Multiplication Principle is used for simultaneous independent choices
• Here: choosing b and c
• Addition Principle is used for separate cases
• Here: different values of a
thats why its (100-k)^2 not k(100-k)^2
#help-36
last quartz
last quartz
wdym
We don’t multiply by k because a is fixed. You only multiply things you’re choosing, and here we’re only choosing b and c.
sacred night
oh
last quartz
Yea
You understand it now?
We fix a = k first. So a is not a choice anymore, it’s already chosen.
Once a = k:
• b has 100-k choices
• c has 100-k choices
Since b and c are chosen independently, we multiply:
(100-k)(100-k) = (100-k)^2
last quartz
glad to help you bro❤️
sacred night
so how does addtion rule apply
sacred night
glad to help you bro❤️
ty
last quartz
so how does addtion rule apply
Addition rule applies because we split the problem into cases based on a.
Each value of a gives a separate case:
• Case a=1: (100-1)^2 triples
• Case a=2: (100-2)^2 triples
• …
• Case a=99: (100-99)^2 triples
These cases cannot overlap (a triple can’t have two different values of a), so we add them:
|S| = (99)^2 + (98)^2 + \cdots + (1)^2
That’s exactly the addition principle:
total = sum of disjoint cases.
Multiplication is for choices made together; addition is for different cases. Different values of a are different cases.
sacred night
i dont get it
last quartz
Think of it like this.
We are grouping triples by the value of a.
• Group 1: all triples where a=1
• Group 2: all triples where a=2
• …
• Group 99: all triples where a=99
A triple can only belong to one group, because a can’t be two numbers at once.
So the groups do not overlap.
When you want the total number of items in separate non-overlapping groups, you add the sizes of the groups.
That’s the addition rule.
Inside each group:
• b has 100-k choices
• c has 100-k choices
→ (100-k)^2 triples in that group.
Total = add all groups:
(99)^2 + (98)^2 + … + (1)^2
sacred night
okay thanks
last quartz
no problem
sacred night
.close
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lime crest
A school sent students to compete in an academic olympiad in $11$ differents subjects, each consist of $5$ students. Given that for any $2$ different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least $4$ different subjects.
final saddleBOT
A school sent students to compete in an academic olympiad in $11$ differents sub...
soft zealotBOT
Copter
lime crest
i dont really know how to think for this problem😓
but the solution is probably by contradiction; assume all students compete in at most 3 subjects?
strange sparrow
It's probably easiest to do this by contradiction
Suppose that every student competes in at most 3 subjects
strange sparrow
Hmm
It's more convenient in this case to use the language of graphs
Represent each subject by a distinct vertex
lime crest
i know nothing about graph theory, sorry ;-;
strange sparrow
between each pair of vertices draw (at least one) edge and label the edge with the person that's common to both subjects
strange sparrow
i know nothing about graph theory, sorry ;-;
It's ok you don't need to know any actual graph theory for this
lime crest
alright
plucky rover
I think you're expected to use pigeonhole principle here?
strange sparrow
Once you have this construction you can ask given some fixed person p how many edges can have the label p?
lime crest
I think you're expected to use pigeonhole principle here?
probably, either that or double counting which im not too good at😭
plucky rover
between each pair of vertices draw (at least one) edge and label the edge with t...
Oh that's neat
See if you can follow what plante is saying
lime crest
okay okay
plucky rover
We can always do it the "traditional" way later
lime crest
Once you have this construction you can ask given some fixed person p how many e...
is this basically equivalent to the problem statement
lime crest
hmmm
strange sparrow
Ok that was phrased kinda poorly
Let's say we fix a vertex (ie. a subject) and we look at the edges incident to that vertex
lime crest
the total number of edges is 11 choose 2
strange sparrow
Let's say we fix a vertex (ie. a subject) and we look at the edges incident to t...
can there be more than 2 edges of the same colour?
strange sparrow
the total number of edges is 11 choose 2
This turns out to be true, but you don't know it yet
You only know that it is at least 11 choose 2
lime crest
can there be more than 2 edges of the same colour?
color?😭
strange sparrow
Sorry, label
plucky rover
can there be more than 2 edges of the same colour?
Wait why not? Suppose all students take all the subjects, that's valid given our constraints
strange sparrow
yes but we are assuming for the sake of contradiction that each student takes at most 3 subjects
tranquil pine
for 3 subjects you'd have a triangle so there would be 3 edges
fossil kiln
<@&268886789983436800>
tranquil pine
for 3 subjects you'd have a triangle so there would be 3 edges
oh you meant from a fixed vertex sorry
lime crest
can there be more than 2 edges of the same colour?
yea
strange sparrow
If there are more than two edges with the same label adjacent to the same vertex what does that mean
lime crest
then theres atleast 4 subjects that the student competed in
strange sparrow
yes but we assumed at the start that each person competes in at most 3 subjects
strange sparrow
Ok for the sake of concreteness let's fix a subject
Say English
it has some edges, at least 10, from itself to other subjects, and each edge is labelled with the name of a person, correct?
We have just reasoned that of these edges, there can be no more than 2 with the same label
strange sparrow
We have just reasoned that of these edges, there can be no more than 2 with the ...
