help-36

[final hedge]

since the text says nabil's age was equal to shuvo's current age in the past, nabil is older than shuvo so let the age difference between them is (N - S) years

and we know nabil's current age is double of shuvo's age back then

right

[cinder vapor]

Okh

[final hedge]

and the difference is (N-S)

right

[cinder vapor]

Okh

[final hedge]

so what was shuvo's age N-S years ago?

this q ong sucks 😭

what is that holy language

[cinder vapor]

Bengali

[final hedge]

no like the framing

[final hedge]

^^^^?

[cinder vapor]

Oh

[cinder vapor]

Dude I m seeing darkness

[final hedge]

wym

like you're confused at a step or

smth?

or message didnt load?

[cinder vapor]

My eyes are twitching

[cinder vapor]

Kinda

[final hedge]

which step

[cinder vapor]

2nd

N is older than s

[cunning ridge]

you going way too fast

[final hedge]

[final hedge]

ight

[cinder vapor]

Then?

[final hedge]

hmm wait

im forming a perfect explanation

[final hedge]

what does this line tell you

[cinder vapor]

Did you say perfect?

[final hedge]

no i lied idk how to explain so im gonna let you understand it yourself

lol

well not like im not gonna help u understand

[cinder vapor]

Still I can't understand shi

[final hedge]

look at the word "was"

so its telling us that nabil in the past was shuvo's present age

right?

[cinder vapor]

Yea

[final hedge]

exactly

so whos older

?

[cinder vapor]

N

[final hedge]

see

you understood it yourself

[cinder vapor]

Yes I m feeling like caveman discovering stone

[final hedge]

lets come back to this

[cinder vapor]

So N is present anv so is S

Right?

[final hedge]

yes

[cinder vapor]

And when N was in past he was equal to to our present S

[final hedge]

read this

and can you form

an equation

N is nabil's current age

S is shuvo's current age

so what can we say from that statement

??

[cinder vapor]

Wait am I tripping

[final hedge]

?

[cinder vapor]

Lemme see my native text for a sec

[final hedge]

i mean

zavier confirmed that

so im using that

lol

[cinder vapor]

Yea I g I m getting word wrong or the placement

[cinder vapor]

(My fault)

[cinder vapor]

Lemme rephrase what I have read

[final hedge]

so umm is the q language im referring to correct or

wrong

[bold turtle]

okay this text looks shit ngl

[final hedge]

yeah

its originally a bengali q

but idk bengali

[cinder vapor]

Fr

[bold turtle]

Srsly I'm trying to translate it by myself and it's a fokken minefield

[final hedge]

we can get an ans tho

ig

[bold turtle]

Like ik relative clauses are weird in Bangla but tf is this even

[cinder vapor]

I can't understand it myself even tho I m a native

[final hedge]

well you can get the numbers to satisfy it

even tho they prob arent logical

[bold turtle]

but the problem is does it satisfy the OG question

I'm not trying to rely on the provided translation fwiw

[final hedge]

we'll only find out after finding

lol

[bold turtle]

Because I'm doubting the translations given

[cinder vapor]

Like I tried watching the native ans vid but I didn't get the logic

[final hedge]

do you have the solution to it

or smth

so i can verify my ans

[cinder vapor]

Ye

[final hedge]

whts the final ans

[cinder vapor]

So S =21

[final hedge]

and N = ?

[cinder vapor]

28

[final hedge]

oh perfect

ight

back to our step

[cinder vapor]

You got it ?

[final hedge]

yep

[cinder vapor]

Ok

[final hedge]

what can you say from this

[cinder vapor]

N is double= when S was

[final hedge]

nope

[cinder vapor]

Then?

[old saddle]

Can I add some details?

[cinder vapor]

Sure

[final hedge]

it says that nabil's current age is double the age of shuvo back then

[final hedge]

you're welcome

[cinder vapor]

Yea that is what I said

[final hedge]

oh

crap

mb

sorry 😭

[cinder vapor]

Isn't N the current age ?

[final hedge]

yes so how can we write that in terms of N and S

[cinder vapor]

Oh that's okay

[final hedge]

where N and S being current ages of Nabil and Shuvo

[cinder vapor]

2N=S+was

[final hedge]

N isnt double

[cinder vapor]

Oh crap

[final hedge]

shuvo's age (N-S) years ago was double

of

nabil's current age

[cinder vapor]

Ye sorry

[cinder vapor]

N = 2(N-S)

[cinder vapor]

?

[final hedge]

nope

N-S is the age difference

[cinder vapor]

Oh sorry

[bold turtle]

"When Nabil's age was equal to Shubho's current age, Shubho's age then was [double (dbiguno)] of what Nabil's current age. When Shubho's age becomes what Nabil's current age is, if the sum of their ages becomes 63, what is each person's current age?"

I feel like this must be a typo

That or idk what dbiguno should mean

[final hedge]

the answers match

tho

[old saddle]

Np - Nabil's present
Sp - Shuvo
Np - d = Sp --> 1
Sp - d = 2Np -->

But the third line "Subho's age is the same as Nabil's present age" doesn't exactly mention when.

To me it should not be the difference of the present times but a totally d years ago

[cinder vapor]

Yea

[bold turtle]

Wait so, 2x?

[final hedge]

it means that shuvo's age (N-S) years later reaches N

i think

[cinder vapor]

Can we talk like

[bold turtle]

Then that must be a typo in the question

[cinder vapor]

Understanding like this is kind ahard

[cinder vapor]

No actually if I m not wrong

[old saddle]

S is Subho's present age?

