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#help-49
rich plaza
How do I do part C of this question?
midnight plankBOT
How do I do part C of this question?
lavish venture
well what is the formula for speed
given the position functions
how would you find the velocity
rich plaza
well what is the formula for speed
distance over time
lavish venture
in calculus terms
lavish venture
oh actually they gave you v=(x,y)
rich plaza
yeah
lavish venture
ahh ok my fault i assumed x and y were functions of position
rich plaza
ohh
lavish venture
were you able to do parts a and b?
rich plaza
yes
lavish venture
right so how did you do part a then
lavish venture
hmm you don’t need to graph
rich plaza
and solved it out
lavish venture
plugged 4 into what
lavish venture
yes but for what equation
rich plaza
both x and y
lavish venture
then you did what
that would give you two numbers
that isn’t the speed
that’s the velocity in each direction
lavish venture
right but that’s not the speed
that’s the velocity vector
speed is a scalar quantity
lavish venture
it gives only magnitude
rich plaza
*?
lavish venture
well given two vectors how would you find the magnitude of the resultant?
you have two vectors
velocity in the x and y direction
how do you find the resultant
hint: you used it in geometry class
the formula is a meme
does pythagorean theorem ring any bells
lavish venture
in general speed is the magnitude of velocity which is just =sqrt(vx^2+vy^2)
lavish venture
wait so the stuff with vectors is this supposed to be calculus?
yes
rich plaza
OH
lavish venture
generally you’ll be given the position in each direction as a function of time
and have to find x’ and y’
lavish venture
then that will be your vx and vy
lavish venture
well im taking ap pre calc rn and they gave me this as my last unit
ohh you’re not in ap calc
rich plaza
yeahhh
rich plaza
i had a feeling this was calc stuff
lavish venture
i remember checking the curriculum in the beginning of last year
there’s a good amount of calc still
rich plaza
yeah
lavish venture
i had a feeling this was calc stuff
well vectors themselves aren’t calculus really
but the x’ and y’ being the derivatives of the position functions is
but that’s not needed here
rich plaza
oh kk
lavish venture
since they give you the velocity component functions
so for part a you need to find the resultant of what you got
and then for part c you set speed=0 then solve for t
using
lavish venture
mhm so you have essentially two legs of a triangle right?
rich plaza
yes
lavish venture
you can treat the vectors as such
then find the resultant
or hypotenuse so to speak
midnight plankBOT
@rich plaza Has your question been resolved?
rich plaza
in general speed is the magnitude of velocity which is just =sqrt(vx^2+vy^2)
if I'm supposed to write this to solve for 0, how do I write a system of equations?
because its asking for a system of equations
lavish venture
if I'm supposed to write this to solve for 0, how do I write a system of equatio...
so if the speed is zero that must mean that both x and y are zero
the only way for the sum of two squares to be zero is if both terms are zero
since squared terms are always positive
lavish venture
you’re solving for the time t such that the speed is zero
rich plaza
so my system of equations would be 0= (x equation) and 0= (y equation)
i dont want to type it out
lavish venture
exactly
rich plaza
it never equals 0
rich plaza
so that means that it doesnt exist
lavish venture
yup
lavish venture
and your reason is because 3^t≠0
rich plaza
that makes more sense now
rich plaza
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midnight plankBOT
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twilit field
Can I have this verified please?
midnight plankBOT
Can I have this verified please?
twilit field
S(a set) denotes all the subsets of that set
unique stream
Ah k so including duplicates?
Nah it'd be the same
So S(...) yields the same as P(...) just as a list instead of set then
unique stream
S(a set) denotes all the subsets of that set
You can always first verify by checking if you have found 2^n subsets where the original set contains n elements
twilit field
I have 8 elements, yeah.
warm basin
Can I have this verified please?
if you mean are those all the ways to choose elements from {R, Q, N}, basic combination verifies that what you wrote is all of them
if you mean smn else i have no clue 😭
twilit field
Is my answer right, is my question.
shell wigeon
That's one too many parentheses
subtle zinc
IS THE LHS A SET
twilit field
sry for delay
I see, thanks!
twilit field
That's one too many parentheses
Sorry
crisp oxide
Not my field, but shouldn’t the rhs be a set?
dreamy lichen
Not my field, but shouldn’t the rhs be a set?
