help-28

[acoustic pebble]

@little fiber

[little fiber]

sry what does sin (a) = cos (90-a) look like again?

[tidal reef]

what do you mean by

what does sin (a) = cos (90-a) look like again?

[little fiber]

when put in right triangle?

[little fiber]

okay I get it

what is it next?

yes I figured it out

the y of a is the x of (90-a) so basically the cos is on the sin

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[void arch]

If two sets are not disjoint and countable, can the union of the sets be countable aswell?

[void arch]

I don’t see how a bijective function will be able to be formed if this was the case

[fast peak]

but you do know how it works if they are disjoint?

[void arch]

Yes

Since you can just make an array that maps to every element from each set?

[fast peak]

what exactly do you mean with array

[void arch]

I mean you still can make an array with if A and B are not disjoint

[void arch]

So listing the elements of A and B in an array?

[fast peak]

again, what exactly do you mean with an array

and how does that give a bijection

[void arch]

Let S_ij denote the jth element in the set i. And S_1=A,S_2=B.
Similar to how you notate a matrix. As A and B are countable you can list them

Let f(1)=s_11, f(2)=s_12 f(3)=s_21 … you can map all the elements from both A and B in a diagonal fashion

[fast peak]

yes

actually writing the bijection out in the case that they are not disjoint is pretty annoying

but essentially in your process you just skip an element if you already had it earlier

that's it

[void arch]

Take for example A={1,3,5} and B={2,3,7}
My listing would of AuB is 1,3,5,2,3,7 written in a 2 by 3 matrix way. Theres two sets of {3} in the union so how can you miss it out?

Wouldn’t my above method still work though. Since my image of f is s_ij not the actual value of the element hence it’s injective?

[fast peak]

your function essentially goes through these two lists in a zigzag. one element at a time

f(1) = first element, then f(2)=next one, f(3) = next one and so on

and if the sets are not disjoint, you just skip the next one if you already had it earlier

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[near mauve]

How do I find out if a boolean function with three variables is monotone or not?

[glass crystal]

what is your function?

[near mauve]

I could get a function but over all how would I determin it?

[glass crystal]

well it depends on your starting set

in your function

what are the properties

is the order relation an easy one to study?

[near mauve]

Q(g, a, q) = (g ⊕ (g | ¬q)) ↓ a

lets say that

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[sterile sigil]

If f(x)=8 for 0<=x<3, and f(x)=2f(x-3), what is the graph of y=f(x) for -12<=x<6

[sterile sigil]

2f(x-3)=8 for -6<=x<0
4f(x-6)=8 for -24<=x<-12

how should I find y=f(x) for -12<=x<6

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@sterile sigil Has your question been resolved?

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[sterile sigil]

<@&286206848099549185>

[torn jolt]

for 3<=x<6, f(x)=2f(x-3)=2*8=16

extend this to the rest of the interval

(backwards)

[sterile sigil]

ohh ok got it (nvm)
ty

[sterile sigil]

how did you get that tho]

[torn jolt]

from the given equation?

3<=x<6

then subtract 3 so 0<=x-3<3

so then from the given f(x-3)=8

[sterile sigil]

ohhh

ok

[torn jolt]

nppm

cant u figure out rest?

[sterile sigil]

Yea I’ll try to

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[tough idol]

Can someone please teach me how to graph this model

[tough idol]

Someone sent me a message but i lost it in one of these channels

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[gritty rose]

Look in your discord history in this server

[tough idol]

How do i do that

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[rocky vale]

Go to the search and do from:(your name)

@tough idol here I searched you and found it

#help-19 message

I think, I found where you posted the same picture, anyway

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[sleek brook]

i need explanation

[torn jolt]

You can only integrate in that range if the function is continuous

[sleek brook]

how do ik ?

[torn jolt]

Asymptote

[sleek brook]

wdym

[torn jolt]

when x = 0 you divide by 0

You can’t do that

[sleek brook]

so i need to replace x with 0?

[woven snow]

is it really the case?

[sleek brook]

i dont get it

i asked someone and he sent me this

[woven snow]

idts that’s correct

You can only integrate in that range if the function is continuous

[torn jolt]

Ok that’s just semantics at this point, but in this case you can’t because the function is undefined at x = 0

[woven snow]

okay now that makes bit sense

perhaps you should take a look at this video https://m.youtube.com/watch?v=dHwrzLDmdT8

[sleek brook]

alright thanks

[normal tree]

this is not correct though

you are allowed to integrate over discontinuities

[woven snow]

thats what i’m saying

like take for example floor function which is very much integrable

[rare dock]

yea it's integrating over unbounded regions that can be troublesome

and even though it's not usually a part of the definition of riemann integral it still makes sense to integrate over regions with finitely many undefined points

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[rare dock]

just define the integral to be the integral of the function extended to the whole interval with whatever values for the undefined points you want

[normal tree]

countably many works, and indeed actually measure zero sets work too

the riemann series still converges okay

[rare dock]

i think that's a little trickier to make sense of actually

$f:\bR\setminus \bQ$, $f(x) = 1$

[glossy valve]

💜𝓁𝒶𝓎𝓁𝒶💜

[rare dock]

the convergence of $\int_0^1f(x)dx$ depends on the extension

[glossy valve]

💜𝓁𝒶𝓎𝓁𝒶💜

[rare dock]

it could be undefined or 1

or did i miss your point?