Just to make sure do you follow everything up to this point
lime crest
wait what were you pointing towards😭
lime crest
Just to make sure do you follow everything up to this point
yeah
strange sparrow
Ok so here's the important part: how many possible labels are there?
lime crest
55 total?
strange sparrow
In total yes but among the edges incident to English?
lime crest
5
strange sparrow
So you have >=10 edges, each edge receives a label out of 5 possible labels, and no label is used more than twice. What does this tell you?
strange sparrow
Indeed, this tells you that there are precisely 10 edges and each label is used precisely twice
Ok, suppose now that "Bob" is one of the labels. Can you now tell me how many edges in the whole graph has this label?
lime crest
3?
lime crest
because theres exactly two edges so theres 3 vertices
strange sparrow
Mhm
Ok I would explain it as follows. Say the two edges labelled Bob are English–Chinese and English–Maths. Then you also have third edge from Chinese to Maths labelled Bob, that gives 3. It is not possible to have more than 3 because that would violate our assumption that each student takes at most 3 subjects
So you know now that this is a graph with exactly (11 choose 2) edges and each label is used exactly 3 times. Can you derive a contradiction now?
lime crest
then the total number of edges is 3x = 11 choose 2 = 55 but this has no integer solution?
strange sparrow
Indeed
lime crest
tranquil pine
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green chasm
Find the volume of the region enclosed here. Use the shell method only.
green chasm
I'm stuck on the part where have to bound the region by pi/4
green chasm
Yup
So far I made 2pi integral_0 ^1 (pi/2 -x)(tanx) dx
I know I am probably wrong on the radius part
barren hound
$2\pi \int_{x=0}^{x=\sfrac\pi4} (\f\pi2-x)\tan x\dd{x}$
soft zealotBOT
clumsy
green chasm
alright, it's all goods now
barren hound
sometimes i like to write them like that just to make sure i know what i'm doing
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vagrant lichen
Hello, could someone check if this proof looks good please?
\begin{Definition}[Divisibility]
A nonzero integer $a$ is said to \emph{divide} an integer $b$, written $a \mid b$, if $b = ak$ for some integer $k$.
\end{Definition}
\begin{Lemma}
Assume $a$ and $b$ are integers. If $a$ and $b$ are both even or both odd, then $a^2 - b^2$ is even.
\end{Lemma}
% --------------------------------------------------------------------------------
\begin{Theorem}
If $n$ is an odd integer, then there exists two integers $a$ and $b$ such that $n = a^2 - b^2$.
\end{Theorem}
\begin{proof}
Let $n$ be an odd integer.
By definition of odd integer, $n = 2k + 1$.
Let $n = a^2 - b^2$ where $a$ and $b$ are integers.
In order for $a^2 - b^2$ be odd, then $a^2 - b^2 = 2l + 1$ for some integer $l$.
By lemma 2, $a$ and $b$ cannot be both even or both odd.
So, it must mean that at least one of $a$ or $b$ is even and the other odd.
Without loss of generality, let $a = 2p$ and $b = 2q + 1$.
Then,
\begin{align*}
a^2 - b^2 &= (2p)^2 - (2q + 1)^2 \
&= 4p^2 - 4q^2 - 1 \
&= 4p^2 - 4q^2 - 2 + 1 \
&= 2(2p^2 - 2q^2 - 1) + 1 \
\end{align*}
which is odd.
Therefore, there exists two integers $a$ and $b$ such that $n = a^2 - b^2$.
\end{proof}
final saddleBOT
Hello, could someone check if this proof looks good please?
\begin{Definition}[...
crystal pawn
soft zealotBOT
Mor Bras
jagged wave
you specify n in terms of k but never solve for k
merely show that the difference is odd
winter lava
You said let n = a^2 - b^2 but didn't show that such a and b exist
jagged wave
it would be a more direct proof to just provide a formula for some odd number (2n + 1)
utilizing the fact that the difference of two consecutive squares is odd, and in fact this generates all odd numbers
vagrant lichen
I see, I'll try that, thanks!
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winter lava
hint || you can find two numbers in terms of k, such that the difference of their squares is 2k+1 ||
vagrant lichen
hint || you can find two numbers in terms of k, such that the difference of thei...
Do you mean this?
\begin{Lemma}
If $a$ is an even integer and $b$ is an odd integer, then $a^2 - b^2$ is odd.
\end{Lemma}
\begin{proof}
Let $a = 2k$ be an even integer and $b = 2k + 1$ be an odd integer.
Then,
\begin{align*}
a^2 - b^2 &= (2k)^2 - (2k + 1)^2 \
&= 4k^2 - 4k^2 + 4k - 1 \
&= 4k - 1 \
&= 2(2k) - 1 \
\end{align*}
which is odd.
Therefore, by definition of odd integer, $a^2 - b^2$ is odd.
\end{proof}
soft zealotBOT
Mor Bras
tired walrus
this is incorrect btw
you can't let a=2k and b=2k+1
this essentially asserts that b = a+1.
which nobody said had to be the case at all.
atomic moon
You need k and k'
tired walrus
you need two different variables, whatever their names may be.
atomic moon
k and k' is cool
tired walrus
at least this is for the lemma as-written
winter lava
following my hint, you can forget about the lemma
vagrant lichen
I'll reformulate the lemma
\begin{Lemma}
Assume $a$, $b$ and $k$ are integers. If $a = 2k$ is an even integer and $b = 2k + 1$ is an odd integer, then $a^2 - b^2$ is odd.
\end{Lemma}
\begin{proof}
Let $a = 2k$ be an even integer and $b = 2k + 1$ be an odd integer.
Then,
\begin{align*}
a^2 - b^2 &= (2k)^2 - (2k + 1)^2 \
&= 4k^2 - 4k^2 - 4k - 1 \
&= -4k - 1 - 1 + 1 \
&= -4k - 2 + 1 \
&= 2(-1)(2k + 1) + 1 \
\end{align*}
which is odd.
Therefore, by definition of odd integer, $a^2 - b^2$ is odd.