S denotes list / sequence of all subsets
I don't think it's a good notation, but it's whatever it is
there is also P that denotes the set of all subsets
Also @twilit field , perhaps you forgot to close this?
urban viper
midnight plankBOT
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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
urban viper
```
What step are you on?
1. I don't know where to begin.
2. I have begun but go...
i was sending it man 😭
urban viper
i worked out the line equation but idk where to go from there
small jasper
Where did this come from
You’re not necessarily drawing a tangent at a point where y=0
small jasper
You’re not necessarily drawing a tangent at a point where y=0
.
urban viper
so then what do i do
small jasper
Write the equation of the tangent for an arbitrary angle and then substitute x=3, y=0
small jasper
Basically find the tangent for any value theta rather than a specific value
b/c you already found that $\frac{dy}{dx}=-\sin \theta$, so the equation is simply $$y=(-\sin \theta)( x-4\sin \theta)+\cos 2\theta$$
grand pondBOT
Civil Service Pigeon
urban viper
oh i see
i found the cartesian equation instead
i got y = 1 - x^2/8
but idk what to do from there
small jasper
i found the cartesian equation instead
Eh you do the same thing
urban viper
Eh you do the same thing
what is 'the same thing'
small jasper
Find the tangent for any point on the curve rather than a specific point
midnight plankBOT
@urban viper Has your question been resolved?
urban viper
Find the tangent for any point on the curve rather than a specific point
ive done that but it was wrong
nvm found it
.close
midnight plankBOT
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dusky cradle
midnight plankBOT
prime hornet
feral sedge
this is a fairly unfortunate integral
dusky cradle
this is a fairly unfortunate integral
Ohh no😂
next bramble
Can you use Laplace transforms for this?
dusky cradle
Can you use Laplace transforms for this?
I don't know actually about Laplace
If it works fine. I will surely use
next bramble
Where did you see that question from?
dusky cradle
Where did you see that question from?
Someone sent it to me @next bramble
feral sedge
i fail to see how laplace transform would help here
next bramble
i fail to see how laplace transform would help here
Yeah, it doesn't really help much. Laplace is my go-to when dealing with hard-looking calculus questions like that.
dusky cradle
So is this not solvable?
feral sedge
it's solvable just not with a pretty process
dusky cradle
Could you tell me the steps? I will try it
midnight plankBOT
@dusky cradle Has your question been resolved?
dusky cradle
@feral sedge
midnight plankBOT
@dusky cradle Has your question been resolved?
dusky cradle
it's solvable just not with a pretty process
If you have no tips, strategy to share then why say so..say NO like others said
Lol
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midnight plankBOT
dusky cradle
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midnight plankBOT
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dusky cradle
@bright shoal
bright shoal
yes
dusky cradle
Yes
grand pondBOT
dusky cradle
Sure
bright shoal
take t^3 common from denominator
dusky cradle
Then?
dusky cradle
Yeah
dusky cradle
Sent
grand pondBOT
bright shoal
Okok
dusky cradle
Use these sign
dusky cradle
🤔
bright shoal
1/t^5=1/t*t^4
dusky cradle
This one is correct
dusky cradle
We have (1/t)*(1/t^4)
But
dusky cradle
We have another term
bright shoal
Integration by parts
dusky cradle
Do u know ibp?
