help-42

[sullen bridge]

so this is completely fun and legal math

as in.. my method of proving it

[rustic osprey]

Don't rlly have time to read what you wrote, also it looks kinda long

$$z\bar{z}w-w\bar{w}z=z-w \implies \left(|z|^2+1\right)w=\left(|w|^2+1\right)z$$
If $z=0$, then $w=0$ and $z=w$. If $z \neq 0$, then $w=tz$ for some $t \in \mathbb{R}^{+}$. Then,
$$|z|^2 (tz)-|tz|^2 z=z-tz \implies (1-t)(t|z|^2-1)z=0$$
Since $z \neq 0$, either $t=1 \implies w=z$ or
$$t=\frac{1}{|z|^2} \implies w=\frac{z}{|z|^2}=\frac{1}{\bar{z}} \implies \bar{z}w=1 \implies z\bar{w}=1.$$

[potent lotus]

Civil Service Pigeon

[rustic osprey]

it's just this

[sullen bridge]

thx, appreciate it

.close

[calm coral]

Channel closed

Closed by @sullen bridge

Use .reopen if this was a mistake.

[sullen bridge]

.reopen

[calm coral]

✅ Original question: #help-42 message

[sullen bridge]

hol'on what is that $\bar z w=1$

[potent lotus]

Hydra_Nuker

[sullen bridge]

oh nvm

.close

[calm coral]

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[calm coral]

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[rancid sphinx]

,rccw

[rancid sphinx]

!status

[calm coral]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[rancid sphinx]

4

[bronze lily]

looks right

why is -1/2 included tho

you just said above that x =! -1/2

since 2x +1 is in the denominator

[rancid sphinx]

Oh my god

I knew something was wrong

I couldn't find it

A minus sign would be the cause of my death one day

[rancid sphinx]

I wrote x ≠ 1/2 instead of -1/2

LMAO

[bronze lily]

happy to help

!done

[calm coral]

If you are done with this channel, please mark your problem as solved by typing .close

[rancid sphinx]

.solved

[calm coral]

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[pale prairie]

\textbf{(c) Prove that $|\mathbb{R}| = |\mathcal{P}(\mathbb{Q})|$ [Hint: Find 1-1 Maps $f: \mathbb{R} \to \mathcal{P}(\mathbb{Q})$ and $g: \mathcal{P}(\mathbb{Q}) \to \mathbb{R}$]}
\\
Let $f: \mathbb{R} \to \mathcal{P}(\mathbb{Q})$. Let $f(x) = {q \in \mathbb{Q} : q < x}$. Note that $f(x)$ is a Dedekind cut, so we correctly map $\mathcal{P}(\mathbb{{Q}}) \to \mathbb{R}$. Now, we want to show $f$ is 1-1. Suppose $f(x) = f(y)$. Then ${q \in \mathbb{Q} : q < x} = {q \in \mathbb{Q} : q < y}$. If $x < y$, there would be a rational number between $x$ and $y$ hence the sets would not be equal. If $y> x$, the same argument holds, thus by trichotomy, $x = y$, hence $f$ is 1-1.
\\
Let $g: \mathcal{P}(\mathbb{Q}) \to \mathbb{R}$

[potent lotus]

toast

[pale prairie]

Hi, i am stuck coming up with a 1-1 function for g

Let $f: \mathbb{R} \to \mathcal{P}(\mathbb{Q})$. Let $f(x) = {q \in \mathbb{Q} : q < x}$. Note that $x$ is a Dedekind cut of $\mathbb{Q}$, so we correctly map $\mathbb{R} \to \mathcal{P}(\mathbb{{Q}}) $. Now, we want to show $f$ is 1-1. Suppose $f(x) = f(y)$. Then ${q \in \mathbb{Q} : q < x} = {q \in \mathbb{Q} : q < y}$. If $x < y$, there would be a rational number between $x$ and $y$ hence the sets would not be equal. If $y> x$, the same argument holds, thus by trichotomy, $x = y$, hence $f$ is 1-1.

minor fixes to part f

[potent lotus]

toast

[pale prairie]

but how do i construct g

or start doing it

[calm coral]

@pale prairie Has your question been resolved?

[pale prairie]

<@&286206848099549185>

[candid crane]

Do you know already that |Q|=|N|?

[calm coral]

@pale prairie Has your question been resolved?

[calm coral]

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Channel closed

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[languid cypress]

The product of some prime numbers, not necessarily distinct, is equal to $10$ times their sum. What are these primes?

[potent lotus]

Weierstrass Obsidian

[languid cypress]

i dont know where to start

[ionic valley]

the product is divisible by 10

[languid cypress]

mmm

Can you explain why the product must be divisible by 10?

[finite oasis]

Ok well u guarantee yourself 5 and 2 so uh

Halfway there lol

[ionic valley]

I don't wanna

[ionic valley]

because it's equal to 10x their sum so it's 10x something and something

[languid cypress]

mmm

[ionic valley]

10 is a factor

[cunning reef]

you can bound it probably

[ionic valley]

there's probably a lot of ways that this is possible

[cunning reef]

||2,3,5||

[finite oasis]

Before you start doing big brain ideas maybe just ooga it and consider cases of how many primes beyond 5 and 2 there could be

[lavish atlas]

guys i need help

[finite oasis]

There needs to be more clearly

[polar kite]

just to be clear, you're not told the number of primes involved in this?

[scarlet cargo]

!occupied

[calm coral]

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).

[ionic valley]

||don't it need to be 100 for that or smth||

[cunning reef]

oh right mb i think im stupid

[finite oasis]

,, q_1 \2 q_2 \2 q_k = 7 + q_1+ \dots q_k

[ionic valley]

because it's 10 times a number so we know 2 and 5 are prime factors
now suppose the remaining prime is p
10(p+2+5)=10p which is NOT true so there are at least 2 other prime numbers
now we can also simplify this to just (7+some primes)=some primes multiplied together

[finite oasis]

You need to have at least two extra primes

So it gotta be at least four

Maybe try solving xy = 7 + x + y and see if there are any suitable solutions

[upper prairie]

It won't be much more than 4 cuz the left hand side is growing far faster than right

[ionic valley]

I am not doing the laplace transform in the channel above this

[finite oasis]

That stuff is like ten times easier than this

[ionic valley]

it looks intimidating

[finite oasis]

Its plug and chug

Anyway besides the point

[languid cypress]

so i need to do laplace transform?