\end{proof}
soft zealotBOT
Mor Bras
tired walrus
you understand that the statement of the lemma still presupposes b = a+1 right
vagrant lichen
Yes
This should be better now:
\begin{Lemma}
If $a$ is an even integer and $b$ is an odd integer, then $a^2 - b^2$ is odd.
\end{Lemma}
\begin{proof}
Let $a = 2k$ be an even integer and $b = 2m + 1$ be an odd integer.
Then,
\begin{align*}
a^2 - b^2 &= (2k)^2 - (2m + 1)^2 \
&= 4k^2 - 4m^2 - 4m - 1 \
&= 4k^2 - 4m^2 - 4m - 2 + 1 \
&= 2(2k^2 - 2m^2 - 2m - 1) + 1 \
\end{align*}
which is odd.
Therefore, by definition of odd integer, $a^2 - b^2$ is odd.
\end{proof}
But I think is something else to what Axe hinted
soft zealotBOT
Mor Bras
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vagrant lichen
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vagrant lichen
This should be better now:
\begin{Lemma}
If $a$ is an even integer and $b$ i...
<@&286206848099549185> Does this proof looks good?
west flame
its good just maybe add a line about why 2k^2-2m^2-2m-1 is an integer
@vagrant lichen
.close
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rain sentinel
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rain sentinel
I thought it might be a sig fig error
old quarry
is E modulus of elasticity? like youngs modulus
rain sentinel
is E modulus of elasticity? like youngs modulus
yes
old quarry
can some one check why I get x10-7
Gpa is 10^6 no?
rain sentinel
isn't giga x10^9?
old quarry
oh ye sorry mb
hmm i have no idea tbh
@rain sentinel
u cant take GPa as 10^9
cuse pascal = n/m^2
and u have mm^2
rain sentinel
oh right so I have to convert to m
old quarry
ye
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robust sequoia
is there any graph plotter for complex functions where I can disregard the imaginary part of the output to convert it from 4d to 3d ?
final saddleBOT
is there any graph plotter for complex functions where I can disregard the imagi...
stone wagon
is there any graph plotter for complex functions where I can disregard the imagi...
desmos 3d
stone wagon
robust sequoia
what does f(x+yi).imag represent
stone wagon
x = real number along x axis
y = real number along y axis
x+yi is how you make a complex number. f is just a function i wrote, it's f(z)=sin(z) in this case. for any number you can access its real part or imaginary part by doing .real or .imag respectively
make sure to turn complex mode on too
stone wagon
is there any graph plotter for complex functions where I can disregard the imagi...
oh sorry you wrote disregard
then it's like this
you can also do this for absolute value plot
and this for argument plot
your options are endless 
robust sequoia
it doesn't work for me. Is it cuz I'm on my phone?
stone wagon
make sure to turn complex mode on too
make sure you have this
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copper roost
can someone explain me how the long bar in de morgan differs to the normal bar
scarlet sequoia
what do you mean? like $\overline{A\cap B}$?
soft zealotBOT
Rafilouyear2026
slender shard
can someone explain me how the long bar in de morgan differs to the normal bar
can you provide an example of this?
scarlet sequoia
The bar means "complement of"
copper roost
**Rafilouyear2026**
yes exactly
scarlet sequoia
so it's the complement of (A intersection B)
copper roost
over A for example
glossy zephyr
if by "bar" you mean the act of negation
Then, yeah, they are the same.
copper roost
so it's the complement of (A intersection B)
so everything outside of A and B
glossy zephyr
But you can negate blocks of logic
scarlet sequoia
so everything outside of A and B
Everything outside of (the intersection of A and B)
copper roost
Everything outside of (the intersection of A and B)
Yes
scarlet sequoia
vs
copper roost
scarlet sequoia
$\overline A \cap \overline B$
copper roost
this for example
soft zealotBOT
Rafilouyear2026
glossy zephyr
I can only advice you pull off the venn diagram
scarlet sequoia
**Rafilouyear2026**
This means "(everything outside of A) intersected with (everything outside of B)"
copper roost
the long bar
copper roost
This means "(everything outside of A) intersected with (everything outside of B)...
thats the same wtf
scarlet sequoia
the long bar
each "bar" or complement applies to the set it's under
scarlet sequoia
thats the same wtf
no
scarlet sequoia
**Rafilouyear2026**
elements in here are outside of A and outside of B at the same time
glossy zephyr
I can only advice you pull off the venn diagram
youll see that $\overline A \cap \overline B \neq \overline{A \cap B}$
soft zealotBOT
scarlet sequoia
**Rafilouyear2026**
elements in here are outside of their intersection
So they're not in (both at the same time)
copper roost
This is the shit that fucks me up
copper roost
lemme just open paint and ill show you
ty
scarlet sequoia
**Rafilouyear2026**
"I haven't eaten apples nor bananas today"
scarlet sequoia
**Rafilouyear2026**
Versus "I didn't eat both Apples and bananas today"
scarlet sequoia
Versus "I didn't eat both Apples and bananas today"
Which can be rephrased as "Out of apples and bananas, there's at least one of them I didn't eat today"
copper roost
uhh
glossy zephyr
A and B
glossy zephyr
My pc crashed 😭
copper roost
The things ones we talked about
glossy zephyr
Gimme a minute pls
copper roost
yes bro
copper roost
oh shit
glossy zephyr
if you think of AND and OR as two operations like add and mult.
NOT (minus), distributes over the elements and it also flips the operation
copper roost
what about this
soft zealotBOT
Rafilouyear2026
glossy zephyr
if you think of AND and OR as two operations like add and mult.