[final hedge]

yeah

[cinder vapor]

Yea

[old saddle]

but why will it be N-S years ago

[bold turtle]

So, like:
"When Nabil's age was equal to Shubho's current age,"
means that Nabil is older than Shubho; so how can Shubho's age, then or now, be double Nabil's age?
Hence my thinking that this must be a typo

[final hedge]

meaning the gap we taking is N-S

[bold turtle]

or maybe I'm really messing with the relative clauses

[final hedge]

even if you consider this q, you can get the correct ans

its solvable

[bold turtle]

"Shubho's which-age[x] it-was, Nabil's current age[y] its [referent, r] double"

okay I can see how r = x here

I'd taken it at face-value and gone r = y (which incidentally is how Yoru's Google Lens translation took it to mean)

okay I'll leave you to it lol, you guys seem to have it under some control

[final hedge]

i meann i have it solved xD

[bold turtle]

-# sure but the question is whether Yoru has it solved

[final hedge]

yah

[cinder vapor]

I have the solved ans

But I can't understand that how this resulted that equation

[bold turtle]

-# No disrespect, I mean that it's meaningless if Velt has it solved if Yoru didn't

[cinder vapor]

What ?

[old saddle]

can you show us?

[bold turtle]

(Velt'er answer jodi ache, labh nai tumi jodi bujcho na)
-# (There's no benefit to Velt having an answer if you don't understand it)

[old saddle]

Is this sum from KC Nag, or other publications?

[bold turtle]

-# sipping into the recesses of what Bangla I can muster lol

[cinder vapor]

The ans is N =28 and S= 21

[cinder vapor]

Actually I have the answer but I don't understand it

[bold turtle]

So, N is Nabil's current age, and S is Shubho's current one, right?

[cinder vapor]

Like I have the value of their

[cinder vapor]

Yes

[bold turtle]

First part - "when Nabil was Shubho's age" - i.e. N-S years ago

Is this part clear?

[sterile marsh]

are we still on this problem 😭

[cinder vapor]

Ye

[final hedge]

yes

[bold turtle]

Yes

now shut

[bold turtle]

@cinder vapor

[old saddle]

is this from KC Nag?

[final hedge]

💔

[cinder vapor]

What is this KC Nag

[sterile marsh]

a publication ig

[cinder vapor]

No

[bold turtle]

Presumably Keshab Chandra Nag

[old saddle]

ooh ok leave it, one of WBs OG books

[old saddle]

ya

[bold turtle]

Indian Bengali mathematician - but the source of the problem is irrelevant

[cinder vapor]

Oh I m not from WB

[old saddle]

oohh

[cinder vapor]

It's from NCTB

[cinder vapor]

Okh

[final hedge]

whts nctb

[bold turtle]

Bangladesh's exam board

[final hedge]

ohh ok

[bold turtle]

well, a government ministry tbh

[sterile marsh]

!redir

[final saddle]

This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.

[bold turtle]

We're on topic fyi

Secondly:

[cinder vapor]

Yea

[final hedge]

dude it was only 1 question, its not that deep

[bold turtle]

@cinder vapor Is this part clear?

[cinder vapor]

Okh

[cinder vapor]

No

[bold turtle]

So for example

Suppose Alice was 24 and Bob was 20, then "When Alice was Bob's age" is 4 years ago, right?

Where'd this 4 come from?

[cinder vapor]

Oh the difference of their current age

[bold turtle]

Right

So Nabil's current age is N and Shubho's is S

[cinder vapor]

Yes

[bold turtle]

So when Nabil was Shubho's age, that's N - S years ago

[cinder vapor]

Okay now I get it

[bold turtle]

Shubho's age at the same time would be S (his current age), minus N - S (how many years ago)

i.e. S - (N - S) = 2S - N

[cinder vapor]

Okay let me write it down first

[cinder vapor]

Didn't get it

[bold turtle]

So if I were to use this example again

[cinder vapor]

Yes as you please

[bold turtle]

At the same time, Bob's age then would be 20 (his current age) minus 4 (how many years ago)

i.e. his age then was 20 - 4 = 16

[cinder vapor]

Okh

Got it

[bold turtle]

Same logic, just with some algebra because we haven't actually got their ages yet

[cinder vapor]

Yea

[old saddle]

@bold turtle
Np - d = Sp
Sp - d = 2Np
d = Ns - Sp

Are these logically correct?

[bold turtle]

Then the bit after, "Whatever Shubho's age then was [ 2S - N, what we've just said ], Nabil's current age is double this"

[cinder vapor]

Okh

[bold turtle]

You're adding far too many variables that don't need to be there

That's just gonna complicate things

[bold turtle]

So since Nabil's age, N, is double that 2S-N age that Shubho was,

We can say N = 2 ( 2S - N )

[cinder vapor]

Okh

[bold turtle]

Try and simplify this?

[old saddle]

@bold turtle

N - ( N - S) = S
S - (N - S) = 2N

Are these logically correct?

[cinder vapor]

Lemme write it down

[bold turtle]

The first line is true but not meaningful as written; the second line I literally just went over

-# And on that last point - I'm literally going over this with the helpee, could you not try and interrupt with this

[cinder vapor]

3N = 4S?

[bold turtle]

ye

We're gonna need to get another equation so that we can solve (simultaneous equations)

Looking at the rest of the question:

[cinder vapor]

Is it like adding two equation together or getting a difference of 2 equation

[old saddle]

ok then what @cinder vapor gave the numbers as answers isn't sitting right

[bold turtle]

We're gonna need to read the rest of the question to find the other equation

"When Shubho becomes Nabil's current age, if the sum of both their ages then is 63..."