[finite oasis]

No

[languid cypress]

why

[finite oasis]

Just ignore what was said about laplace

[ionic valley]

you actually do the weierstrauss substitution

[finite oasis]

Do this

[ionic valley]

I'm joking

[upper prairie]

Why would you seriously consider it in this context at all

[polar kite]

also ideally you would also want to keep x and y small for hopefully obvious reasons.

[languid cypress]

because it's a mechanical thing

[upper prairie]

Consider 5 primes case

[languid cypress]

but there isnt a finite limit

[polar kite]

you'll still ideally want not too many primes because there's a product on one side and a sum on the other.

[ionic valley]

funny

[upper prairie]

2^p ≤ 7 + 2p will hold only up to some finite p

[upper prairie]

Yes

[languid cypress]

?

[upper prairie]

Substituting anything higher than 2 will only cause the LHS to grow larger

[ionic valley]

I found out the answer

[upper prairie]

You'll find that this fails for p = 4

So you really only need to check for 2 cases: 4 primes and 5 primes

[languid cypress]

how

why not 8 primes

[upper prairie]

Are you listening to my explanation at all

[ionic valley]

because of what the ant lemonade fan said

[languid cypress]

I don't understand what's written there

[plucky oak]

simply put, consider a product and a sum. which grows faster: the product or the sum?

[ionic valley]

they're substituting all the primes as some p amount of 2 since that's easier to interpret and gives us a rough guess as to how many primes we have to place

[languid cypress]

product

[plucky oak]

if the product grows faster then as you add more elements the product will blow up beyond what the sum can reach.

(with the same elements.)

so the more primes you have in your list the higher your odds of having your product be higher than 10 times your sum simply because the growth rate of a product is faster than that of a sum.

[ionic valley]

and all primes afterwards would be larger than 2 o this is the absolute max number of primes

[vital cosmos]

Notice:
For prime numbers 3
We have: 3k + 1 3k + 2

[ionic valley]

is 1 a prime

[vital cosmos]

NO

[ionic valley]

is 4 a prime

[languid cypress]

yes

[polar kite]

what??

[vital cosmos]

huh???

[languid cypress]

because one wasn't

[pure breach]

uh?

[finite oasis]

This might be a troll

[languid cypress]

1 can only be divided by one and 0

[vital cosmos]

Alright 3k+1 for even and 3k + 2 for odd

no

[ionic valley]

even primes?

[lyric ravine]

1 is by convention not a prime because then every other number would not be a prime, and also 1 cannot be divide by 0

[languid cypress]

ok ,thanks

[vital cosmos]

So 3k + 1 for odd so k must even 3k + 2 k must be odd

k

2 3 5, 5 5, 3 7 ?

[ionic valley]

I wonder what u were trying to say hmm

[ionic valley]

going back to this, you can further this by making it
xy-x-y+1=8
(x-1)(y-1)=8 x and y have to be prime numbers

[languid cypress]

isnt just 9?

[calm coral]

@languid cypress Has your question been resolved?

[languid cypress]

<@&286206848099549185>

[open jolt]

Is this the question?

The product of some prime numbers, not necessarily distinct, is equal to $10$ times their sum. What are these primes?

[potent lotus]

Matt V

[languid cypress]

yes

any idea?

[open jolt]

I have not tried this before, but I would use the "Series" formula/method.

[languid cypress]

what formula

[open jolt]

Such as: P1 x P2 x ... x Pn = 10 x ( P1 + P2 + ... + Pn )

Left side - product
Right side - sum

And since 2 x 5 = 10 and 2 and 5 closest to 0 are prime. I would think it is the first number.

I think that's as far as I can get it.

• So at least we know that the prime numbers can be 2 and 5.

[languid cypress]

yes

[nimble flume]

@languid cypress i think i got it

the answer is 4 prime numbers

2,3,5,5

[languid cypress]

why

[nimble flume]

check them

product=10(sum)

[languid cypress]

why are there 2 fives?

[nimble flume]

there can be

cause it's prime

[languid cypress]

why not 3

[nimble flume]

3 is also a prime number

[languid cypress]

i mean why not only 2,3,5

but 2,3,5,5

[nimble flume]

2,3,5 dont prove the equation

[languid cypress]

?

[nimble flume]

2,3,5,5 proves that

[languid cypress]

???

[nimble flume]

the question didnt restrict to use a prime number once

[languid cypress]

why not 2,3,3,5

[nimble flume]

let me check it for that

[languid cypress]

how do you check?

[chrome reef]

2*3*3*5=90
2+3+3+5=13
13*10=130
so not possible

[nimble flume]

@chrome reef help him I need to go

[chrome reef]

well you can just check it

like by finding the product and 10*sum

[languid cypress]

no i just close the channel

I don't trust much

[chrome reef]

!done

[calm coral]

If you are done with this channel, please mark your problem as solved by typing .close

[languid cypress]

It seems a bit suspicious to me: the solution came out immediately, but when they asked you how you controlled it, you disappeared.

[rain portal]

Probably AI usage.

[languid cypress]

@nimble flume

[chrome reef]

aren’t you asking for a possible primes where the product = 10(sum)

2*3*5*5=150
2+3+5+5=15
10(15)=150

[languid cypress]

Ok, but without redoing the whole solution: why after removing 2 and 5 do we end up looking for primes with product = sum + 7?

[chrome reef]

i think it’s cuz u already know 2 and 5 must be in the equation

[languid cypress]

Yeah but that only explains why 2 and 5 are included. I’m asking why removing them changes the equation into “remaining product = remaining sum + 7”. Can you write that algebra step? pls

[chrome reef]

sure

[languid cypress]

can you write in latex pls

[chrome reef]

yeah

[languid cypress]

?

where is the code

[chrome reef]

latex

it’s on my server

[languid cypress]

server?

[chrome reef]

discord server

[languid cypress]

can i join?

[chrome reef]

no

[languid cypress]

why

[chrome reef]

private

[languid cypress]

but i need the code

so i can copy it

[chrome reef]

why

[languid cypress]

to put on my paper

overleaf

[chrome reef]

shouldn’t you write it…?

[cedar oyster]

you intend to just copy solutions?