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[rare dock]

i guess as long as you pick an extension that makes the integral converge that's ok

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[tough idol]

Does anyone here do tutoring

[ivory ruin]

Couldn't you get a real tutor?

outside of discord?

[gaunt lynx]

it’s okay, just ask ur qn and ppl who can help will :)

[tough idol]

Ok

well im trying to make a trig function mimic a straight line

with certain conditions

which makes it hard to say over messages so anyone will to help can they jump in a call with me

willing*

[tardy quiver]

u can send a picture

[tough idol]

Ok ill just write the conditions down and send a pic

sorry for bad english

<@&286206848099549185>

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[tough flower]

i can't read most of it can you write the conditions down on discord? i can see the picture fine

[tough idol]

Well the condition is that it has have a base of 200

Walls that are 50m high

The midpoint has to be 100m

The angle of the roof has to be 30 degrees no less or more

The max heigh of roof is 30 meters higher than the walls (so 80)

And it can be any equation but the area under the curve has be an equation that maximizes it’s area

So trig equation would be best

So if you know how to mimic a straight line using a trig function please teach me

[tough flower]

what do you want the equation to represent @tough idol

[tough idol]

I want the equation to represent the roof

[tough flower]

do you want a curve instead of a trapezoid? is that what you are asking for?

[tough idol]

Yes

That maxmises the area underneath the curve

I did a quadratic and got 8414

And I did a trig which gave me 11 thousand

Someone else got 15 thousand which I can’t see how that is possible

We also need to show proof on how we got the equation so it can’t be a guess

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[ruby yarrow]

what is 0-(-6)

[void swift]

6

[void swift]

also go to #help-9

to get help

go there and send your problem and they will help you

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[fallen charm]

for the second part, why can't we do (5 choose 3) * (7 choose 4) ?

i.e. ways of choosing 3 questions from the first 5, then choosing 4 questions from the remaining 7 unanswered questions

[wild sleet]

it's the same as choosing 3 out of 4 by doing (4c1)(3c1)(2c1)

it creates a distinction

you count choices where the question is picked at step 1 and step 2 as separate

[fallen charm]

sorry I don't quite follow?

[wild sleet]

i confirmed that it doesn't work, and gave an example where it doesn't work either

we can't choose 3 questions out of 4 by choosing 2 and then choosing 1 out of 2 remaining, this is a different operation, it gives the correct answer if you actually do care if the question was picked when you were choosing 2 or at the second step

you don't care, you don't want this answer, it's too big, you want a smaller one

[fallen charm]

I see

so how are you supposed to do a question like this?

[wild sleet]

you're supposed to add up 3 cases probably

[fallen charm]

what if there are a lot of cases? is there a general approach?

[wild sleet]

can.t think of anything

[fallen charm]

ah ok. & thx for clarifying the above!

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[lone flint]

Find solution set of $24-e^{-2x} = 10$

[glossy valve]

ahmed349

[lone flint]

,rotate

[lone flint]

I got the answer correct

But the answer guide says it should be negative while I got it positive

[shrewd hamlet]

So $24-e^{-2x} = 10$

[glossy valve]

Stephen

[lone flint]

Yeah

[shrewd hamlet]

14 = e^ (-2x)

[lone flint]

Yeah

[shrewd hamlet]

ln(14) = ln(e ^ (-2x))

ln(14) = -2x

[lone flint]

-2x = ln(1/14)
x = -1/2 ln(1/14) but why is my way wrong

[shrewd hamlet]

How’d u get 1/14?

The - sign?

[lone flint]

From the -2 in -2x?

Divide both sides by -2

?

[shrewd hamlet]

So x = -ln(14)/(2)

[lone flint]

The heck

[shrewd hamlet]

?

[lone flint]

How

[shrewd hamlet]

What does the answer say

[lone flint]

This

[shrewd hamlet]

Do u understand up to this step

[lone flint]

Yeah

[shrewd hamlet]

So to solve for x, divide both sides by -2

[lone flint]

Divide both sides by -2

[shrewd hamlet]

Are u sure the answer doesn’t say 1/2 (ln (1/14))

[lone flint]

It does

It says that

[shrewd hamlet]

Yep

So with properties of logs

See the last property

[lone flint]

Yeah

[shrewd hamlet]

The - sign in front becomes the exponent of 14

So it becomes 14^-1

Which is 1/14

[lone flint]

Yes

Shit

Oh

Ohh

[shrewd hamlet]

So what part aren’t u understanding?

