this isn't topology
#help-39
rare scaffold
fine, here's a different approach that's actually the same
determine all functions h that do not satisfy g R h
rare scaffold
@stoic imp
stoic imp
determine all functions h that do not satisfy g R h
how do I count
rare scaffold
how many functions {...6} -> {...16} are there? ignore R
stoic imp
16 P 6
rare scaffold
wait, no
what about 1,2,3,4,5,6 all map to 12? that's a function you didn't count
the outputs don't have to be unique
stoic imp
P is permutation
rare scaffold
n P k is n!/(n-k)!
stoic imp
16^6
rare scaffold
is that a different answer?
stoic imp
16 possibilities for the first out of 6 spots
16 possibilities for the second out of 6 spots
rare scaffold
it's harder to follow if you're spontaneously changing arguments. is 16^6 different from 16 P 6?
rare scaffold
ok
now that I'm caught up, yes your new answer makes sense
the key is that you have 16 choices for each of the 6 entries
now count the number of those functions that are not related to g
rare scaffold
it's harder to do the opposite, i.e. count the number of functions that are related to g
stoic imp
so, directly
6 x 16^5
rare scaffold
no, but explain?
stoic imp
we just need them to share at least one element in their images
stoic imp
it's harder to do the opposite, i.e. count the number of functions that are rela...
f R g iff im f intersects with im g
stoic imp
a R b iff im a n im b ≠ {}
rare scaffold
now count the number of those functions that are not related to g
my mistake for disliking the first version of that lol. I did ask for the number of functions not related to g, though
stoic imp
no, but explain?
10 choices for first spot 10 choices second spot etc
the image of g is {15,14,13,12,11,10}
1,2,3,4,5,6,7,8,9,16
rare scaffold
you see how every function h that is not related to g is precisely a function {1,2,3,4,5,6} -> {1,2,3,4,5,6,7,8,9,16}?
stoic imp
yes, so?
rare scaffold
this was the intended answer to my first question, X = {1,...,9,16}
stoic imp
10^6?
rare scaffold
it's not topology
rare scaffold
only if you think so
you should try to enjoy finding multiple solutions to the same problem, it's good for you
rare scaffold
if I thought it wasn't appropriate for a newbie, I wouldn't have mentioned it
a function h that is not related to g is precisely a function {1,...,6} -> {1,...,9,16}, and the latter space is easier to work with
amongst these functions, how many are related to f?
stoic imp
amongst these functions, how many are related to f?
eh
the image of f is {2,4,6,8,10,12}
2,4,6,8
4^6
stoic imp
this does not count the function h(1) = 1, h(else) = 2
4 x 10^5
rare scaffold
this overcounts
stoic imp
I need more hints
rare scaffold
what class is this for?
stoic imp
what am I overcounting
stoic imp
what class is this for?
it's first year gen eds
rare scaffold
suppose 17 people run a race, including a guy named Kevin. how many top-five placements are there, featuring Kevin?
have you seen a counting problem like this before?
in case it's unclear, I want the number of top-5 placements in which Kevin is present
a,b,Kevin,c,d
stoic imp
( 17 P 1 ) x 16 x 15 x 14 x 13
rare scaffold
17P1 is just 17, and you deleted that message last time
have you or have you not seen a question like this before?
<~> do you know this counting trick?
stoic imp
I guess i have, is just that counting is hard
stoic imp
<~> do you know this counting trick?
no
rare scaffold
counting is hard, genuinely
ok
so I actually alluded to this before, sometimes it's easier to count the opposite thing
every top-5 configuration falls into one of two sets: that which sees K, and that which does not
so to count all of the placements with K, we count every placement, then subtract the ones without K
that's (17 P 5) - (16 P 5)
stoic imp
so to count all of the placements with K, we count *every* placement, then subtr...
nice
rare scaffold
a similar idea applies to the question of how many functions are related to f
rare scaffold
no, only because remember we want the total number of functions related to f that are not related to g
stoic imp
the image of g is {15,14,13,12,11,10}
rare scaffold
every function that does this is obviously not related to g, so we restrict our search to functions {1,...,6} -> {1,...,9,16} that are related to f
stoic imp
every function that does this is obviously not related to g, so we restrict our ...
how to use the complement principle
rare scaffold
mmmm so you did know the trick after all
stoic imp
I am bad at counting
rare scaffold
unless if you just came up with that name
stoic imp
mmmm so you did know the trick after all
it was taught in class but I'm bad
rare scaffold
it really helps to see other examples of counting problems you've done in class or on other assignments, especially those you've actually solved yourself
stoic imp
help
rare scaffold
it was taught in class but I'm bad
you get better by doing it
I have a question
when do you normally ask for help on a math assignment?
how much time do you allow yourself to work things out on your own?
rare scaffold
I sympathize with this, especially for combinatorics, but that wasn't my question
rare scaffold
ok, sure, just checking
rare scaffold
I thought you might tend to ask questions the moment you're uncertain about how to proceed, i.e. without trying random stuff first, that's all
rare scaffold
sorry for not being a combinatorics pro
you're not supposed to be lol
stoic imp
it might seem easy to you, but sometimes things go south in the middle of the exercise
rare scaffold
no I get it, and no it's not easy for nearly anyone when they first start
stoic imp
using this compliment principle should be your last resource, in class teacher said it's useful when what you are trying to count is very hard
rare scaffold
that's weird to say. counting complements can be really useful
it shouldn't be treated as a last resort lol
actually it's good because you immediately consider the complement and pick which one seems easier at the time
stoic imp
she didn't said it should be the last resource, that's on me, but she did said the last sentence
rare scaffold
yes
so to recap
every function h : {1,...,6} -> {1,...,16} that is related to f and not g is, first, a function that is not related to g
so we can count amongst the collection of functions that are not related to g, how many are related to f
since a function not related to g is just a function -> {1,...,9,16}, we want the number of functions {1,...,6} -> {1,...,9,16} that are related to f
rare scaffold
she didn't said it should be the last resource, that's on me, but she did said...