Obviously
bright shoal
So see
dusky cradle
grand pondBOT
dusky cradle
,w d/dt (1-1/t^3)
How?
dusky cradle
It is
bright shoal
Now check on your tab
dusky cradle
But i am doing
dusky cradle
Yeah
bright shoal
,w int 1/[t^4*(1-1/t^3)^3]
grand pondBOT
bright shoal
Got it?
bright shoal
Oh k
dusky cradle
bright shoal
Where did the 1 go
bright shoal
This image
dusky cradle
Here
bright shoal
dv and u
grand pondBOT
Requested by oppailol.
odd lagoon
wolfram alpha letdown
bright shoal
Lol
dusky cradle
You are confusing me at these steps u and du
And dv
Let's back to it again
And i know how the integrating by parts
bright shoal
Okom
@dusky cradle
See imma tell simply
Integrate the 1/t^4 and stuff
And differentiate the 1/t
bright shoal
Integrate it
feral sedge
If you have no tips, strategy to share then why say so..say NO like others said
so i just went away for some time
bright shoal
Noo
feral sedge
also "NO" is wrong
bright shoal
U should type in a formal way
feral sedge
so i am not going to tell you "no" when the answer is "yes"
bright shoal
That sry for the inconvenience but I don't have sufficient info on this topic to help u
feral sedge
also you are listening to someone who does not know how to integrate 🤷♂️ so your choice here
bright shoal
U talking bout me?
feral sedge
yes
dusky cradle
he is helping perfectly
bright shoal
We are just using ibp
dusky cradle
Why you are saying such things sir?
dusky cradle
Why being toxic
Nevermind, we will continue
bright shoal
Ok
dusky cradle
I guess that is why they are saying you wrong
viral dagger
Given that subset $A={a_1, a_2, a_3, \dots, a_{2001}} \subset \mathbb{N}$, let m be the ammount of subsets of $A$ that has $3$ elements ${a_i, a_j, a_k}$ where $1\leq i < j < k \leq 2001$, such that $a_j=a_i+1$ and $a_k=a_j+1$, for every possibility of set $A$, what is the highest value of m?
midnight plankBOT
Given that subset $A={a_1, a_2, a_3, \dots, a_{2001}} \subset \mathbb{N}$, let m...
viral dagger
im not sure if i translated this correctly, but i gave it my best shot
grand pondBOT
Skill_Issue
visual tiger
!status
midnight plankBOT
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
1
visual tiger
Alr
So the first hint I would give you
Is that integers a,b,c such that b = a+1 and c=b+1 have a name...
(If you write them all in terms of a you will maybe see better)
visual tiger
Yeah
feral barn
midnight plankBOT
modern sapphire
The equivalent way of writing AF = BG is OF = OG
so it would be great if you could prove the second statement instead
try to create new triangles with one having OF as a side and other having OG as its side
and prove they are congruent
midnight plankBOT
@feral barn Has your question been resolved?
feral barn
try to create new triangles with one having OF as a side and other having OG as ...
Yep thanks
midnight plankBOT
midnight plankBOT
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midnight plankBOT
subtle zinc
1:2 isn't a number
it is simply an idea, however you may also treat it like a number
this is just a definition thing, not worth worrying about
no it's never true
let's say alice and bob both have apples in the ratio 1:2
that means for every apple alice has, bob has 2 apples
so bob has twice as many apples as alice
midnight plankBOT
@last slate Has your question been resolved?
dusky cradle
|ax|+|by|=1 it will draw a square always?
midnight plankBOT
|ax|+|by|=1 it will draw a square always?
dusky cradle
Is there any rhombus possible?
Square is special case of rhombus when all amgles 90°
So only case?
viral dagger
didnt you post this question before
dusky cradle
Yes I posted but i didn't get my desried answer properly
It will be always a square?
dreamy lichen
rhombus would be if if a != b I think
dusky cradle
rhombus would be if if a != b I think
What is that?
a!?
viral dagger
a=/=b
chilly cobalt
a is not equal to
viral dagger
$a\neq b$
grand pondBOT
Skill_Issue
dreamy lichen
dreamy lichen
It will be a rhombus right by your condition?
