molten bay
Thanks
#help-49
midnight plankBOT
glacial slate
it's still under specified, even for abelian groups; a = (1,0) and b = (1,1) in C3xC4 have the required orders, and ab = (2,1) has order 6
west iron
~~it's still under specified, even for abelian groups; a = (1,0) and b = (1,1) i...
(1,1) has order 12, no?
glacial slate
yeah I'm checking it over
mm I shouldn't mix multiplicative and additive notation 😭
mm nvm, yeah you can just check the factorisations of any power less than 12 and it doesn't work
west iron
ok it should always be true
i can't see how to get a second isomorphism theorem argument to work though
unless you can prove that hk generates <h><k>
but consider hk^n = 1
h^n and k^n need to be inverses, check when their orders are the same for the first time
@molten bay
midnight plankBOT
Available help channel!
jade magnet
midnight plankBOT
jade magnet
the irony is i also got a different answer that doesn't make sense
i actually integrated t^2
so it's t^3 / 3
((2x)^3)/3 - ((x)^3)/3
8x^3 / 3 - (x^3)/3
= 7x^3 / 3
and then when you differentiate that that's 7x^2
which doesn't make sense
lyric charm
why doesn't it make sense?
jade magnet
why doesn't it make sense?
i mean I'm pretty sure i didn't use the FTC here
the student actually does
lyric charm
you went a different route.
jade magnet
ok
lyric charm
i can tell you that actually your route gives the correct answer despite not being the one suggested in the question.
so the correct conclusion to draw from this is that the student is indeed wrong somewhere.
think about where!
jade magnet
so the correct conclusion to draw from this is that the student is indeed wrong ...
yeah for sure
i mean the way the student integral with the limits 2x to 0 - the integral with the limits x to 0 surely that is an integral with the limits 2x to x
so that looks fine
but then if it's not the limits x to 0 can you not differentiate it then
idk how to explain it
lyric charm
you can but it requires the chain rule as well.
jade magnet
was his mistake that he can't use FTC when there isnt a limit of 0
jade magnet
you can but it requires the chain rule as well.
yeah ok
lyric charm
no, the lower limit can be any constant.
jade magnet
was his mistake that he can't use FTC when there isnt a limit of 0
this seems to be the issue
oh
lyric charm
the upper limit has to be specifically x for a direct FTC application -- otherwise, you'll have another factor in there from the chain rule.
the derivative of $\int_0^{b(x)} f(t) \dd{t}$ is going to be $f(b(x)) b'(x)$.
grand pondBOT
Ann
jade magnet
the upper limit has to be _specifically x_ for a direct FTC application -- other...
oh
jade magnet
**Ann**
ohh
that explains it then
so either way he couldn't use the FTC rule
so the error is on line iii
lyric charm
i don't think "couldn't use FTC" is a good way to put it
jade magnet
i don't think "couldn't use FTC" is a good way to put it
ok
either way the upper limit was not x
on line iii
lyric charm
yes thats the mistake
lyric charm
i just didnt like your absolutist phrasing
jade magnet
yes thats the mistake
yeah
jade magnet
i just didnt like your absolutist phrasing
yeah sorry
mb
that clears it up then
but uh
I'm not gonna close the channel yet
because
I'm now stuck on the very next question
the first set has amin + amax = 22
and the second set has bmin + bmax = 27
so i just need to minimise bmax - amin
the issue is how
my first thought was to make a min and amax as close as possible considering the median
amin < 8 so i thought why not try amin = 7
so amax = 15
and then do the same thing with bmin which is less than 9 but then it cant be 8 or 7
and then suddenly bmax is probably bigger than i want and not necessarily giving the smallest range i feel
idk
midnight plankBOT
@jade magnet Has your question been resolved?
violet rivet
I'm now stuck on the very next question
the first set has a sum of 10×3=30 and median is 8 so the other 2 terms have a sum of 30-8=22
same for the 2nd set with sum of 36 and sum of other 2 terms of 27
for the range to be as small as possible the difference of the largest number and the smallest number must be minimum
in the first set the smaller number must be 7 and thus the larger number is 22-7=15
in the 2nd set the smaller number must be 6 cuz both 8 and 7 are already used thus the larger number is 27-6=21
so the smallest range is 21-6=15
jade magnet
the first set has a sum of 10×3=30 and median is 8 so the other 2 terms have a s...
uhhh
it said the answer was 14
not 15
violet rivet
it said the answer was 14
ohh
gloomy python
the first set has a sum of 10×3=30 and median is 8 so the other 2 terms have a s...
The smaller number could be 8 too
It could be 8,8,14
violet rivet
The smaller number could be 8 too
"...set of DISTINCT integers..."
violet rivet
the first set has a sum of 10×3=30 and median is 8 so the other 2 terms have a s...
so the method is still the same like this one but reverse the order
do it for the 2nd set first then for the 1st
so the 2nd set is 7,9,20
and the first set is 6,8,16
hence the range is 20-6=14
midnight plankBOT
@jade magnet Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
viral dagger
whats even the motivation to this
midnight plankBOT
whats even the motivation to this
viral dagger
i legit cant get the motivation to most of these "let this be A and this be B" with like 10 variables and you will eventually get the answer
midnight plankBOT
@viral dagger Has your question been resolved?