[lone flint]

I got it it moved the - to the exponent

[shrewd hamlet]

Yes

Exactly

[lone flint]

Yeah the answer says -1/2 ln(14) my bad

Thanks

[shrewd hamlet]

Sure, no worries

[lone flint]

.close

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[zealous lion]

Need help for this question

[scenic wren]

can you post your work?

[zealous lion]

yes, one second.

[deft zodiac]

one long second

[zealous lion]

basically stuck on part i), and also don't know what ii) is talking about

[scenic wren]

what you did at the beginning

[devout inlet]

To correct it: write it as (y^2 - 1)/y then it's easy to see what the division comes to.

[scenic wren]

instead you can do this

and to solve this, you can use u substitution

u=y^2-1 works fine

[zealous lion]

oooh i see

thanks I'll try that

thanks!

.close

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[agile vigil]

can someone check whether my NFA is correct?

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[hot mango]

the rate of change is the slope of the graph

this can also be interpreted as the amount that changes between each number on the table

[tight monolith]

Yeah but i don’t know how to solve the two problems

[hot mango]

What do you not understand from what I said so I can go into more detail

[tight monolith]

Like i just don’t understand how to solve it like where do i begin how do i tell what to start with and to begin my equation

[hot mango]

alrighty

wpould you like to start with prolem one or two first

[tight monolith]

One

[hot mango]

ok so rate of change is how much y is changing between when x changes by 1

so in this case how much is y(cost) changing when x(Tickets) increases by one?

[tight monolith]

One ticket is 15 dollars so it’s increasing by 15 every time

[hot mango]

and this is the rate of change

do you need to write an equasion or just find the rate of change

[tight monolith]

Just to find it

[hot mango]

alright well problem 1 is done

you all good with it?

[tight monolith]

15 dollars per ticket?

[hot mango]

yes

best way to do this is find y/x

so 75/5 is 15

and so on

[tight monolith]

Ok i’m ready for 2 now

[hot mango]

ok so have you learned about y=mx+b

[tight monolith]

Yes

[hot mango]

alrighty

can you derive an equation from the graph?

[tight monolith]

Idk how

[hot mango]

alrighty

so

So b = 0 as it starts at the origin

no we need to find m

as x is any number and our imput

best way to do this is plug in the out put and use algabra to figure it out

so

at point (1,400)

y = 400 and x = 1

400 = m*1

so m=400

this means the equasion is

y=400x

and because the rate of change is the slope, it is 400

[tight monolith]

So my answer would be 400

[hot mango]

yes

[tight monolith]

400 calories per hour

[hot mango]

yes

[tight monolith]

Ok thx

[hot mango]

does this make sense?

[tight monolith]

Yes

[hot mango]

alrighty

[tight monolith]

Ok i am done thx

.close

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[little river]

Is $f: \mathbb{N} \longrightarrow \mathbb{N}, \ n \mapsto \Big{\begin{array}{cc}
\frac{n}{2} & \text{n \ even}\
\frac{n+1}{2} & \text{n \ uneven} \
\end{array}$ bijective?

[glossy valve]

Levens

[little river]

I dont think it is

But

[rare dock]

try evaluating f at some small n's

[little river]

is my prove for injectivity correct, cause i lowkey dont think it is:

The mapping is called injective if for every $y \in \mathbb{N}$ there is at most one $x \in \mathbb{N}$ with $f(x) = y$. Let $n_1, n_2 \in \mathbb{N}$. So:
$$
f(n_1) = f(n_2) \Longrightarrow n_1 = n_2.
$$
Case 1; $n_1, n_2$ both even:
$$
\frac{n_1}{2} = \frac{n_2}{2} \Longrightarrow^{| \times 2} n_1 = n_2
$$
\
case 2; $n_1, n_2$ both odd:
$$
\frac{n_1 +1}{2} = \frac{n_2 + 1}{2} \Longrightarrow^{| \cdot 2} n_1 + 1 = n_2 + 1 \Longrightarrow^{| - 1} n_1 = n_2
$$
And so $f_2$ is injective.

[glossy valve]

Levens

[rare dock]

i can tell without reading it that it's not correct

[little river]

yeah

i dont think its injective either but like idk how to show that it isnt

[rare dock]

✨

[little river]

yeah thats why i dont think its injective

[paper thistle]

lol

[little river]

but idk how to prove it

[rare dock]

so... you've proved it? lol

[little river]

no i only proved it for some n, not all

wait what

[rare dock]

injective says for all a,b in N, f(a) = f(b) implies a = b

[paper thistle]

wait I messed up sorry

[little river]

yeah

[rare dock]

so to prove f is not injective

you just have to show there exist a,b such that f(a) = f(b) and a =/= b

[little river]

oh right

counterexample

mb

ok i got that down

what about surjectivity

[rare dock]

what do ya think?

[little river]

i think so?

i mean i think it is surjective

[rare dock]

why?

[little river]

i tried it for the first few n

[rare dock]

ok so if i give you any m at all

what n would have the property f(n) = m?