does this make intuitive sense?
do you see how you'd get to this point on your own?
pearl pondBOT
@stoic imp Has your question been resolved?
stoic imp
since a function not related to g is just a function -> {1,...,9,16}, we want th...
how to count those
rare scaffold
how to count those
did you try it on your own first?
stoic imp
did you try it on your own first?
first output out of the six outputs must be either 2 4 6 8
rare scaffold
@stoic imp
rare scaffold
@stoic imp
rare scaffold
I'm literally still here
stoic imp
help
stoic imp
10^6 - 10 P 4
stoic imp
are you changing your answer again?
well since you didnt confirmed it was correct I started self doubting myself
rare scaffold
well I was trying to think of a reason why it's wrong
and I think it has something to do with your designation of a "first" input that's required to map to one of the four options
stoic imp
how to correct it
rare scaffold
explain your other answer
stoic imp
explain your other answer
which one
stoic imp
the only other one that came after your first
i tried using compliment princepol
@rare scaffold
rare scaffold
i tried using compliment princepol
you don't need to ping after a reply. replies ping.
explain how you used the complement principle
@stoic imp second ping
stoic imp
you don't need to ping after a reply. replies ping.
explain how you used the co...
welll
10 possibilities for each output
since a function not related to g is just a function -> {1,...,9,16}, we want the number of functions {1,...,6} -> {1,...,9,16} that are related to f
from 1 to 16 theres 10 numbers
10 possibilities for each of the 1 to 6 inputs
stoic imp
then we subtract the ways that out of the 10 possible outputs we pick one of the 4
rare scaffold
what is the quantity you're subtracting?
stoic imp
10 p 4
rare scaffold
what does it represent?
stoic imp
2 4 6 8
rare scaffold
these are the number of functions that do what, exactly?
stoic imp
the image of f is {2,4,6,8,10,12}
stoic imp
these are the number of functions that do what, exactly?
when the images have non empty intersections
aka when they are related
rare scaffold
that's not what you want
if X is a finite set and A is a subset of X, we have |A| = |X| - |X-A|
stoic imp
i want when they have empty intersecgion?
rare scaffold
in this case, X is the collection of functions {...6} -> {...9, 16}, and A is the collection of functions that are essentially related to f
characterize a function in X-A
@stoic imp still here
stoic imp
characterize a function in X-A
say what
stoic imp
don't just say "say what", tell me what you're confused about
what do you mean by
"characterize f x a"
rare scaffold
what do you mean by
a function in X-A, with X and A defined as above, is just a function that is not related to f
the whole point is that #{functions related to f} = #{all functions} - #{functions not related to f}
@stoic imp do you really just lock your phone every time a message is sent to you or something lol
pearl pondBOT
@stoic imp Has your question been resolved?
pearl pondBOT
@stoic imp Has your question been resolved?
stoic imp
a function in X-A, with X and A defined as above, is just a function that is not...
why do you abstract everything with this notation, "X", "A"
stoic imp
<@1187916384341078070> do you really just lock your phone every time a message i...
i had to help my parents with something
I am back and reloaded
stoic imp
a function in X-A, with X and A defined as above, is just a function that is not...
the idea is that you should try to help me because i am new to math, but you using this " a function in X-A" without defining X and A makes things a little bit confusing ngl, im new with that notation
stoic imp
the whole point is that #\{functions related to f} = #\{all functions} - #\{func...
yes, this is the what so called, complements principle of counting, how do I count the total number of functions ?
stoic imp
<@1187916384341078070> do you really just lock your phone every time a message i...
sorry, I had to help my parents with something, is just that this counting exercises are hard
stoic imp
.close
pearl pondBOT
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Available help channel!
exotic gale
In base $b$, we have $r = 0.\overline{57}_b$ and $3r = 1.\overline{06}_b.$ What is the value of $r$ in base ten? Express your answer as a common fraction.
jolly parrotBOT
7
exotic gale
i think we divde 3r by r or do we subtract 3r by r to get 2r and then do something with that?
1.06 repeating over 3 equals 0.57 repeating
base b
slender viper
Multiple approaches here
- Write r in terms of b
- Trying to do arithmetic in base b (pretty much equivalent to approach 1, but possibly requires more experience to do properly)
exotic gale
Multiple approaches here
1. Write r in terms of b
2. Trying to do arithmetic in ...
how would i write r in terms of b
doesn't the question give that already?
slender viper
how would i write r in terms of b
r = 5/b + 7/b^2 + ...