NOT (minus), di...
refer to this.
copper roost
if you think of AND and OR as two operations like add and mult.
NOT (minus), di...
so this flips it
and its or now
So "W Not U M not“
or
glossy zephyr
yea and you get notW and M
If i have to give you a little advice about it
this type of logic is analog to boolean algebra
copper roost
oh so i can just memorize if there r 2 bars it gets normal
glossy zephyr
oh so i can just memorize if there r 2 bars it gets normal
Not even have to memorize it
copper roost
its just like - and - turns to plus
glossy zephyr
everything that doesnt have a bar, it now has
everything that does, now doesnt
If i might give you some advice
This type of logic is completely analog to boolean algebra
and in boolean algebra we use + for OR and * for AND
- is also the negation of an argument
so
-(a+b) = (-a) * (-b)
copper roost
shit i understand it
glossy zephyr
you can use the obvious rules of basic arithmetics to see that - (-a) = a
glossy zephyr
yeah
They teach De Morgan Laws because in reality its a really broad concept
They appear in a LOT of fields of mathematics
copper roost
oh yeah
misty axle
hii guyss, can i ask help please? ive been wondering since earlier if this is correct (image)
And another one is im not sure if which one here is appropriate for circuits in our moving lab: The term “lead” can refer to either the metal wire terminals of electronic components used for connection, or solder, a fusible metal alloy used to create permanent electrical bonds between components and circuit boards.
final saddleBOT
hii guyss, can i ask help please? ive been wondering since earlier if this is co...
misty axle
this was the instruction for the diagrammm: In the space provided below, draw a simple circuit that includes at least one battery, resistor, capacitor, and LED connected by conducting wires. Label each component properly.
severe hawk
doesnt the capacitor behave like an open circuit when charged under dc?
also, im not sure what are you actually asking?
misty axle
doesnt the capacitor behave like an open circuit when charged under dc?
ohh im not sure where to put it 💔
misty axle
also, im not sure what are you actually asking?
"lead" was one of the materials we had to define but im not sure which definitin should i use for itt
misty axle
doesnt the capacitor behave like an open circuit when charged under dc?
but the rest of the parts are fine already right?
glossy zephyr
doesnt the capacitor behave like an open circuit when charged under dc?
eventually, yeah
misty axle
eventually, yeah
would capacitor be facing resistor? like on the bottom line of the rectangle? 🥹
glossy zephyr
Depends on what you wanna do
My obvious guess is that this should turn the LED on as long as current flows
misty axle
My obvious guess is that this should turn the LED on as long as current flows
OHH okk
glossy zephyr
if so and if i remember correctly, for that to happen you put a resistor and capacitor in parallel to the LED
because if it isnt made that way, the capacitor will eventually cut off current
It will turn on and then fade out
Depending on the capacitor-resistor pair, slower or quicker.
misty axle
thank youuu! idk much about this topic so it helpssss
glossy zephyr
thank youuu! idk much about this topic so it helpssss
being in parallel literally does the opposite, it remains on after current stops because the capacitor starts feeding the circuit itself
and then fades
anyways, hope that helped
misty axle
being in parallel literally does the opposite, it remains on after current stops...
i seeee, it would look like thiss?
misty axle
anyways, hope that helped
thank youuuu
glossy zephyr
🥀 do you know what series and parallel is?
misty axle
🥀 do you know what series and parallel is?
nnooo 😭
glossy zephyr
gimme a sec
misty axle
okayy
glossy zephyr
Two resistors in each portion of a circuit
The top two are said to be in series
the bottom two are said to be in parallel
This is what i meant with resistor-capacitor combo parallel to the LED
stray prairie
Bro this one is rc Circuit
Rc circuit series
It is solved by how much you current pass through it
glossy zephyr
also, the resistor here is just basically a filler that we do irl to be able to tune better the circuit
stray prairie
If current is in steady state capacitor didn't pass any current through it
Tell me what you need help
misty axle
thaknk u guysss i am currently trying to udnerstand itttt
misty axle
Tell me what you need help
i msggeddd
stray prairie
You dm me
severe hawk
hm, i'm not sure you can have the led alone without the resistor, isnt the resistor supposed to be in series with the branching being only the capacitor and the led?
severe hawk
This is what i meant with resistor-capacitor combo parallel to the LED
if we're talking ideal components all current would go through the led on this setup iirc
misty axle
hm, i'm not sure you can have the led alone without the resistor, isnt the resis...
this making my head spin ehheuerur
glossy zephyr
if we're talking ideal components all current would go through the led on this s...
starting from an empty cap., it will split based on the resistance at t = 0, and then slowly opt out for the LED side more and more
Once you cut current then capacitor will start discharging
and the LED will fade out
you generally will use a really small capacitor, iirc the idea of this whole thing is that the LED never has a sudden peak or drop in current which is bad
glossy zephyr
hm, i'm not sure you can have the led alone without the resistor, isnt the resis...
dont take my word for it, im realy bad at electronics, but since youre working with relatively small V and I, the internal resistance to the LED is enough to keep it somewhat leveled.
lights in general are just a fancy resistor.
misty axle
i rlly appreciate this guys fhfhf
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wary solar
I got 5 but my textbook says its 1
The og expression is (u-v)•(u+2v)
With u = 2 v = 1 and the angle between them = 2π/3
final saddleBOT
I got 5 but my textbook says its 1
The og expression is (u-v)•(u+2v)
With u = 2 ...
vital crag
just looks like you read your own handwriting wrong. there should be a v.v term on the right
slender shard
u and v gotta be the worst letter pair
wispy grove
just looks like you read your own handwriting wrong. there should be a v.v term ...