[little river]

well for even n's, its 2m = n and for uneven, its 2m - 1 = n

[rare dock]

you're kinda saying too much

[little river]

oh

so what

[rare dock]

it's just

for any m, f(2m) = m

like that's the whole proof

not the only proof but that's all you need

[little river]

oh

idrk how to prove it all from that

[rare dock]

what?

[little river]

im still stuck basically

[rare dock]

but

i just gave the whole proof

f is surjective if for every m in N, there exists n in N such that f(n) = m

[rare dock]

and this is what that's saying

[little river]

wait whys that the whole proof

[rare dock]

it shows this

[little river]

why was that too much

[little river]

how does that apply to the uneven n's

[rare dock]

i don't care about the uneven n's because i don't have to 💅

[little river]

lol

i care about them🥰

[rare dock]

we also have f(2m-1) = n

but i don't care about that

[little river]

$f_2$ is called surjective if for every $y \in \mathbb{N}$ there is at least one $n\in \mathbb{N}$ with $f(n)=y$. So we choose an arbitrary $y \in \mathbb{N}$ such that:\
\

1. straight n's:
   $$
   f(n) = y = \frac{n}{2} \Longrightarrow 2 \cdot y = n
   $$
2. odd n's:
   $$
   f(n) = y = \frac{n+1}{2} \Longrightarrow 2 \cdot y - 1 = n
   $$
   Thus $f_2$ is surjective.

is this legal

[rare dock]

it's not like we have to find them all

[little river]

oh it cuts off

[rare dock]

can you rewrite this with consistent variable names?

[little river]

ohhh wait thats my old version

sry hold on

like this?

[rare dock]

why are there 3 variables?

[little river]

ok ig ill get rid of x

or n

u choose

[rare dock]

your proof, your choice

[little river]

lol okay

less work

[rare dock]

what are these implication arrows?

[little river]

uh

should i get rid of some

[rare dock]

yes, every single one

[little river]

aw okay

[rare dock]

the proof is backwards

[little river]

Oh

[rare dock]

no omg ok

i think this is worse 😭

[little river]

lol

[rare dock]

i didn't mean like

just delete them and leave everything else exactly as it was lol

[glossy valve]

Levens

[little river]

why cant i just have it like this

do i write it backwards now

[rare dock]

yes it's still backwards

[little river]

okay

[rare dock]

and is saying more than you need to

[little river]

$f_2$ is called surjective if for every $y \in \mathbb{N}$ there is at least one $n\in \mathbb{N}$ with $f(n)=y$. So we choose an arbitrary $y \in \mathbb{N}$ such that:\
\

1. straight n's:
   $$
   n = 2 \cdot y \Longrightarrow \frac{n}{2} = y = f(n)
   $$
2. odd n's:
   $$
   n = 2 \cdot y - 1 \Longrightarrow \frac{n+1}{2} = y = f(n)
   $$
   Thus $f_2$ is surjective.

is this legal

[glossy valve]

Levens

[little river]

like that?

[rare dock]

much better

but still saying way more than you need

[little river]

am i really saying too much

why do i cut off the odd n's part

[rare dock]

ok look

if someone says

"prove there exists a real number x such that 4x^11 + 5x^5 + x^4 + 2x^3 + 9x = 0"

you can just say 0 is a valid x and you've proved it

you don't need to find them all or even explain how you came up with 0

[little river]

oh okay.. so i just take the first case which coincidentially was for the even n's

but i could also only do it for uneven

[rare dock]

literally just this is enough

[little river]

okay

thanks layla

..are u up for another one

[rare dock]

maybe 😳

[little river]

yay

okay

my problem with this one

is the notation of it

so

[rare dock]

ok, what's confusing?

[little river]

the whole pair thing with x,x^2

idk how im supposed to show/contraprove surjectivity with that

[rare dock]

what about the notation is confusing?

[little river]

like usually for surjectivity

you rewrite for y right

so

f(x) here is (x,x^2)

but how do i rewrite y = (x,x^2)

[little river]

wait not y

bruh

idk man

[rare dock]

f_1 is surjective would mean: for every pair of real numbers (a,b), there exists a real number x such that f_1(x) = (a,b)

[little river]

okay

i still dont know how to visualise it

[rare dock]

the image of f_1 is the exact same as the graph of x -> x^2 (as in the graph of the single variable real function)

and i'm using a kind of technical definition for graph when i say that

that's how i'd 'visualize' it

but can you think of a pair of real numbers that can't be written like (x,x^2)?

[little river]

I guess (1,2)

[rare dock]

ok that works

[little river]

i see

ok let me have a go and ill get back to u

[rare dock]

alrighty 🙂

[little river]

The mapping $f_1$ is called surjective if for every $(a,b) \in \mathbb{R}^2$ there is a $x \in \mathbb{R}$ with $f_1(x) = (a,b) $ gives. We assume that this is not the case and create a counterexample.\
There is a tuple $(a,b)$ such that $f(x) \neq (a,b)$ holds.\
\
Let $a=1$ and $b=2$, then:
$$
f(x) = (x,x^2) = (1,2)
$$
This is wrong because $1^2 \neq 2$ and thus $f_1$ is not surjective.\

Is this... okay

[glossy valve]

Levens

[rare dock]

holds... for all x in R?