Think of something like this, but try simplifying it
exotic gale
Think of something like this, but try simplifying it
like 5/b+7/b^2 is something with denom b^3?
slender viper
like 5/b+7/b^2 is something with denom b^3?
erm... you could put it on a denominator b^3, but b^2 is enough
Also there's extra terms to deal with on account of r having a repeating base-b expansion
pearl pondBOT
@exotic gale Has your question been resolved?
grand sun
AEDC is a square with side 17. AB=8, BC=15. Find FD/AG. Can someone give me a hint on this problem
glossy blade
Gimme a min
ancient sapphire
Use your trig functions
pearl pondBOT
Gimme a min
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
iron basin
?????
glossy blade
Wattahel?
grand sun
is there a solution without trig
ancient sapphire
I don't think so
glossy blade
Similar triangles
grand sun
i did it with similar triangles but the solution was too long
glossy blade
Oh ok
grand sun
in which triangles can we use trig
glossy blade
Drop perpendiculars and apply
ancient sapphire
If you know 1 side of a triangle and an angle you can use trig functions to figure out your missing pieces
grand sun
i think if we draw a diagonal AD then we can try to prove that the 2 triangles that are formed are congruent
iron basin
similar
grand sun
yes, but i got answer 1 after manually calculating the side FD
so they are congruent
but we gotta prove it
.close
pearl pondBOT
pearl pondBOT
Available help channel!
crimson nebula
Can somebody explain the proof please
pearl pondBOT
Can somebody explain the proof please
crimson nebula
Be precise
glass meadow
I wouldn't say that's a proof, it's a definition
crimson nebula
I doubt too .
But ..
neon fossil
It says that its assumed that xn doesn't converge to x and so we are basically just using the definition of convergence to prove the second sentence...?
glass meadow
A sequence (x_n) converges to x if, for all epsilon > 0, there exists a natural N such that for all n >= N, |x_n-x| < epsilon
That's the definition of "sequence X converges to x"
The negation is that there exists an epsilon > 0 such that for any natural N, there exists n >= N such that |x_n-x| >= epsilon
(if you've done a bit of logic it should be clear how that's the negation)
crimson nebula
what part doesn't make sense?
The second line ,
"That is , for each k.."
Does it mean , that for every natural number , we can't guarantee that this inequality holds.while It may hold for some natural numbers ...
glass meadow
Yes
Well, not quite, it's guaranteed it does not hold for at least one number
|x_n-x| >= epsilon is true for at least one n >= k
crimson nebula
Ok thanks
crimson nebula
|x_n-x| >= epsilon is true for at least one n >= k
Isn't it the same as my statement
glass meadow
No, your statement is like "we don't know"
The correct statement is "that sequence definitely does not converge (to x)"
glass meadow
"such that |xn-x| ≥ ε holds", yes
crimson nebula
Okkhhayy
glass meadow
We say "for all k in N, there exists n >= k (such that) ..."
,, \forall k \in N (\exists n \geq k ...)
jolly parrotBOT
Nel
crimson nebula
I dont know , for past few days , it takes me a lot of time , to figure out statments
I was stuck on this "proof" for 2 days
Trying to make sense and logical flow
glass meadow
Maybe find a course on first-order logic
crimson nebula
It seems like , real Analysis is like a task for me now ,not the enjoyment of math
crimson nebula
Maybe find a course on first-order logic
Whats that
glass meadow
Basically logic with quantifiers ("for all" and "there exists")
crimson nebula
Do you have some recommendations
glass meadow
No, sorry, but there is #book-recommendations
glass meadow
The convergence statement written symbolically looks like this:
,, \forall \varepsilon > 0 \left(\exists N \in \mathbb{N} \left(\forall n \in \mathbb{N} \left(n \geq N \implies |x_n - x| < \varepsilon \right)\right)\right)
crimson nebula
Ah yes the definition of converges
jolly parrotBOT
Nel
glass meadow
That's what first-order logic looks like, and it makes negating statements very easy without having to think about what every word means
Of course, words are still useful, nobody writes everything in logic symbols, but for precise semantics it can help
crimson nebula
Of course, words are still useful, nobody writes everything in logic symbols, bu...
I found a course !
29 lectures , each 1.5 hour
Automata is there
Too
glass meadow
You don't need that much to better understand general mathematical statements
crimson nebula
My primary aim is real Analysis
Only want to do what's relevant but in detail
Can you suggest which lectures should I watch
glass meadow
Just a few results from searching "introduction to first order logic":
https://builds.openlogicproject.org/content/first-order-logic/introduction/introduction.pdf
http://intrologic.stanford.edu/lectures/lecture_16.pdf
https://people.math.ethz.ch/~halorenz/4students/LogikML/Nutshell.pdf
Again I'm not the one to ask for recommendations but feel free to drop a message in #book-recommendations , even if you're looking for video content
pearl pondBOT
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Available help channel!
eager jewel
Let Γ denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to Γ at a point P intersect the y-axis at YP. If PYP has length 1 for each point P on Γ, then which of the
following options is/are correct ?
eager jewel
eager jewel
i wrote the tangent as y=x(dy/dx) + Yp
and i dont know how to use that length = 1 condition
i thought ill take a point (x,y) on the curve and do x^2+(y-yp)^2 = 1
oh bruh
i can use y-yp = x(dy/dx) 😭
ok but then i get the condition
dy/dx = +- sqrt((1-x^2)/x^2)
how to choose between + or -
or can we take both?
calm wing
Gamma is in the first quadrant
calm wing
well consider what happens as you approach (1,0)
you can't approach it from below smoothly (gamma is in the first quadrant) so you have to approach it from above
feral olive
if the tangent line is always 1 you can use the distance formula for any point on the function and let that be true for all x
eager jewel
you can't approach it from below smoothly (gamma is in the first quadrant) so yo...
uh i dont understand what this means..one sec let me send a diagram of the situation im imagining
feral olive
you are able to find f'(x) using that
eager jewel
i thought ill take a point (x,y) on the curve and do x^2+(y-yp)^2 = 1
👍 yes i got that thank u tho
eager jewel
you can't approach it from below smoothly (gamma is in the first quadrant) so yo...
