato**
if you really want to use logic, though: expand out to $$(q \vee q) \wedge (\neg p \vee q).$$ if $q \equiv \textbf{T},$ what can be said about $\neg p$?
boreal girderBOT
haseeb
strange pendant
we dont care about truth value of neg P
if q is True then this is a tautology
regardless of the truth value of p
upbeat gorge
we dont care about truth value of neg P
why not? we want to show that if q is true, p is not always true
upbeat gorge
regardless of the truth value of p
so, does that mean p is always true? or can p be false
strange pendant
why not? we want to show that if q is true, p is not always true
q and (p -> q)
simplifies to p -> q
strange pendant
so, does that mean p is always true? or can p be false
can be both
and still p -> true holds
upbeat gorge
simplifies to p -> q
yeah, but the question is asking about q -> p
upbeat gorge
and still p -> true holds
possibly, yes, but it's not guaranteed. hence we cannot deduce q -> p
this is about as much as you can manipulate with the variables. after that, you'd have to look at truth values
strange pendant
possibly, yes, but it's not guaranteed. hence we cannot deduce q -> p
how so?
ye of course, if that would be the case then it would be an iff and not an implication, is just hard to phrase it in prop logic
ye
upbeat gorge
how so?
because q = true and p = false makes our situation, (q ^ (p->q)), true. at the same time, it makes q -> p false
disagree >.> for example, you should understand that (A -> B) -> C is different to A -> (B -> C), and you should be able to say which one you are specifically proving
im not gonna defend everything that you are tasked to do in a uni course, but stuff like this builds your logic muscle, and allows you to avoid saying logical fallacies when it's not just propositional symbols
for example, every maximal ideal is prime (if the ring is commutative), but that doesn't mean that every prime ideal is maximal. recognizing the fallacy becomes easier when you've had practice with arbitrary symbols
but this is probably not the best place to discuss pedagogy, sorry
i believe that is used to prove that every nonempty ring has a maximal ideal
(not rng)
upbeat gorge
possibly, yes, but it's not guaranteed. hence we cannot deduce q -> p
sorry for the clutter. here's the final bit of our reasoning we left off at
so the actual way to solve this question would be to show that, if Q is true and P is false, P ^ (P -> Q) is true, but Q -> P is not
thus, the converse cannot always be assumed
strange pendant
what?
upbeat gorge
you can think of the statement we wanted to prove like this: "for all p, q, if p->q and q is true, then q->p"
if we wanted to negate that, it would be "there exists p, q, if p->q and q is true, then q ^ not(p)"
(when i say 'for all' or 'there exists', i mean some combination of true/false for p and q)
so we picked a specific case that disproves it, based on our propositional stuff
strange pendant
Exercise 12. Whenever Laly and Leo sleep at their grandmother's house, they return home with a package of cookies. If one day we saw them arrive with the package of cookies, we might be inclined to conclude that they slept at their grandmother's house. Can you identify the error in the reasoning above? Hint: It is known as the fallacy of affirming the consequent.
strange pendant
you can think of the statement we wanted to prove like this: "for all p, q, if p...
can we think it in terms of words?
IF LALO AND LEO SLEPT AT THEIR GRANDMAS HOUSE, THEY ATE COOKIES IS P -> Q
LALO AND LEO SLEPT AT GRANDMAS IS P
ATE COOKIES IS Q
NOW
WE WANT TO SHOW THAT Q -> P IS A FALLACY
NOT BECAUSE THEY ATE COOKIES THEY WENT TO GRANNIES HOUSE
NOT NECESSARILY AT LEAST
upbeat gorge
what's the negation of Q -> P, in words?
strange pendant
NOT(IF LALO AND LEO ATE COOKIES THEN THEY WENT TO GRANMA)
strange pendant
what's the negation of Q -> P, in words?