[little river]

wdym

[rare dock]

what are you trying to say there?

[little river]

there’s a tuple (a,b) with f(x) =/= (a,b)

mb, my original is in german - google translate probably butchered it

[rare dock]

you don't "assume that's not the case"

you're showing it's not the case by creating a counterexample

[little river]

true

good point

makes more sense

and for injectivity?

hmm let me think

no right? since it’s real number

[rare dock]

wait i wasn't done commenting

[little river]

numbers

[little river]

lol okay

mb

[rare dock]

you're trying to say tuple such that ... holds for all x, right?

cus you didn't really explain why f(x) \neq (1,2) for all x

[little river]

mm right right

is there anything else?

[rare dock]

that's all ig

[little river]

do we disprove injectivity with a counter example again?

[rare dock]

nope ✨

[little river]

😔

is it injective

[rare dock]

wdy think?

[little river]

ig it is, at first i thought bc of negatives and all.. but now i think it’s injective. cause even if it was a negative, the output would be different anyway

[rare dock]

ok sounds good, try to write the proof

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[polar nacelle]

1. A six sided die with labels {1, 2, 3, 4, 5, 6} is loaded such that, for each i ∈ {1, 2, 3, 4, 5, 6}, the probability
   of seeing i when rolling is directly proportional to i. What is the probability of rolling an even number?

[polar nacelle]

this would just be event space 2,4,6 right?

1/3?

that's how you'd solve it?

event space 3 divided by sample space 6?

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@polar nacelle Has your question been resolved?

[polar nacelle]

or 1/6+1/6+1/6+1/6+1/6+1/6=1

[vapid herald]

seems like 1/3 to me

[polar nacelle]

oh yea i meant for sample space

cuz it has to equal 1

for probability

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[vapid herald]

3/6

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[hard phoenix]

It depends on the context

Generally it would, since negative 5 x negative 5 = 25

But in cases where a negative value might not be possible (e.g. if you're calculating the length of a line), you can always assume that the square root of 25 would be positive 5

[tropic condor]

square root is a function and can only have one value

[rare dock]

hmm i'd say

generally

sqrt(25) = 5 and that's all

[hoary valve]

its called a principal root

basically if you take the square root of any positive number the root must be positive

[rare dock]

sqrt(25) = 5 by definition

what other way would you want it?

[glossy valve]

Yousuf

Yousuf

[rare dock]

because working with functions is nice and it's still easy to capture both the square roots of a number

the square roots of a (nonnegative) number x are sqrt(x) and -sqrt(x)

wdym?

ok so

ugh nvm what i was going to say is too complicated

by a square root of a number x i mean a number y such that y^2 = x

positive x's have two y's such that y^2 = x

so like in your example

if x = 4

2^2 = 4 and (-2)^2 = 4

those are "the square roots" of 4

and 0 just has one square root, 0, and negative numbers have none

not right now but yea those could be different words in other places

ok so

when x is positive, x has a positive square root and a negative square root

sqrt(x) is just, by definition, the positive square root of x

as a function that's how it's defined

the positive square root is 2

sqrt(x) is not "the square roots of x" it's "the positive square root of x"

the stuff i said before was just like

background information

i'm so sorry 😭

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[sleek moat]

My professor wrote this, but I can't understand how he got from the first equation to the last.

Can someone please elucidate the process? Is it some sort of Taylor expansion?

[full forum]

@sleek moat Has your question been resolved?

[sleek moat]

I understand that 2.95e-2 is (0.632/0.368)^2/100 (my professor used different sig figs).

However, I don't understand why we are assuming that $10K=\frac{0.63004}{0.36996}N_{O_2}^{1/2}$.

[glossy valve]

PhysMan

[sleek moat]

I'm pretty sure the limit is $\infty$.

[glossy valve]

PhysMan

[sweet ingot]

Should be (N_O_2)^(-1/2) or smth

rn the difference between the expression here for 10K and the one you wrote gets larger and larger as N_O_2 gets smaller (approaches inf)

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[little river]

hello, hello, is this function surjective?

[glossy valve]

Levens

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@little river Has your question been resolved?

[little river]

<@&286206848099549185>

[onyx glen]

this sounds weird

like, this function IS surjective but it sounds like you were way off in actually establishing that

[fast peak]

the function is surjective if for every z in R there exists a pair (x,y) with f(x,y)=z

aka the equation x^2-y^2+1=z has to always be solvable for all z in R

[little river]

it is

right

2 solutions

uh

mm

[little river]

ye let me scratch that

[little river]

i tried showing this but i need some help

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@little river Has your question been resolved?

[little river]

mm it's not injective right? should i show that w a counterexample?

<@&286206848099549185>

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[weak silo]

hi yall, uhm is this a quadratic inequality?