these r the 2 scenarios im thinking of
calm wing
huh? neither contains the point (1,0) tho
calm wing
hmm
calm wing
note that sqrt(1-x^2) is undefined for x > 1
so i assume we shouldn't consider points for x > 1
calm wing
well now that we are restricted to [0,1] basically, no, it can't
eager jewel
feral olive
use your formula for the tangent line and combine it with the distance formula
clear shore
why this problem looks kinda like the one in my book lmao
eager jewel
use your formula for the tangent line and combine it with the distance formula
already done bruh now im trying to figure out the sign of the slope
grave mirage
Slope is not defined for x>1 right?
feral olive
already done bruh now im trying to figure out the sign of the slope
you should already have the answer tho
i got one of the options
clear shore
i got one of the options
what did you get
eager jewel
Slope is not defined for x>1 right?
ye just remove the parts for x>1
feral olive
what did you get
B
eager jewel
you should already have the answer tho
im not able to figure out the sign of the slope
clear shore
Yeah same
eager jewel
B is wrong
clear shore
really?
feral olive
how though
clear shore
istg it's B
eager jewel
@calm wing whered u go 😭
feral olive
$f(x)-y_0=f'(x)x$
jolly parrotBOT
ImOakley
grave mirage
It's prolly D and whatever's the solution for D
eager jewel
It's prolly D and whatever's the solution for D
what do u mean "prolly"?
feral olive
$1=\sqrt{(f(x)-y_0)^2+x^2}$
jolly parrotBOT
ImOakley
grave mirage
what do u mean "prolly"?
Probably
feral olive
$1=\sqrt{(f(x)-f(x)+f'(x)x)^2+x^2}$
jolly parrotBOT
ImOakley
grave mirage
no bro like based on what can u say that
I can see a decreasing function acting like that
feral olive
$1=\sqrt{(f'(x)x)^2+x^2}$
jolly parrotBOT
ImOakley
grave mirage
dy/dx = +- sqrt((1-x^2)/x^2)
Right got it
eager jewel
I can see a decreasing function acting like that
ok well i cant, and why is it a decreasing function?
eager jewel
uhh why
eager jewel
yeah
grave mirage
x is always positive as it is in the first quadrant
feral olive
hang on
feral olive
@eager jewel is this select more than 1 answer
grave mirage
It can't be positive
Because x=1 corresponds to y=0
feral olive
thats why b wasnt right
grave mirage
Because x=1 corresponds to y=0
So it has to be the other option
feral olive
because as apoorv said its either only increasing or only decreasing
grave mirage
Which is the negative sign
grave mirage
Welcome
eager jewel
.close
pearl pondBOT
eager jewel
ty @feral olive and @calm wing
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Available help channel!
crimson nebula
Suppose I want to prove that this sequence is divergent.
Am i allowed to use such language .
"It can easily be seen that..."
plush bramble
sure if it follows from theorem 3.4.5
pearl pondBOT
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sharp smelt
pearl pondBOT
sharp smelt
Let $A$ be generated by $(a_1,a_2 \dots , a_n)$, each being distinct and no power of any $a_i$ is any power of $a_j$.
This is as if it were then the group would be generated by a subset of $(a_1,\dots,a_n)$
Then $A={a_1^{k_1} \dots a^{k_n} \mid k_i \in \mathbb{Z} , 1≤i≤n }$. Let $H≤A$. Then every element of $H$ must be of the form $a_i^{h_i} \dots a_k^{h_l}$. This being a representative element of $H$, $h_j$ are fixed . Suppose an element that's not of this form were in $H$, call it $t$. Then $a_1^{b_1} \dots a_{I}^{b_{i}} \dots a_{l}^{b_l} \dots a_n^{b_n}=t$.
We now multiply across by $a_i^{-h_i} \dots a_{l}^{-h_l}$.
Then $a_1^{b_1} \dots a_{i}^{-h_i+b_{i}} a_{l}^{-b_{l}+h_l} \dots a_{n}^{b_n} \in H$.
Then $a_1^{b_1} \dots a_n^{b_n} \in H$.
But this is a contradiction to our assumption that $H$ is of the form $a_i^{h_i} \dots a_l^{h_l}$ for some fixed $i$ to $l$. |
we're done.
I feel there's something wrong with this, but I can't quite put my finger on it.
jolly parrotBOT
wai
sharp smelt
I see the error now
Is this salvagable though
<@&268886789983436800>
I did think of induction, and my book verifies that one way to prove this is indeed induction
brisk scarab
Does n-generated just mean "can be generated by a set of size n"
sharp smelt
1≤i≤n
sharp smelt
I did think of induction, and my book verifies that one way to prove this is ind...
But I was thinking of avoiding induction
brisk scarab
how have you gotten existence of such a_i
like
even in Z if we have any two generators a and b (e.g. 2 and 3) then a*b is a "power" (where a*b is additive notation) of both
unless I'm being slow
jolly parrotBOT
wai
brisk scarab
yes, but in multiplicative notation (for the group Z with addition, which is confusing and alas nothing can be done about it) then $a_j^{a_i}=a_i^{a_j}$
jolly parrotBOT
Edward II
brisk scarab
in the usual notation for integers, both are equal to $a_i\times a_j$
jolly parrotBOT
Edward II
sharp smelt
then the group can be generated by 1 element , right
brisk scarab
yeah sure; are you also taking n minimal then
I suppose you can with no issues but then the n in your proof may not be the same n as in the question technically I suppose you can with no issues but then the n in your proof may not be the same n as in the question technically (i.e. where you're being asked to prove the generalisation of "a subgroup of the 2-generated group Z is also 2-generated", and your proof would say "well actually Z is 1-generated, so it's subgroups are 1-generated, which means they're 2-generated"
brisk scarab
what you sent has nothing to do with being n-generated though
sharp smelt
oh , right I just need to show if $A$ has a generating set, then so does any of its subgroups
right
jolly parrotBOT
wai
rustic gate
what the blud
sharp smelt
?