First, can you see how many years in the future this will be?

[cinder vapor]

Pls tell

[bold turtle]

"When Shubho becomes Nabil's current age" - start from this

Shubho's currently S years old, and Nabil is N years old

[cinder vapor]

Wait so we got 3n is equal 4S

[bold turtle]

Yes that's one equation; we're trying to make another

[bold turtle]

How much longer does Shubho have to wait to become Nabil's current age?

[cinder vapor]

Wait the the answers are scattered lemme write all the info together

[cinder vapor]

Sorry my phones battery is literally close to be dead

Can we talk about it tomorrow

I am expressing my gratitude to all that tried to help me to today. I will not forget your help that have benefited me

[bold turtle]

sure; .close this channel and pick it up later

[cinder vapor]

Thanks bro

[final hedge]

waes help 42 has to be a troll right

[cinder vapor]

I appreciate your help brother

[final hedge]

nah you did everything yourself

[bold turtle]

(restate the problem and the translation;

"When Nabil's age was equal to Shubho's current age, Nabil's current age is double what Shubho's age then was. When Shubho's age becomes what Nabil's current age is, if the sum of their ages becomes 63, what is each person's current age?"

)

[final hedge]

i only showed you how

[bold turtle]

when you open the new channel @cinder vapor tomorrow

[cinder vapor]

Okh thanks for your help also

Bye

.close

[final saddle]

Channel closed

Closed by @cinder vapor

Use .reopen if this was a mistake.

[bold turtle]

He's very resilient to any aid, yes

[final hedge]

i have a very short temper so sorry if i had crashed out in there 😔

[bold turtle]

nah dw I can see why

[final hedge]

its open for 2 hours and they reacted to the bot twice saying their prob isnt solved

do i modping

and they arent even telling whats the exact doubt

[bold turtle]

I was about to lol

[final hedge]

oh go on

[final saddle]

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[hardy beacon]

I am wheezing out...
I have only studied annihilator approach to solve these types of differential equations...Its a bit lengthy pls help😭

[willow cliff]

well first off you wanna solve for the roots of the $D$ poly right

[soft zealot]

TestTickler

[hardy beacon]

Yes lemme send you a ss

[willow cliff]

i also only know the annhilator method but lets see what i can do

[hardy beacon]

Too complex

[willow cliff]

be careful in the begining

[hardy beacon]

I feel like this question uses an another approach

[willow cliff]

a repeated root means that it has general solution $y=(a+bx)e^{2x}$, a and b are constants

[soft zealot]

TestTickler

[willow cliff]

oh wait i think u got it later on, nvm

[hardy beacon]

I did this question while being stressed

[trail mango]

[hardy beacon]

I literally have my finals two days after

Let me do it again

[final saddle]

@hardy beacon Has your question been resolved?

[hardy beacon]

No

Annihilator doesnt seem right for this type of question

.close

[final saddle]

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[supple jolt]

<@&268886789983436800>s

Mr. Least

[final saddle]

Channel closed

Channel closed due to the original message being deleted.
If you did not intend to do this, please open a new help channel,
as this action is irreversible, and this channel may abruptly lock.

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[floral moth]

how to solve it using synthetic geometry

i already did a proof showing that the area of FEG is equal to the area of EDC, but using coordinate bash

[final saddle]

@floral moth Has your question been resolved?

[severe verge]

no need coord bash for this

notice that triangles FGD and FCD have the same area

[floral moth]

damn

[severe verge]

(since they share the base FD and their altitudes are the same length)

[floral moth]

yeah im blind

yeah buts trivial now

thank you

.close

[final saddle]

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[vital crag]

<@&268886789983436800>

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[opaque pagoda]

Hello! Can someone please help me explaining how can I find the order and degree of an equation?? 🙁 I really dont get it

[vital crag]

have you learned these definitions

[opaque pagoda]

So to identify order I need to find the highest derivative?

I’d guess is 3… and for the degree it must be also 3 right???

Wait I think I got it

Tyyyy!!!

.close

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[shadow smelt]

how do i do

[ornate token]

What is the total angles of pentagon.

This is a regular pentagon and OA=..=OE, can appeal to the fact that OA,OB,..., OE they all dissect A,B,...,E by half.

[shadow smelt]

6

[ornate token]

Total measurements of those angles?

Then for each angle, what is the measurement?

Eh, not in this case, total angles of pentagon is not 360 though.

[cyan sandal]

i meant around O

but u got it

[ornate token]

Well there are 2 approaches.

1, if you go from bases angles of A,B,...,E.

Or 2, just argue that the lines OA,..,OE cut angle O equally.

[broken field]

I thought there was a formula for it

[final saddle]

@shadow smelt Has your question been resolved?

[broken field]

Nvm this is just viewing center as a circle

[final saddle]

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[shadow smelt]

how do i do this

[broken field]

Ur just finding middle right

Meaning u can view O as a part of a whole circle

[shadow smelt]

yes

[broken field]

So it would be the circle total angle dividing the number of center angles right

[shadow smelt]

so 72 degrees

[broken field]

Yes should be

[shadow smelt]

how do i do

[broken field]

Ok u first find the angle of the hexagon

[timber leaf]

Find measurement of each angle in hexagon

Then take 180 - that angle to get B,C in triangle BCG

[stable dew]

or

extend ED

and apply the parallel lines rule thingy(i forgot the name)

[shadow smelt]

how

[timber leaf]

it is a regular hexagon

total is 720

[broken field]

n-2 x 180

[shadow smelt]

ok

so

180 degrees for each angle then

[broken field]