[vast fossil]

In terms of x yes

But technically it's third degree I think

[weak silo]

why 0-0

[vast fossil]

You can determine that by adding powers in each of the terms

In this case you have x^2 y^1

And nothing higher

[weak silo]

Oh oki thanks

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[full forum]

Please don't occupy multiple help channels.

[rapid laurel]

.close

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[torn jolt]

.close

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[median beacon]

is this cayley table correct?

[simple totem]

looks good

[median beacon]

so i need to show if its a group or not now

there are three properties, associativty, inverse and identity element

how do i use the table to show associativty

?

[onyx glen]

are you instructed to (dis)prove assoc SPECIFICALLY using the table and nothing else

[median beacon]

no

but

in the question, cuz it says cayley table, i just thought we would have to soely use the table

but then again i guess that doesnt have to be the case

[simple totem]

you can

associativity is annoying to deal with here

consider if there’s an identity element

[onyx glen]

it isn't really all that annoying

$3(3xy)z \overset?= 3x(3yz)$

[glossy valve]

Ann

[simple totem]

oh true

[median beacon]

is z this identity element we assume?

[onyx glen]

no

[median beacon]

oh yh forget that

yh ok cool

[median beacon]

am i supposed to sub values in to show it holds?

[simple totem]

you need to show this isn’t a group

check if there’s an identity element in the table

[simple totem]

ann showed why associativity holds pretty much

[median beacon]

why this isnt@

is it not a group>

?

[simple totem]

.

[median beacon]

theres no 1's through the diagonal

so no

[simple totem]

this may be a group sorry

[median beacon]

how tho

[simple totem]

that’s not how you check for an identity

[median beacon]

there arent any idenity elements

[simple totem]

you need to check if there’s an element e so that e * g = g * e = g for all g in the group

[median beacon]

ok

lets do

associavitity first

ive done closure already

[simple totem]

ok how do you check for associativity

do you know the definition @median beacon ?

yes, so how would you check if the operation is associative?

[median beacon]

sub in values for x y and check if LHS = RHS

?

[simple totem]

there’s an easier way

show 3(3xy)z mod 10 = 3x(3yz) mod 10

use the fact that multiplication on the reals is commutative

[median beacon]

hmmm

and how would i show this

[simple totem]

3x(3yz) = 9x(yz) = 9xyz
3(3xy)z = 9(xy)z = 9xyz

[median beacon]

@simple totem

[simple totem]

yea looks fine

[median beacon]

cool

[simple totem]

now find the identity (there actually is one)

[median beacon]

wat about commutative?

[simple totem]

you don’t need to show commutative for a group

[median beacon]

wait

how about closure?

[simple totem]

it’s implied in the table

every product gives you an element in the set

[median beacon]

my module notes doesnt say anything about proving closure

can we do identity element first plz

before inverse

[simple totem]

they’re saying the . operation is a binary operation

that’s only the case if the set is closed under it

you can take that as granted, or show it

it’s up to you

[median beacon]

ok

can we prove G2 next plz

weve done G1

[simple totem]

yes

read above we already discussed how to find the identity

[median beacon]

so all i need to say os

theres 1

in the table

so

theres an indentity element

?

[simple totem]

no

[median beacon]

1 in the set?

[simple totem]

1 is not the identity

look at the definition of the identity again

notice that 1 * 1 =/= 1

[median beacon]

is it 7?

[simple totem]

yes

[median beacon]

cuz 1 x 7

= 1

[simple totem]

and 3*7 = 3

and so on

[median beacon]

^^

[simple totem]

you also need to show left multiplication

[median beacon]

wdym

[simple totem]

you showed g * 7 = g

you also need to show 7 * g = g

[median beacon]

^^

[simple totem]

looks fine

[median beacon]

imma have a go at inverse

then show u

[median beacon]

do all the elements of S need to be inverses?

for it to be a group?

[simple totem]

they need to have an inverse

[median beacon]

so all of them

need to have inverses?

[simple totem]

yes

[median beacon]

well

only two

[simple totem]

look at your definitions

[median beacon]

have invereses

3 x 3 = 7

and 7 x 7 = 7

thats it no?

[simple totem]

no

do you know the definition of an inverse

[simple totem]

and no

[median beacon]

oh 1 x 9 = 7

[simple totem]

yes

and 9 * 1 = 7

[median beacon]

@simple totem

[simple totem]

yes

[median beacon]

cool and lastly, would u mark this 7/7?

[simple totem]

not sure, i’m not a grader

[median beacon]

but is this along the write lines

for these kinda questions?

[simple totem]

i think so

[median beacon]

thanks for the help

appreciate it

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[thorny inlet]

Hey, I am building a minuture windtunnel and for the contraction I need some help I dont know what they mean by these angles. For example a third degree polynomal curve

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[sleek moat]

It appears that $\alpha/2$ is the angle formed because of contraction of $H_{ent}$ to $H_{exit}$. $\alpha/2=\tan^{-1}\frac{(H_{ent}-H_{exit})/2}{d},\ \beta/2=\tan^{-1}\frac{(W_{ent}-W_{exit})/2}{d}$ where $d$ is the distance between the inlet and the exit.