shell wigeon
Imagine a smaller board (needs to be 2(2p+1) by 2(2q+1), so 6x10 for example)
You want to count the numbers on the white squares
The red and green lines is a way to partition the space so that white squares are only counted once, and black squares are counted 0 or 2 times
So if you sum all the squares along the red and green lines, and then subtract twice the squares with blue dots, you get your white sum
Since the problem is only about parity, subtracting twice is easy to account for
All that's left is to check that the red lines yield an odd sum, and similar for the green lines
midnight plankBOT
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Available help channel!
tidal turret
midnight plankBOT
tidal turret
need help with 33)
runic hamlet
how often are you gonna post a non english picture without a translation and any indication of your progress
dawn dagger
seems like you have to find a function that is only surjective
tidal turret
Let f : X -> Y be a function
Injectivity: f is said to be injective if ∀ x,y ∈ X . f(x) = f(y) ⟹ x = y
Surjectivity: f is said to be surjective if ∀ y ∈ Y . ∃ x ∈ X . y = f(x)
Bijectivity: f is said to be bijective if f is both injective and surjective
tidal turret
seems like you have to find a function that is only surjective
how did you realized? i dont follow
dawn dagger
because f has only a right inverse
tidal turret
what is the difference between left and right inverse?
hushed mauve
if g(f(x)) = x, g is a left inverse of f
if f(h(x)) = x, h is a right inverse of f
tidal turret
g is the right inverse of f
but f is not the right inverse for g
similarly, f is the left inverse of g
but g is not the left inverse of f
dawn dagger
So I said to find a function f that is only surjective, you could but also just find a function g that is only injective, since g is only left invertible
I think that is easier
dawn dagger
?
Can you think of an easy example that maps not all natural numbers e.g. maps only to a subset of N?
tidal turret
oh, g needs to N->N and be injective and not surjective
dawn dagger
yes
dawn dagger
x-1 maps to all naturals wdym
tidal turret
x-1 maps to all naturals wdym
1-1 not in N
dawn dagger
but that example doesn't make sense in the first place
you can't pick a function whose range is greater than its codomain
you can divide the natural numbers into two simple subsets
the even naturals and odd naturals
so you can pick a map that only maps to even naturals for example
tidal turret
g(x) = 2x
dawn dagger
you can in fact pick any map that only maps to naturals numbers with a special property
that way you manage to not map to all natural numbers
tidal turret
now what?
dawn dagger
find a left inverse that isn't a right inverse
tidal turret
in other words now we have to find a surjective function that is not inyective
dawn dagger
no
tidal turret
f(x) = x + 1
dawn dagger
if we do some algebra we can find something like f(n) = sqrt(n) for g(n) = n^2 but that wouldn't work because that is the actual inverse
but since that satisfies f o g = id we would need something similar to that
i think something that could work is using the ceil or floor function
dawn dagger
this wouldn't change f o g since ceil(n) = n = floor(n) so might try f(n) = ceil[sqrt(n)] for example and see
dawn dagger
i gave an example with n² try to apply the samr strategy on 2n
dawn dagger
yea that's the actual inverse of 2n so try to play around with ceil(y/2) or floor(c/2)
tidal turret
yea that's the actual inverse of 2n so try to play around with ceil(y/2) or floo...
help
like
g(n) = 2n right
dawn dagger
yes
tidal turret
then, let u = g(n)
dawn dagger
ok
dawn dagger
yes
tidal turret
g(u) = ?
dawn dagger
well thats id_N obviously
so we have to find a similar but different f
and we can take advantage that ceil(n)=n=floor(n)
tidal turret
wait a second dude
g(n) = 2n
then, let u = g(n)
u = g(n) = 2n
u = 2n
n = u/2
g(?) = ???
dawn dagger
g(?) = 2?
dawn dagger
what's your question because i am confused of your confusion
dawn dagger
what are you trying to do
the actual inverse of 2n is trivially n/2
can we get past that?
tidal turret
the actual inverse of 2n is trivially n/2
how?
dawn dagger
didn't you just do that 1 or 2 help channels ago?
tidal turret
yes but
dawn dagger
but maybe you should take a break atp cause idk whats going on
tidal turret
g(n) = 2n
then, let u = g(n)
u = g(n) = 2n
u = 2n
n = u/2
g(u/2) = 2(u/2)
g(u/2) = u
similarly
g(n/2) = n
the actual inverse of g is n/2 then, but we want it to not be surjective?
we need to restrict the range for not to be N
my bad, we want it to be surjective but not injective
dawn dagger
yes which you can do by applying ceil/floor
tidal turret
ok
dawn dagger
ok it seems that doesnt work with n/2 actually
torpid garnet
Renato, ¿qué te dijo de las curvas de nivel y la traza?
tidal turret
ok it seems that doesnt work with n/2 actually
why
torpid garnet
today I dont have anal class
Y dale con anal class, jajaja. Mañana te pregunto entonces.
tidal turret
oki
chilly cobalt
Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s,t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer
midnight plankBOT
Find all polynomials $P(x)$ with integer coefficients such that for all real num...
grand pondBOT
Copter
chilly cobalt
status is 1😓
midnight plankBOT
@chilly cobalt Has your question been resolved?
chilly cobalt
<@&286206848099549185>
midnight plankBOT
@chilly cobalt Has your question been resolved?
last slate
Find all polynomials $P(x)$ with integer coefficients such that for all real num...
this is hard
I remember solving it 1-2 months ago
uhh lemme see if I remember how I did it
idk why you're asking here and not on like mods
last slate
**Copter**
ok firstable what do you think the solutions are?
