Like from 1st matrix, we get x as -3-6s
#help-42
mental tree
atomic slate
ok
atomic slate
yes
mental tree
This would give us 3t + 6s = -19
atomic slate
thats what im saying with the matrix in my first message, yes
mental tree
So we have 1 equation in 2 variables
atomic slate
$\begin{bmatrix} -6s & -3t & 19 \ -6s & -2t & 18 \ -3s & t & 7 \end{bmatrix}$
potent lotusBOT
amber
mental tree
Okay yeah now I get it
Mb
So basically all you have to do is
Solve the 1st 2 equations
You’ll get value of s and t
Done ?
atomic slate
-8/3, -1
mental tree
And then you put it in last equation to check if it follows the variables
mental tree
So now you have the values of s and t
Just put them in any one of parametric form
Either of s variable or t variable
You get 3 coordinates
Which represent the point of intersection
atomic slate
13, 15, 13?
mental tree
Yep
atomic slate
beautiful
mental tree
ACC to equations you have yes
atomic slate
thank you so much
mental tree
And also just note that
When you are putting values of s and t in last equation and it doesn’t follow the last equation
atomic slate
then they dont intersect
blazing coyote
Prove that every subgroup of a cyclic group is cyclic.
calm coralBOT
Prove that every subgroup of a cyclic group is cyclic.
blazing coyote
So essntially I want to prove every element of the set can generate every other element of the set
Defn : A Cyclic group is a group that can be generated by a single element
Does this element need to be part of the sub group
fiery dove
Defn : A Cyclic group is a group that can be generated by a single element
Right. But not every element will generate the group.
velvet osprey
Does this element need to be part of the sub group
maybe it would be a lot easier if you put some notation to it instead of trying to fumble like this
call your group G.
G is cyclic, meaning that there exists x ∈ G such that <x> = G.
let H be a subgroup of G. you want to show H is also cyclic.
fiery dove
Does this element need to be part of the sub group
If the subgroup contains the generator for the whole group, then it will be equal to the whole group.
velvet osprey
if it so happens that H contains x, what can be said of it
blazing coyote
It generates the entire group
blazing coyote
H is the group G
velvet osprey
correct
so in fact generally x need not belong to H
simply because there exist cyclic groups that have subgroups other than themselves
hmm i am actually not that sure if this problem is feasible to rawdog from the defns alone
without any sort of mental picture
blazing coyote
I think a proof by contradiction is in order here.
The book does mention use exponentiation
"Prove every subgroup of a cylic group is cyclic. Do this by working with exponentiation and use the description of the subgroups of $\Z^{+}$ "is the Question in my book
potent lotusBOT
What a wonderful world !
blazing coyote
However, I was hoping to do the general case
velvet osprey
general case as opposed to what
blazing coyote
nvm, my bad
Let $H$ not be a cyclic subgroup, there then exists an element that cannot be written as $a^n$, for some arbitrary element in $G$ and $n \in \Z$
potent lotusBOT
What a wonderful world !
blazing coyote
I have to show that this somehow violates the conditions for a set to be a group/subgroup
I suspect it would violate closure , but how do I show that
velvet osprey
mmmmmmm nope.
non-cyclicity doesnt look like that.\
im gonna say contradiction will not cut it here.
blazing coyote
Contrapositive then? If an subgroup is non-cyclic, then the group it's a subgroup of is non-cyclic too
pseudo wedge
I feel like you can do a constructive proof here
Consider that any element of H looks like a^k for some integer k, where a is a generator for G
blazing coyote
I feel like you can do a constructive proof here
That would just prove existence, no?
pseudo wedge
That would just prove existence, no?
No I mean you can prove the statement directly
Prove that any subgroup H of G is cyclic
blazing coyote
I feel like a contrapositive will be easier here
blazing coyote
No I mean you can prove the statement directly
hmm, okay
Let me think over this
I'm confused
i'll close it for now
sorry
.clsoe
.close
sharp current
I don't really understand how to figure out OP. At first I tried OT - TP and got 6.8m but I think it was wrong. (I am new to this please be patient with me😔)
fathom shuttle
im sorry to tell you this but the diagram is impossible
sharp current
Wait what
sharp current
Ohh
fathom shuttle
i would bring this up with your teacher
sharp current
Thank you. I will ask my teacher that gave it to me tommrow about it
sharp current
.close
calm coralBOT
calm coralBOT
Available help channel!
rare raft
hello
winter elbow
Stop pinging random people
calm coralBOT
calm coralBOT
Available help channel!
pallid pond
in this question, i've solved a using cosine rule, and when i tried doing question b, i found the angle size of BCA but dont really know how to find the bearing
pallid pond
upper sparrow
if you know angle BCA, which is the alpha, and you found theta as part of part a, the purple angle you have, you should be able to get easily 
pallid pond
so i do sum like 360-theta?
and the alpha
ok ima tyr
try
it doesnt work
i think im dumb
i did 360 - value for theta - value for alpha
flat orchid
Yeah
pallid pond
i dont know what to do next
i got 96...
oh wai
im so dumb
sorry
got it
thanks guys
much appreaciated
.close
calm coralBOT
calm coralBOT
Available help channel!
remote mural
calm coralBOT
tawdry gust
yeah
echo jacinth
Hello Guys
calm coralBOT
Hello Guys
echo jacinth
In need help so urgent
velvet osprey
why urgent
echo jacinth
echo jacinth
why urgent
I have an exam after 1 hour
Well i did evaluate the double integral and got it right
now the other way to evaluate is using greens theorem backwards
so i get the line integral
<@&286206848099549185>
calm coralBOT
@echo jacinth Has your question been resolved?