[chrome reef]

!noans

[calm coral]

The purpose of this server is to help you learn; please don't ask for direct answers. Ask for guidance, explanations, or feedback instead.

[languid cypress]

i need the code

just copy and paste here

@chrome reef

[cedar oyster]

you have no honor

this is nothing more than a blatant shameless attempt at plagiarism

[languid cypress]

in reality because the photo (even deleted) was quite obviously chatgpt

[cedar oyster]

the source of the material you claim as your own is not important

the shameful feature is that you'd like to claim something you did not write as your own

[languid cypress]

why are you telling me these things

[cedar oyster]

because you ought to hear it

[languid cypress]

I don't really care what you're writing, honestly.

[cedar oyster]

lol

[chrome reef]

you’re telling me that latex is chatgpt?

[languid cypress]

yes

[chrome reef]

interesting

[languid cypress]

<@&268886789983436800> sorry to ping, I just want clarification. A solution image was posted and then deleted after I asked for the code/source. He also said it was hosted on his own server, but no source was provided. I’m not accusing anyone; I just thought it looked suspicious and wanted to know if it can be checked.

[stark sparrow]

Asking for peoples code to copy because you want to copy a solution is against the rules btw

[languid cypress]

What's the difference between copying from a photo or having the code?

[drifting seal]

both are against the rules

[balmy bane]

You're essentially asking for a solution, either way.

[languid cypress]

I am not understood

[stark sparrow]

Coming onto this server to get solutions you can copy is against the rules

[balmy bane]

When asking a question, show what you tried to solve the problem. We are not here to do your homework for you, but we are here to help you learn how to do your homework.

[stark sparrow]

We're here to help you learn not help you cheat

[languid cypress]

also I don't think it was the solution

[hoary quarry]

👀 dra,ma

[balmy bane]

People may help assist you with a solution; but that's not the same as you directly requesting one

Now you did mention it as requesting LaTeX code, but that's not functionally distinct

Admittedly, sniper didn't really help steer that discussion in a productive direction

[languid cypress]

I wasn’t asking to copy anything. I asked because the deleted image and the “own server” explanation sounded like excuses to me, and I don’t like being misled. But I understand the rules, so I won’t ask for source/code anymore.

[balmy bane]

It's more this message I think some ppl are grappling with, were I to hazard a guess

[restive geyser]

hi

[languid cypress]

That message was meant as a check to see if he understood what he posted, not as me asking for the solution. I already knew the step; I was trying to verify whether the deleted image/source was AI/copypasted.

[tiny lark]

-# lets js stick to the Qsn

[languid cypress]

I am not understood

[chrome reef]

mb

[chrome reef]

yes i do understand

i can help explain it to you

but i can’t just outright give you the answer

[languid cypress]

but before it was possible

[chrome reef]

?

[languid cypress]

up to this point the solution could be written

[chrome reef]

then it should be written by you

[languid cypress]

you send the photo first and then write this, it doesn't make sense

[chrome reef]

yeah i forgot i cant give you the answer so thats my bad on my part

[cedar oyster]

well I would comment, but...

[languid cypress]

no context

[cedar oyster]

imo you were fine in writing what you did, it didn't directly answer the original question after all

[chrome reef]

that’s true

i think OP should write the latex code though

[languid cypress]

Fair enough. I’ll stop asking for source/code. I just find it funny how the deleted image suddenly became “not the answer” only after people started questioning it.

[chrome reef]

it’s not the answer but it’s steps towards the answer

[languid cypress]

I just want to understand whether you actually followed the reasoning

[cedar oyster]

no, seeing the raw tex does not give any indication for the legitimacy of the argument

you're supposed to just read the argument and think for yourself

[languid cypress]

i understand. I’m not asking for the LaTeX anymore. I just wanted to know whether he understood what he posted, since the image was deleted and the later explanation was unclear.

[chrome reef]

i understand what i posted

[cedar oyster]

up to an abuse of notation (the number of primes ought to change when you remove two of them), the conclusion that the resulting product was 7 more than the resulting sum seemed fair

and indeed something you could recreate

[calm coral]

@languid cypress Has your question been resolved?

[calm coral]

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[soft nova]

What should be the thinking process in solving this

[tiny lark]

,rccw

[tiny lark]

well, are you familiar with cot(x) = tan ( pi/2 - x)

its called complementary angles

[soft nova]

Yes

[tiny lark]

Okay and are you also familiar with the fact that tan(x) * cot(x) = 1

[soft nova]

Now that I think about it, I get why tan(x) times cot(x) is 1

[tiny lark]

so you understand this statement right?

[soft nova]

Yes

[tiny lark]

nice

Well can you see that the first term and the last term,

they can be manipulated using the statements i mentioned above

[tiny lark]

.

[tiny lark]

.

lmk if it doesnt strike you yet

No?

[soft nova]

Trying

[tiny lark]

Oh sorry sure try!

[soft nova]

Oh I see

tan(pi/2 - pi/20)
is tan(9π/20) and the last term is cot (9π/20) so they become 1

[tiny lark]

Yup!!

nice one

[soft nova]

Same with the 2nd and 4th term

So answer is cot(5π/20) ?

[tiny lark]

Yeah!!

Good job

[soft nova]

TYSM

[tiny lark]

and this is a standard value no?

no using calcs

[soft nova]

It would be cot(9 x 5) or cot(45) which is 1

[tiny lark]

Ayy

W

[astral hornet]

holy tuff 🤑

tryna be positive to random ppl can't even do that 😭

[tiny lark]

So is that all @soft nova

Or anymore?

[soft nova]

no

I have more

[tiny lark]

Okay then post

Yeup

Butt,

first try on your own

[soft nova]

Any one of these

[rustic osprey]

.pin

[soft nova]

.close

[calm coral]

Channel closed

Closed by @soft nova

Use .reopen if this was a mistake.

[calm coral]

Available help channel!

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• Show your work, and if possible, explain where you are stuck.
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[still terrace]

the xy plane can be defined as E: x + y = 0 ?

[solid pasture]

?

x + y = 0 is a line
y = -x

[still terrace]

x + y + 0z = 0

[marsh valley]

This still wouldn't be the xy-plane.

It's a plane

[solid pasture]

the equation of the xy-plane is z=0

[still terrace]

wait why

actually yes

wtf

yes exactly

because z = 0

[marsh valley]

z is not 0

The coefficient is 0

You have no restriction on z in your equation.