yeah u have messed up in this step
u urself got confused in bet u and v
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rain sentinel
Can someone help me rearrange the top line to find d
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Can someone help me rearrange the top line to find d
shell condor
sorry I don't get this, it's $\frac{d^4}{2^4}$ and $\frac{d}{2}$ right?
soft zealotBOT
doctorstrangejr
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rain sentinel
**doctorstrangejr**
yes
thorny otter
$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \times \frac{d}{c}$
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Varixiuqlhfbgraijbzjnqghppxnqmvw
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gentle vector
yo, Just started my MTH241 calculus class, wanted to get someone to check this out to make sure i dont embarrass myself before posting it to my group's discussion forum. we were told to each pick a problem I picked the second one heres what I've written so far: The second problem is a demand function rather than a cost function, so were determining the consumer behavior rather than production cost.
2a) we have the points (P,Q) as (20,900) and (60,300) so using the slope formula: m=300-900/60-20 which simplifies to -600/40 = -15 Slope
2b) For every $1 increase in price per crate, the quantity demanded is decreased by 15 crates.
2c) Plugging the point (20,900) into Q=mP+b to find the intercept we get 900=-15(20)+b, so b=1200, can now plug that into the demand function Q= -15P+1200
2d) we plug the price "40" into our demand function and get Q=-15(40)+1200=600, so at the price of $40, 600 crates are demanded.
2e) Same process as 2d), plugging this time "100" and we get Q=-300 but that doesnt make sense right? you cant have negative demand, unless people want to return crates of oranges for a refund? lol I think the right way to intrepret this would be to say that the price would be too high for consumers, so the demand would be zero.
final saddleBOT
yo, Just started my MTH241 calculus class, wanted to get someone to check this o...
@gentle vector Has your question been resolved?
slender shard
yo, Just started my MTH241 calculus class, wanted to get someone to check this o...
yes your answers make sense to me
good job
final saddleBOT
@gentle vector Has your question been resolved?
proper dagger
final saddleBOT
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.
final saddleBOT
Available help channel!
elfin glacier
sorry not a math question, who has a chinese learning server if anyone
leaden moon
sorry not a math question, who has a chinese learning server if anyone
Search up
elfin glacier
i kinda dont wanna do one that you can find on the global search
leaden moon
sorry not a math question, who has a chinese learning server if anyone
Discord can let you discover servers
leaden moon
i kinda dont wanna do one that you can find on the global search
Well then do it
elfin glacier
Well then do it
thats not how that fucking works
leaden moon
thats not how that fucking works
What's the problem with that?
final saddleBOT
leaden moon
im too tired for this shit
Cmon don't be lazy
elfin glacier
Cmon don't be lazy
blocked
leaden moon
💀
final saddleBOT
Available help channel!
flint hemlock
Does anyone know if this symbol like a percent symbol exists or not
final saddleBOT
Does anyone know if this symbol like a percent symbol exists or not
flint hemlock
My teacher said it means ‘between’ but I cannot find this symbol on the internet
tired walrus
probably sth your teacher invented
final saddleBOT
heady moon
My teacher said it means ‘between’ but I cannot find this symbol on the internet
b/w dis shit is between
keen scroll
you can just assume what it means from reading the whole sentence really
flint hemlock
aint no way im writing that in public exam
he literally did not write b/w but % with 2 dots instead if 2 circles
I should take a ss in school next time
final saddleBOT
Available help channel!
fluid blaze
how do i convert a general ellipsiod to a standrad ellipsoid?
final saddleBOT
how do i convert a general ellipsiod to a standrad ellipsoid?
fluid blaze
i mean i get how to do it for ellipse. consider a circle with center at the center of ellipse and find the ones which only intersect at 2 points but its way harder to think about in 3d
like i can manage after i have of ellipse at orgigin
stone wagon
how do i convert a general ellipsiod to a standrad ellipsoid?
what is the standard ellipsoid
isn't that a sphere
or is it where one of the axes = 1
final saddleBOT
final saddleBOT
Available help channel!
terse folio
are these equal?
final saddleBOT
are these equal?
terse folio
summing from j then i vs i then j
woven ledge
Help please
I get 85 to 95 in maths usually 2 exams ago i didn't even study yet got 85 now i study hard and am getting grades below 60 for 2 exams despite studying more and when I study I see solved examples first see their logic and then solve q my exam went worse than my timed sample paper every peer of mine is getting ahead I'm scared all this happened in 1 year
gritty drift
For finite sums, yes. For infinite I believe there are some edge cases.
terse folio
For finite sums, yes. For infinite I believe there are some edge cases.
is this related to fubini's theorem in some way?
gritty drift
Help please
I get 85 to 95 in maths usually 2 exams ago i didn't even study yet ...
!occupied . Probably also not appropriate for a help channel
final saddleBOT
Help please
I get 85 to 95 in maths usually 2 exams ago i didn't even study yet ...
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).
gritty drift
is this related to fubini's theorem in some way?