calm coralBOT
@echo jacinth Has your question been resolved?
fickle flume
how to find the antiderivative of 1/(x^2 + 1)
calm coralBOT
how to find the antiderivative of 1/(x^2 + 1)
flat orchid
Integrate it
fickle flume
But then I would have to know the antiderivative
flat orchid
Interesting
Find the function whose derivative is 1/(x^2+1)
That's your antiderivative
topaz raft
how to find the antiderivative of 1/(x^2 + 1)
this is a special one its arctan but u can derive it'
fickle flume
Yes I know it is arctan but I don't understand
where it comes from
it isn't intuitive at all
I tried to u-sub but it didn't work for me
is that when you trig-sub?
flat orchid
No?
flat orchid
You're just trying a method you know will or will not work
fickle flume
that*
or wait no im confused
so i would have to know the derivative of tan(theta)?
i dont know that either
😿
flat orchid
Interestung
Do you know definition of derivative by first pr9nciple?
@fickle flume
fickle flume
If I do not in English
Is it like the definition of a derivative? Like where you use a limit
and let h go to 0
fickle flume
Oh ok
I remember it, I only used it when we were introduced to derivatives the first time, then we got the power rule
At least I think it is called the power rule
flat orchid
Well you should first prove the derivative of tan x by that rule
Assuming you know what limits are
flat orchid
Then do it first
Then you'll get a good idea of what it's antiderivative will be
fickle flume
Okay that's sick
flat orchid
Yes
fickle flume
Thanks
flat orchid
But also quite tedious to do
So it's better to memorise these formulas rather than applying it again and again
Wastes a lot of time
fickle flume
Ok so most people do end up memorising?
fickle flume
but they come from the definition
flat orchid
They end up using them so much while splving
flat orchid
but they come from the definition
Yes
fickle flume
ok ty that answers my questions i think, and so i use trig-sub after (which also looked tedious, at least to me)
flat orchid
Yep
steel kraken
hi
steel kraken
i need help with the last part of this question
calm coralBOT
@steel kraken Has your question been resolved?
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
@steel kraken Has your question been resolved?
steel kraken
.close
calm coralBOT
calm coralBOT
Available help channel!
lofty gale
(I made this myself) This is a square, E is the midpoint of DA. Each side of the square is 2 units long. What is the area of EFBA? (Idk how to do this, I need help)
boreal cloud
EDC and CFB are similar, and you can find the ratio
you can find the area of EDC so you can also find CFB
then then minus both from the square
unkempt drift
EDC and CFB are similar, and you can find the ratio
nice helping work btw
prisma warren
Hi, if anyone need help, message me.
unkempt drift
you said just enough to help them do the question on their own
lofty gale
It was so simple turns out lol
lofty gale
you said just enough to help them do the question on their own
Yeah
To close this is just .close right?
.close
calm coralBOT
prisma warren
(I made this myself) This is a square, E is the midpoint of DA. Each side of the...
sEFBA = 3/4 - 1/4 * 4/5 = 11/20
boreal cloud
nice helping work btw
thanks lol
prisma warren
(I made this myself) This is a square, E is the midpoint of DA. Each side of the...
sECB = sCED * 4/5 = 1/4 * 4/5 = 1/5 therefore 11/20
calm coralBOT
Available help channel!
prisma warren
Ohhh, I get it now
how old are you?
calm coralBOT
how old are you?
frozen rampart
... is this worth opening a whole help channel
potent smelt
calm coralBOT
calm coralBOT
Available help channel!
tame furnace
how do i do this 🙏
calm coralBOT
how do i do this 🙏
boreal cloud
show your working
tame furnace
boreal cloud
$(2x)^2 - (x^2)^2$ is not $(2x - x^2)^2$
potent lotusBOT
ashy!
cunning reef
$(2x)^2 - (x^2)^2$ is not $(2x - x^2)^2$
(its \neq btw i think)
tame furnace
$(2x)^2 - (x^2)^2$ is not $(2x - x^2)^2$
is it supposed to be together?
4x^2-4x^3+x^4
boreal cloud
you are just expanding out the wrong thing
dull wagon
the answer key is wrong then
boreal cloud
oh hell i was looking at the cancelled parts
calm coralBOT
Available help channel!
dusky flax
is this correct notation? won't it just be 1?
calm coralBOT
is this correct notation? won't it just be 1?
flat orchid
Notation?
swift laurel
no, because the [] denotes rounding down
flat orchid
No the answer is correct it should be 0
dusky flax
my doubt is
swift laurel
when they say "greatest integer function" they mean "round down to the nearest integer"
flat orchid
when they say "greatest integer function" they mean "round down to the nearest i...
No
nimble harbor
the rounding is outside?
dusky flax
the question looks like they are asking to floor down the result of the limit
but they are limiting the floor of the function
flat orchid
Greatest integer function means 'greatest integer less than or equal to the value'
dusky flax
idk is it an actual monstrous notation
velvet osprey
is this correct notation? won't it just be 1?
no it's not correct notation
flat orchid
Oh wait nvm i didn't read your statement properly my bad
swift laurel
actually i think you're right, the [] should be inside the limit for that result to be true
velvet osprey
they ac put the floor brackets in the wrong place
yes it should be inside thel imit
velvet osprey
the question looks like they are asking to floor down the result of the limit
bu...
yes
theres a BIG difference between these
especially since floor is a discontinuous function
so you cannot be lackluster about where it goes relative to the limit symbol, notation-wise
dusky flax
they ac put the floor brackets in the wrong place
thank you
this solution further gaslights me 
flat orchid
Yep it seems I'm stupid as well 
full drum
Are you sure
dusky flax
thank you
this solution further gaslights me <:pandaohno:513070260471988226>
should I just confirm this solution as a mistake then
flat orchid
should I just confirm this solution as a mistake then
This is cengage right?