[still terrace]

yeah so

in Plane notation thing

[marsh valley]

So it's not the xy plane.

[still terrace]

how do i notate the xy plane

[marsh valley]

z=0

[solid pasture]

z = 0

[still terrace]

??

or were u talking aboutwhat i said

[marsh valley]

x+y + 0z=0 is a plane, it's just not the xy-plane/

[still terrace]

yeah

ty

also it could aslo be z = 5 or something right

or

no nvm

nah

the d thing confused me

[solid pasture]

d?

[still terrace]

ax+bx+cx = d

[solid pasture]

ohh

[still terrace]

so ur basically saying z is literally 0 so your plane is literally in the xy thing

yeah

[solid pasture]

yes

noi restriction on where x and y is so they can be anywhere as long as z = 0

[still terrace]

so wait can i post another proble in here

[solid pasture]

yes

[ionic valley]

u been studying for a while bro

[solid pasture]

hes locked in

[still terrace]

hell nah man i can barely do like 50% of the content

[solid pasture]

what content are you doing

[solid pasture]

oh okay

[still terrace]

analysis, linear algebra and probability

[still terrace]

so for the 1.2

do i just calulate the intersecting point with Xy plane and E

and then use the cosine formula for the angle

[ionic valley]

1.1 use distance formula for
1.2 think in xy, yz, and xz plane

wait

oh ok

nvm I misunderstood this

[solid pasture]

yes

[ionic valley]

for 1.2 you wanna think just the xy plane, so z=0
you're going to get the linear intersection between them which is makes it fairly easy to tell what the angle is

[still terrace]

shit okay yeah

i gotta relearn how to do that rq

[marsh valley]

(You may also just take normal vectors to E and the xy-plane and compute the dot product to solve for the angle)

[still terrace]

btw,

for the angle thing

there r 2 formulas

one with sine and one with cosine

when do u use which

[marsh valley]

The sine one is for the magnitude of the cross product.

[solid pasture]

sine to find area

[ionic valley]

there's an easier way to think about this

it intersects with the xy plane

so we know z=0

plug it into your plane

[still terrace]

what

z = 0

?

[ionic valley]

z=0 is the xy plane

if you can imagine it

[still terrace]

yes

i can

[ionic valley]

so what is the intersection of the plane described with the xy plane

[marsh valley]

At the end of the day finding the intersection may help in some way, but the normal vectors is usually how you find the angle between planes.

[ionic valley]

^ yeah

[still terrace]

ty

ill finish this tomorrow im tired

[calm coral]

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[calm coral]

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[pale prairie]

Let $S$ be the set of finite sequences of $X$. What is the best approach of showing $|S| \leq |X|$?

[potent lotus]

toast

[pale prairie]

Assuming X is infinite

@trail epoch

<@&286206848099549185>

[rancid sphinx]

blud. you really ask the most absurd questions

are you pursuing something in mathematics?

like a UG?

[pale prairie]

yeah ug

[pale prairie]

btw the issue with this is ordering

like $(abc) \neq (acb)$ but for subsets, ${a,b,c} = {a,c,b}$

[potent lotus]

toast

[pale prairie]

so perhaps i need to come up with an injection from S to X

i guess

is a sequence just a cartiesian product?

[mortal orbit]

Could you show the set of sequences of length n in X is in bijection with X?

[pale prairie]

i think a bijection is hard

[mortal orbit]

well, injection

[pale prairie]

We finally want to show that $|S| \leq |X|$. Observe that for any $s \in S$, $s \in X^n$ for some $n \in \omega$. That is, $S = \displaystyle \bigcup_{n \in \omega} X^n$. So $|S| = \displaystyle|\bigcup_{n \in \omega}X^n |$. Observe that for any $n \in \omega$, $|X^n| =|X|$. Then $\displaystyle|\bigcup_{n\in\omega}X^n| \leq \sum_{n \in \omega}|X^n| = \aleph_0 \cdot |X| = |X|$. Thus $|S| \leq |X|$.

this is what i did

[potent lotus]

toast

[mortal orbit]

well that's basically it

[pale prairie]

kk

I think

the two things i kind of assumed is the $\leq$ part

[potent lotus]

toast

[pale prairie]

but can i just say it comes from the definitino of cardinal arithmatic

\textbf{Definition - Cardinal Addition:} \
$\kappa + \lambda = |X \cup Y |$ whenever $|X| = \kappa$, $|Y| = \lambda$, and $X \cap Y = \emptyset$\

[potent lotus]

toast

[pale prairie]

since we dont know whether the sets are disjoint

then $|X \cup Y| \leq \kappa + \lambda$?

[potent lotus]

toast

[pale prairie]

\subsection{Problem 1.} \textbf{Prove that for any infinite set $X$, the cardinality of $X$, the cardinality of finite subsets of $X$, and the cardinality of the set of finite sequences from $X$ are all equal.}
\\
\textit{Proof:} Let $|X| = \kappa$. Let $C = {Y \subset X : |Y| < \aleph_0}.$ We want to claim that $|X| \leq |C|$. We first want to find an injection $f: X \to C$. Let $x \in X$. Let $f(x) = {x}$. Observe that ${x}\in C$, since ${x} \subseteq X$. Now let $f(x) = f(x')$ for some $f(x'), f(x) \in C$. That is, ${x} = {x'} \iff x = x'$ by Axiom of Extensionality. so $f: X \to C, f(x) ={x}$ is an injection, so $|X| \leq |C|$.
\\
Let $S=$ finite sequences of $X$. We now want to show that $|C| \leq |S|$. We first want to find an injection $g: C \to S$. Let $A \in C$ be any finite subset of $X$ with cardinality of $n$. We can uniquely order $A = {a_1, a_2, a_3, ..., a_n}$ since $A$ is finite, so fix a well-ordering $\leq$ on $A$ where $a_1 \leq a_2 \leq ... \leq a_n$. Then let $g(A) = (a_1a_2....a_n)$. Because $a_1, a_2, ..., a_n \in X$, this is a valid sequence of $X$, hence $g(A) \in S$. It also is injective since our ordering is unique for any subset.
\\
We finally want to show that $|S| \leq |X|$. Observe that for any $s \in S$, $s \in X^n$ for some $n \in \omega$. That is, $S = \displaystyle \bigcup_{n \in \omega} X^n$. So $|S| = \displaystyle|\bigcup_{n \in \omega}X^n |$. Observe that for any $n \in \omega$, $|X^n| =|X|$. Then $\displaystyle|\bigcup_{n\in\omega}X^n| \leq \sum_{n \in \omega}|X^n| = \aleph_0 \cdot |X| = |X|$. Note the $\leq$ comes from the definition of cardinal addition, where we are not guaranteed disjointedness. Thus $|S| \leq |X|$.
\\
Therefore, $|X| \leq |C| \leq |S| \leq |X| \implies |X| = |C| = |S|$. $\blacksquare$