Fubini's is a bit out of my depth but the gist looks similar
terse folio
Fubini's is a bit out of my depth but the gist looks similar
Fundamentally integration is just the sum right
woven ledge
!occupied . Probably also not appropriate for a help channel
Sorry but I'm lost so what should I do which channel should I seek
terse folio
riemann sum
plucky rover
Sorry but I'm lost so what should I do which channel should I seek
#study-discussion probably
terse folio
are these equal?
and if this is true then that means you can swap order of integration..?
trail mango
they are formally the same
gritty drift
riemann sum
Yeah basically. You can interchange multiple integrals in most cases (as long as bounds dont depend on each other etc), but the infinite upper bds on the sums do make it a little different.
terse folio
Yeah basically. You can interchange multiple integrals in most cases (as long as...
hold on but in integration isnt upper bd on the sum supposed to be infinite?
im trying to consolidate my understanding rn
gritty drift
Yes but this sum converges by definition, our "free" sum in your example might not necessarily converge
trail mango
but treating the x_i’s as functions if e.g. one of the sums doesn’t converge it doesn’t make sense to write this
trail mango
a more interesting question is if they are equal so long as one of them converges
terse folio
a more interesting question is if they are equal so long as one of them converge...
wat if other doesnt
trail mango
well that’s kinda part of the question. can that happen?
i can tell you that they are equal if one of them converges absolutely
so that might help you look for a counterexample
final saddleBOT
@terse folio Has your question been resolved?
junior token
is this related to fubini's theorem in some way?
It is Fubini’s theorem on the measure space Z x Z
junior token
and if this is true then that means you can swap order of integration..?
also, consider the sequence of scaled characteristic functions $\chi_{(0,1)}, 2\chi_{(0,1/2)}, 3\chi_{(0,1/3)}, 4\chi_{(0,1/4)}, …$, which converge pointwise to 0 but whose integrals converge to 1
soft zealotBOT
soup_norm
broken slate
lusty mason
How do I solve x²>2x
final saddleBOT
How do I solve x²>2x
lusty mason
I know i asked a similar question yesterday but idk how to do it like this
slender shard
find the roots and try drawing a rough sketch
lusty mason
x²-2x>0 first?
slender shard
yes
lusty mason
x(x-2)>0?
slender shard
yes
lusty mason
Roots are 0 and 2
slender shard
yes
now draw a rough sketch
the most important part is knowing if it makes a smily face or a sad face
lusty mason
Smily!
slender shard
yes!
slender shard
yes!
final saddleBOT
pliant shore
x²-2x>0 first?
you can test x = 100
100^2 - 2 * 100 > 0 is true
that's how you know the region lies outside the roots
the only other possibility is inside the roots
pliant shore
the most important part is knowing if it makes a smily face or a sad face
I do this too but there's another way of thinking about it
<@&268886789983436800>
final saddleBOT
Available help channel!
tulip panther
what is this question asking me to do? i dont want the answer i just want to know what the question means
final saddleBOT
what is this question asking me to do? i dont want the answer i just want to kno...
next thorn
-# dw, no one gives answers here!... or atleast- no one is allowed to 👍
what is a congruent triangle?
tulip panther
oh okay hahah
its a triangle that has the exact same angles and sides as another triangle
proud igloo
what is a congruent triangle?
better to ask what congruency between traingles is ❤️
congruent and equilateral are sometimes used interchangeably
cuz english is awful
next thorn
-# it doesnt matter anyways, I tried solving the question and failed
-# I concluded that (x-1) is positive and (x-5) is negative but- thats every choice 😭
great luck to keqae!
tulip panther
😭 oh dear
final saddleBOT
@tulip panther Has your question been resolved?
winter stag
excuse my shitty ms paint work
my interpretation is that theyre asking you to find the range of x where you can have 2 triangles like this
proud igloo
yes!
proud igloo
😭 oh dear
lets think about a case where there would be 2 congruent trinagle x-1=-x^2+6x-5
tulip panther
my interpretation is that theyre asking you to find the range of x where you can...
im sorry i dont understand 😭
proud igloo
excuse my shitty ms paint work
we dont want this we want something like this were i could say oh x-1 should be greater than the height of the triangle
tulip panther
how are the two triangles you shown congruent?
proud igloo
they are not
tulip panther
oh okay
proud igloo
this is what we want yes?
tulip panther
why is this what the question wants?
proud igloo
yes
tulip panther
no as in
proud igloo
2 non congruent triangles
proud igloo
well the question want 2 non congruent triangles
tulip panther
right
with the side lengths and angle shown in the original triangle
wait, is it asking for 2 non congruent triangles that are composed of exclusively the lengths
x-1
and the other one
and an angle of 30 degrees
proud igloo
hmmmm
im struggling to see how the base of the triangle fullfill this
we could just let is equal like this
big hmmmmmmm
i think this is just wording
like given these sides lengths and this angle find the values of x for which there are 2 non congruent triangles
@tulip panther can we agree with this?
tulip panther
like given these sides lengths and this angle find the values of x for which the...
hmm okay
tulip panther
<@630785569269612555> can we agree with this?
yeah
proud igloo
now we need to draw a relationship between the x^2 term and the x-1
could say that x-1<-x^2+6x-5
...
tulip panther
how does this let us find 2 non congruent triangles?
proud igloo
well we have drawn them
and I can see here that in order for my 2 non-congruent property to hold
that $h<x-1$
soft zealotBOT
Nyxzore
proud igloo
and that $x-1<-x^2+6x-5$
soft zealotBOT
Nyxzore
proud igloo
<@630785569269612555> can we agree with this?
thats just putting this diagram into a bounding equality
i dont use <= because ...
proud igloo
we would degrade into this
tulip panther
well we have drawn them
ohhhh
i see it
they both have the same lengths and angle
proud igloo
good good
proud igloo
<@630785569269612555> can we agree with this?
they are not congruent
tulip panther
we can just slap the h onto the left of the 2nd inequality?
proud igloo
$h<x-1<-x^2+6x-5$
soft zealotBOT
Nyxzore
proud igloo
exactly
and now using my knowledge of trig how can i represent h in terms of x
hint
tulip panther
oh oh oh oh
proud igloo
oh oh oh
proud igloo
mb king
tulip panther
h = sin30(insert quadratic here)
proud igloo
and sin30 is 0.5?