full drum
Both limits approach a certain value
dusky flax
This is cengage right?
yes..
full drum
So the functions floor value will be 0
dusky flax
i thought they are quite reliable..
full drum
I don't think the book is wrong
flat orchid
I don't think the book is wrong
No but limit is always a fixed vslue
swift laurel
should I just confirm this solution as a mistake then
you're correct, this solution is wrong
flat orchid
Not tending to
full drum
No but limit is always a fixed vslue
Hmm
flat orchid
If limit was outside gif then it would've been correct
velvet osprey
thank you
this solution further gaslights me <:pandaohno:513070260471988226>
yeah this is BS
swift laurel
lim [sin x/x] = 0 but [lim sin x/x] = 1
velvet osprey
trash book
flat orchid
i thought they are quite reliable..
A lot of Indian books copy paste from old books
dusky flax
lim [sin x/x] = 0 but [lim sin x/x] = 1
yeahhh
calm coralBOT
calm coralBOT
Available help channel!
full drum
In proving real numbers to have a greater cardinality than natural numbers by Cantors diagonalisation, what all axioms do we use? Can someone help me list them exhaustively?
full drum
I mean which of the axioms from the ZFC are required, and which are not.
jolly furnace
You only need ZF's nine axioms, no choice
full drum
So how do we start enumerating the list we construct
That list will be a subset of the reals right
How are we claiming to create a list
jolly furnace
We cannot enumerate the list, we don't even contruct it, since it's impossible
We only pretent we can contruct it
That subset will indeed be a subset of the real numbers
full drum
That pretending to construct it doesn't invoke choice?
jolly furnace
Because we only define a function to get that list
$f: \mathbb{N} \to [0, 1]$
potent lotusBOT
street
full drum
Arent we assuming the existence of such a function
jolly furnace
Yes, that is correct, a mistake in my wording
Such a function cannot exist, same as with our list of real numbers betwen 0 and 1
full drum
Right, so in assuming this (non existent) function, we aren't invoking choice?
jolly furnace
Why would we? We aren't constructing a well-ordered list
full drum
Yea this was my intuition too
jolly furnace
We do not need to know how exactly the function maps a natural number to a real number, we just assume that it does
When using proof by contradiction, the first step is to assume something exists, in this case, our list of real numbers
full drum
**street**
What if we were to define a function
From [0, 1] to N
Then we will be invoking choice to choose those elements from [0, 1] right? Which we then connect to each of the naturals?
jolly furnace
( f : [0,1] \to \mathbb{N} )
To define a function without providing the formula of the function, we must use AC to select an element ( x \in [0,1] ) for each value of ( f(x) ).
potent lotusBOT
street
jolly furnace
In short terms, yes, you are correct
jolly furnace
You only need AC when making an arbitrary choice over an uncountable set
Natural number are countable, which is why we don't need AC
full drum
But we aren't making a choice over the range?
jolly furnace
The function simply maps each natural number to a real number in that range, we don't need to "choose" any real number between that range
We don't need to choose, since the function's formula clearly defines the relationship between those numbers
full drum
But we don't have any formula right?
Can we do diagonalisation with the list just being the natural numbers
The function being the identity function
Can we just say that, "let's start listing the reals, first Well list all naturals (which are subset of reals) then we'll move on to others", and then "oops we exhausted the naturals while listing the naturals"
Seems wrong because of the same reason this doesn't work for proving integers larger than naturals
We can't just "exhaust" the naturals like that
Since they are an infinite set
full drum
We don't need to choose, since the function's formula clearly defines the relati...
So like idk how we are constructing this function definition/formula you speak of
jolly furnace
To clarify, in the context of Cantor's diagonalization we do need AC because we assume that there already exists a set of real numbers, if we wanted to explicitly construct said set without a specific rule ( without the formula of the function ) for how to pick those numbers we would need AC
This is how i understand this problem
full drum
You're saying we do need choice?
jolly furnace
I'm saying that if we wanted to construct such a function we would need it, yes
But in the context of Cantor's diagonalization the function is assumed to already be constructed
full drum
Choice only speaks of the existence of a function right
So assuming it to have already been constructed, still means we believe in such a functions existence
In fact if a function is constructible, we won't need ac
Only because we aren't sure of its constructibility, but are sure of its existence, am I suspecting the invoking of ac
So either it's a constructible function, or we are using ac
jolly furnace
Yes
calm coralBOT
@full drum Has your question been resolved?
calm coralBOT
@full drum Has your question been resolved?
calm coralBOT
@full drum Has your question been resolved?
calm coralBOT
calm coralBOT
Available help channel!
blazing coyote
Quick question about theorm 5.9. we consider say sigma^i until i=n, right
velvet osprey
no
we consider sigma^k(1) for all natural k.
but it has to loop back on itself eventually.
you should be able to prove it
blazing coyote
but it has to loop back on itself eventually.
fair, this makes sense
velvet osprey
let $\sigma : X \to X$ be a bijection and let $x_0 \in X$. prove there exists a positive integer $k$ such that $\sigma^k(x_0) = x_0$.
potent lotusBOT
Ann
velvet osprey
this is not a terribly hard proof in my opinion.
but you should ensure you're able to write it out properly.
lyric ravine
Alternate: use the fact that ~ on [n] given by a ~ b <=>there exist integers k,l such that sigma^k(a) = sigma^l(b) is an equivalence relation
blazing coyote
but you should ensure you're able to write it out properly.
let me try
lyric ravine
Its just 1 line after that💀
velvet osprey
yes but conceptual overhead.
i dont want wai to flounder and fumble trying to prove something is an ER.
unkempt drift
using a nuke to kill a mosquito
velvet osprey
@blazing coyote i will give you one bit of advice off the bat: contradiction is NOT a good route here.
lyric ravine
I would argue proving stuff is equiv relation is an important part of algebra in general(and in this case proving equivalence is not even hard)
lyric ravine
I mean sure you can do whatever you want i was just pointing out there are different formulations to do the same problem
blazing coyote
Thanks a lot!