Here is my full proof

[potent lotus]

toast

[pale prairie]

.solved

[calm coral]

Channel closed

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[calm coral]

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[marble isle]

Hello, I am studying for my math test but am confused. I don’t want to resort fully to chatbots to help out so I’d like some human help, haha.

Lately I’ve had to miss some of my classes due to personal issues.

Here is the first page if anyone wants to help and we can see how the rest of things go.

[minor osprey]

Whixh part is confusing you? Is it the definitions of the mean, median and mode?

[marble isle]

Generally all of it. But that’s probably where to start, yes

[minor osprey]

So for the mean, you have to add all the values together and divide by how many there are

[marble isle]

in totality, its 100 right?

[minor osprey]

Yep

[marble isle]

and i divide by 6?

thanks, kai!

[minor osprey]

Yeah🥰

[azure geode]

np

[marble isle]

i got 16.6666666667, i forget what to do in circmstances like this

[azure geode]

js do it in this form x/y

dont devide to get exact number or wtv

[minor osprey]

If i read the sheet write doesnt it say to round to whole numbers? Or do they just mean the values the gave are rounded

[marble isle]

it does say round answers to the nearest whole number yes

[marble isle]

i dont understand this haha, i should clarify this is like the bottom level college math class, never been my strong suit

[lone schooner]

OK so in this case, as it is smth.5 or higher (smth.6666667), you should round up

if it were lower than smth.5, round down

so 16.666666667 rounded up is ...

[marble isle]

ok so its 16.6 which is 16.7?

[lone schooner]

no

whole numbers, remember?

[limpid lagoon]

whole numbers means no decimals. numbers like 1, 2, 3, etc.

[azure geode]

if it says give the exact value just write it as a fraction if it says round to cosest whole number or decimal or something like that just do as the other guy says

[limpid lagoon]

the question states to round to the nearest whole number

[marble isle]

so its just 17?

[lone schooner]

yes

[azure geode]

the whole discord server is in this channel imma stop bothering bye

[marble isle]

your contribution was still appreciated, thanks!

[azure geode]

np

[marble isle]

ok so now the median

[limpid lagoon]

https://youtu.be/e3uY2LraXts?si=HYJpw6YGgWMWk-Ap

watch this video to help you remember all the definitions

[lone schooner]

use this, it genuinely helped me (ik it sounds stupid):
hey diddle diddle, the median's the middle, you add and divide for the mean. The mode is the one that appears the most and the range is the difference between.

Its like to the tune of a nursery rhyme or smth

[marble isle]

i see, what does the middle mean in this context?

[lone schooner]

its like if u arrange them in order of lowest to highest, you pick the middle one

e.g

(from lowest to highesr)

a,b,c,d,e,f,g

d is in the middle

(1,2,3,4,5,6,7)

(4 is in the middle)

but if u have an even number of items:

h,i,j,k,l,m

The middle is j and k

so u take the mean of those

(j+k)/2

[limpid lagoon]

something that helps is arrange them lowest to highest, then cross one off from each side until you get to only one or two left

and if there's 2 left then do what was mentioned above and find average

[lone schooner]

just take the number of items, add one, divide by two.

Hope i helped, slide into dms if u need more help cuz i gtg @marble isle

[marble isle]

you did, thanks!

so its 18?

[azure geode]

yoo krish can u hop on my channel quickly pls

[marble isle]

and the mode is most, so itd be 18 too

for 2., am i divding 601 by 7 because its in a week?

Here’s what I got

[rustic osprey]

this all looks fine

though you can also check this yourself in #bots

,w 83, 88, 86, 87, 88, 84, 85

[potent lotus]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by civil_service_pigeon

[rustic osprey]

example for mean and median

[marble isle]

oh i didnt know that, interesting. thank you

[rustic osprey]

[calm coral]

If you are done with this channel, please mark your problem as solved by typing .close

[marble isle]

Need help on the lower half of this page

[minor osprey]

Who is responsible for printing that💔 💔

[marble isle]

haha it is a bit awkward isnt it

[minor osprey]

You first find the renevue for both 2000 and 1998

And substact to find the difference

[marble isle]

2000 - 1998 is 2 right?

[minor osprey]

I mean you lok at the graph to find the revenue for both eyar

[marble isle]

ohh

[minor osprey]

Years

[marble isle]

240 - 210?

[minor osprey]

Uhh i can hardly read that graph 🥀

[marble isle]

a little closer

[minor osprey]

Isnt it 230-200😭

[marble isle]

im not sure!

[minor osprey]

This is how i assume it was meant to be

[marble isle]

i see i see, thank you, i appreciate it.

then yes its 230 - 200 haha

30

[minor osprey]

Remmebr that its in milliosn

[marble isle]

so 30 million?

[minor osprey]

Yes🥰

[marble isle]

im confused on what to do for c as well

[minor osprey]

Uhh

Idk what they would expect you to write 😭

[marble isle]

same!

[minor osprey]

Like, just look at the graph bro💔 💔

[minor osprey]

It’s obvious when it’s decreasing…

Yeah idk about that one bro sry gang

[marble isle]

all good

Second to last page

[calm coral]

@marble isle Has your question been resolved?

[rain portal]

Which question?

Question 7?

[marble isle]

thatd be where to start yeth

[rain portal]

Which one do you think is greater, -3 or -7?

[marble isle]

-3 is closer to positive, so -3

[rain portal]

Correct.

Now, let's go with b.

[ember orchid]

so the alligator eats -3 😄

[marble isle]

for b, does it cancel out because theyre both negative?

and they become positive

[rain portal]

They are not equal, are they?