[glossy valve]

PhysMan

[sleek moat]

It seems they do not specify $d$. Instead preferring that you choose a suitable $d$ based on the resulting contraction half-angles.

[glossy valve]

PhysMan

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[obsidian lodge]

https://talkmath.ams3.digitaloceanspaces.com/8dqijcc73f6bf81.png

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[spice orchid]

Ask a question

[obsidian lodge]

How do I solve g)

[wild sleet]

you need to say what's the lowest and highest number you can write

that's range

the smallest is 0

[obsidian lodge]

For unsigned, that'd be 0000 … 0000

For 16 bits

And highest is 1111 … 1111

Do we consider the 0 as well?

16 bits mean we have 2^16 combinations

[wild sleet]

you're kinda supposed to just give it in decimal

i think

not 0000 0000 0000 0000, but just 0

[obsidian lodge]

Integer, 16 Bit Unsigned: Unsigned whole or natural numbers ranging from 0 to +65535. Integer, 16 bit Unsigned data type is used for numerical tags where only positive variables will be used. Integer, 16 Bit BCD: Unsigned Binary Coded Decimal value ranging from 0 to +9999.

[wild sleet]

yeah you basically do 2^16 and subtract one

[obsidian lodge]

So here they don't consider 0 as a natural number, so they took 2^16. -1

Right?

[wild sleet]

no

think in decimal, what's the range of one digit decimal numbers

it's 0-9

[obsidian lodge]

yep

[wild sleet]

why do you subtract 10−1, it's not because you don't consider 0 to be a natural number

you just do

[obsidian lodge]

because 10 is 10^1*1 + 10^0*0

And 9 is 10^0 * 9

It's a two decimal digit numeral

[wild sleet]

i'll go somewhere else

[obsidian lodge]

🥲

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[boreal iris]

hello

[boreal iris]

in this question, i tried solving by taking cases of 113, and 122

but the websites that i refer to, use this formula for a shorter calculation

can sm1 help me with the logic behind this one?

I understand combinatorics by translating stuff into "words", if it makes sense

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@boreal iris Has your question been resolved?

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@boreal iris Has your question been resolved?

[full forum]

@boreal iris Has your question been resolved?

[tough flower]

um your website and problem don't match because the website says that one person gets no ball while your problem says each person gets a ball so you have the wrong equation

@boreal iris

[boreal iris]

No, the solution image is one part of the formula, they subtract what theygot there with the total number of ways of distributions

I need help with how they derived the formula

[tough flower]

what did they get as the answer? it should be 60 because 5x4x3

the first person has five balls to choose from, the second person has 4 after the first chose, and the last person has 3 after both of them chose, leaving 5x4x3 possibilities

classic combinatoric problem, im assuming they don't care about repeats

@boreal iris are you understanding this?

[boreal iris]

The answer is 150

[tough flower]

?

[boreal iris]

You're not accounting the distribution of the remaining balls

[tough flower]

oh wait, i assumed each person only gets one ball, oops, need to read more carefully

right

then you can use sticks and stars method

or whatever its called

[boreal iris]

What is that?

Can you give a gist?

[tough flower]

so like, 5 balls, 3 people, the people are bars, the balls are the stars, and the bars stand between the balls, and you can arrange them in many ways, and you find the amount of ways

like | | | o o o o o

and then you arrange them

but for your problem, its more complicated because each bar has to have one between each other

I've done these types of problems so many times and i keep forgetting how to do it 😭

[boreal iris]

Oh no, that word be way too tedious, even with multinomial

Ik it's inclusion-exclusion, but can't keep my finger on the logic

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[vast cove]

Hi everyone

[vast cove]

I have a question in combinatorics

Here is the question
We choose randomly a number with 10 digits (0 can be the left most digit as well).

a. What is the probability that the sequence 89989 doesn't appear in the number?

b. What is the probability that only 6 different digits will appear in the number?

[floral perch]

think about a logically

wait i mis read

ok so in ten digits 89989 (5 digits) can appear 5 times

[vast cove]

something needs to be multiplied by (10-5+1) 6

6 times isn't it?

[floral perch]

yeah my bad

6

just look at a single case

the numbers to the side of the sequence will always all be available

it's just for 5 digits we can't

[rare dock]

i think it's harder than it looks because you'll double count things if you aren't really careful

if you do it like that

[vast cove]

Yup me too

But I think I can say 10^10 is the total options

so I can deduce 6 options that this sequence appears in every point

but it seems too easy

[floral perch]

i feel like that might just be it

oh wait

8998989989

that would also make the sequence appear

[rare dock]

8998999891 for example would also get double counted

well that depends what you do but i'm just trying to say that's an example of what makes the problem harder than it looks

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[near ether]

thats not the correct derivative

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[bold temple]

I have a math test for tomorrow and and I do not understand anything , can someone help me?