(the bot closed the channel but if you want help you can dm me of you want)
twilit field
Suppose that $H$ is a subgroup of $G$ such that whenever $Ha≠Hb$, then $aH≠bH$. Prove $ghg^{-1} \subset H$ for all $g \in G$
midnight plankBOT
Suppose that $H$ is a subgroup of $G$ such that whenever $Ha≠Hb$, then $aH≠bH$. ...
grand pondBOT
wai
twilit field
I need a hitn
*hint
😔
I instead to have more to work with assumed the contrapositive
if $aH =bH$ then $Ha=Hb$
grand pondBOT
wai
twilit field
Proving $ghg^{-1}$ forms a group isn't too bad
grand pondBOT
wai
twilit field
proving it's contained in $H$ is proving to be a bit of a challeneg
grand pondBOT
wai
twilit field
h\in H btw
visual tiger
h\in H btw
so, take h in H and g in G. Try to apply your contrapositive to a = g and b = ... something else
twilit field
gH=g^{-1}H
tribal temple
visual tiger
gH=g^{-1}H
and why would that be
visual tiger
H has the inverse of all elements in H...
and gH is not a group
gH is the set of all elements of the form gx, where x is any element in H
twilit field
Got it
visual tiger
I don't think there's a way to bring g^(-1) immediately into this
what you could instead bring...
twilit field
should I sleep on this problem
like is it worth sleeping on ( I'm quite ahead of my class rn in group theory)
visual tiger
if you... want to?
This problem requires you to spot the relevant b to use
if you find it, it's over
twilit field
where b is an element of G
visual tiger
basically the only property you have is that H is a subgroup
and that's precisely how to find B
I won't say more
midnight plankBOT
midnight plankBOT
Available help channel!
dusty portal
I got $U=\begin{pmatrix}4&3\0&5\end{pmatrix}$, and I get why it's right, but how do you even get that for $U$? And why doesn't $3R_1-R_2\to R_2$ work in this case?
midnight plankBOT
I got $U=\begin{pmatrix}4&3\\0&5\end{pmatrix}$, and I get why it's right, but ho...
grand pondBOT
;(
sharp coral
for finding LU factorization the only row operation you need to be doing is adding a multiple of one row to another row. here you're sneaking in multiplying a row by a scalar as well
sharp coral
if you do that then you lose the property of having the diagonal of L being all 1s
proud ermine
You should remind yourself what an elementary row operation is.
This is a common mistake because it often doesn’t matter
dusty portal
if you do that then you lose the property of having the diagonal of L being all ...
Ohhhhhhhhhhh
I forgot lmao
You have to be able to apply that
The thing is that it yielded the correct result
So is that supposed to happen?
dusty portal
You should remind yourself what an elementary row operation is.
Ok
sharp coral
**;(**
what do you mean by the correct result? the U posted here differs from the U given in the answers
dusty portal
what do you mean by the correct result? the U posted here differs from the U giv...
Correct result for x
sharp coral
well yes, it still provides a valid factorization of A into lower and upper triangular matrices. the only difference is that it doesn't follow the convention that the diagonal of L should be all 1s
dusty portal
Ok
It clicked now, thank you
I can only have a coefficient that isn’t one on one row
dusty portal
well yes, it still provides a valid factorization of A into lower and upper tria...
Thank you as well
Is this generally a faster method than inverses? Just because of the lower triangular/echelon form?
sharp coral
it's less operations but more bookkeeping in general, which is part of why it's preferable in computer implementations (where bookkeeping is easy)
it's also more numerically stable (less liable to be thrown off by limited precision error)
midnight plankBOT
@dusty portal Has your question been resolved?
tidal turret
midnight plankBOT
tidal turret
can I get some help with 33
cedar mason
pas parle espangol
tidal turret
midnight plankBOT
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
1
tidal turret
f(g(n)) = n
g(f(n)) ≠ n, find f,g
cedar mason
....what's n
lavish venture
....what's n
bro what
tidal turret
∀n ∈ N
lavish venture
do you know any results about left and right inverses?
tidal turret
do you know any results about left and right inverses?
you are asking me or him? (or both)
lavish venture
so i’ll take that as a no then
tidal turret
I dont understand what does g having a left inverse imply?
in terms of surjectivity or injectivity
Let f : X -> Y be a function
Injectivity: f is said to be injective if ∀ x,y ∈ X . f(x) = f(y) ⟹ x = y
Surjectivity: f is said to be surjective if ∀ y ∈ Y . ∃ x ∈ X . y = f(x)
Bijectivity: f is said to be bijective if f is both injective and surjective
lavish venture
I dont understand what does g having a left inverse imply?
f has a left inverse iff f is injective and f has a right inverse iff f is surjective. try using your statements here to prove this
lavish venture
yes
tidal turret
f doesnt have left inverse => f is not injective
lavish venture
probably easier to deal with f
tidal turret
probably easier to deal with f
we need to find both btw
lavish venture
yea i’m saying to think of it as f being surjective but not injective
you can do either
tidal turret
yes, f has a right inverse, thus f is surjective
so, we gathered that f is surjective but not injective
lavish venture
probably easiest to use some piecewise definition
tidal turret
,, f : \begin{cases} 1 &\iff x \leq 2 \ x - 1 &\iff x > 2 \end{cases}
this is surely not injective
grand pondBOT
Renato
lavish venture
that should work
lavish venture
now just find the appropriate g
tidal turret
∀x ∈ N
f(g(n)) = n
g(f(n)) ≠ n, find f,g
g has left inverse => g is injective
g doesnt have right inverse => g is not surjective
tidal turret
now just find the appropriate g
g(x) = 2x
lavish venture
you need f(g(n)) = n
is f(2n) = n for all n?