[ember orchid]

you can do that but then the alligator also flips

[rain portal]

2/3 and 3/10, which one is closer to the origin?

[marble isle]

wym origin

[rain portal]

The point (0,0).

Or, the position.

[marble isle]

well when you divide -3/10, its -0.3, so i suppose thats closer

[rain portal]

Correct.

-2/3 is -0.6666 and -3/10 is -0.3

[marble isle]

for c, 1/3 is already positive so its that right

[rain portal]

Perfectly correct.

How about d?

Notice they are all decimals.

[ornate hornet]

Want a challange? Calculate the median. 😰

Q.6

[marble isle]

.632?

[rain portal]

Correct.

[marble isle]

haha ill try once the other stuff is over with

[rain portal]

Wait.

[ornate hornet]

Np mate!!

[rain portal]

0.632 is greater than 0.63

Since 0.632-0.63=0.002

So the answer would be opposite @marble isle .

[marble isle]

Oh ok ok

-5.5 for e?

[rustic osprey]

.pin

[marble isle]

Appreciated. I do need help with the stuff above as well

[azure geode]

i recommend u to learn a bit independantly so u can be confident with ur answers having someone help on every question wont make u develop problem solving skills

[marble isle]

I agree! I often do, but like I said personal issues have gotten in the way from me studying a lot this time and my test is soon.

[azure geode]

aha makes sense im in the same boat

but yes ur answer is correct

anyway goodnight

[marble isle]

Night 😄

[calm coral]

@marble isle Has your question been resolved?

[rain portal]

If you are done, please close the channel.

By typing .close

[rain portal]

Good luck!

[marble isle]

.close

[calm coral]

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[pale prairie]

(k) $|{f : \mathbb{R} \to \mathbb{R} : f$ is continuous}|

[potent lotus]

toast

[pale prairie]

hmm

how much does the continuity part really matter

evidently it should be less than R^R

but is it also equal to R^R?

i.e is $|\mathbb{R}|^{\mathbb{R}} \leq |F|$

[potent lotus]

toast

[toxic stump]

i think continuity matters a lot just by density of Q in R

[pale prairie]

hmm

[toxic stump]

i think you'd get knocked down to $\mathfrak c$ instead of $2^{\mathfrak c}$

[potent lotus]

blanketism

[toxic stump]

but im a little rusty

[pale prairie]

o

interesting

what would be the intuition for that

[toxic stump]

well if we didn't care continuity, we can just slap on whatever- but if we cared about continuity and reals are limits of rational sequences, then we must take $2^{\aleph_0}$ into consideration i suppose

[potent lotus]

blanketism

[pale prairie]

hmm

[calm coral]

@pale prairie Has your question been resolved?

[calm coral]

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[radiant rune]

What has he done?

[radiant rune]

How has he solved this problem

[tiny lark]

wasnt there a shortcur for this

smth like add the numerators and the denominators?

[radiant rune]

Yes

[tiny lark]

so thats done then?

7+11= 18

(x+y) + ( z-y) = x+ z

brings us to 18/(x+z)

[radiant rune]

I understand that

But

[tiny lark]

do you want the proof for this maybe.

[radiant rune]

How can he write 7/x+y = k/x+x = 11/z-y = the sum of 7/x+y and 11/z-y

You can't write 2 = 2 = 2+1

[tiny lark]

so you do want the proof

[tiny lark]

it would actually be 2/1 = 2/1 = (2+2)/(1+1)

[tiny lark]

?

[radiant rune]

No

so is a/b = c/d = a+c/b+d?

Because they are adding numerator

And denominator

[tiny lark]

yeah

famous property hai ye

kaafi jagah use hoti hai

mera matlab apke standard mein atleast

[radiant rune]

Oh

Anyways speak english, i think it's the rule

[tiny lark]

not in help channels

[radiant rune]

Is it componendo and dividendo?

Oh

[tiny lark]

but haan its somewhat related to C-D

[radiant rune]

So did he used (a+b)/(a-b)?

[tiny lark]

oh wait pura statement nahi likha

a/b = c/d
(a+b)/(a-b) = (c+d)/(c-d)

[tiny lark]

Nahi

[tiny lark]

ye use kiya

[radiant rune]

Yeh konsi property hoti hsi

[tiny lark]

but iska proof uses C-D i think

[radiant rune]

It is not componendo and dividendo

[tiny lark]

iska naam nahi malum mujhe but its a property

[tiny lark]

idts ruko dekhta

[radiant rune]

Have you qualified IOQM/RMO? btw

Or INMO also

[tiny lark]

Haan this is componendo

[tiny lark]

nahi am in 11th rn stidying for jee🥀

[radiant rune]

I see

[tiny lark]

this property is called componendo

[tiny lark]

If we combine both C and D, we get this

[radiant rune]

Bruh componendo and dividend was a+b/a-b = c+d/c-d

[tiny lark]

Yes thats both of them applied

[tiny lark]

I am saying only componendo applied is this

[radiant rune]

Componendo is a+b/b = c+d/d not a+c/b+d

[tiny lark]

weird i thought it was componendo

either the proof for this property is

let a/b = c/d = k (k is a constant)
a = bk
c = dk
(a+c) = k(b+d)
therefore, k = (a+c)/(b+d)

iska naam mujhe nahi malum then but this is the proprty ig

[radiant rune]

Oh

It is being manipulated from componendo

[radiant rune]

How could I think of doing such thing

How will it came to my mind

After seeing the problem

And those three equation in fraction

[opaque field]

It's a common process whenever you have something=something=something then you equate them to some k to write in one parameter

[calm coral]

@radiant rune Has your question been resolved?

[rugged eagle]

-# what the blud

[calm coral]

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[radiant rune]

What to do after wiritng (ab + bc + ca)/abc = 6?

[finite oasis]

Take the reciprocal

And multiply by 24

[radiant rune]

Oh got it

Thanks

.close

[calm coral]

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[calm coral]

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[simple musk]

how to realize the drawing of this surface

the lad probably did traces?

can someone help me?

[ember orchid]

do you mean how do you understand what this surface is

?