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[halcyon sleet]

How do I isolate y in this?

[gritty rose]

Why do you need to

[halcyon sleet]

I need to find tangent line

and do so I would have to isolate y and then derive it to find the slope

right?

[gritty rose]

Differentiate

And no. You should have learned implicit derivatives

[halcyon sleet]

oh ok

ill try that instead

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[robust wasp]

Is it possible to calculate the time it takes for water to drain from an object that isn't cylinder? I only know of torricelli's' law, but that's with cylinders.

[wanton gorge]

yeah

using the chain rule

however you need another factor

such as ds/dt or dr/dt

really depends on the object

[robust wasp]

Hyperbola..

I found the volume

[harsh carbon]

What parameters do you know

[wanton gorge]

^

[robust wasp]

I just know height, width, length and volume

However the hyperbola is uneven

One side is larger than the other.

[wanton gorge]

find the derivative of the volume, in terms of 1 variable, replace.

then if you want how much the volume is decreasing

find how much the radius/side/anything is decreasing

for ex 4cm/s

and then multiply

[wanton gorge]

send the picture

[wanton gorge]

oh shiz that i cant do

[robust wasp]

Yeah..

[wanton gorge]

im not that lvl

[robust wasp]

Damnit.

Well, thanks for your help so far.

[wanton gorge]

np

also scratch what i said

it'd be wrong in this case

since it'd need a cst shape

[robust wasp]

Alr.

[harsh carbon]

Can we have the question too?

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[robust wasp]

It's kinda my own question, simply: Find the time it takes for water filled structure to drain.

[harsh carbon]

Oh okay, I'll think about it. Not sure you can get to a definitive answer with volume,height,width and length.

[robust wasp]

Alr, thanks.

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[torn jolt]

pretty sure u should calculate the volume that can be held in that
then check area of like the circle shaped exit
and in each of 1m^2 of areas 1m^2 will be drained i guess
but you still need to know the time so basically this msg is useless

[robust wasp]

I know the volume of the shape and how much water is in there.

[robust wasp]

Also you need to account for different speeds, water depths is greatest at the beginning so water drains faster when its full than when its almost empty.

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[torn jolt]

also when the water ia draining air needs to flow inside so this might slow down the draining depending on the way the air is entering
your question is so dependent of many different conditions so im incapable of helping

[robust wasp]

Alr, thanks for giving some input. I was going to assume there not to be wind for this problem anyway.

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[stiff swan]

why is pi raised to negative infinity =0?

[stiff swan]

<@&268886789983436800>

[reef sparrow]

do you mean in a limit?

otherwise raising something "to negative infinity" doesnt make much sense to me

[stiff swan]

kindaaa

something like this

so the way we were taught to do it was

first do limit of the 1/w first

so that's negative infinity

now evaluate pi raised to negative infinity

[reef sparrow]

in this case that works

[stiff swan]

next evaluate cot 0

then you got the answer

[reef sparrow]

but what youre doing is looking at the LIMIT as pi^[stuff] goes to pi^-infinity

in other words, take pi

and keep taking "more negative" powers

pi^-1, pi^-2, pi^-3

(they dont have to be integers but its just for the sake of example)

you can see its getting successively smaller

1/pi, 1/pi^2, 1/pi^3...

and indeed, by taking "larger negatives" (i.e. closer to -infinity), you can make the value as close to 0 as you want

so the limit is 0 / 0 is an asymptote / whatever phrasing you like to use

[aside: i do caution you that this only works because "w" only occurs in the expression once]

[your technique would give an incorrect result for, say, the limit of (1+1/n)^n as n → infinity]

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[stiff swan]

what's the correct way of doing it then?

[reef sparrow]

well, that limit is precisely euler's number "e"

you probably wont be able to evaluate it with techniques of introductory calculus

[stiff swan]

dang we got even harder stuff than this

I don't know how they do it

let me show you another example

[stiff swan]

here

so same thing

they sub t

[reef sparrow]

yeah, the technique works there as well.

[stiff swan]

they get tan t, then they evaluate ^e

what's this technique called I can't find it on youtube

[reef sparrow]

the point is that it may not work if the variable (in this case, t) appears multiple times in the limit

uh i am not familiar with a name

like the mathematical justification is "limits commute with continuous functions" but i dont think thatll help you much

[stiff swan]

okay so tan(pi/2 from the right) that's negative infinity right?

[reef sparrow]

yes

[stiff swan]

e^-infinity = 0?

[reef sparrow]

yes

[stiff swan]

we sub 0

lnx 0 is 1

cosh (1) is 0

correct?

limit is 0?

[reef sparrow]

ln(0) is 1?

news to me

[stiff swan]

oh frick

ln1 is 0

[reef sparrow]

the limit of ln(x) as x→0 from the right is -infinity

[stiff swan]

the graph of ln is the same with log?

right?

yep right

okay but what is cosh(1/-infinity)?

also is 1/-infinity = -infinity?