just use your example
how would you find the inverse of n - 1
tidal turret
x = n - 1
x + 1 = n
lavish venture
now verify it works
find f(g(n)) and g(f(n)) for n = 1, 2 and then for n > 2
conclude that f(g(n)) = n but not the other way around
tidal turret
when x = 1
f(1+1) = f(2) = 1
when x = 2
f(2+1) = f(3) = 2
when x > 2
f(x + 1) = x - 1
lavish venture
when x = 1
f(1+1) = f(2) = 1
when x = 2
f(2+1) = f(3) = 2
i thought you were using N to exclude 0
tidal turret
my bad
lavish venture
i mean you can fix it
tidal turret
this is hard dude, tips?
lavish venture
just change your definition of f accordingly
lavish venture
this is hard dude, tips?
proving the results you’re using
tidal turret
just change your definition of f accordingly
domain of f is N
lavish venture
don’t try and be a memory champion and remember things you don’t understand
tidal turret
g will be another piecewise? or no?
g(n) = n + 1
g(n+1) = 1 when n + 1 <= 2
g(n+1) = n when n + 1 > 2
,, f : \begin{cases} 1 &\iff x \leq 2 \ x - 1 &\iff x > 2 \end{cases}
grand pondBOT
Renato
tidal turret
f(n+1) = 1 when n + 1 <= 2
f(n+1) = n when n + 1 > 2
g(f(n+1)) = g(1) when n + 1 <= 2
g(f(n+1)) = g(n) when n + 1 > 2
∀x ∈ N
f(g(n)) = n
g(f(n)) ≠ n, find f,g
g has left inverse => g is injective
g doesnt have right inverse => g is not surjective
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
@lavish venture help please dude
lavish venture
g will be another piecewise? or no?
you could just do f as you defined it on N = {1, 2, …} then g = n + 1 so that f(g(n)) = f(n + 1) which implies that f(g(1)) = f(2) =1, f(g(2)) = f(3) =2 and then for n > 2 this holds since f(g(n)) = f(n + 1) = (n + 1) - 1 = n. but g(f(1)) = g(1) =2, so g(f(n)) ≠ n for all n
tidal turret
you could just do f as you defined it on N = {1, 2, …} then g = n + 1 so that f(...
damn dude, you got a good grasp on intro to proofs
lavish venture
just practice
midnight plankBOT
midnight plankBOT
Available help channel!
quaint shoal
Guys does this seem correct? I’m confused about the questions that have the massive jumps, I’m thinking it’ll either be -oo/oo or DNE?
midnight plankBOT
Guys does this seem correct? I’m confused about the questions that have the mass...
quaint shoal
2b I’m also quite confused.. it seems like it’s approaching 0.333?
fallow scarab
yes 2b looks right
quaint shoal
yes 2b looks right
okay how abt 3c,d,e
I don’t get how these massive jumps impact it
like 3.999 is -12998
while 4.001 is 13002
does that just mean it’ll go forwards -oo?
fallow scarab
I don’t get how these massive jumps impact it
try plotting the points
quaint shoal
try plotting the points
Seems like there’s an asymptote potentially?
fallow scarab
right
quaint shoal
hmm alright then, so as it approaches the asymptote it’ll get infinitely closer
so I mean it approaches oo
or -oo depending on the question
right?
fallow scarab
too vague to understand what you're talking about. which problem?
quaint shoal
too vague to understand what you're talking about. which problem?
sry I was thinking out loud 😅
brave berry
I solved for x using the discriminant equation for two distinct roots and got b > sqrt(4ac) but im a bit confused as what to do. Challenge question part A
midnight plankBOT
I solved for x using the discriminant equation for two distinct roots and got b ...
torpid garnet
I solved for x using the discriminant equation for two distinct roots and got b ...
,rotate
grand pondBOT
torpid garnet
I solved for x using the discriminant equation for two distinct roots and got b > sqrt(4ac) but im a bit confused as what to do. Challenge question part A
karmic trench
I solved for x using the discriminant equation for two distinct roots and got b ...
Analyse both cases
b^2-4ac > 0
Ac>0 and ac<0
brave berry
Ac>0 and ac<0
So when ac is positive vs negative?
brave berry
Yes bro
karmic trench
Best of luck
brave berry
This question is pure torture
karmic trench
The challenge questions are fun
brave berry
Best of luck
Did u do it asw?
karmic trench
Actually gets u thinking
karmic trench
Did u do it asw?