tulip panther
yesh
proud igloo
,w sin30deg
soft zealotBOT
proud igloo
so now we have $\frac{-x^2}{2}+3x-\frac{5}{2}<x-1<-x^2+6x-5$
tulip panther
okay so we can just sub for h and double
?
is it not..
h = quadratic/
?
instead of less than
soft zealotBOT
Nyxzore
proud igloo
yep
tulip panther
yep yep
proud igloo
break it into 2 seperate ineq
tulip panther
soft zealotBOT
Nyxzore
proud igloo
then solve $-x^2+6x-5<2x-2$
soft zealotBOT
Nyxzore
proud igloo
i just multiplied both sides by 2
tulip panther
for the first inequality i have x < 1 or x > 4
proud igloo
yes!
proud igloo
yes
winter stag
should be quite doable 
soft zealotBOT
Nyxzore
soft zealotBOT
Nyxzore
proud igloo
what an awful question
winter stag
i recognised it cuz ive played these games before
literally impossible during an exam setting
tulip panther
i really hate to break it to you
winter stag
this is why i dont miss uni entrance tests
tulip panther
hahahahahahah
its such a nightmare
it ends tmmr
and ill never be as stressed like this on anything ever again
winter stag
gl my friend
winter stag
ayy wait a sec was it just a sign error for the 2nd inequality
proud igloo
holy silly
winter stag
bruh
winter stag
i think we just solved it wrong
tulip panther
i made a sign error on the second ineq but it doesnt change anything
proud igloo
100%
soft zealotBOT
winter stag
tulip panther
gng didnt trust my ahh
winter stag
yes yall were on the front door
tulip panther
ayyy
ive been looking at the question
what was your thought process behind creating the 2 inequalities? @proud igloo
next thorn
-# AND I TRIED TO SOLVE IT LMAOOO
tulip panther
how do you do that translucent font 😭
tulip panther
-# woah
toxic tree
Im watching a video on indeterminate forms, he says if you can turn the fraction into 0/0 or infinity/infinity you can 'just use lhopitals rule', what do I do with either of those using lhopitals rule? The derivative of 0 is 0 in the numerator and then the derivative of 0 is 0 in the denominator so we just get 0/0 again, no?
final saddleBOT
Im watching a video on indeterminate forms, he says if you can turn the fraction...
zenith barn
you're taking the derivative of the original functions in the numerator and denominator
formal trail
you take the derivative before plugging in
final saddleBOT
final saddleBOT
Available help channel!
true hemlock
Hello can someone help me with question 5 because i have no idea how to solve it and if it possible can you explain me how did you do it Write the formula for f(x).
final saddleBOT
Hello can someone help me with question 5 because i have no idea how to solve it...
atomic moon
Think piecewise
vital crag
Hello can someone help me with question 5 because i have no idea how to solve it...
you need to show what question 5 is
true hemlock
uhhh Write the formula for f(x).
vital crag
final saddleBOT
final saddleBOT
Available help channel!
rain sentinel
final saddleBOT
final saddleBOT
@rain sentinel Has your question been resolved?
final saddleBOT
@rain sentinel Has your question been resolved?
bold zenith
You can ping helpers if you want, idk mechanics so can't help
What is B and D in the formula?
final saddleBOT
final saddleBOT
Available help channel!
restive sinew
Order of element (5,3) in Z_30×Z_12?
final saddleBOT
Order of element (5,3) in Z_30×Z_12?
restive sinew
so here is my attempt
order 5 elements in z30
Phi(6)?
Or phi(5)?
So 4 elements?
30=2×3×5
6,12,18,24
Same for phi(3)=2
So we have 4×2=8?
desert mantle
instead of getting lost in formulas you can also just try and compute it
restive sinew
Yeah i am computing
desert mantle
also, think about what the operation even is
restive sinew
Cross
desert mantle
no
restive sinew
Internal product
desert mantle
no
desert mantle
what operation is in Z_30 and Z_12
restive sinew
External product
?
desert mantle
no
desert mantle
external product is how you combine the groups
but what is the operation of the individual groups
desert mantle
Z_30 is a group, yes?
desert mantle
what is the operation in that group
restive sinew
+?
restive sinew
Can it not be ×?
restive sinew
what does order of 5 mean
5+5+...
desert mantle
until what
restive sinew
Ohh
desert mantle
infinitely often?
restive sinew
Z30 is not group under multiplication because
restive sinew
what is the multiplicative identity
1
and we don't have
So it doesn't form a group am I right
Correct reason?
I'm just asking
desert mantle
and we dont have .... ?
desert mantle
the word you are looking for is multiplicative inverses
which is correct
why does 5 not have a multiplicative inverse
restive sinew
I meant i can check many ways to discard
Multiplication identity is 1
Which is present in z30
desert mantle
is that supposed to be a full answer?
desert mantle
I guess yes
it isnt
restive sinew
Why not?
desert mantle
what does the existence of 1 have to do with the non existence of 5^-1
restive sinew
We have seen multiplication inverse not present so z_30 doesn't form group in ×
desert mantle
you have not shown that multiplicative inverses arent there
restive sinew
1/5 is not integer
desert mantle
thats not what multiplicative inverse means
the multiplicative inverse is another element x in the set such that 5*x=1
restive sinew
1/5 is not persent in z30
restive sinew
Z30 is group of integers
jagged wave
shouldn't we be looking for 5^n = 0?