rustic gate
they probably don't mean for X to be a set then 
sharp smelt
they probably don't mean for X to be a set then <:monkey:651816857153699840>
what 😭
rustic gate
they probably mean a multiset
sharp smelt
I see, that just makes stuff so much worse
rustic gate
no it makes it better
rustic gate
because now n-generation just means surjection Z^n -> A
sharp smelt
I dont' quite flllow
rustic gate
well if you have a generating set g1, ..., gn allowing duplicates, then you get a map Z^n -> A which takes (0, ..., 1, ... 0) -> gi
where the 1 is in the ith spot
the map is surjective because the set generates
sharp smelt
I see
that makes sense
Okay, I think I'll just come back to this tomorrow 🥀 , not clicking
pearl pondBOT
@sharp smelt Has your question been resolved?
pearl pondBOT
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Available help channel!
cedar depot
pearl pondBOT
cedar depot
i found that f(1) f(2) make a series as 1 , 2 , 4 , 7 and so on
i tried to find f100 by this but its getting complicated and i wasnt able to find the summation of that series
steep saddle
looks familiar 
what if you subtract 1 from the series
is this series a well known sequence
well i mean that as in
it may have a formula already
viscid shale
If you consider how the function is constructed its pretty obvious what the series is
steep saddle
aside from the 0 it's the triangle numbers
cedar depot
>If you consider how the function is constructed its pretty obvious what the ser...
i didnt get what u mean
steep saddle
i didnt get what u mean
look up triangle numbers
cedar depot
aside from the 0 it's the triangle numbers
triangle numbers? what
midnight haven
try writing f(x) + f(x+1) .. f(100)
viscid shale
Lets assume we want to know the value of f(5) without prior knowledge
steep saddle
cedar depot
try writing f(x) + f(x+1) .. f(100)
thats wild. 1 step ans
viscid shale
```
f(5) = 5 + f(4)
>f(4) = 4 + f(3)
f(5) = 5 + 4 + f(3)
>f(3) = 3 + f(2)
f(5) = 5 + 4 + 3 + f(2)
>f(2) = 2 + f(1) = 2 + 1
f(5) = 5 + 4 + 3 + 2 + 1
mb, im used to the tex markdown, lmao
midnight haven
```
f(5) = 5 + f(4)
>f(4) = 4 + f(3)
f(5) = 5 + 4 + f(3)
>f(3) = 3 + f...
yeah basically that is the outcome
<@&268886789983436800>
viscid shale
<@&268886789983436800>
midnight haven
sniped
viscid shale
thanks
cedar depot
so i got it
viscid shale
If you want to know the value of f(X), then it will just be X + (X-1) + (X-2) ... 2 + 1
cedar depot
If you want to know the value of f(X), then it will just be `X + (X-1) + (X-2) ...
oh
viscid shale
Which is a pretty well known series, as already has been stated
cedar depot
reals
cedar depot
```
f(5) = 5 + f(4)
>f(4) = 4 + f(3)
f(5) = 5 + 4 + f(3)
>f(3) = 3 + f...
ohhh even this series is simple
its a bit hard to notice the patterns like this
can yall help me or give tips on the patterns
viscid shale
Some of them you can actually construct on your own
Others just come from practice/prior knowledge
steep saddle
can yall help me or give tips on the patterns
Experience
cedar depot
Some of them you can actually construct on your own
like this one?
i had the idea of writing x = f(x) - f(x-1)
and adding all of it
this gets the ans but there seem to be more ways to do this
viscid shale
That requires you to know f(x) and f(x-1), so you probably cant do much about it as is.
cedar depot
That requires you to know f(x) and f(x-1), so you probably cant do much about it...
i didnt rlly get what u mean
we can put x from 0 to 100
and we have a way to find f(-1) too by the relation thats how i got it like rn
does this not work?
viscid shale
Like, yes, you can do so, but that would take significantly more time
viscid shale
Precisely, if you want to add all the values from 0 to 100
You first have to compute them individually
cedar depot
f (x) - f(x-1) summation till 100 will cancel all the terms ecet f100 - f-1
viscid shale
You still dont know what f(100) is in that case
cedar depot
You still dont know what f(100) is in that case
so in lhs
we have 1 + 2 + 3 ... 100
this is n(n+!)/2
and in rhs we will have f(100) - f(-1)
if we put x as 0 in the eq we get f(-1) is -1
sorry its 1 mb
viscid shale
ah, i see your idea
cedar depot
so 5050 +1
viscid shale
Yea, ig you can do that
cedar depot
nice
viscid shale
Really elaborated compared to just replacing a few values and noticing its 1+2+3...(n-1)+n, but it works
cedar depot
Really elaborated compared to just replacing a few values and noticing its 1+2+3...
yea lol
tysm for the help
weary ferry
I can’t find the channel where I needed the help, as it put me in the discussion one after it closed the channel
pearl pondBOT
@cedar depot Has your question been resolved?
jovial harbor
pearl pondBOT
mortal seal
u=sqrt(x-1)
steep saddle
How to do with substitution
u = x - 1
mortal seal
Yeah probably works as well
mortal seal
Yup
mortal seal
!done
pearl pondBOT
If you are done with this channel, please mark your problem as solved by typing .close
jovial harbor
nah
I got more
just grabbing more paper
can we do 8?