Yes I got A* this summer
brave berry
The challenge questions are fun
They’re satisfying to get but im just horrendously bad at proof
brave berry
Yes I got A* this summer
Congrats thats sick
karmic trench
It’s good ur giving it a go Ngl I was too lazy and just went straight to exam questions
In hindsight it would’ve been better to develop my problem solving skills
brave berry
Im trying to self teach it so anything helps
karmic trench
Bicen maths is great if you aren’t using him already
My teachers were abysmal so I practically self taught maths and further maths
brave berry
I really want to do fm but my head of maths said my average grade was like 0,5 off what I should have gotten
I can still do it but im on a waitlist so its bugging me
brave berry
6 8s 3 7s 1 6
karmic trench
That should be good enough
karmic trench
6 8s 3 7s 1 6
Congratulations that’s amazing
brave berry
Congratulations that’s amazing
Thank u man I expected better but im happy w them
karmic trench
They’ll give you a good shot at top courses for sure
brave berry
That should be good enough
Well ive got to write a personal statement as to why i should be part of the fm cohort and its annoying
karmic trench
That seems super extra I’m sure they’ll cave in and let you
brave berry
They’ll give you a good shot at top courses for sure
Yea but i feel as if ill be behind if they start the course without me
brave berry
That seems super extra I’m sure they’ll cave in and let you
Hopefully, im going to send email in tomorrow and im praying they let me in 💔💔💔
karmic trench
best of luck
brave berry
thank you bro
Also
If ac > 0
I dont know how to word it
Like
What does it mean for b and the fact it can be any value
Like if its a positive number does it mean that sqrt4ac is positive asw and that it meets the criteria?
karmic trench
if 4AC>0 say it’s 5,6,7,8 etc etc there are infinitely many numbers such that B is greater 9,10,11,12 etc etc
karmic trench
Well if ac<0
brave berry
Nvm i realised
karmic trench
b^2 - 4ac > 0 —> b^2 > 4ac
brave berry
My question was silly
brave berry
b^2 - 4ac > 0 —> b^2 > 4ac
Yea
brave berry
Yea i realised mb 😭
karmic trench
No worries
brave berry
How do u get good at proofs like that
brave berry
Okay
karmic trench
Next time you see a question like this you’ll spot what to do
brave berry
Okay thank you
So to re iterate
If ac > 0
And b^2 > 4ac or b > sqrt4ac
there are infinite numbers bigger than 4ac or sqrt 4ac
Therefore b can be any of them?
And if ac < 0 b is automatically bigger than it so it fits the criteria
karmic trench
Yes also you know of solution banks right
brave berry
Uhh ive heard of them but idk what they r
karmic trench
If you search up pmt pure msths 1 solution bank
It’ll have method for every question in the book
brave berry
Ohhhhhh alright
karmic trench
very useful
karmic trench
Yes
brave berry
That’s useful thank you
karmic trench
No prob
brave berry
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tidal turret
midnight plankBOT
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
Let f : X -> Y be a function
Injectivity: f is said to be injective if ∀ x,y ∈ X . f(x) = f(y) ⟹ x = y
Surjectivity: f is said to be surjective if ∀ y ∈ Y . ∃ x ∈ X . y = f(x)
Bijectivity: f is said to be bijective if f is both injective and surjective
prime hornet
sorry, could you translate these?
I think I have an idea as to what they want, but I'm not totally sure
tidal turret
prime hornet
okay, cool! which one do you want to start with?
tidal turret
i
prime hornet
sure!
so, we want to prove that g is injective, knowing that f o g is injective
i.e. we need to prove that if g(x) = g(y), then x = y
so we need to start by assuming that g(x) = g(y) 
any ideas on how we can obtain x = y from here?
tidal turret
if fg is injective then
f(g(x)) = f(g(y)) => x = y?
prime hornet
yes!
all we needed to do was apply f to both sides, and out falls x = y
that proves i) immediately
any questions, or shall we move on to ii)? 
tidal turret
what?
prime hornet
do you have any questions about the proof we just gave, or is everything clear? 
tidal turret
it is not
lavish venture
maybe you’re confused why applying f maintains equality?
tidal turret
yes
lavish venture
it’s a function so it’s well defined
x = y implies f(x) = f(y)
passes the vertical line test so to speak
lavish venture
x = y implies f(x) = f(y)
replace x and y with g(x) and g(y)
tidal turret
passes the vertical line test so to speak
you mean, like that one input is to one outout and we csnt have one input and two different values meaning like one single input should always give the same outout and not have other outputs or smthg
lavish venture
yes if the inputs are equal then the outputs must be equal
lavish venture
you’re done
tidal turret
fog(x) injective, fog(x) = fog(y) <=> g(x) = g(y)
what?
Let f : X -> Y be a function
Injectivity: f is said to be injective if ∀ x,y ∈ X . f(x) = f(y) ⟹ x = y
Surjectivity: f is said to be surjective if ∀ y ∈ Y . ∃ x ∈ X . y = f(x)
Bijectivity: f is said to be bijective if f is both injective and surjective
ah my bad, FOG is injective
not f
fog(x) injective, fog(x) = fog(y) <=> x = y
lavish venture
whenever you’re trying to prove a function is injective you always start out with "assume h(x) = h(y)" and try to show that x = y. you could also show that if x ≠ y then h(x) ≠ h(y). so to show g is injective you should start with something like let x, y be in A such that g(x) = g(y) then deduce x = y
lavish venture
fog(x) injective, fog(x) = fog(y) <=> x = y
yes
tidal turret
assume g(x) = g(y)?
lavish venture
g(x) = g(y) —> f(g(x)) = f(g(y)) —> x = y
tidal turret
g(x) = g(y) —> f(g(x)) = f(g(y)) —> x = y
this
lavish venture
assume g(x) = g(y)?