[simple musk]

the geometrical intuition for this region dawg

[ember orchid]

if so I can explain that

okay

so let’s take the simplest case and evaluate the 1<x^2+y^2<=2 for z=0

notice the x^2+y^2 ?

this is just a circle

and then since it’s 1<= we only include values with radius greater than 1

so this simple region looks something like a circle with a whole in the middle

now extend upward, evaluate at z=1

1<x^2+y^2<3

our outer circle is bigger and our hole remains the same

that is all the intuition you need

it’ll be circles continuously stacked on top of each other

where the outer radius increases as the magnitude of its height increases

and the center hole remains at a constant radius of 1

[ember orchid]

does that make sense ?

[simple musk]

no

[ember orchid]

what are you having trouble with?

[simple musk]

everything

[ember orchid]

do you understand what i mean when I say evaluate the function @ z=0

?

[simple musk]

its <= in both sides dawg

[ember orchid]

i’m aware

[simple musk]

why are you doing strict

[ember orchid]

i just had a typo

doesn’t mean too much

[simple musk]

?

[ember orchid]

do you understand what i mean when i evaluted the function at z=0?

[simple musk]

that's what we call a trace

[ember orchid]

yes

are you able to visualize that that trace looks like?

[simple musk]

no

[ember orchid]

so yk how x^2+y^2=r^2 generates a circle of radius r?

[simple musk]

center origin

[ember orchid]

now, just place limits on it

@ z=0

the trace is

1<=x^2+y^2<=2

this generates a circle where r is now bounded

[simple musk]

if you place limits is a disk.

[ember orchid]

1<=r^2<=2

?

[simple musk]

disk.

[ember orchid]

uhhh huh

as i said earlier a circle with the hole cut out in the center

[simple musk]

it's a disk not a circle

[ember orchid]

do you understand why this trace @ z=0 looks as such?

[ember orchid]

why are you so caught up on semantics?

a circle with a cutout hole is an apt and fine description

if you prefer the term disk then that’s suitable for you

[simple musk]

why

[simple musk]

why

hole cut..?

[ember orchid]

but if you are able to come to the conclusion that it means disk then that means my description did a fine job at conveying the info

[ember orchid]

radius has a lower boundary of 1

this means that any value r<1 is excluded

[simple musk]

can u do a drawing

[ember orchid]

i’m on my phone so it’s low quality

but here

since we have bounded our region such that 1<=r^2<=2 @ z=0

all values of r<1 are excluded

this leaves a circular cutout at the center of r=1

[ember orchid]

once again, semantics.

[simple musk]

yeah

[ember orchid]

If you are able to come to the conclusion that i’m referring to a disk, then my description was fine

[simple musk]

sure and everything in between r < 1

[ember orchid]

The conveying of information still occurred and there’s no reason to harp on it.

[ember orchid]

so does this now make sense why there’s a central hole?

[simple musk]

sure

[ember orchid]

and then you can see just by intuition that the outer radius grows as z increases in magnitude

[simple musk]

wat

[ember orchid]

the outer radius of your disk is defined by z^2 + 2

as the |z| increases so will the outer radius of the disk

[simple musk]

ah yes

[ember orchid]

so that’s why you end up with a sort of hourglass shape

the smallest radius occurs @ z=0 with a radius of sqrt(2)

[simple musk]

what

[ember orchid]

then increases in both the positive and negative directions

[simple musk]

,w hourglass

[potent lotus]

Failed to get a response from Wolfram Alpha.
If the problem persists, please contact support.

May 5, 2026, 8:09 AM

[ember orchid]

https://tenor.com/view/hourglass-hourglass-in-space-gif-300053338439957676

except this shape would continue growing infinitely as z increases

[simple musk]

you mean a double cone?

[ember orchid]

if you prefer that description sure.

[ember orchid]

👍

[simple musk]

i domt get it

[ember orchid]

what exactly are you struggling with?

[simple musk]

everything

[ember orchid]

you just seemed to grasp the trace fine

[simple musk]

dont get it

[ember orchid]

goodluck bro i hope you’re able to find it out

[simple musk]

so hard

.close

[calm coral]

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[calm coral]

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[iron ore]

How do I find the height of the box if it’s not given

[west viper]

intuitively

[ionic valley]

it's x

[west viper]

when you fold the sides up

the side's width is now the box's height

so yeah its x = 2

[iron ore]

Wait how did you know it was x

I th ought x was just the cut out corners?

[west viper]

ay

pay attention

x is the length of the cut out corner right?

[iron ore]

Yea

[ionic valley]

what happens when you fold that weird section upwards

[green mica]

Whats going on?

[iron ore]

Ohh

[west viper]

its a square so the side are all equal to x

so the box's current side has the same width as x

[iron ore]

And for this one I just don’t know what to do next at all

[ionic valley]

can you zoom in closer to the two questions

[finite sedge]

,rccw

[vivid geyser]

Note that l=w

[iron ore]

Oh yea

[vivid geyser]

62 <= w+l+h, not w.l.h

[ionic valley]

ok for the first one you have l+w+h__<__62
let's assume the maximum since we want the maximum volume so l+w+h=62, now since the base is a square we can assume l=w so let's use 2w, you get 2w+h=62 and then you optimize

[iron ore]

Wait why would it not be multiplication I thought the volume equation was L•W•H

[ionic valley]

it specifically said length+width+height

[vivid geyser]

length+width+height must less than 62, it is the first sentence of the question

[iron ore]

Oh okay but I’m still not sure what to do once I got h by its self

[vivid geyser]

Well, have you learned derivative?

[iron ore]

I don’t think so but I maybe might just don’t know the name

[vivid geyser]

Okay, firstly, let's have w=l, right?

Now, we have $w+h+l\leq 62$, can you simplify it?

[potent lotus]

❤ Neferia ❤

[iron ore]

I made it into h=62-L-W

[vivid geyser]

But l = w?

[iron ore]

Yea I know

[vivid geyser]

So h = 62 - 2l

right?

[iron ore]

Yea

[vivid geyser]

Now, let's do $V=w \cdot h \cdot l$

[potent lotus]

❤ Neferia ❤

[vivid geyser]

Can you substitute h in?

[vivid geyser]

we knew $w = l$, so you can write it as: $V= l \cdot (62-2l) \cdot l$

[potent lotus]

❤ Neferia ❤

[iron ore]

Oh yea

[vivid geyser]

Can you keep simplifying them?

[vivid geyser]

Not quite right, it would be $(62L-L^2) \cdot L$

[potent lotus]

❤ Neferia ❤

[vivid geyser]

You are missing a bracket.