NOT(Q -> P) = NOT(NOT(Q) OR P) = Q AND NOT P
LALO AND LEO ATE COOKIES AND THEY WERENT AT GRANNYS HOUSE
upbeat gorge
right, and our question is: is this possible, given what we know
upbeat gorge
if p -> q and q is true
strange pendant
q is cookies
strange pendant
right, and our question is: is this *possible*, given what we know
are you asking if its possible that lalo and leo ate cookies and they werent at grandmas house knowing that they ate cookies?
like
I dont get it dude
why dont we do it using propositional logic
,, q \land (p \to q) \land (q \to p) \ q \land (\neg p \lor q) \land (\neg q \lor p) \ ((q \land \neg p) \lor (q \land q)) \land (\neg q \lor p) \ ((q \land \neg p) \lor q) \land (\neg q \lor p) \ (((q \land \neg p) \lor q) \land \neg q) \lor (((q \land \neg p) \lor q) \land p)
boreal girderBOT
Renato
upbeat gorge
why dont we do it using propositional logic
because, if you prove that p->q and q->p is never true, then you are proving that p <-> q cannot exist
we are trying to prove it is possibly false, not always
strange pendant
dude is hard
upbeat gorge
are you asking if its possible that lalo and leo ate cookies and they werent at ...
yes
upbeat gorge
let's do it like this then
$$\underbrace{q \land (p \to q)}{\text{true}} \land \underbrace{(q \to p)}{\text{false}}$$
boreal girderBOT
haseeb
upbeat gorge
pick a truth value for q and p that makes this happen
and conclude that the whole thing is false in that specific case
strange pendant
pick a truth value for q and p that makes this happen
p =. false q = true
upbeat gorge
you raise a good point
strange pendant
like only because they went to grandma does not necesarilly mean they had cookie
upbeat gorge
that is what p -> q is
but if they are holding a package of cookies, it's possible they bought it from a shop
or got them from a friend
there are many ways to get cookie, so we cannot backtrack and say it was definitely grandma
strange pendant
there are many ways to get cookie, so we cannot backtrack and say it was definit...
ye
upbeat gorge
the thought process is: we want $$q \land (p \to q)$$ to be true. then both $q$ and $p \to q$ must be true, so $q$ is true, and we can't say anything about $p$ yet (true and false both work). \
so now, turn your attention to $$q \to p.$$ This is only false if $q$ is true, and $p$ is false. Ergo $p$ is false.
boreal girderBOT
haseeb
strange pendant
i guess
I kinda get it haseeb @upbeat gorge
is it possible to find a contradiction using propo logic?
compact pewterBOT
@strange pendant Has your question been resolved?
strange pendant
.close
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pliant canopy
Is this right so far this is my second time trying to do this and i feel like so far it might be okay and i remember learning something about when u reach the same thing as the original equation ur supposed to do something but idk i could be wrong(sorry if its hard to read)
compact pewterBOT
Is this right so far this is my second time trying to do this and i feel like so...
paper anchor
You can just take your initial Integral as 'I'
You reach a simplification after applying IBP twice
That is, you get 'I' on the RHS too, so 'I' = x + y'I', hence 'I' = x/(1 - y), something like this
Let me know if I make sense
Wait I'll solve integral of xsinx so that you get the gist, one minute
Or ykw, e^x sinx, I just realised xsinx is too simple for this
paper anchor
Let me know if it's not visible
You get what I mean? IBP for your initial Integral, then IBP for the integral you get from the first IBP, and now the integral you get is same as the initial one
And after that it's just basic algebra, basically you end up on the RHS with the constants from both the IBP and the same Integral as the one you were trying to find, you have to simplify it with LHS and you get the answer
paper anchor
Whoops, my answer is incorrect IG, messed up in the sign, but you get what I mean
compact pewterBOT
@pliant canopy Has your question been resolved?
oblique rapids
how do u convert cm to leters
compact pewterBOT
how do u convert cm to leters
oblique rapids
letres
viscid canopy
do you mean cubic cm?
inland nacelle
1 l = 1000 cm^3
inland nacelle
1 l = 1000 cm^3
1 cm^3 = 1 ml as a result
viscid canopy
liter is a unit for volume
strange ermine
m is like litres?