"if g(x) = g(y) then…"
so assume g(x) = g(y)
show that it must be the case that x = y
otherwise we would have distinct inputs mapping to the same output which clearly isn’t what we mean by injective
lavish venture
how would you start 2
lavish venture
Let f : X -> Y be a function
> ***Injectivity:*** f is said to be **injectiv...
refer to this
tidal turret
f o g surjective, then ∀c ∈ C, ∃a ∈ A, c = f o g(a)
lavish venture
perfect
tidal turret
assume there exists a k ∈ C
lavish venture
use that to show f is surjective
lavish venture
assume there exists a k ∈ C
🤔
f is surjective if for all c in C there exists b in B such that f(b) = c
this is what you’re trying to show
go in order of the quantifiers from left to right
"if for all c in C…"
translates to
let c be in C
in your proof
tidal turret
well
lavish venture
f o g surjective, then ∀c ∈ C, ∃a ∈ A, c = f o g(a)
how can you use this to find such a b in B?
lavish venture
yes
tidal turret
c = f(b)
lavish venture
👍🏻
tidal turret
assume f o g surjective, then ∀c ∈ C, ∃a ∈ A, c = f o g(a)
lavish venture
if you want to write it formally just say something like "let c be in C. then since fog is surjective there exists a in A such that f(g(a)) = c. choose b = g(a) in B so that f(b) = c which implies that f is surjective"
notice how the order of the quantifiers follows the structure of the proof
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
.close
severe grail
I'm not sure if I'm on the right track for 2
midnight plankBOT
I'm not sure if I'm on the right track for 2
severe grail
severe grail
I redid the q
lavish venture
$\frac{1 + \cos y}{1 + \sec y} \cdot \frac{\cos y}{\cos y} = \frac{1 + \cos y}{\cos y + 1} \cdot \cos y = \cos y$
grand pondBOT
knief
scenic wyvern
sometimes you need to see the bigger picture to see why we do stuff
it's like asking a construction worker why he's pouring cement on the foundation when there's no building yet
lavish venture
🗣️
severe grail
oh oh
I get it
yeah I understand what he just did but
where did you get this term from
lavish venture
1?
severe grail
yea
lavish venture
i don’t see what’s unclear here
the motivation is that cos * sec = 1
if you really are unsatisfied then do the boring way
turning sec into 1/cos and combining the denominators however you do it
but i mean i think this is quite obvious
scenic wyvern
the idea is that we want to multiply the denominator by cos
but to do that we also have to multiply the numerator by cos, otherwise we'd be changing the value of the expression
hence that fraction
midnight plankBOT
@severe grail Has your question been resolved?
severe grail
but to do that we also have to multiply the numerator by cos, otherwise we'd be ...
oh so we could basically multiply by anything when it gives you a value of 1 such as 100/100 or tan/tan
I thought we were not allowed to pull up random numbers
scenic wyvern
theoretically yes, but ofc we're gonna want to multiply by something that simplifies shit
severe grail
theoretically yes, but ofc we're gonna want to multiply by something that simpli...
yep
scenic wyvern
I thought we were not allowed to pull up random numbers
but this is NOT random
severe grail
but this is NOT random
wdym?
severe grail
I meant how is that not random?
scenic wyvern
because we set that up
severe grail
u mean its not random because it was from the question?
ur referring to cosy right?
scenic wyvern
i mean the fraction is cos / cos, so
scenic wyvern
u mean its not random because it was from the question?
it's not random because it was set up specifically to simplify the expression
scenic wyvern
if it helps your cause, i suppose
scenic wyvern
that is not even correct
solar quail
You can't cancel like that - the $\dfrac{1}{\cos{y}}$ term is part of a sum.
grand pondBOT
lavabucket6200
solar quail
You would need to get it over a common denominator.
scenic wyvern
or you can (and should!) multiply termwise
solar quail
or you can (and should!) multiply termwise
But this is probably a simpler path.
severe grail
or you can (and should!) multiply termwise
example?
lavish venture
sir i strongly suggest reviewing your laws of arithmetic
you don’t seem to have a firm grasp of how arithmetic works
much less with variables involved
scenic wyvern
cos y * (1 + 1/cos y)
= (cos y)(1) + (cos y)(1 / cos y)
= cos y + 1
lavish venture
also notice how i didn’t multiply through in the numerator
severe grail
sir i strongly suggest reviewing your laws of arithmetic
what is the laws of arithmetic again??
lavish venture
i kept the cos outside
because i was trying to cancel the 1 + cos
associativity, commutativity, additive inverses, multiplicative inverses, additive identity, multiplicative identity, distributive property
understand all of these for real numbers
distributive property especially
i’m not sure you know what it even is
lavish venture
you made the mistake twice here
or who knows what you even did in the denominator
i don’t know how 1/cos * cos became cos^2
scenic wyvern
distributive property especially
on this:
scenic wyvern
where’s this from? an old help channel?
no, this is my old guide that i'm working to put in docs
scenic wyvern
more than just my students actually
scenic wyvern
hanako reference guide when
ongoing
scenic wyvern
oh a lot
lavish venture
yea i'm planning to sit down tmr night and work on it some more
what are you studying
scenic wyvern
probably best continued in another channel
severe grail
associativity, commutativity, additive inverses, multiplicative inverses, additi...
what are these properties for? is it better to understand the operations
lavish venture
if you don’t understand these properties of real numbers then how are you supposed to be able to translate it to expressions involving variables
these skills are required to manipulate expressions
properly at least
severe grail
properties of real numbers?