[iron ore]

Ah ok

[vivid geyser]

Can you keep multiplying L in?

[rain portal]

Yes, in addition, I agree that derivative could be useful here.

Then we could set derivative = 0.

When you are done, feel free to send your solutions here.

[iron ore]

62L^2-2L^3?

[rain portal]

Correct!

[iron ore]

I’m not sure what ti means to set the derivative to 0

[rain portal]

Now, have you studied derivative?

Have you seen this before? @iron ore

Any of these notations?

[rain portal]

No problem.

[iron ore]

we haven’t don’t that we have been doing at some point in the problem finding the max or min and using those coordinates points

[rain portal]

Oh, really? It won't be as fast as derivative then.

But... Hmm, this kind of problem do require derivative.

Can you elaborate on what does it means "coordinate points"?

[iron ore]

This is the answer key

[rain portal]

I am not sure what does he/she means by 2nd -> Trace -> Max

This is interesting...

[rain portal]

Ahh, by calculator?

Yeah, it would solve the problem smoothly.

[iron ore]

If I did it with the later version of the problem I did like the one with the l^3 would I still be able to put that in the calculator?

[rain portal]

Ah, yes.

It is still correct.

[calm coral]

@iron ore Has your question been resolved?

[calm coral]

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[calm coral]

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[wise nacelle]

Hi

[foggy topaz]

Do you have a specific question?

[scarlet cargo]

-# ig they just wanted to say hi lol

[drifting jay]

what is the question?

[foggy topaz]

Its a possible troll he did it in another channel too

[plucky oak]

!done if you do not have a question, or please send one!

[calm coral]

If you are done with this channel, please mark your problem as solved by typing .close

[wise nacelle]

Sorry

[plucky oak]

so do you have a question?

if not, you can type .close to close the channel.

[finite oasis]

.close

[calm coral]

Channel closed

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[honest compass]

A 3D Cube, ABCD (Bottom Face) EFGH (Top Face).
P = Middle of EH
Q = Middle of FB
R = Middle of GC

T = Middle of EA
U = Middle of HD
V = Middle of BC

All the sides are 8cm distance, so point A to E is 8cm, B to C is 8cm, etc.
Find the closest distance from face PQR to face TUV...

[honest compass]

I am genuienly stuck ;-;

[thin hill]

Lets call P having a height of 8, therefore Q,R,T and U all have a height of 4. And V has height of 0

If we look at it, both faces are going down the same total distance (4) over the same length (8). Therefore, they have the same slope. So if they have the same slope, then the distence between them is constant. So we just need to find the distence between two convient points that are directly above eachother.

We can use P and the midpoint of UT. We have P=heright 8 and hieght of midpoint of UT is 4, then is distance is 8-4=4

Does that make sense?

[honest compass]

The face of PQR to face of TUV isn't 4cm because it's not the closest distance

If it's that easy i wouldn't have asked here brochacho ;-;

If we have a center point of PQR and we take it straight down till it hits the face of TUV, then it'll be 4cm, but it's not the closest distance because its not perpendicular.

[limber elm]

First off, PQR//TUV so distance from any points in PQR to TUV are the same, agree?

[honest compass]

Yes

[limber elm]

Now, can you see how BC is a line lie on plane TUV

[honest compass]

Yes

[limber elm]

The same for EH, you can see how EH is also on plane PQR

Now we find distance between E to TUV

Notice TU is perpendicular to EFBA

Which mean TUV is perpendicular to EFBA

[thin hill]

Also treating the planes like lines on a coordinate plane then using a perpendicular transversal works, but I see you guys have got it covered

[limber elm]

So is PQR

Which mean d(E, TUV) is just distance between E and TB

Did I go too fast?

Honestly, why not try coordinate and vectors, that might be easier

[honest compass]

I know! but the test is about geometry ;-;

[limber elm]

Okay fair

so far so good?

[honest compass]

Nah, ill just fail this test, it's fine, thanks for the help tho <3

[limber elm]

If you change your perspective to be directly looking at the face EFBA, you would see plane PQR as just the line EQ, and plane TUV as just TB

So their distance should be distance between EQ and TB

Or just distance from E to TB

Very intuitive way to think about it

[honest compass]

.close

[calm coral]

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[tiny pike]

I’m having trouble with the solution to question 2, specifically the step where the equation found for “a” is substituted and then related to log base 6 of 16. The question asks: if log base 12 of 27 equals a, calculate log base 6 of 16. (I would love a detailed explanation😇)

[uncut thicket]

Is this your solution?

[uncut thicket]

He wants to write log_6 16 in function of log_12 27

[calm coral]

@tiny pike Has your question been resolved?

[tiny pike]

Yup

But i cant finish the rest

[uncut thicket]

Português?

[tiny pike]

Yes

[uncut thicket]

Essa questão parece ser dificil kkk

[tiny pike]

Ué

Sim, ela é

[tiny pike]

Vc consegue fazer?

[uncut thicket]

Tá

Invés de usar base 10

Usa base 6 no log 27

[rancid sphinx]

gang

i did this exact question sometime ago

wait

[uncut thicket]

Save us

[tiny pike]

Plis plis

[rancid sphinx]

ok. should i send the sol

or just hint

see. log12(27) = 3/ log12(3)

are we on the same page?

[rancid sphinx]

?

[tiny pike]

Do you mean this should be the answer?

[rancid sphinx]

no. this is step 1

[tiny pike]

It says I have to consider it as equal to a

[rancid sphinx]

can you send your work (if you did any)

[calm coral]

Channel closed

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[tiny pike]

.

[rancid sphinx]

also @tiny pike

do this

well you did nearly everything

i see

[calm coral]

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[rancid sphinx]

can you express log4(3) as a / 3-a

[tame forge]

Can you express your question more unambiguously?

[rancid sphinx]

brother. we are solving this question #help-42 message

it got timeout

[tiny pike]

I have to put the two equations together

[rancid sphinx]

yeah ik

i am tryna guide you

[tame forge]

ok

[rancid sphinx]

can you put log2(3) as 2a/ 3-a

[tiny pike]

Ig but in this case wouldn't really make sense to do so

[rancid sphinx]

it will

you will ned to break log6(16) as log 6(2^4) which is 4log6(2)

which then becomes