I think the idea is we move ateast one place to the right each time we compose
so we eventually cycle back to x_0
blazing coyote
I mean in cyclic notation
blazing coyote
and X is a finite set I suppose
lyric ravine
I would argue proving stuff is equiv relation is an important part of algebra in...
Not to mention the fact ||that the equivalence classes give you all the disjoin cycles at once||
velvet osprey
Not to mention the fact ||that the equivalence classes give you all the disjoin ...
ok sure, but again, one thing at a time.
im deliberately giving wai a small but manageable thing to prove.
velvet osprey
mmmm
blazing coyote
oh
velvet osprey
not really any theorems as such no
blazing coyote
As $\sigma$ is a bijection, the image of each pre-image is unique.
potent lotusBOT
What a wonderful world !
blazing coyote
That's to say it's one-one lol
velvet osprey
(in a bad way.)
blazing coyote
Would using induction here be a bad idea
I'm not sure of how to put my idea in words
Let $\abs{X}=n$, we then have $\sigma^i(x)=x$ for some $1≤i≤n!$ as each composition is bijective too, thus after atmost $n!$ compositions, we would $\sigma^i(x)=x$
potent lotusBOT
What a wonderful world !
blazing coyote
this feels wrong
Let $\sigma(x_0)=x_i$ for some $0≤i≤n$, say $\alpha$ we then have $\sigma(\sigma(x_0))=\sigma(x_i)=x_{\beta}$, such that $\alpha ≠ \beta$, and so on, eventually we find that $\sigma(\sigma(\dots \sigma(x_0)\dots )) = x_l$, where $l$ is not an index for which $x_i$ has been mapped to before, composing once more we find that for $\sigma$ to be a bijection $\sigma x_l$ must map \textbf{ not sure what comes here }
something still feels off
potent lotusBOT
What a wonderful world !
lyric ravine
While it is true that $\sigma^{n!}(x_0)=x_0$ forall $x_0\in [n]$, your reasoning isnt correct. If you have proven that the order of any element in a finite group is finite, you can pretty much copy paste that proof here
potent lotusBOT
Asteroid
blazing coyote
I don't think I've proven the order of any element of a finite group is finite
.
let me think about this for a while longer
.close
calm coralBOT
blazing coyote
thanks a lot everyone
calm coralBOT
Available help channel!
lyric ravine
let me think about this for a while longer
,texsp
I will leave the proof here in case you want to come back:\
||We must have $\sigma^{k}(x_0)=x_0$ for some $k$, since if not, $\sigma^i(x_0) \neq \sigma^j(x_0)$ forall $i,j\in \mathbb{Z}$ which would mean there are infinitely many elements in $X$||
potent lotusBOT
Asteroid
lyric ravine
.close
calm coralBOT
calm coralBOT
Available help channel!
full drum
calm coralBOT
full drum
I'm pretty much completely new to this topic
I went over the sentential logic chapter yesterday. But I'm kinda stumped at this parameters thing
What r these predicate symbols and how r they different from function symbols
stray tundra
the n-place predicate symbols are relations of n elements.
Think of them as operations that require n arguments
glass heart
I would guess if you keep reading you'll see some examples
full drum
Ok but are the logical symbols like -> just special forms of predicates?
stray tundra
i guess that could be argued to be so
full drum
the n-place predicate symbols are relations of n elements.
Think of them as oper...
So can we say -> is a 2 place predicate of its own? Which takes p, and q as argument and gives p->q's truth value?
stray tundra
but a very easy example are the usual operations, like addition being a binary operation, or a 2-place predicate
full drum
Any value right
stray tundra
that would depend on the variable
full drum
True or False is just something we do for our purposes
full drum
that would depend on the variable
No like if we are talking purely of syntax
stray tundra
you're gonna need to put an specific example there, but no, purely of syntax they are not necessarily equivalent
full drum
?
stray tundra
since some variables can take only T/F value (1/0) and others can take other values, they are not necessarily valid on all the n-place predicates
because some of the predicates have restrictions in the variables, like division not being defined if the second argument is 0
stray tundra
it would probably help if you think of the first-order languages as if they were programming languages
full drum
If you have a relation undefined for a certain element, then is this "being undefined" a matter of syntax or of semantics
stray tundra
logical symbols would be the parenthesis you use to group stuff.
sentential connective symbols would be the reserved keywords, like if
variables would be the names you assign to stuff
predicate symbols would be operations defined in the language, like addition, multiplication, bit shifts, and so on
full drum
😭I've not worked with programming
stray tundra
but you've heard of it, havent you
stray tundra
anyways, as Denascite said, it's probably worth reading a bit further for examples
full drum
Fair enough
full drum
If you have a relation undefined for a certain element, then is this "being unde...
What is the answer to this
stray tundra
im not sure tbh
but it usually doesnt really come up
since we're usually working with valid and sound arguments
stray tundra
where question
full drum
I saw this on a wikipedia
"There are many deductive systems for first-order logic which are both sound, i.e. all provable statements are true in all models; and complete, i.e. all statements which are true in all models are provable."