ah well I gtg
so Imma close
.close
pearl pondBOT
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random flower
Hi there, i am working on this question "The points (–2, 4, 5), (4, 5, –3), and (1, –1, 6) are three of four vertices of parallelogram ABCD. Explain why there are three possibilities for the location of the fourth vertex and find the three points. " I wrote that 3 possible points exist, as there could be a parallelogram in each plane
random flower
however, i am not sure if this is correct, as when i graph the points in each plane in desmos, the XZ plane does not look like a proper parallelogram
I am having similar trouble with a question giving the 3 points of a triangle and asking for it's shape
<@&286206848099549185>
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@random flower Has your question been resolved?
random flower
.close
pearl pondBOT
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pine jacinth
Heres an equation that repreents the amount of bacterias in function of the time past in minutes : N(t) = 10(2)^(t/40). Express by an equation the time in function of the amount of bacterias.
heavy onyx
find the inverse function
pearl pondBOT
bleak sonnet
HEY EVERYONEEE!!!!!!
MY NAME IS HEAVEN
IM SO GLAD TO BE HERE
I HAVE A SERVER TOO!!!
Its for those who wanna make music but dont kno where to start
heavy onyx
HEY EVERYONEEE!!!!!!
!redir
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This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
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hazy escarp
Hey
cinder flower
fixed vector?
hazy escarp
Let’s say I have an unknown which is point m and I wanna get rid of it in these vectors
sharp vigil
can you show the context in which you saw that phrase?
hazy escarp
Yes
hazy escarp
can you show the context in which you saw that phrase?
Uh I don’t have a problem with me
But he mentioned it in class and my exam is soon
It’s like in 2 hours
So let’s say point M is any point
And I wanna remove it from
Let’s say I have
Vectors MB+MA+MC
I wanna remove m
To make it a fix vector
What do I do
pearl pondBOT
@hazy escarp Has your question been resolved?
hazy escarp
Okay I have another question about function
When I see a function and I solve it to find the image (y)
Let’s say I see the sign >< (alligator)
Do I solve it and take everything to the left making it 0 and solve it
Or do I just solve for x
And do I do a sign table or an interval
Or not and how do I know
.close
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cedar depot
pearl pondBOT
cedar depot
so first i take x = y = 1
from this we get a quadratic in f(1)
for which the values of f(1) i get as 3 and -2
now i put y = 1/x
so f(x).f(1/x) = f(1) +2(1/x + x +1)
but at this point im stuck
i tried doing x = 1/x
cedar depot
try x=x and y=1
aight
oh what
i got it
but im getting 2 diff functions for the diff values of f(1)
clear shore
yeah I think there is a way to reject one of them
I haven't figured it out tho
Gimme a min
cedar depot
they took both values and both of them are the ans
its an example problem, in ques they prolly say f(1) >0 or sum
for a single value
tysm for the help i got it
clear shore
I guess?
cedar depot
u gor it in a single step tho-
clear shore
lucky guess
cedar depot
so thank you
clear shore
np
pearl pondBOT
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sharp smelt
pearl pondBOT
sharp smelt
so $p(x) \to p(x) + ((f(x))$ right
jolly parrotBOT
wai
sharp smelt
Yeah
the form is confusing me here
((f(x)) is a set, so all elements are of the form p(x)+((f(x))
feral sedge
Yeah
sharp smelt
I suppose (f(x)) could absorb some part of p(x) but still
desert wagon
that was a silly mistake
do you know what else was SILLY mistake??
feral sedge
Yeah this is going to act like reduction modulo f(x)
sharp smelt
ah, lemme work a bit then
sharp smelt
Yeah this is going to act like reduction modulo f(x)
$(f(x))= { \sum a_ix^{nb_i} \mid a_i \in \Z, b_i \in \N }$
jolly parrotBOT
wai
pearl pondBOT
feral sedge
no worries
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Available help channel!
sharp smelt
no worries
.oh wait, here all I need to do is use the fact that it's a homomorphism , do I not
That's literally it
pearl pondBOT
That's literally it
sharp smelt
.close
pearl pondBOT
sharp smelt
and how does that give you the answer
well, it tells me that any polynomial in R[x] iis mapped to a polynomial of that forms
anyways class now
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Available help channel!
sterile crypt
went on chatgpt and it told me that the entire rombous does not fit within each sector (DAB and DCB)
solar ember
Is this an igcse question?
solar ember
I did this exact question 8 years ago haha
sterile crypt
lol what a coincedence
solar ember
You have to use the sector area formula: $Area = \frac{1}{2}r^2\theta$
jolly parrotBOT
Ajay
solar ember
So $$Area_{sector} = \frac{1}{2}(4^2)\left(\frac{2\pi}{3}\right) = \frac{1}{2}(16)\left(\frac{2\pi}{3}\right) = \frac{16\pi}{3}$$ You got the correct are for the rhombus
jolly parrotBOT
Ajay
unborn abyss
that's not a circle it's an ellipse right?
solar ember
No, it is circular
sterile crypt
**Ajay**
i understand how the area of the sector equation works
my question was
is the rombous a part of the sector
and if so we should do sector - rombous
am i correct
and if not why
solar ember
yes, to get the segment area
sterile crypt
okay
intersting
gpt was lying to me ig
ty
i was full tweaking for 4 hrs abt this
solar ember
Can you solve it on your own now?
pearl pondBOT
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Available help channel!
coral pecan
I do not understand vectors and need help solving this
pearl pondBOT
I do not understand vectors and need help solving this
dire tapir
I do not understand vectors and need help solving this
for 36(a) think of what path you can take to get from A to C, there's two paths but one of them isn't useful
coral pecan
-4a+3b+2a+b
coral pecan
for 36(a) think of what path you can take to get from A to C, there's two paths ...