I thought it was properties of multiplication, addition, subtraction, division
lavish venture
I thought it was properties of multiplication, addition, subtraction, division
these are operations we use for real numbers
the objects themselves are the real numbers
the operations are functions that take in real numbers and spit out another one
severe grail
I get it
severe grail
associativity, commutativity, additive inverses, multiplicative inverses, additi...
so am I watching the wrong vid
severe grail
anyways i have to sleep
gnn
severe grail
.close
prime swallow
I need help with someone to explain the problem and how to solve
midnight plankBOT
I need help with someone to explain the problem and how to solve
prime swallow
Dont worry about my answers
pearl hull
,rccw
grand pondBOT
prime swallow
Like i said dont worry about my answers i just need help woth how to figure this question out
Ok so
What your saying is when x=4 find f(x) which is really just y
prime swallow
Ok and so when its f(x)=12 how do you explain that?
prime swallow
So can i change the f(x) to y so y=12 so what is x
pearl hull
So can i change the f(x) to y so y=12 so what is x
Just check the table
I’ve already marked it for you
prime swallow
Ok gimme a sec
pearl hull
prime swallow
Ok so i understand the f(x)=12 but now the ones with f(#) is confusing me
pearl hull
Whyy
ok so, let’s look at this
We first replace y with f
for every x input, you’ll find a value y
and now we tell everyone what we input to get the result
prime swallow
Thats the problem if we change it to y then would y=-1
pearl hull
For example, When we input 3, we’ll get 4
f(3) = 4
prime swallow
Ah ok
And so if the output is 12 what is the input?
And so are all my answers correct?
@pearl hull
scenic wyvern
is that a 4 or a 9 in a)
prime swallow
9
scenic wyvern
looks all good
fringe pine
I did something wrong I just don’t know what I did wrong 
midnight plankBOT
I did something wrong I just don’t know what I did wrong <a:spongebob_sadge:1021...
fringe pine
Could it be an algebra mistake?
lyric charm
you fucked up your long division i think
-2x - (-x) = -x not -3x
,w long division (x^3 - 2x + 4)/(x - 1)
grand pondBOT
lyric charm
yeah so x^2 + x - 1 + 3/(x-1)
fringe pine
what the fart
midnight plankBOT
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
pearl hull
Stop
fringe pine
you fucked up your long division i think
i see it now 😭
lyric charm
if you need help with an actual question of your own, then open your own channel.
fringe pine
yeah so x^2 + x - **1** + **3**/(x-1)
ty
lyric charm
see #❓how-to-get-help for how to do that
fringe pine
.close
midnight plankBOT
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Available help channel!
jade magnet
midnight plankBOT
jade magnet
for i i think i need to find the angle AEG and AGF or something
idk exactly
idk how to actually find the angle
vivid remnant
for i i think i need to find the angle AEG and AGF or something
no need
lethal path
i definitely need to find another angle
part i) right?
vivid remnant
i definitely need to find another angle
well yeah but since y and z are same the opposite sides must be equals so AE = AF
yeah
lethal path
it comes from the question: GE and GF are the perpendiculars to AC and AB
jade magnet
part i) right?
yes
jade magnet
well yeah but since y and z are same the opposite sides must be equals so AE = A...
wait so we know y and z are the same because both EAG and GAF are the same and because ge and gf are perpendiculars?
is that how you deduced y = z
vivid remnant
is that how you deduced y = z
yes
lethal path
wait so we know y and z are the same because both EAG and GAF are the same and b...
yeah, so both EGA and AGF must equal 180 - x - 90
jade magnet
it was clearly obvious
vivid remnant
yes
lethal path
yeah it was a "read the damn question" moment
midnight plankBOT
last slate
💀
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Available help channel!
ionic void
hey there how do i work out the area of this? i lowk need to fix my grades if im gonna get anywhere in maths 💔
midnight plankBOT
hey there how do i work out the area of this? i lowk need to fix my grades if im...
midnight plankBOT
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Available help channel!
hot current
This might be an open question: given a symmetric positive definite matrix A with its inverse A^{-1} and a matrix B with full rank, is it possible to compute the inverse of B^T A B efficiently?
midnight plankBOT
This might be an open question: given a symmetric positive definite matrix A wit...
@hot current Has your question been resolved?
cedar coral
define efficiently?
gilded viper
the calculation of inverse matrices is always problematic 
torpid garnet
define efficiently?
Efficiently has an standaed definition which is that its computational complexity is polynomial on the size of the input
cedar coral
well, naive algorithms of multiplication and inverse matrixes are O(n^3), so that's "efficient" by that definition
midnight plankBOT
@hot current Has your question been resolved?
hot current
Basically, we know that B^T A B is invertible because B is full rank so the whole product is symmetric positive definite. A^{-1} is already computed. So maybe there exists a formula to compute (B^T A B)^{-1} using A^{-1} but I might be wrong.
Basically, is it possible to avoid any matrix inversion algorithm?
midnight plankBOT
@hot current Has your question been resolved?
midnight plankBOT
@hot current Has your question been resolved?
midnight plankBOT
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analog carbon
So im wanting to take a math course over the summer. Idk if i should do it online or at community college. I go on my phone a lot when doing school stuff and just half do things.
midnight plankBOT
So im wanting to take a math course over the summer. Idk if i should do it onlin...
merry skiff
i feel like you should do it at community college if you get easily distracted
analog carbon
I also did a math course online this year before school and didnt pass the final exam
Yea
I was using like AI
In online
A teacher is better than no teacher
merry skiff
yeah
analog carbon
I mean
I want to self study in khan academy first
To get a gist of honors alg 2
Then if i dont feel a good understanding i might take community college
The online course was $500 and i failed
I feel like i let everyone down
So yea im definitely gonna take community college
merry skiff
ok!
lavish venture
The online course was $500 and i failed
are you in high school?