How many angles are there to find in the hexagon

Eh I will just count for u 720 dividing 6 is 120

[timber leaf]

think for once buddy

[shadow smelt]

excuse me

[timber leaf]

6 angles, total is 720

[broken field]

My first time seeing a helpful acting out

[shadow smelt]

bro if u divide

by 6

you get for each angle

[timber leaf]

yes

[shadow smelt]

ok

so am I NOT THINKING

ok

[timber leaf]

Unfortunately yes

[shadow smelt]

good

[broken field]

[shadow smelt]

so what happens after that

[broken field]

180 - the answer u got

Since ABG or DCG is a straight line

It means 180

[shadow smelt]

ok

so bgc is

[timber leaf]

You want to find GBC and BCG first

[shadow smelt]

bro what

its the same thing

[timber leaf]

what i mean here is the angle

[broken field]

On this one the triangle is the same luckily

[timber leaf]

wdym same?

[broken field]

I meant

It's equal

So what's ur answer better dogr

[timber leaf]

you mean equilateral

[broken field]

Yea I'm tryna find that auto correct

It didn't appear so I don't dare say it

[shadow smelt]

ok

so

180-144

[broken field]

Where did u get 144

[shadow smelt]

so if

[timber leaf]

what is 720 divides by 6.

[shadow smelt]

120

[timber leaf]

Then where did that 144 come from

[shadow smelt]

ok so

ok its 120

so 60 degrees

[broken field]

Yesir

[shadow smelt]

for BC

its 60 degrees SIR

[broken field]

Ye so u would add both B and C

And take the triangle full angle and minus both

To get x

So what do u get

[shadow smelt]

how do i get the triangle full angle

[broken field]

Using the formula (n-2) x180

Triangle has 3 angles to be found

So 3-2 is 1 which x 180 means 180 angle

[timber leaf]

A triangle is 180 degree

Re-read your notes please, if you cant get this then harder problems will be unsolvable

[shadow smelt]

ok so basically

if i add 120 + 120

240 - 180

60

degrees

[broken field]

Ok so 180-120 means 60

U got 60 for each B and C

A triangle full 180 means u minus B and C to get x which in this case 180-120 which u would get 60

[shadow smelt]

ok ty

[broken field]

[shadow smelt]

imma do this one rq

[timber leaf]

Same concept, but instead of hexagon it is pentagon

[broken field]

Able to find the Pentagon angle using (n-2) x 180?

[shadow smelt]

yes

ok so i got

180 degrees

[broken field]

For what

Ok u tell me how many angles are there to find in pentagon

[shadow smelt]

5

[shadow smelt]

for BFC

[broken field]

Well BFC is in a triangle so it would be impossible for itself to be 180

Ok so u got 5 meaning u input it into (n-2) x180 u get?

And after u got the total angle for pentagon calculate one of it's interior

[shadow smelt]

that would be

108

for an interior

[broken field]

Correct now using that try to find B or C

At the triangle

[shadow smelt]

72 degrees

[broken field]

Nice so now u take the triangle full angle and minus both B and C

To get x

Which in this case both ur B and C is 72 rn

[shadow smelt]

so

x would be 36 degress then

[broken field]

Yes

[shadow smelt]

wow...

😃

[broken field]

If ur finished make sure to close

[shadow smelt]

.close

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[night basin]

Hello, can someone explain why this method is incorrect for solving this problem?

[loud sundial]

Forgive my idiocy, but where is 1 coming from

[night basin]

Umm cause like the big square is cut into three parts so it also divided the side of the tilted square into three parts

[timber leaf]

I think you should find the area of 4 full small squares first

[night basin]

Right Yoh mb

Thanks

.close

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[ruby crypt]

hi, idk where to begin

[ruby crypt]

what is even measure 0 intuitively speaking

[final saddle]

@ruby crypt Has your question been resolved?

[timber leaf]

A null set

[ruby crypt]

null as in empty?

[timber leaf]

Yes

[ruby crypt]

wth

there is no way right?

[timber leaf]

Well but i read here, it probably mean that the set is so small such that its length is considered 0

[ruby crypt]

yes the length, so set shouldn't be empty

but what I dont understand is actually the concept of measure 0 itself

[final saddle]

@ruby crypt Has your question been resolved?

[sand barn]

List all numbers of the set in sequence. Chose a number such as 0.1.

Place a interval of 0.05 around the first element

Then 0.025 around the next and so on

Its called measure

Understand @ruby crypt

[ruby crypt]

hi im eating rn

so u mean, by placing interval of x, this means [0.1-x, 0.1+x]?

[wintry mantle]

Measure 0 means if you plug S into a function which meaures volume of a set, you get zero

You can prove a) by contradiction using the fact that x^2 is bijective

[ruby crypt]

u plug elements in set S into a function that measures volume?

just got back

[wintry mantle]

You plug in the whole set S

https://en.wikipedia.org/wiki/Measure_(mathematics)

[ruby crypt]

ok this is just basic vocabulary meaning of measure

lol

what am I even questioning

but what does it mean when a set is of measure 0? Does it mean the distance between points/values inside the interval is very very close to 0?

[whole halo]

do you know what the measure is for this problem

[ruby crypt]

0

[whole halo]

bro

Im talking about the function

do you know what measure function youre using for the problem

like what would the measure of [0, 1] be

[ruby crypt]

um 1...?

[whole halo]

you dont know then

go look in your notes or the book and find what measure youre using

[ruby crypt]

what?