Can you help me dissect this statement
These systems they speak of
stray tundra
not really, since i'll have to leave really shortly
stray tundra
sorry
full drum
It's alright 
calm coralBOT
devout vortex
bro i am new myself, is this about AI ?
AI model?
one variable, linear reggesion AI model?
then?
acoording to chatgpt, no.
full drum
I think this needs more context. What are you modelling
velvet osprey
acoording to chatgpt, no.
!nogpt
calm coralBOT
acoording to chatgpt, no.
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).
full drum
What would be an example of a post event variable
You are gonna use this model for predicting flight delays right
While predicting flight delay, will you know the data on these variables
Yea
A model is used for prediction
You can't possibly know a flight will have maintenance issues beforehand(say while booking)
hmm. If ur looking for explanatory (causation) analysis then u shud be able to use these as well. R u like trying to measure the strengths of different factors?
full drum
I think it's logical to include something like maintenance issues if yout goal is explanation
Make sure you aren't using endogenous variables tho
devout vortex
i acutally did my research , suppose the Say you're modeling the effect of a training program on income:
Income (2024) = Bias(0) + income(2023) . weight() + income(2025) . weight()
calm coralBOT
@junior patrol Has your question been resolved?
remote mural
Hii
calm coralBOT
@weak kraken Has your question been resolved?
gray flame
Hello
How can I evaluate this?
(E): cos(π/9) × cos(2π/9) × cos(3π/9) × cos(4π/9)
velvet osprey
are complex numbers allowed
gray flame
I'm not sure
It just gave me (E) and the 2nd question was evaluate
Just this single word
velvet osprey
ok so have you done as part a instructed
gray flame
Yes
velvet osprey
and the result?
gray flame
The answer was
(1/8)Sin(8π/9)
velvet osprey
ok so then E = that/(2 sin(pi/9))
gray flame
Ye
velvet osprey
you may also want to rewrite sin(8pi/9) as sin(something else)
velvet osprey
Like this?
no
gray flame
Ow
velvet osprey
do you know that sin(theta) = sin(pi - theta) always
gray smelt
The answer was
(1/8)Sin(8π/9)
sin(pi/9)
gray flame
do you know that sin(theta) = sin(pi - theta) always
I'm not sure
This year was confusing overall
velvet osprey
this is a trigonometric identity
gray flame
Sorry I got distracted
I did try this tho
Will this work
Cos3π/9 is 1/2
And there's an identity which is;
cosx cos2x cos4x = 1/8
velvet osprey
I did try this tho
Will this work
well it kind of works
but it also isn't really what i was trying to get you to do
flat orchid
Cos3π/9 is 1/2
And there's an identity which is;
cosx cos2x cos4x = 1/8
It's not an identity, just a specific case for this instance
calm coralBOT
@gray flame Has your question been resolved?
bronze adder
calm coralBOT
bronze adder
Can someone check my proof here for (a)
I also need help getting started in (b)
I was thinking of considering subsequences that exist only in A or L
But I don't know how to justify that such subsequences would exist
calm coralBOT
@bronze adder Has your question been resolved?
pseudo wedge
Can someone check my proof here for (a)
is {x_n} a sequence in L?
and also when you have multiple indices just write lim_{m\to\infty} it's not that hard 🙏
bronze adder
and also when you have multiple indices just write lim_{m\to\infty} it's not tha...
No
.close
calm coralBOT
calm coralBOT
Available help channel!
flat orchid
calm coralBOT
flat orchid
Some help here would also be appreciated
calm coralBOT
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).
sweet stag
serene talon
oh interesting, i was going to suggest a different approach completely
flat orchid
oh interesting, i was going to suggest a different approach completely
Please do tell
serene talon
use sin²t+cos²t=1 to manipulate the integrals to arclengths of sections of circles
serene talon
use sin²t+cos²t=1 to manipulate the integrals to arclengths of sections of circ...
don't think this works lol theres no squared
serene talon
nevertheless the approach does give another way from Goethia'a, its just not as interesting a difference
first, use > sin²t+cos²t=1, then , use > 1+cos t = 2 cos²(t/2)
sweet stag
don't think this works lol theres no squared
it will work just long
flat orchid
first, use > sin²t+cos²t=1, then , use > 1+cos t = 2 cos²(t/2)
Where would this come? During solving
sweet stag
Where would this come? During solving
write the expr in terms of sin & cos
serene talon
Where would this come? During solving
I'm intentionally skipping steps to make you think about it ;)
flat orchid
I'm intentionally skipping steps to make you think about it ;)
Right. Thanks
Just tell me this
By using what you mentioned here, are you converting above integral to below integral times some constant or are you finding both the integrals individually
serene talon
I did the latter, but you could perhaps do the former too
flat orchid
Ah okay
serene talon
I also might have rewrote the integrand with a squared that wasnt there , i dont have my work anymore but im suspicious @flat orchid
flat orchid
I also might have rewrote the integrand with a squared that wasnt there , i dont...
Interesting
Have you mistakenly taken tan^2x ?
Also that goetia guy said your method is viable but long
So should I assume it's to be done the same way as what he did but with sin and cos instead of tan?
serene talon
Have you mistakenly taken tan^2x ?
I think I might have done that
flat orchid
I think I might have done that
Oh no that makes the problem infinitely easy i think
serene talon
So should I assume it's to be done the same way as what he did but with sin and ...