lavish venture
taking courses just because and not taking them seriously could hurt you
don’t take courses unless you’re going to take it seriously
lavish venture
if colleges look and see you failing courses you’re taking at the college level it doesn’t reflect well by the time you apply
analog carbon
I will be taking honors alg 2 this next summer
lavish venture
also at the level you’re at there really isn’t much of a benefit to taking those courses
just learn it on your own
analog carbon
Wym
lavish venture
it’s not like you’re going to get credit for it
analog carbon
I just wanna be ahead
lavish venture
what math are you in in school?
copper mason
I eat wood
lavish venture
ok so wdym take algebra 2 in the summer?
what would you take the year after?
in school
lavish venture
if you could accelerate your math by taking courses earlier then yes i would suggest it so long as you can handle it
is this something kids at your school do?
analog carbon
if you could accelerate your math by taking courses earlier then yes i would sug...
Yes ima self study alg 2 during when im relearning honors geo
analog carbon
is this something kids at your school do?
Yes
lavish venture
do you know that your school would even let you skip it?
analog carbon
do you know that your school would even let you skip it?
Ye my counselor said yes
I really like math. Its one of my fav subjects so i wanna excel at it
lavish venture
just make sure that when you’re in a class you’re actually doing the work
college courses go much quicker
especially summer courses
analog carbon
just make sure that when you’re in a class you’re actually doing the work
Yea i used ai a lot during online
It felt like crap
lavish venture
if you slack off for even a few days or a week you could find yourself far behind
analog carbon
Yes i will truly try to not be distracted
lavish venture
Yea i used ai a lot during online
please do not use ai for problems
analog carbon
What about for checking
lavish venture
because it’s almost a certainty that you’re using it in a way that’s hurting you
lavish venture
What about for checking
i mean this isn’t so bad but it’s probably better to develop the skill of being able to check yourself
knowing if your work is valid
lavish venture
if not you could always ask here
chatgpt should really be more of a search engine
lavish venture
So i should take community college rather than online
if that’s better for you and keeps you disciplined then yes
being in class away from the distractions can help
lavish venture
I really like math. Its one of my fav subjects so i wanna excel at it
try going through some books
analog carbon
Wym
lavish venture
like if you’re serious about wanting to progress
use books
don’t just learn passively with videos
lavish venture
you can learn the kind of stuff you’re learning with just online resources and videos but you really need a large source of problems
especially for something like algebra
lavish venture
which is just developing that muscle of computation bashing with proper technique
lavish venture
What books
i think the AOPS algebra books are excellent
analog carbon
Lemme see
analog carbon
Oh alright
lavish venture
depending on your level prealgebra might be a better start but you can look through it to see if youre comfortable with everything in it or not
if you are then go to intro to algebra
lavish venture
did you look it up?
analog carbon
Yes there are multiple
lavish venture
they have a diagnostic test at the bottom for each of these books
to test if you need it
and if you’re ready for the book
lavish venture
lavish venture
U think geometry is easier than alg 2
uhh depends
geometry can get pretty challenging
analog carbon
Im more of an algebra guy
lavish venture
i would say algebra 2 is easier
anyways
have a look at the table of contents, "are you ready for this" and "do you need this" for both of these books
and decide which one you need
analog carbon
Oh alright
lavish venture
they also have videos btw
aops on youtube
and alcumus has a lot of good problems
lavish venture
You’re really took thought into this
no worries
analog carbon
I might have some questions to ask you if i need help rather than AI
lavish venture
I might have some questions to ask you if i need help rather than AI
you could always ask on the server
i’m not always available
analog carbon
Oh ok
lavish venture
there are a lot of helpful people here
analog carbon
Alright
lavish venture
i’m a student myself though
lavish venture
So i should probably do khan academy for now
use the books too
to learn math you have to do a lot of problems
khan academy should be a more of a supplement to your learning
analog carbon
Oh ok
lavish venture
U still in hs
no i’m in my 2nd year of undergrad
analog carbon
So i should get books
lavish venture
So i should get books
i can dm you where to find them for free
if you don’t mind using pdfs instead of a hard copy
midnight plankBOT
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vivid fjord
.close
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frosty plume
The translator kinda sucks so I'll fix it:
Task 8: give a regular tetrahedron ABCD w/ edges legth=2, let M and N be the midpoints of AC and CD respectively, these clauses are true or false?
Help me plss idk how to explain it specifically 😭
midnight plankBOT
The translator kinda sucks so I'll fix it:
Task 8: give a regular tetrahedron AB...
scenic wyvern
,rccw
grand pondBOT
zenith snow
Can u show the original question,the one without translation?
midnight plankBOT
@frosty plume Has your question been resolved?
sullen bolt
is there any way to quicksolve for x²+y²+z²?
midnight plankBOT
is there any way to quicksolve for x²+y²+z²?
@sullen bolt Has your question been resolved?
lethal path
is there any way to quicksolve for x²+y²+z²?
So x^2 / 35 = 1/2 * (x^2 / 5 - x^2 / 7) for example
Same trick works on all of them
sullen bolt
what a nasty trick
dire ridge
Looking for statistics teacher immediately,even students with good command are welcome
frosty plume
.reopen
midnight plankBOT
Available help channel!
frosty plume
Can u show the original question,the one without translation?
Here
jagged zephyr
have you drawn the figure? and what does the problem ask?
frosty plume
have you drawn the figure? and what does the problem ask?
Yes, abcd) true or false
My figure