[whole halo]

"a set of measure 0" only means that you put it through the measure and you ge t0

[ruby crypt]

all I was taught is lebesgue

[whole halo]

well yea

thats your measure then

the lebesgue measure is a function, it tells you how big things are

[ruby crypt]

is it stupid if i ask there are more than one measure

[whole halo]

🤔 well given we have to name this one after someone's name...

what do you think, is it just the one?

[ruby crypt]

no i was saying like cm, mL, and stuffs are measure right

[whole halo]

no

those are called units

a measure function and a unit are not the same thing

[ruby crypt]

ok this goes deep

[whole halo]

bro did you even learn what the lebesgue measure is in school

[ruby crypt]

oh is it like length, weight and stuff

[whole halo]

no it isnt

[ruby crypt]

yes wait lemme show

this is for lebesgue measure 0

[whole halo]

ok look around for the word "lebesgue" or "borel"

do you see anything like that in your school book or the school notes

[ruby crypt]

and I dont understand this theorem

[whole halo]

just googling things then showing them to me doesnt count, I want to see if you need to know this stuff for the course or not

[ruby crypt]

no borel

[ruby crypt]

thats from the lecture notes

[whole halo]

any lebesgue?

[ruby crypt]

ok so in my lecture notes

they didn't really say "Lebesgue measure 0 set is..."

the only thing that defines lebesgue measure 0 set is this

in my lecture notes ofc

and heck how am I supposed to understand

[whole halo]

you can swear bro

they didnt tell you shit about lebesgue

here watch this:

see the 0 in "measure 0"?

and the "< epsilon" in the last line?

[ruby crypt]

ye

[whole halo]

if you change those to "measure L" and "< L + epsilon", thats how you find how big something is

the L needs to be as large as possible while still allowing for any epsilon > 0 like that

[sand barn]

List all numbers of the set in sequence. Chose a number such as 0.1.

[whole halo]

do not do that

[sand barn]

Place a interval of 0.05 around the first element

[whole halo]

oh you mean the set in the problem, I see

[sand barn]

Then 0.025 around the next and so on

[whole halo]

generally you cant go around and "list all numbers" for most sets, but for this one you can

[ruby crypt]

he is just repeating what he said

nice job teach

[sand barn]

That's an intuitive explanation

[ruby crypt]

@sand barn

[whole halo]

thats not generally how you write down an interval around x

[ruby crypt]

so is it with x/2 instead?

[whole halo]

should instead be [x - 0.1, x + 0.1] or (x - 0.1, x + 0.1)

remember, this is around x

[ruby crypt]

oh around x got it

[whole halo]

so for example if x is 5, this would be (4.9, 5.1)

[sand barn]

That's what I said

[ruby crypt]

so is that measure 0.1

[whole halo]

oh I see

[ruby crypt]

and btw I just found out measure is not covered for my exam bruh

[whole halo]

measure of x woud be x/2 yea

[ruby crypt]

but its good to kno

[sand barn]

Then next element is [x -0.05,x+0.05]

[whole halo]

man this is messy

[ruby crypt]

right

[whole halo]

we're going to stick to using x for "elements of the set" from now on

not "what the measure/length is"

(4.9, 5.1) = (5 - 0.1, 5 + 0.1) has measure 0.2 for example

[ruby crypt]

alr

[sand barn]

I see

[ruby crypt]

so by measure 0, we want the number around x to be as small as 0

like very close to 0

[sand barn]

Yes

[ruby crypt]

?

I don't get the purpose

[whole halo]

yea, the left side tells you that "S can be completely covered up by intervals"
then the right side tells you those intervals have a total length that can be made as small as possible

[ruby crypt]

is it to list all real numbers

[ruby crypt]

I see yes ok

[whole halo]

now because of that, its not like S can contain an interval

if it contained [0, 0.1] for example, its measure is at least 0.1

because of that [0, 0.1] there

[ruby crypt]

I see

[sand barn]

But can't we divide x by anything like 3 or 1679?

[whole halo]

well I mixed "x being the 'epsilon'" with "x being the element"

you could do that, but we're just not calling it x anymore

call it delta instead or something

[ruby crypt]

huh

[sand barn]

Ok we'll call it e

[whole halo]

not a popular choice

[ruby crypt]

call what e?

[sand barn]

e/2 or e/3

[sand barn]

Epsilon

[whole halo]

ok vins remember when we had (4.9, 5.1) = (5 - 0.1, 5 + 0.1)?

[ruby crypt]

yse

[whole halo]

you started off calling x the 0.1

[ruby crypt]

AAA

yes

[whole halo]

this is a weird decision, and I misread what you said

[ruby crypt]

so now we call e

[whole halo]

x is intended to be used for points, or for parts of the set

not for how big something is

we're calling it e now, so (x - e, x + e)

this has length or measure 2e

(e > 0)

[ruby crypt]

ofc

[sand barn]

Correct lets stick to 2e

[whole halo]

now you can always do e/3 or e/whatever as coder said

this makes the set smaller...

[ruby crypt]

aslongas 2>0

[whole halo]

but youre making an assumption here

[ruby crypt]

e

[whole halo]

now a common example of a set of lebesgue measure 0 are sets that have finite or countable elements

have you heard of this before

[ruby crypt]

nope

[whole halo]

think about the set {0, 1, 2}

this set has a grand total of 3 elements

it is lebesgue measure 0

can you see why

[ruby crypt]

what

ooo

[whole halo]

go to the definition and think about what coder is saying

[ruby crypt]

uh because pick a number

[sand barn]

We use curly brackets for finite sets and intervals are []/()

[ruby crypt]

then pick a small e > 0, then it can't have anything other than that number

[whole halo]

and more importantly, see here you can only pick a countable amount of these intervals at a time

for example, youre picking 3 intervals because theres 3 elements

[ruby crypt]

well I can pick more no? try with different e

[sand barn]

Of course

[whole halo]

well you need to be able to do this for all e > 0

not just "different e thats easy for me to do"

[ruby crypt]

so its not only 3 intervals

[whole halo]

thats not what I said

[ruby crypt]

whats this picking 3 intervals

[whole halo]

the number of intervals is separate from e

[sand barn]

Nope as long as you can feasibily calculate

[whole halo]

well you picked one interval per element right

its a normal choice to do that

[ruby crypt]

one interval means one e?