If you want to do that analogy, I believe so, yes
But I'd rather say you just have to remember that 1 dm³ = 1 L
Or, if you prefer, 1 cm³ = 1 mL
compact pewterBOT
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oblique rapids
compact pewterBOT
rigid sparrow
is the question to find its total surface area?
oblique rapids
yep
rigid sparrow
do you know what is surface area of a cylinder
oblique rapids
yes
oblique rapids
so a quater cyclinder plus the who rectangle
rigid sparrow
yeah
there is another way of think of it
individually finding every surfaces surface area
you can see you have a total of 5 surfaces
2 quaters of a circle
2 rectangular
and another curved surface whose area can be solved as simple as of a rectangle
but if you find the first method easier please go with it
oblique rapids
so to find like a half or quarter= a quater or half of the normal surface area of a cyclinder then u plus the 2 rectangles or one
right?
.close
compact pewterBOT
rigid sparrow
so to find like a half or quarter= a quater or half of the normal surface area o...
umm yeah if you dont understand it completely i could explain better
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Available help channel!
molten badger
Each triangle is randomly colored with one of 7 colors: Red, Orange, Yellow, Green, Blue, Indigo, Violet. Calculate the probability that two adjacent regions are colored with 2 different colors
compact pewterBOT
Each triangle is randomly colored with one of 7 colors: Red, Orange, Yellow, Gre...
molten badger
It's a 20 sided regular polygon
kind viper
do you mean like the probability that there are no two adjacent same colored regions ANYWHERE
molten badger
do you mean like the probability that there are no two adjacent same colored reg...
yeah I believe so
wiat
graceful kelp
Each triangle is randomly colored with one of 7 colors: Red, Orange, Yellow, Gre...
ok so there are 20 pairs of adjacent regions
TSR, SRQ, ..., ATS
molten badger
do you mean like the probability that there are no two adjacent same colored reg...
yes
My brain is fried
kind viper
i don't see a way without nasty incl excl bashing
molten badger
i don't see a way without nasty incl excl bashing
could you like, sketch the idea for me
kind viper
not sure if i have it fleshed out rn
hushed magnet
I also dont see anything else
set A_i,c = {all colorings so that triangles i, i+1 have color c}
essentially you want to find the complement of the union of all of those
but thats ewww
I wonder whether there is some induction type argument
hmm I think we can do sequence counting
wlog suppose one of the triangles has color red
royal vale
maybe we let the first triangle have 7 colors right, the triangles next to them would have 6 possible colors
then ig u just repeat
hushed magnet
we are enumerating them clockwise
royal vale
ya
hushed magnet
so the colors are a sequence with distinct adjacent terms
then just need to figure out the end condition
cause last triangle has to fit
royal vale
im thinking is it possible to just think of the first triangle have 7 colors that’s possible, the next triangle has 6 possible colors, and the triangle after has 6
and so on
molten badger
I'm processing, my brain is so fried rn I even have hard time understanding english 
royal vale
like think of how many colors can the triangle have
molten badger
yeah your way is 7*6^{18}*5 right?
royal vale
yes
molten badger
I mean I thought of that but it seems wrong
hushed magnet
you cant just do that
the last triangle does not have the same 6 choices
cause it borders the first one
royal vale
the last triangle does not have the same 6 choices
hence *5
hushed magnet
it could still have 6
just not always
for each color c, count the number of sequences with length n which start with red and end with color c
molten badger
can we split it into 2 cases to count
royal vale
oh wait i get why my method is wrong i misread the question im sorry
compact pewterBOT
@molten badger Has your question been resolved?
molten badger
Is inclusion and exclusion the only way?
random mica
the last triangle does not have the same 6 choices
this is the problem with normal counting
graceful kelp
how about recursion
yeah actually
random mica
so instead if you have a_n denote the number of colourings with n triangles (and also a_n-1 for n-1 triangles) then find the number of colourings for n+1 triangles
hushed magnet
Is inclusion and exclusion the only way?
we just did something without incl excl....
molten badger
so instead if you have a_n denote the number of colourings with n triangles (and...