I think so
dreamy lichen
Sides will be same
yes
viral dagger
by the way, you can use a graphing calculator and experiment with it
dusky cradle
by the way, you can use a graphing calculator and experiment with it
I have already done
That is why i am here there is no calculator who can give about angles
viral dagger
That is why i am here there is no calculator who can give about angles
cough cough eyes cough cough
dreamy lichen
y2-y1/x2-x1
you can either arctan this, or use geogebra instead
viral dagger
you can calculate it yourself with trig
dusky cradle
cough cough eyes cough cough
No issues blindly leave 😑
viral dagger
No issues blindly leave 😑
whar
dusky cradle
you can calculate it yourself with trig
Am I not doing?
viral dagger
Am I not doing?
huh
dusky cradle
you can either arctan this, or use geogebra instead
I don't use websites/calculators
I only know is slope formula
viral dagger
you said you used a graphing calculator 😭
dusky cradle
And i used it and i got angles 90°@dreamy lichen
dusky cradle
No issues blindly leave 😑
.
dusky cradle
(1/3/-1/2)=-2/3
dreamy lichen
so the slope is (1/3) / (1/2)
dreamy lichen
(1/3/-1/2)=-2/3
or this
dusky cradle
Of course
dreamy lichen
2/3 and -2/3, depending on which line we take
dusky cradle
Aren't m1m2=-1 here?
grand pondBOT
Result:
-0.44444444444444
not perpendicular
dusky cradle
Let me check another slope
dreamy lichen
in this pic there is no other slope
dreamy lichen
parallel lines have same slope
dusky cradle
Wait let me check
dreamy lichen
the top right and down left have -2/3
the top left and down right have 2/3
Btw to calculate the exact angle, we can calculate angle of the bottom-right line relative to x axis
,w arctan(2/3)
grand pondBOT
dreamy lichen
,w arctan(2/3) to deg
grand pondBOT
dreamy lichen
that gave us the blue angle
note that blue = green
and yellow = 2*green
so ,calc 2*33.69
,calc 2*33.69
grand pondBOT
Result:
67.38
this is the yellow angle
orange + blue = 90
so orange = 90-blue
and 2*orange = purple
,calc 90 - 33.69
grand pondBOT
Result:
56.31
,calc 56.31*2
grand pondBOT
Result:
112.62
sure, if you dont undestand some step, just ask
midnight plankBOT
Available help channel!
last slate
in this book it says the monotonic law holds for addition of integers
midnight plankBOT
in this book it says the monotonic law holds for addition of integers
last slate
it says
monotonic law - from a < b it follows that a + b < b + c
is this a misprint
associative law : (a + b) + c = a + (b + c)
monotonic law : from a < b it follows that a + b < b + c
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midnight plankBOT
last slate
.reopen
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bright shoal
Ok
bright shoal
!status
midnight plankBOT
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
I dont know where to begin
bright shoal
Ok
1.Commutative law
It says that if there are two numbers a and b
And if we add a and b keeping anyone as the first number
The addition is equal
like
a+b=b+a
last slate
yes
bright shoal
You know both the laws right?
last slate
yeah
bright shoal
You are mainly asking for the monotonic law
last slate
i dont know monotonic
last slate
okay i was like confused
bright shoal
Okay
last slate
is this a bad book then
bright shoal
Now is your doubt resolved
last slate
should i write a bad review on amazon
midnight plankBOT
If you are done with this channel, please mark your problem as solved by typing .close
midnight plankBOT
bright shoal
Welcome
last slate
thanks a ton
bright shoal
Np
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dense arch
i'm not stuck but somewhat confused as to what to do here, since r2,r3,r4 are all similar
midnight plankBOT
i'm not stuck but somewhat confused as to what to do here, since r2,r3,r4 are al...