[whole halo]

(-0.1, 0.1) U (0.9, 1.1) U (1.9, 2.1)

no?

[ruby crypt]

yes that true

[whole halo]

read this

[ruby crypt]

but thats for measure 0.2

[whole halo]

this is saying "the length of the intervals add up to a number less than e"

it doesnt tell you how big each interval is

just that they have to add up to be less than e

so (-0.1, 0.1) U (0.9, 1.1) U (1.9, 2.1) would be good for e = 0.7, 0.8, 0.60001, etc.

[ruby crypt]

u mean epsilon right

[whole halo]

yea

[ruby crypt]

I see

[whole halo]

coder is using e to be short for epsilon

typing out the whole word gets tedious

I am too

[ruby crypt]

so this interval works for epsilon > 0.3?

[whole halo]

vins uh

[ruby crypt]

0.6

[whole halo]

how long is (-0.1, 0.1)?

there you go

[ruby crypt]

0.6

0.6

[whole halo]

this interval works for epsilon > 0.6

now keep in mind you know what e is in advance

you then use this e to make intervals as small as you need

[ruby crypt]

ofc

[whole halo]

that still cover up S = {0, 1, 2} here

so what intervals would you go with

[ruby crypt]

err 0.05

[whole halo]

as a reminder you dont know how small e is

[ruby crypt]

or e/2

e/3

e/100

[whole halo]

lol

just e/6 is enough

[ruby crypt]

ok

why 6 tho

[whole halo]

🤔

lets see

[ruby crypt]

oh

u wann add it up to epsilon?

[whole halo]

what else would I be doing lol

[ruby crypt]

ik ur trick buddy 😛

[winter birch]

wot

[ruby crypt]

eo actually we let e = eps/6

[whole halo]

you were already supposed to know the trick, but Im glad you feel happy about this

[ruby crypt]

gotta cherish every moment

[whole halo]

e and epsilon are the same thing, I dont know what you mean by this

again we're just having e be short for epsilon

[ruby crypt]

ok

[sand barn]

e = e/6 or eps= eps/6

[ruby crypt]

ok lets do e = epsilon

[whole halo]

vins

[sand barn]

That is what we did

[whole halo]

we're not having e mean how big an interval is from its center anymore

we're having e just be short for epsilon, which is how big the intervals are in total

[ruby crypt]

yes ok

[whole halo]

e/6 = epsilon/6 will be how big the intervals are

alr I want you to write down the intervals that:

• cover S = {0, 1, 2} up entirely
• have lengths that add up to be < e

[ruby crypt]

entirely?

im dead

[whole halo]

???

S only has 3 elements

did you forget about (-0.1, 0.1) U (0.9, 1.1) U (1.9, 2.1)?

doesnt that cover S up? entirely?

[ruby crypt]

but thats only one

um yea

ok

oh

the word "entirely" sounds scary

[whole halo]

bro

you sound like youre in high school this is crazy

[ruby crypt]

(0 - e/6, 0 + e/6) U (1 - e/6, 1+ e/6) U (2-e/6, 2+e/6)

[ruby crypt]

mb bro

[whole halo]

yea thats the idea

[whole halo]

now notice here all you had to do was (x - e/(2n), x + e/(2n)) around each x of S

[ruby crypt]

ok

[whole halo]

so for example S has 3 elements, and you just - e/6, + e/6 each of them

if S had 10 elements, youd do - e/20, + e/20

[ruby crypt]

wow nice touch of 2n

[whole halo]

you can see we have something that works here

because of this, if S only has a finite number of elements

it has lebesgue measure 0

it makes sense, S should have 0 length, and see here the measure agrees with what we think

[ruby crypt]

cuz by definition, we take n -> inf

?

[whole halo]

no

n here is just how many elements S has

think about what we just used n for

[sand barn]

Which can go till inf

[ruby crypt]

oh right yea its number of elements

[whole halo]

it can still go towards inf though, but be careful

some infinities are bigger than others

this will only work for the smallest infinity, which is the one coder is assuming

[sand barn]

That's not related here

[whole halo]

coder you need to confirm that S is countable first

[sand barn]

It is already confirmed it has measure 0

[whole halo]

youre saying to go through every element

[ruby crypt]

wait what im confused now, how did u get this conclusion

isn't it supposed to be measure e/3

[whole halo]

do you remember how the notes defined "measure 0"?

[whole halo]

it has measure e, not measure e/3

remember, e is how big it is in total, not how big each one is

[sand barn]

Total is 3*e/3

[whole halo]

now see here the first half of the definition

we were able to cover S up with intervals, with lengths that added up to e, for any e > 0

that means you could make e as small as you want

and the length just gets smaller

[ruby crypt]

so our (a1, b1) = (-0.1,0.1) and so on?