No I understood this part I feel
But when they ask a mid point
Like where 2 lines intersect
dire tapir
But when they ask a mid point
Ah okay. Say for an example, in question 11 we know M is the mid point of OA. It also shows you know OM is a. What would be the ratio be for OM:MA?
dire tapir
Close, it's a half of OA because M is the midpoint so it means OM and MA are equal distance
coral pecan
Close, it's a half of OA because M is the midpoint so it means OM and MA are equ...
Ohh midpoint
dire tapir
yeah
dire tapir
How could I miss that 🤦♀️
don't worry, my dumbass missed these type of stuff during gcse 😭
coral pecan
So ratio is 1:1 ya ?
dire tapir
basically
coral pecan
Ok so
coral pecan
So ratio is 1:1 ya ?
Is this correct for OM AND MA
dire tapir
Is this correct for OM AND MA
yes
dire tapir
yeah
dire tapir
yes
coral pecan
What does vector terms mean 🥲
dire tapir
What does vector terms mean 🥲
wdym?
karmic flower
use section formula for vectors?
dire tapir
a and b in terms of vector
it just means what the vector equals to using the variable a and b
coral pecan
use section formula for vectors?
What's section formula for vectors 😭😭
dire tapir
What's section formula for vectors 😭😭
it's similiar to what we are doing, but putted in a formula
dire tapir
Then the answer would just be this or
for (a), yeah
karmic flower
What's section formula for vectors 😭😭
wait that's the name right
karmic flower
The one used in coordinate geometry
dire tapir
wait that's the name right
yeah
coral pecan
I just wanna go through it rather than using the formula, so you understand it
I don't even think I learnt the formula but yes makes sense
karmic flower
its applicable here as well as in complex numbers
dire tapir
What's the formula
as an example
karmic flower
as an example
yeah ts
coral pecan
Yes I never learnt this formula
I think it's better if I go step by step or I'll probably make some mistakes
dire tapir
It'll help you understand what's going on if you do step by step
but using the formula can be useful for saving time
coral pecan
I don't even understand the formula 🥲
Guys sorry I'm really frustrating to teach maths to
coral pecan
But i don't think sector formula vector is in gcse
karmic flower
do you need help with 2nd part of the problem?
coral pecan
no it isn't
Are you saying it's there or that it isn't there ?
coral pecan
do you need help with 2nd part of the problem?
Yes ig
dire tapir
Are you saying it's there or that it isn't there ?
it isn't there
coral pecan
Yes that's what I'm also saying
So would they like reduce marks if I use section vector formula ?
dire tapir
anyways i'll let @karmic flower take over, i have to go
dire tapir
So would they like reduce marks if I use section vector formula ?
no, as long as you show your working and get it right
coral pecan
anyways i'll let <@1372083254580084817> take over, i have to go
Ok take care
coral pecan
no, as long as you show your working and get it right
hmm
@karmic flower your still here right ?
karmic flower
Yes im thinking of how would I solve it without using the above result
ts is tuff without it. im gonna try again after a while. Sorry man. im not getting it rn
pearl pondBOT
@coral pecan Has your question been resolved?
coral pecan
ts is tuff without it. im gonna try again after a while. Sorry man. im not getti...
It's fine I get it
coral pecan
still this was the method I originally had in mind
Using the formula ?
karmic flower
Using the formula ?
Yeah. I expressed P two times assuming lambda and nu as ratios in which the lines are divided. then compared both equation and coefficient of a and b. It gave two equations. Solving them I got value of lambda and nu. Plug them into P to find its position vector wrt O. Now you can find the required ratio
pearl pondBOT
@coral pecan Has your question been resolved?
coral pecan
Yeah. I expressed P two times assuming lambda and nu as ratios in which the line...
Gng I haven't learned lambda and annol and nu🥲🥲🥲
karmic flower
Gng I haven't learned lambda and annol and nu🥲🥲🥲
eh. lambda and nu are just variables I assumed to be the ratios in which sides are divided. you could take them x and y if you're comfortable with that.
pearl pondBOT
@coral pecan Has your question been resolved?
ebon basin
help how to integrate tanxsinxdx?
pearl pondBOT
help how to integrate tanxsinxdx?
toxic lichen
$\int \tan(x) \sin(x) \dd{x}$
jolly parrotBOT
Ann
toxic lichen
this?
ebon basin
yes
toxic lichen
ok, progress so far?
ebon basin
i managed to recall the chain rule. i dont know how to use it tho
toxic lichen
that's for differentiation...
ebon basin
or was i supposed to make it into sin2x/cosx
toxic lichen
kinda useless for this integration question
ebon basin
sin^2x/cosx
toxic lichen
sin2x
do you mean sin^2(x) or do you mean sin(2x)?
anyway
writing the stuff inside the integral as sin^2(x)/cos(x) and then (1 - cos^2(x))/cos(x) is the way to go imo
ebon basin
kinda useless for this integration question
oh damn
hybrid ivy
you can probably split that fraction you got into two fractions
toxic lichen
you'll need to either know or rederive the integral of sec(x) dx
ebon basin
oh then i can split it into two integrals sum
how do i integrate sec x
isnt it tan x
west sail
do you have access to a common integral table?
karmic flower
Convert into sin and cos function. then cosx=t
rustic atlas
Alternatively subbing u=sint works
karmic flower
you know the result for integrating √a^2-x^2 dx. That's be useful later on
toxic lichen
$\int \frac{\dd{x}}{\cos(x)} = \ln\absv{\sec(x) + \tan(x)} + C$
jolly parrotBOT
Ann
toxic lichen
int sec**^2**(x) = tan(x) is what you were thinking of earlier
ebon basin
int sec**^2**(x) = tan(x) is what you were thinking of earlier
how can i derive like sec x and csc x on the go?