@dense arch Has your question been resolved?
dense arch
<@&286206848099549185>
dense arch
after going through RREF, the 2 rows obtained do not correspond with the answer
nvm got it
.close
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rich plaza
midnight plankBOT
rich plaza
So they give us the 55%, would I put that in the bottom right box as .55
and then for the other two boxes I dont really know what to do
Like since its percentages, I was thinking that maybe u have to subtract 100-55 and that answer goes in the other 2 boxes or the top right one
or smth like that
idk tho
I've never done a matrix with percentages
or just a matrix like this
<@&286206848099549185>
<@&286206848099549185>
lean spade
Assuming that there's a 100% chance a customer will buy some sort of item during their visit, and that you only buy either a food item or a non food item, then I believe you can think of the first column of the matrix as representing the "probability vector" of a person who purchased a food item originally
fathom knoll
can you first tell what the NN entry should be?
lean spade
so the top left would be 0.6 and the bottom left would be 0.4
rich plaza
can you first tell what the NN entry should be?
i thought it would be .55
rich plaza
Assuming that there's a 100% chance a customer will buy some sort of item during...
oh kk
lean spade
i thought it would be .55
yeah
rich plaza
so the top left would be 0.6 and the bottom left would be 0.4
so the other column would be .55 and .45
fathom knoll
i thought it would be .55
good
rich plaza
yes
fathom knoll
now it depends on whether the rows represent the first time they bought or the columns
but if the rows do
the the rows need to sum to 1
so you can solve for the other 2
rich plaza
how do I know if the rows are the first time or the columns r
rich plaza
the the rows need to sum to 1
oh
so for the first row I would add .6 and .45
and the second one i would do .4 and .55
but they dont add up to 1
fathom knoll
how do I know if the rows are the first time or the columns r
i think that's something you explicitly say
no no
I think maudran is doing it as columsn instead of rows
rich plaza
oh kk
lean spade
I think maudran is doing it as columsn instead of rows
yeah
since it's standard to multipy vectors to the right of matrices
rich plaza
so the columns would be the first time?
lean spade
so the columns would be the first time?
that's how I would interpret it
rich plaza
ohh
lean spade
but maybe either way is acceptable
rich plaza
so like if I chose the row f and column n block
the top right one
I would say that they bought a non food item the first time and bought a food item the second time
and thats 45% of them
How would I do this>
*?
do I just plug in the same matrix twice, multiple them
and then thats my answer?
lean spade
yeah but you'd need to specify which entry corresponds to the probability they're asking for
rich plaza
oh ok
lean spade
I would say that they bought a non food item the first time and bought a food it...
You'd say that the probability a person chooses a food item the second time if they bought a non food item the first time is 45%
rich plaza
so would that be the top left box then?
lean spade
so would that be the top left box then?
ye
rich plaza
You'd say that the probability a person chooses a food item the second time if t...
oh ok
tyyy
.close
midnight plankBOT
lean spade
np
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twin crow
i was solving this problem where i have to prove RHS = LHS :
cos(2pi/7) + cos(4pi/7) + cos(8pi/7) = -1/2
but then i gave up on solving and checked the solution where they just multiplied and divided by 2sin(pi/7) and used product formula to expand.
my question is why would i even think of multiplying and dividing by 2sin(pi/7)??? how can someone think of approaching the problem this way? like what is the intuition behind multiplying and dividing by 2sin(pi/7)?
midnight plankBOT
i was solving this problem where i have to prove RHS = LHS :
cos(2pi/7) + cos(4...
midnight plankBOT
@twin crow Has your question been resolved?
visual comet
Can some pls check
midnight plankBOT
@visual comet Has your question been resolved?
small jasper
Can some pls check
Yeah it’s fine
rancid kestrel
how would i prove this
midnight plankBOT
how would i prove this
rancid kestrel
hard shard
SAS
rancid kestrel
how would i prove oc and ob congruent
feral sedge
how would i prove oc and ob congruent
they are both radii
rancid kestrel
how do you know
visual tiger
how do you know
they both go from O to a point on the circle
by definition, points on the circle are which distance away from O?
rancid kestrel
yeah but what if they just are really close to the circle but theyre not actually on it
if it was just that i wouldve already solved it
is there no other method to prove the triangles congruent
visual tiger
yeah but what if they just are really close to the circle but theyre not actuall...