[whole halo]

yea

[ruby crypt]

and making e as small as possible, aaaa

yea S is still inside that union

[whole halo]

the measure of the set is how small we can make it

here, the measure is 0

[ruby crypt]

ah got it

got it

[whole halo]

alr now for another interesting example

have you heard of a countable set yet

[ruby crypt]

yes a set is countable if a set is countably infinite or finite

iff actually

[whole halo]

yep, now this definition only allows you to use a countable number of intervals

you gotta be able to count them from (a1, b1) to ...

so its not like you can do this () trick with any uncountable set

you can do it for countable sets, like {1, 2, 3, 4, ...} or the rational numbers

[ruby crypt]

yseh

[whole halo]

so as a result, any countable set has lebesgue measure 0

[ruby crypt]

I understand it now

[whole halo]

however, you dont necessarily need countable to get lebesgue measure 0

[ruby crypt]

I wanna move to one more topic after this, do u still hv time

[whole halo]

maybe

I first gotta show you how you do this problem, though

[ruby crypt]

ehh i think I prioritize my other topic rather than this, cuz this other topic is included in exam

[whole halo]

alr

[ruby crypt]

just quick question

very quick

why uniformly continuous and continuous everywhere is a different thing

[whole halo]

uniformly continuous is being specific on the kind of continuity you have

continuous everywhere is just saying "its continuous everywhere", doesnt really tell you which kind

those are separate things, you can have one, the other, both, or neither

[ruby crypt]

wdym which kind

[whole halo]

uniformly continuous you can think is a better or a stronger version of continuity

[runic needle]

doesn't unfiormly continous imply continunuos everywhere

[ruby crypt]

yes but not the converse

[west orbit]

[runic needle]

continous means continuous at x for all x in the domain

[ruby crypt]

thats where im confused

[whole halo]

sign(x - 1) √x is uniformly continuous on [2, 4] but not continuous everywhere on its domain

remember youre using the word "everywhere"

you could yknow just leave it out

[ruby crypt]

sign(x-1)?

[whole halo]

function that can cause a discontinuity at x = 1

this function is:

sign(positive number) = 1
sign(0) = 0
sign(negative number) = -1

so sign(x - 1) would be:
1 if x > 1
0 if x = 1
-1 if x < 1

[ruby crypt]

ok

[whole halo]

also,
uniform continuity => continuity
uniform continuity everywhere you want => continuity everywhere you want

but you cant say uniform continuity => continuity everywhere you want

[ruby crypt]

ok so here comes the confusion: what if the domain is now (2,4), its not uniform continuious is it

[whole halo]

it still is

[ruby crypt]

what

[whole halo]

it doesnt need to be a closed set

who told you that

oooh did you think that one rule was all there was to it

theres more to this than just "continuous on compact set means uniformly continuous on that compact set too"

[ruby crypt]

ok I know nothing, carry on

[whole halo]

you could just have uniformly continuous for other reasons

ok you should relearn what uniform continuity is

you know continuity has epsilon-delta?

[ruby crypt]

yes

uniform cont. too

[whole halo]

now an issue is that the delta depends on epsilon and on x

hold on there Im still going

so for example for √x, you need to make your delta smaller as x gets smaller

you cant just make a delta based on epsilon

wait a minute

screwed up, √x is uniformly continuous

the usual counterexample is 1/√x instead

1/√x is continuous everywhere but not uniformly continuous everywhere

youre gonna have to swap out all the times I said √x with 1/√x

[ruby crypt]

uh

[whole halo]

anyway 1/√x depends on epsilon and x for the delta
for example if you want to confirm 1/√x is continuous at x = 0.001, youre gonna have to pick a much smaller delta than with x = 1000

the value of delta then doesnt stay the same depending on the x

uniform continuity means the value of delta is uniform, you pick an epsilon and theres a delta that always works, regardless of x

you can think of it as "if you change x by delta, then f(x) will never change beyond epsilon"

[ruby crypt]

oh so one delta works for all eps?

[whole halo]

I didnt say that

one delta works for all x

[ruby crypt]

nvm

[whole halo]

one delta works for all x and, at least, this epsilon we got that delta from

(and all bigger epsilon than that too)

[runic needle]

the difference just comes down to switching the order of quantifiers lol

[whole halo]

thats reductive

[ruby crypt]

can u tell me where a function is continuous but not uniform and a function that's uniform

[whole halo]

1/√x is continuous but not uniformly continuous

[ruby crypt]

I haven't seen the clear picture of the difference

[whole halo]

https://en.wikipedia.org/wiki/Uniform_continuity

see here in this picture for example

say you know epsilon = 0.5

you pick delta = 0.2

now when you look at g(x) = √x, which is uniformly continuous on [0, ∞),

youll notice the red curve never crosses the black lines on the red rectangle

however, when you look at f(x) = 1/√x, which is not uniformly continuous on [0, ∞),
youll notice the blue curve crosses the black line atop the blue rectangle

[ruby crypt]

oh yea it is

oh oh oh

interesting

[whole halo]

now these rectangles represent the epsilon and delta that was picked

the rectangles are 0.4 wide and 1 tall

(2 delta and 2 epsilon)

[ruby crypt]

ah yes ofc

[whole halo]

so you can think of them as confirming that, if you only move x by less than 0.2, then y in that area only moves by less than 0.5

for the red curve, this always happens

[ruby crypt]

yes

[whole halo]

for the blue one, you just move left enough and it will shoot up through the ceiling

doesnt matter how small your delta is

so we cant just settle with a uniform choice of delta = 0.2 like we did for √x

[ruby crypt]

DANNGGGGGG

[whole halo]

so 1/√x is not uniformly continuous

[ruby crypt]

interesting

[whole halo]

uniform continuity is really convenient because then we dont need to care about the x anymore

you want an epsilon, you just need a delta for it

that way you can be more global about the facts you can get out of the function

[ruby crypt]

but then seeing this blue line, since small changes in delta can cause the value to change more than epsilon, how is it still continuous 🤔