I'm not sure
gonna try a Weierstrass substitution
flat orchid
What's that
flat orchid
Oh no that's gonna worsen further
Cuz you can't cancel the tan x/2 term
It will come in the root i suppose
sweet stag
you gotta make use of the symmetry of the problem @flat orchid
flat orchid
you gotta make use of the symmetry of the problem <@1356904580633596075>
I understand that's what you've done here.
But the other user @serene talon managed to calculate both the integrals individually it seems.
serene talon
no, I didnt
I rewrote the problem as tan^2 instead of tan
originally
so, ignore my attempt, sorry for the mistake
sweet stag
you can exploit what gfaux said and solve it too
flat orchid
you can exploit what gfaux said and solve it too
But he said that under the assumption that it was tan^2x no?
sweet stag
idk
calm coralBOT
calm coralBOT
Available help channel!
narrow hamlet
for this question
calm coralBOT
for this question
humble slate
for this question
use crammer's rule
narrow hamlet
use crammer's rule
i have no idea what that is
humble slate
i have no idea what that is
...
humble slate
a difficult concept
let me read the question
narrow hamlet
alg
humble slate
well...u dont know cramer's rule right?
narrow hamlet
nah
humble slate
u know matrix method?
narrow hamlet
well i might know the concept now that iv googled its just never heard the word, might know meaning it looks fairly simply from a visual standpoint
or title
crammers
humble slate
it can be solved by matrix method too
narrow hamlet
aight idk what that method is also lel
specfically the name
but my method was valid i assume?
humble slate
...'
humble slate
but my method was valid i assume?
flawless
calm coralBOT
@narrow hamlet Has your question been resolved?
calm coralBOT
calm coralBOT
Available help channel!
kind summit
sigge works at a company with 112 employees, he has gotten the task to make a quiz at their party and he has to make groups (all equal sized)
he tries a few different grop sizes (groups of; 2,3,4,5) and there is always one person left over
how many people from the company are going to the party
kind summit
13 is apparently not right
like i get how its unrealistic but why is that wrong lol
sry for the bad translation of the question, it was done manually but i am pretty sure i didnt miss anything out
heavy seal
It says how many PEOPLE are going to the company. There are 112 people, so 13 definitely isn't right. You also wouldn't be able to get 13 if all group sizes must be the same
kind summit
that might be a translation error
ill retranslate with ai
sorry
Sigge works at a company with 112 employees. He has been tasked with organizing a quiz at the Christmas party and needs to divide the company into equally sized groups.
He tries out different group sizes and notices that no matter if he divides them into pairs, groups of three, groups of four, or groups of five, there is always one person left over.
How many people from the company are at the party?
heavy seal
It tells you that there is always 1 person left over, what do you think you can do with that?
kind summit
well it just means that the amount of people that are going isnt divisable by them numbers
nvm kind of but it needs to be 1 person left over
heavy seal
well it just means that the amount of people that are going isnt divisable by th...
No, that is not correct, it means that you can't have ALL employees and be in equal group sizes
If there is always 1 person left over, then that means there'll always have to be 1 person that does not show up
kind summit
i dont get it
heavy seal
Ohhh wait, I THINK I get it, but either way this is a horrendously worded question.
kind summit
okay
heavy seal
Okay, so, we're trying to find the number of people who go right? Let's just call this n for the time being
kind summit
yeah
heavy seal
And the group size options are 2, 3, 4, and 5
Wait no that would not work, ignore me, it's the other way around
heavy seal
Yes, I know, gimme a second
kind summit
the max people could be 112, cus there is 112 people at the company, and the number has to be 1 off divisible by one of them numbers every time
i think
heavy seal
Okay, let's have A, B, C, and D be the total number of GROUPS you end up with for 2, 3, 4, and 5 person groups respectively
heavy seal
the max people could be 112, cus there is 112 people at the company, and the num...
The max can not be 112 as they said there is always 1 left over
kind summit
The max can not be 112 as they said there is always 1 left over
no like there is 1 person without a group
heavy seal
Yes but they can't go to the quiz if they're not part of the quiz, they need to be in a group
Otherwise the answer would just be 112 and all the other information would be pointless
kind summit
i dont get it i think im just gonna put in nothing and see what it suggests, since its just an excersise anyway and i can do similar ones after anyways
this is the translation of the suggested method
i honestly have no clue of what that meant
oh nvm i think i get it now
.close
calm coralBOT
kind summit
<@&268886789983436800>
calm coralBOT
Available help channel!
fickle flume
hi
fickle flume
trouble finding the particular solution for
y'' + 2y' - 3y = 64e^x
i guessed y in the form of y = Ce^x
but as you can see that is equal to 0
i got stumped so i looked at the notes for this
and they add an x
y = Cxe^x
as a new guess
why does it work?
geometrically if possible
hmm
for a first order I should get solutions on a line
no i dont understand this help please 😭
the right hand side can never equal 0 for real numbers
untold summit
fickle flume
this isnt for repeated roots
i already got that answered for me
this is for the particular solution
like for the homogenous i understand it better
untold summit
yeah i get it but its similar
marsh valley
I'll look into more geometrical/intuitive explanations, but from what I remember from ODEs, you should usually guess a particular solution that looks like the RHS in a way. But if you just guess Ce^x, that "clashes" with one of the homeogeneous solutions (since the roots of the characteristic polynomial are -3 and 1), so your ansatz has to be different, and usually mulitplying by x gets rid of that
marsh valley
Yes
fickle flume
okay i see
marsh valley
Because either way you'll have to add those solutions to the particular one
untold summit
yeah i get it but its similar
*look at the variation of parameters bit on that post
marsh valley
Yeah I can't really come up with much as to why this works, but it's essentially just a way of making a "good
ansatz
fickle flume
maybe this is beyond waht i can understand at the moment, for the x that appears for double roots i did try to imagine a vector space and the fact that we should be able to describe all solutions and we cant do that with just one vector
fickle flume
Yeah I can't really come up with much as to why this works, but it's essentiall...