i dont think writing it as (sinx)^-1 and derivation wont work
or should i just memorize
toxic lichen
if you don't have access to u-sub then you don't have access to any method to rederive those integrals im afraid
ebon basin
if you don't have access to u-sub then you don't have access to any method to re...
dont be afraid im here
but ok i got it!
west sail
...lol?
ebon basin
thank you for your help
karmic flower
or should i just memorize
memorise?
ebon basin
yea
karmic flower
maybe
ebon basin
but memorization is just for the weak
i derive it because im too lazy to bother otherwise
thank you once again for help👍
.close
pearl pondBOT
karmic flower
Proof for int secx dx
toxic lichen
op doesnt have access to substitution
karmic flower
that's sad:( is there any other way then
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Available help channel!
coral pecan
eh. lambda and nu are just variables I assumed to be the ratios in which sides a...
Ohh
coral pecan
What is lambda and nu
sharp vigil
lambda and nu are just greek letters
they can be variable names like any other letter
glacial bluff
here is a picture of the Greek alphabet that I can find for OP.
toxic lichen
it's even shorter than the english alphabet
west sail
are my eyes deceiving me or are there two forms for lowercase sigma?
oh word-final position sigma, okay. interesting.
pearl pondBOT
@coral pecan Has your question been resolved?
coral pecan
What would learn these be useful for
west sail
a lot of variables in math are expressed in Greek letters.
for example, angles are usually expressed with theta.
the two roots of a quadratic are usually expressed as alpha and beta.
the change in a variable is usually expressed as delta.
and the summation notation is essentially just a really big sigma.
midnight haven
Upsilon is new
west sail
oh and in statistics there's a kind of distribution called the chi-squared distribution, which is exactly the chi in the Greek alphabet.
in general, in STEM (so science, tech, math, engineering) people use Greek letters a lot, for better or worse. might as well get the ball rolling.
toxic lichen
What would learn these be useful for
they are symbols that mathematicians use all over the place
knowing them makes formulas seem less alien
pearl pondBOT
@coral pecan Has your question been resolved?
west sail
you should indeed refer to your syllabus, but most math and science syllabi use Greek letters to various degrees.
some more, some less.
pearl pondBOT
coral pecan
More than half are useful for hs
Ok Ty GUys
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vernal minnow
Sorry das es deitsch ist aber wie macht man die schar bei der 3
glossy blade
Can you translate?
vernal minnow
Yeah
Investigate the common position of the plane E and the family of lines gt as a function of t:
For which t is the line gt parallel to the plane E? Does it lie on the plane E?
For which t does the line g intersect the plane E orthogonally? Also state the point of intersection. (6 points)
The coreect answers on tze notes but i dont get how you find out waht to do
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Available help channel!
green meteor
Is this right i dont have answers
pearl pondBOT
Is this right i dont have answers
iron basin
Is this right i dont have answers
righto
green meteor
Is the working fine
pearl pondBOT
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Available help channel!
stoic imp
pearl pondBOT
jolly parrotBOT
Renato
cinder flower
since these are surjective functions from {1,…,8} to {1,…,8} they must be bijections, which correspond to permutations of {1,…,8}
i think that will be easier to work with than thinking about functions
stoic imp
since these are surjective functions from {1,…,8} to {1,…,8} they must be biject...
who is talking about injectivity
cinder flower
nobody
stoic imp
by bijection you mean an isomorphism, aka injective + surjective?
cinder flower
a surjective function from {1,…,8} to {1,…,8} must be a bijection. because every one of the 8 elements in the codomain must be mapped to. since there are only 8 elements to send into the codomain, each must be sent to a different one
toxic lichen
by bijection you mean an isomorphism, aka injective + surjective?
yes
though those usually are not called isomorphisms
the word "isomorphism" is reserved for shit like groups or rings or whatever
cinder flower
bijections are isomorphisms in the category of sets 
stoic imp
a surjective function from {1,…,8} to {1,…,8} must be a bijection. because every...
no
we can send 1 to 8 to 1
cinder flower
what
cinder flower
can you write down a surjective function for me
from {1,…,8} to {1,…,8}
or even just from {1,…,4} to {1,…,4}
i promise you it will be injective
stoic imp
or even just from {1,…,4} to {1,…,4}
f(1)=1
f(2)=1
f(3) = 1
f(4) =1
toxic lichen
that's not surjective
toxic lichen
f(1)=1
f(2)=1
f(3) = 1
f(4) =1
this function fails to hit 2 as an output
stoic imp
this function fails to hit 2 as an output
why is that a dealbreaker
cinder flower
do you know what surjective means?
toxic lichen
your function's image is just {1} and fails to be {1,2,3,4}
stoic imp
so in a subjection the cardinality of the image needs to be equal or greater than the cardinality of the domain
modern talon
cinder flower
hello blanket
stoic imp
do you know what surjective means?
no
cinder flower
no
then why are you telling me “no you’re wrong” when i say things about surjective functions
lmao
stoic imp
well I thought i knew, i just noticed i didnt
cinder flower
okay
cinder flower
f(1)=1
f(2)=1
f(3) = 1
f(4) =1
anyway, you notice how 1 is the only value gets mapped to here?
there is no x with f(x) = 2 for example
nothing maps to 2 or 3 or 4
surjective means that everything in the codomain gets mapped to by something
f(1)=1
f(2)= 4
f(3) = 3
f(4) = 2
is an example
stoic imp
ok