I'm not understand what you said
Could you explain it more
molten badger
sorry my brain is cooked today
random mica
is that we dont know how many choices there are for the last triangle
and if we split into cases
we dont know how many colourings there are per case
unless we use recursion
so $a_n:=$ number of colourings ( with the restriction ) with n triangles
boreal girderBOT
CherryMan
random mica
and find a relationship between $a_n$, $a_{n-1}$ and $a_{n-2}$
boreal girderBOT
CherryMan
molten badger
hmm
random mica
so a_n+1 is not exactly 7a_n-1
you realise that right?
theres some correction to be done with a_n-2 @molten badger
molten badger
sorry give me a min, my head's legit ached
random mica
its alright
molten badger
my brain give up on me i can't process anything how can we find the relationship between an , an-1 and an-2
random mica
alright i guess we can walk through it
(also i might be wrong so please forgive me)
say we have a disk with n triangles
molten badger
say we have a disk with n triangles
yes
random mica
say the first one is red
the nth one is yet to be coloured
IF the n-1'th triangle is not red
then we have 5 options for the nth one
IF the n-1'th triangle is red
we have 6 options for the nth one
random mica
sorry 5
molten badger
oh okay
molten badger
i'm processing 😭
random mica
i'm processing 😭
yeah i got it
should i restart
@molten badger
so
CASE I:
colour of triangle 1 is not equal to colour of triangle n-1
notice how in all a_n-1 colourings the colour of the first and n-1 triangle cannot be equal
so our number for this case will be a_n-1 * 5 options for the nth one
CASE II:
colour of triangle 1 = colour of triangle n-1
notice how in all a_n-2 colourings the color of the first and n-2 triangle are not equal
so we can colour the n-1 triangle with the colour of the first
and then 6 options for the nth one
so here we have a_n-2 * 6
SO
we get $a_n = 5a_{n-1} + 6a_{n-2}$
boreal girderBOT
CherryMan
molten badger
my brain fail me, I will get some sleep and back tomorrow, thank for the help @random mica
I legit forget what i processed 2 mins ago after I pull out the calculator to calculate
.close
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mystic crater
compact pewterBOT
mystic crater
how does 3cosec^2 (1x/2) for example turn into -6cot(1x/2)
instead of -3cot(1x/2)
icy nymph
how does 3cosec^2 (1x/2) for example turn into -6cot(1x/2)
what 😭
why are you writing
1x/2
mystic crater
wdym
icy nymph
instead of -3cot(1x/2)
because of the inside
chain rule
divide by half
so multiply by 2
so ur -3 becomes -6
if that makes sense
mystic crater
no point in typing 1x/2
because that's just x/2
yeah i just wanted to make it look like the question
mystic crater
chain rule
my class so far only covered simple chain rule integration like (3x^2 + 6)^3 dx
icy nymph
my class so far only covered simple chain rule integration like (3x^2 + 6)^3 dx
you can't integrate that by reverse chain rule
icy nymph
my class so far only covered simple chain rule integration like (3x^2 + 6)^3 dx
wait but
ur doing trigonometric
before reverse chain rule?
that's
strange
mystic crater
you can't integrate that by reverse chain rule
sorry, without the ^2 in the brackets
we also did things like
integrating 3 / 2x + 7 dx
icy nymph
do u understand why it's -6 now?
mystic crater
do u understand why it's -6 now?
nah
icy nymph
nah
ok so like
imagine differentiating cot(x/2)
it becomes
1/2 csc^2(x/2), right?
@mystic crater
mystic crater
it becomes
1/2 csc^2(x/2), right?
yes
mystic crater
yes
icy nymph
so we essentially multiply by 2 because of the inside, right?
icy nymph
how does 3cosec^2 (1x/2) for example turn into -6cot(1x/2)
that's why this becomes -6?
mystic crater
wdym multiply by 2 bc of the inside