????
I thought the drawing was obvious
rancid kestrel
what id the point isnt actually on the circlew
visual tiger
"what if A isn't actually on the circle"
rancid kestrel
in class sometimes they say u cant assume its on the circle
visual tiger
in class sometimes they say u cant assume its on the circle
then what's the point of the circle being here
rancid kestrel
idfk
midnight plankBOT
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digital crag
midnight plankBOT
digital crag
Why is this the graph for the parametric equation?
elder fulcrum
The graph is y with respect to x
e^t is always positive
Express y in terms of the x here
How do u say e^2t in terms of e^t
?
digital crag
is that like eliminating the parameter
elder fulcrum
Yes you can say that
digital crag
so y= e^2x?
elder fulcrum
But more than "eliminate" you can rather say you are expressing one variable in terms of another
elder fulcrum
so y= e^2x?
No
What is x?
x is e^t
And y is e^2t
So u want to express y in terms of x
It's like you know a foot is how many cm, and yard is how many cm, so now you want to "eliminate" the cms to express how many feet is a yard
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twilit field
midnight plankBOT
twilit field
let $(a,b) \in (\mathbb{R} - \mathbb{Z}) \times \mathbb{N}$. It then follows , from definition, that $a \in (\mathbb{R} - \mathbb{Z}) \implies a \in \mathbb{R}, b \in \mathbb{N}. $ so we have $(a,b) \in \mathbb{R - Z} \times \mathbb{N}.$ However $ a\notin \mathbb{Z}$
grand pondBOT
(why am i here )= idk
twilit field
This is my working so far?
Is it fine until now?
So what we essentially want to do is remove all points with integral abscissa
Which is what the operation on the right hand side achieves
but is this possible using set theory too?
(Like can I construct a proof using set theory ?
modern sapphire
Yeah, distribution works here too
twilit field
I know
but don't I have to prove that?
or is that an Axiom in set theory ?
yes/no please
modern sapphire
Well when you ask it like that, I remeber distributive law applied to only intersections and unions
twilit field
I mean we maybe could derive it using propositional logic
hard shard
do you think the statement is true or false?
prime hornet
well, if you think it's true you'll need to show equality of sets, which I think you've started to do but got stuck?
twilit field
Yes, pretty much
twilit field
Yes, I got that too
twilit field
It Is NOT , but how do I show that
prime hornet
there is nothing to express wdym
twilit field
I meant I told the exact same thing in words
prime hornet
"since a \in R \setminus Z, a \notin Z, so (a,b) \notin (Z x N)"
midnight plankBOT
prime hornet
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past pulsar
How would I draw a graph that makes this proof easier to understand?
midnight plankBOT
How would I draw a graph that makes this proof easier to understand?
bright shoal
!nopdf
midnight plankBOT
Please post images (such as PNGs or JPGs) of the question rather than other filetypes such as PDFs which have to be downloaded. Non-image downloads can potentially contain viruses or other security risks.
midnight plankBOT
@past pulsar Has your question been resolved?
midnight plankBOT
@past pulsar Has your question been resolved?
midnight plankBOT
@past pulsar Has your question been resolved?
mighty zodiac
Can anyone take photo and send solution for integration of x sin inverse x?
midnight plankBOT
Can anyone take photo and send solution for integration of x sin inverse x?
hard shard
it takes time
pearl hull
It takes time
quiet hinge
It does take time indeed
slender walrus
searching will be faster than someone doing it and sending it
because they most likely won't
if however you want people to guide you, they'll be more than happy to help
mighty zodiac
.close
midnight plankBOT
quiet hinge
Yeah, you can send your attempt at a solution or something of that sort
pearl hull
lol
slender walrus
or less happy now
clear pumice
/can someone solve this question?