harsh sorrel
whys this a channel😰
restive sinew
Nope
desert mantle
shouldn't we be looking for 5^n = 0?
yes but wrong notation which will confuse
jagged wave
alr
desert mantle
inverse 5 is 31
1=31 in Z_30
so 31 is in Z_30
but 5*31=5 is not 1 mod 30
so the inverse of 5 is not 31
final saddleBOT
Hello
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).
jagged wave
wait so we're using 1 for identity?
restive sinew
I am lost what kind of thing you want me to show
tired walrus
wait so we're using 1 for identity?
we're a long way
jagged wave
got it
restive sinew
By modular arithmetic
tired walrus
popking asked why Z_30 doesn't make a group with multiplication
dena is trying to explain
restive sinew
I said 1/5 is not integer
jagged wave
oh, mb
tired walrus
I said 1/5 is not integer
flawed argument
desert mantle
in the usual integers that would be true
tired walrus
in Z_30, the number 7 has an inverse but 5 doesn't
desert mantle
but we are not in the usual integers
we are doing modulo 30
things are different
1/7 is not an integer, but 7*13=1 mod 30
so 7^-1=13 in Z_30
thats what the expression 7^-1 means
7^-1 is whatever x satisfies 7*x=1
restive sinew
1/7 is not an integer, but 7*13=1 mod 30
Ahha now i understood
What we are doing
Inverse 5 is
5x=1 mod 30
tired walrus
yeah so. can you explain why 5x=1 (mod 30) does not have a solution
restive sinew
x=inverse 5 mod 30
desert mantle
can you tell me what 5*x mod 30 is for x=1,2,3,4,5,6,7,8,9,10,11,....
restive sinew
5,10,15,20,25,30,35,40,45,50,55,60...
desert mantle
and mod 30 ?
tired walrus
30 mod 30 is?
desert mantle
does any of the numbers equal 1?
restive sinew
0
tired walrus
35 mod 30 is?
restive sinew
5
tired walrus
so you should have written
5,10,15,20,25,0,5,10,15,...
not what you put
you ignored the "mod 30" bit until you were asked about it a second time
restive sinew
0,5,15,25,0,5,10,15,25..m
tired walrus
does any of the numbers equal 1?
.
restive sinew
Nope
tired walrus
so there you have it
5^-1 mod 30 does not exist
as you can see WA agrees with me
it also tells you the inverse of 5 mod 30 does not exist
anyway this is far removed from the original q
desert mantle
so we see that 5 doesnt have a multiplicative inverse
desert mantle
which now means that Z_30 is not a group with respect to multiplication
tired walrus
this answered why Z_30 is definitely meant as a group with + and not with *
desert mantle
I want to note that we can see this much faster by looking at 0
restive sinew
Yeah for + it is
desert mantle
0 also has no multiplicative inverse
restive sinew
Yeah it doesn't have
tired walrus
so can you now tell us
order of 5 in (Z_30,+)
tired walrus
and order of 3 in (Z_12,+)?
restive sinew
4
tired walrus
so in the direct product of groups, (5,3) has order...?
restive sinew
(6,4) elements
restive sinew
so in the direct product of groups, (5,3) has order...?
No idea
desert mantle
can you show us the definition of this in your notes?
tired walrus
are you ok going back into theory
restive sinew
I'm not finding formula
I don't know the meaning of order of it
As u can see how i stucked
jagged wave
what is the zero element in Z_30 x Z_12
restive sinew
(5,3)+(5,3)...
tired walrus
first, tell me:
if some element g of a group G has order 3, then g^6 equals?
desert mantle
can you show us the definition of this in your notes?
.
tired walrus
also yes give definition from notes
restive sinew
first, tell me:
if some element g of a group G has order 3, then g^6 equals?
gcd of 6 and that group
I guess you are asking me in very other way ma'am
i know how we find elements and order of that element
jagged wave
define the order of an element
tired walrus
gcd of 6 and that group
absolute nonsense
restive sinew
Order of group /gcd(a,a^r)
jagged wave
?
desert mantle
i know how we find elements and order of that element
sorry but clearly no
tired walrus
diagnosis: you kinda don't know the necessary shit here
restive sinew
I am not okay with writing proper definition in English
tired walrus
I am not okay with writing proper definition in English
do you have notes
restive sinew
I don't have any notes
tired walrus
I am not okay with writing proper definition in English
honestly this means you don't know it
if you are not able to write the definitions of things then you don't know what those things are
restive sinew
Can you rewrite your question in any example form
tired walrus
no debate
restive sinew
Okay no problem it is your opinion
desert mantle
I don't have any notes
you dont have anything written down from your course?
jagged wave
if an element x has order n, then ||x^n = e.||
desert mantle
no material at all?
desert mantle
that is material
jagged wave
have you considered writing notes at all? it helps to
restive sinew
if an element x has order n, then ||x^n = e.||
It depends on the operation
jagged wave
It depends on the operation
no?
desert mantle
so you were given this question without any lecture?
jagged wave
by x^n we simply mean x * x * x * ... n times for the group operation *
restive sinew
About the exam and notes
jagged wave
Because I can't explain much
study.
restive sinew
by x^n we simply mean x \* x \* x \* ... n times for the group operation \*
Do you think I don't know it?
I am simply saying it depends on the operation
jagged wave
the notation is operation-agnostic
jagged wave
yeah, it's pretty general
desert mantle
do you really want to tell us that you have had a course during which you have not written down anything at all and you have an upcoming exam?
restive sinew
Bruh? This is completely ridiculous. We have an exam which have no syllabus
They only wrote syllabus until graduation
desert mantle
so you are going to be tested on stuff that you have not been taught?