could i make this ansatz, and have the x not be needed, but for it to still work?
like can i always add that x and it disappears later, it just makes the algebra a smidge more tedious
marsh valley
Well the method of undetermined coefficients (i.e. making a good guess) usually checks whether any part of your guess looks suspiciously like the homogeneous solution and multiplies those by x when they do
Idk what your ansatz would be otherwise
They have a whole example on here
fickle flume
okay ty i will check that out
marsh valley
fickle flume
the complementary equation is the same as the characteristic equation?
marsh valley
The complementary equation they are referring to is the homogeneous problem associated with the equation
fickle flume
so for my problem it would be y'' + 2y' - 3y = 0?
that's the complementary equation?
marsh valley
complementary and homogeneous are synonymous
fickle flume
ah ok
marsh valley
In this context
fickle flume
and then the complementary has a general solution
i look at that general solution
it will be two terms for this order of ODE
and my guess was Ce^x
initially
i havent solved the homogenous yet but
if i did
one of the terms would be Ce^rx?
but r_1 or r_2
but that doesnt look the same as Ce^x
so i wouldnt multiply by x
oh wait but i do find r
does that imply one of the r's are 1?
fickle flume
I'll look into more geometrical/intuitive explanations, but from what I remember...
you said it here
my bad
okay that makes sense then
one of the terms is Ce^x
for the general homogenous solution
and that tells me I should multiply by x
wild, thank you for helping me, I don't understand why it works but that helps me to guess correctly in the future
.close
calm coralBOT
calm coralBOT
Available help channel!
blazing coyote
Let $\varphi: G \to G'$ be a surjective homomorphism. Prove that (a) if $G$ is cyclic, then $G'$ is cyclic, and (b) if $G$ is abelian , then $G'$ is abelian
\
Proof
(a) Let $a$ be a generator of $G$. Let $a^n$ be an arbitrary element of $G$, for some $n \in \N$.Then $\varphi(a^n) = (\varphi(a))^n$. Thus the image of $G'$ forms a cyclic subgroup. As It's a subjective homomorphism, every element in $G'$ has a pre-image. Thus $G'$ is cyclic
\~\
(b) As $G$ is abelian $(ab)=(ba)$. We then have $\varphi(ab) = \varphi(a)\varphi(b) = \varphi(ba) = \varphi(b)\varphi(a) \implies \varphi(a)\varphi(b)=\varphi(b)\varphi(a)$. Thus the image of $\varphi$ is abelian Moreover, as $\varphi$ is surjective, every image has a pre-image. Thus $G'$ is abelian.
potent lotusBOT
What a wonderful world !
blazing coyote
Forgot to mention, for arbitrary a,b \in G
civic dirge
Masterfully done
calm coralBOT
blazing coyote
.reopen
calm coralBOT
✅
blazing coyote
yea?
civic dirge
I was going to suggest using g' as an arbitrary element in G', and showing there's some element that cycles to it. But on second thought I'm not sure that's cleaner. So I decided not to say anything.
Your proof is good and displays that you understand the concept
blazing coyote
I was going to suggest using g' as an arbitrary element in G', and showing there...
Hmm, that makes sense too
thanks
.close
remote mural
calm coralBOT
remote mural
Heres my answer for number 9 and the answer provided
The difference is it substituted x=sqrt(t) before taking the derivative, i dont think it works that way , am i or the book wrong
white urchin
if you sub x = sqrt(t) before taking the derivative
you will have to use the chain rule when you do take the derivative
for example
in the function $f(x) = x^2 - 5x$, where we wish to take the derivative $\frac{df(\sqrt{t})}{dt}$
potent lotusBOT
_Kookie
white urchin
you want to perform the chain rule, which will look like as follows
$$\frac{df(\sqrt{t})}{dt} = \frac{df(x)}{dx}\cdot \frac{dx}{dt}$$
white urchin
But i didnt use the t before taking the derivative
yes, you didn't
but
you computed $$\frac{df(x)}{dx}$$, then you just subbed in each $x$ with $\sqrt{t}$
potent lotusBOT
_Kookie
white urchin
which is not wrong
but you are forgetting to multiply the entire result with $\frac{dx}{dt}$
potent lotusBOT
_Kookie
remote mural
Isnt x= wtv but wtv would be a constant
white urchin
what is wtv?
potent lotusBOT
_Kookie
white urchin
just made a correction to one of the formulas I put in here
white urchin
nice
remote mural
😀so whenever determining dy/dt, while y is in terms of x, i can jus directly convert x to t in the original expression
white urchin
yes you can
or if converting x to t in the original expression makes it too messy
go ahead and use the chain rule
and introduce t later
remote mural
I understand now. Before this conversation i was seeing t as a irrelevant constant so i thought id get the f(sqrt(t)) value as a constant as well if i used t in the first step
Another question, do i have to memorize derivatives of trigs...