quiet hinge
Also even if you claim your own help channel, no, people wont solve questions for you
slender walrus
and post embeded pics instead of files
clear pumice
ok
midnight plankBOT
Available help channel!
radiant vector
midnight plankBOT
radiant vector
I checked answers and the 2/5 being probabilty of B was instead replaced with probabilty of chem if physics and I dont get why'
real pivot
I checked answers and the 2/5 being probabilty of B was instead replaced with pr...
do you remember the formula for conditional probability?
real pivot
alright so in the second image you are given
P(A), P(B), P(B | A) respectively
real pivot
P(B|A) = P(A n B)/P(A)
now solve for P(A n B)
and you will see why they plugged in P(B | A) instead of P(B)
real pivot
wdym which one
radiant vector
like between p(B) and P(B|A)
real pivot
well the third case in the second image is clearly a conditional prob case so you know that you are given P(B | A) and P(A) (case 1) so all you have to do is multiply those two
P(B | A)P(A) = P(A n B)
or
P(A | B)P(B) = P(A n B)
radiant vector
ohh
real pivot
ye so if they instead gave u the prob of students studying chemistry given they chose to study physics (P(A | B)) then you would have multiplied P(B)
midnight plankBOT
@radiant vector Has your question been resolved?
radiant vector
.close
midnight plankBOT
midnight plankBOT
Available help channel!
twilit field
midnight plankBOT
twilit field
The definition of union states that a given element belongs to the union if it belongs to any of the sets. On the other hand the definition of intersection states that an element belongs to the intersection if and only if it belongs to all the sets. The only way for this to be true is if all the sets have the same elements.
would this suffice as a proof?
visual tiger
as always you need a mathematical proof
unique stream
would this suffice as a proof?
Hm I'd at least write the last sentence formally
"The only way for this to be true is if all sets have the same elements." Prove that directly
visual tiger
you have intuited the result, you should formally prove it
twilit field
OK.
zealous schooner
The definition of union states that a given element belongs to the union if it ...
I mean this doesn't have any reasoning in it
this is basically just a fancy way of saying "it's true because obviously"
twilit field
So I basically have to prove that they are subsets of one another
zealous schooner
I think if you write your statement more mathematically then it's better
let $S=\bigcup_{
twilit field
By Definition $x\in \bigcap_{\alpha \in I} A_{\alpha} \iff x\in A_i \forall i\in I $
on the other hand $x\in \bigcup_{\alpha \in I} if x\in A_i \text{ for some } i\in I$.
So thus if $x\in \bigcap A_{\alpha}$ it's also in $\bigcup A_{\alpha}$.
\
so $\bigcap A_{\alpha} \subseteq \bigcup A_{\alpha}$
would be the proof in one direction
zealous schooner
\iff
twilit field
ah
grand pondBOT
(why am i here )= idk
visual tiger
By Definition $x\in \bigcap_{\alpha \in I} A_{\alpha} \iff x\in A_i \forall i\in...
again, you need to say that I isn't empty
we went through the proof earlier
twilit field
I'm taking that to be obvious because that's mentioned in my book
visual tiger
nothing is obvious to the one that's going to correct you
visual tiger
if I is empty then both sides are empty sets
only union
intersection would be "for every i in empty set..."
which is always true
so it would be the universe set in which you're in
zealous schooner
what??
twilit field
OK. Added that
zealous schooner
that just broke my brain
visual tiger
that just broke my brain
I can talk about it more in DM
zealous schooner
sure!