[whole halo]

continuity lets you also have it depend on x

youd get to change the delta as the box moves

uniform continuity wouldnt let you do that

1/√x is still continuous because you just make the box thinner as you move further to the left

[ruby crypt]

I actually don't remember doing practice questions and letting delta depend on x, I will find some later

but they should have delta in terms of x and e right

[whole halo]

yea

[ruby crypt]

ok ok

[whole halo]

many problems you do work with uniformly continuous functions to keep the load easy on you

so thats why you didnt need to know what x was

for 1/√x, youd have to

for example polynomials are uniformly continuous everywhere (on a bounded set)

[ruby crypt]

u mean we need to know where x is continuous at?

like from the question

[whole halo]

you didnt need to use what x was for the delta, you could pick the same delta = # for any epsilon and have it work

ok thats not really true nvm

usually you do practice on straight lines

those ones are uniformly continuous

for parabolas and other polynomials though, theres a hidden way you do use the x in your delta

[ruby crypt]

so something like f(x) = 100x^2 is uniformly continuous?

[whole halo]

yea I messed up again

[ruby crypt]

i dont think so

[whole halo]

you have to restrict f(x) to only be "-100 ≤ x ≤ 100" or "-10000 ≤ x ≤ 10000"

this is because f(x) slopes steeper and steeper

[ruby crypt]

thats very demanding

well anyways, ik the difference now lel, got the full pic

[whole halo]

alr

[ruby crypt]

and maybe one more b4 i close

how to prove that a function is continuous everywhere

usually in Qs they just say "prove it's continuous at 2"

[whole halo]

alr

"continuous at x = 2" has you check for continuity at x = 2

"continuous on [1, 2]" has you check for continuity at every x in [1, 2]

so continuity at x=1, at x=2, at any x in between

you dont look further than [1, 2] though I believe

well thats hard to say actually, it might depend on who you ask

[ruby crypt]

alr go on

[runic needle]

why would you look further

[whole halo]

because itd be disingenuous to say "this function is continuous at A and at B, so its continuous at A U B"

[ruby crypt]

u mean (A,B)?

[runic needle]

wat

[whole halo]

A and B are sets here, you can see that with the U

the U is union

,,\cup

[soft zealot]

mtt

[whole halo]

man I cant believe I typed that

[ruby crypt]

why would a set be continuous

[whole halo]

oh whoops

should be "on A" not "at A"

[runic needle]

idk this seems valid to me

[ruby crypt]

oh

ok

[whole halo]

consider a function thats 1 for nonnegative numbers and 0 for negative numbers

this function is continuous on nonnegative numbers and on negative numbers, but its not continuous on both

[ruby crypt]

alr that makes sense

[whole halo]

now if youd prefer this to be true, we have to be more specific on this "continuous on" stuff

[runic needle]

umm it's not cts at 0 which is non-neg

[whole halo]

you uh

moved the goalposts

[whole halo]

see:

now thats why we would look further

but thats not really the same thing as saying "on" is it

more of "in"

[ruby crypt]

oml

[runic needle]

idk dude

[ruby crypt]

oh bruh im stupid, I just need to show its uniformly cont.

[whole halo]

yea continuous everywhere doesnt have anything to do with uniformly continuous

continuous everywhere just means its continuous everywhere you care about

[ruby crypt]

uniform continuous => cont everywhere

[whole halo]

uniform continuous => cont

uniform continous everywhere => cont everywhere

ok to be more specific

usually when we use on, we dont look further

[ruby crypt]

on an interval right

[whole halo]

that results in stuff like this, but thats alr

on a set, could be an interval

when you say "continuous on something", that something is a set

then you look within that set and see if theres anything discontinuous

[ruby crypt]

like the previous ones we talked abt? (x-e, x+e)?

[whole halo]

you dont look outside the set, it can cause unexpected behavior but thats not what we need

(x - e, x + e) is an interval, I can tell you that

[ruby crypt]

yes

[whole halo]

dont know what you mean specifically beyond that

its an interval, thats for sure

[ruby crypt]

nope yep thats what I meant

[whole halo]

now "everywhere" is sort of handwavy, its short for "no need to watch out for discontinuous stuff"

it really just is the word everywhere

for example if I have √x, I can call that "continuous everywhere"

here, its "continuous on its domain [0, ∞)"

[ruby crypt]

yes how to prove that

[whole halo]

because I dont really have a reason to put in x = -1 and go "aha its discontinuous at x = -1" when I dont even have √x be real there

everywhere just cuts words out

so people then usually add whatever interval or set they mean right after it anyway

"continuous everywhere on [0, ∞)"

just means "continuous on [0, ∞)"

everywhere here is just a feeling kind of word, its to tell you the scope of what we're talking about

sometimes people use this in vague statements like

"1/x is continuous everywhere"

[ruby crypt]

exactly

[whole halo]

1/x is continuous on (-∞, 0) U (0, ∞)
1/x is not continuous on all of R

but you can feasibly consider either of those everywhere

because you dont have a good reason to do 1/0, so why consider it

1/x is continuous everywhere
1/x is not continuous everywhere

[ruby crypt]

so can I be sure that this typa question asking continuity everywhere would have a low chance to come out in exam 😅

[whole halo]

its just a word, itll be clear when you read it

[ruby crypt]

ok

[whole halo]

they should tell you an interval, or just tell you the one point they care about

[ruby crypt]

ok ok

thanks bro

[whole halo]

np

[ruby crypt]

whats that pfp btw