I know dsinx/dx is cosx, and dcosx/dx is -sinx, but wat abt the other 4
calm coralBOT
@remote mural Has your question been resolved?
hearty tusk
how do I solve this? I don't remember this
calm coralBOT
how do I solve this? I don't remember this
unkempt drift
how do I solve this? I don't remember this
the POI (y'' = 0) and y = 0 are two derivatives apart
but then f(x) is the derivative of h(x), so only one derivative apart
what's going on with h(x) is that h(x) keeps on decreasing (negative area is being added for x in [0, 6) )
hearty tusk
f is like h' right
unkempt drift
but then after x = 6, positive area is now being added
hearty tusk
yea
unkempt drift
but then after x = 6, positive area is now being added
so now h(x) is increasing, but the value of h(x) is still going to be negative
hearty tusk
if f graph is h' then h have rel min at x=6 so its the smallest and then we got f=h' and for h''=f' the tangent line is prolly above f graph? so h'' > h' > h?
is this correct reasoning?
unkempt drift
if f graph is h' then h have rel min at x=6 so its the smallest and then we got ...
yeah so h''(6) is f'(6) that's correct
and f'(6) is positive
hearty tusk
wait i have to consider the area below that right
unkempt drift
if f graph is h' then h have rel min at x=6 so its the smallest and then we got ...
so it's h < h' < h'' yes
hearty tusk
i dont need to consider the area cuz if i the 0 is gone its js x right
unkempt drift
i dont need to consider the area cuz if i the 0 is gone its js x right
what do you mean?
hearty tusk
h(6) = F(6) - F(0)
unkempt drift
ah
hearty tusk
am i ignoring this
wait the area is negative
so it doesnt matter
i get it
thanks 🙂
.close
calm coralBOT
unkempt drift
h(6) = F(6) - F(0)
yeah just to clarify, you don't know what F(0) is nor what F(6) is exactly
but you do know for sure that F(6) is less than F(0), so F(6) - F(0) < 0
cause of the area yep
calm coralBOT
Available help channel!
night lion
calm coralBOT
night lion
In my lecture teacher gave 2 min to try out this question. Look at red part and box part in my notes. The particular bx term in quad eqn is different. I made the same mistake 1 time before this. What could be the reason I am repeating this mistake
unkempt drift
In my lecture teacher gave 2 min to try out this question. Look at red part and ...
they're the exact same thing
night lion
Teacher factorized x(3y +3) but I 3x(y+1)
unkempt drift
x(3y + 3) = x * 3(y + 1) = 3x(y + 1)
night lion
But answer would be different then?
unkempt drift
4y - 4 = 4(y - 1)
unkempt drift
ah wait I know what happened
unkempt drift
because you set x to be the independent variable and y to be the constant, your quadratic for the discriminant is in terms of y, right
wait
no they're both correct sorry
night lion
Would the answer be same,?
night lion
because you set x to be the independent variable and y to be the constant, your ...
Yes he taught to do this
No
See the blue box in my copy
I factorized 3x(y+1)
And he did x(3y+1)
unkempt drift
ahhhhhhhhh your coefficients are wrong
night lion
They are same but
unkempt drift
it's algebraically equivalent yes
but you must choose b = 3(y + 1), a = (y - 1), and c = 4(y - 1)
tidal oriole
who can solve x(x-1) = 0 solutions, help me.
unkempt drift
who can solve x(x-1) = 0 solutions, help me.
!help
calm coralBOT
who can solve x(x-1) = 0 solutions, help me.
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
unkempt drift
but you must choose b = 3(y + 1), a = (y - 1), and c = 4(y - 1)
because you need to match ax^2 + bx + c
night lion
Oh
unkempt drift
so x^2 (y - 1) + x * 3(y + 1) + 4(y - 1)
night lion
Do I have to keep ax²+bx+c back in mind?
night lion
I couldn't notice this thing yet
unkempt drift
that's okay! we're here to help
remote mural
yes
calm coralBOT
@night lion Has your question been resolved?
calm coralBOT
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).
frozen rampart
no spam pls this isn't your help channel
frozen rampart
lmao fr
calm coralBOT
calm coralBOT
Available help channel!
sand raft
How does cos inverse of under root 1 - x sq become sin inverse x
devout burrow
How does cos inverse of under root 1 - x sq become sin inverse x
Use the trigonometric identity of sin^2(x) + cos^2(x) = 1
tall moon
How does cos inverse of under root 1 - x sq become sin inverse x
draw a triangle
sand raft
Use the trigonometric identity of sin^2(x) + cos^2(x) = 1
Yeah cos x = root of 1 - sin square x
But that will reduce it to cos inverse of cos x, making it x
sand raft
draw a triangle
Okk I’ll try that
devout burrow
Yeah cos x = root of 1 - sin square x
But that will reduce it to cos inverse of ...
What would cos(x) be?
devout burrow
Yeah cos x = root of 1 - sin square x
But that will reduce it to cos inverse of ...
That's cos^2(x), not cos(x)
sand raft
draw a triangle
I didn’t understand how to draw a triangle from it ;-;
devout burrow
Lol no worries
sand raft
It becomes sin inverse x once I put x as sin theta yea
devout burrow
Everyone makes mistakes
sand raft
Tysm
devout burrow
All done?
calm coralBOT
calm coralBOT
Available help channel!
vernal oxide
vernal oxide
my attempt
wheat pike
yup ✅
calm coralBOT
@vernal oxide Has your question been resolved?
remote mural
can anyone help me with this question? i dont want you to solve the question for me, i just want some guidance, i cant think of anything in order to solve this question
flat orchid
can anyone help me with this question? i dont want you to solve the question for...
Where are you stuck
remote mural
i want to solve it using the subtraction principle but i dont know the statement
flat orchid
Well I have no idea what subtraction principle is
For the very first question abt number of committees, i would suggest taking cases
unreal isle
i want to solve it using the subtraction principle but i dont know the statement
youll have to do part 1 normally, and then for part 2 you can do total - female majority for male majority
do keep an eye out for a case where neither is in majority
by normally i mean chalking out the cases