zealous schooner
**(why am i here )= idk**
Let $S=\bigcup_{i\in I}A_i=\bigcap_{i\in I}A_i$\
$$\forall x\in U: x\in S\iff x\in A_i\ \forall i\in I$$
grand pondBOT
kheerii
zealous schooner
something like this
grand pondBOT
(why am i here )= idk
twilit field
I think I have another idea
$x\in \bigcap_{\alpha \in I} A_{\alpha} \implies x\in A_{\alpha} \forall i \in I$
grand pondBOT
(why am i here )= idk
twilit field
Now $x \in \bigcup_{\alpha \in I} \implies x \in A_{\alpha} \text{ for some } i \in I$
grand pondBOT
(why am i here )= idk
zealous schooner
yeah this is basically what I wrote
twilit field
oops
sorry
now if they are equal
that is $\bigcap_{\alpha \in I} A{\alpha} = \bigcup_{\alpha \in I} A_{\alpha}$
grand pondBOT
(why am i here )= idk
visual tiger
wait what are you setting out to prove
aren't you trying to prove that A_alpha are all equal to each other?
twilit field
That all $A_{\alpha}$ are equal
grand pondBOT
(why am i here )= idk
visual tiger
ok
twilit field
yeah, going to argue that now
zealous schooner
just prove A_alpha_1 = A_alpha_2
can you check dm's?
exotic pelican
try showing that $\bigcap_{\alpha \in I} A_{\alpha} \subseteq A_{\beta}$ and $A_{\beta} \subseteq \bigcup_{\alpha \in I} A_{\alpha}$ for all indices $\beta$
grand pondBOT
rbit
twilit field
now for a certain $x$ to belong to the union, it has to belong to atleast one of the sets. But if the same is true for the intersection, it must belong to all the sets. As this is true for all x, the sets are equal
grand pondBOT
(why am i here )= idk
twilit field
is this fine?
pure shore
is this fine?
I would recommend that you start with a specific statement you wanna prove, write it down in mathematical terms then prove it.
zealous schooner
you want to prove $$x\in A_i\iff x\in A_j\ \forall\ i,j\in I$$
grand pondBOT
kheerii
pure shore
Simplest way to state it in my opinion would look like this:
for any two indicies $\alpha$ and $\beta$ where $\alpha \neq \beta$, $A_{\alpha} = A_{\beta}$
grand pondBOT
Mohamed Mohsen
twilit field
mhm
twilit field
**kheerii**
yup, and I've explained that in words, have I not
zealous schooner
i mean if you just want an insight as to why this is true then your thing is good enough
visual tiger
yup, and I've explained that in words, have I not
I think that this is your recurrent problem
For you proving = explaining in words
Problem is that proofs follow a very particular "syntax"
Such that explaining in words or visualising something isn't gonna fly
twilit field
I could try writing it mathematically, I guess?
pure shore
I could try writing it mathematically, I guess?
write it in such a way a 2 year old baby could understand. if you read it and you feel any ambiguity, it's not rigorous.
twilit field
Got it
zealous schooner
do 2 year old babies even know how to read?
twilit field
I did , If I remember right
pure shore
do 2 year old babies even know how to read?
do people ever hear of metaphors ?
zealous schooner
do people ever hear of metaphors ?
do people ever hear of not taking things too seriously?
pure shore
it clearly meant make it simple and clear
pure shore
do people ever hear of not taking things too seriously?
bro i don't want to but you seem too serious idk why
zealous schooner
i am the most unserious person you'll meet
zealous schooner
i guess that's my bad xD
you're good lol
twilit field
OK. So $x\in \bigcup A_{\alpha} \implies x\in A_{\alpha}$ for some $\alpha \in I$. However, we already know that $x \in \bigcap A_{\alpha}$ which means that $\bigcup \subseteq \bigcap $ in this case
jagged saffron
[Fyi its not inherently the fact that your "proof" is written in words that makes it not a proof (see hlounge discussion). It's that your argument written in words is missing logical steps]
twilit field
is this better?
grand pondBOT
(why am i here )= idk
visual tiger
**(why am i here )= idk**
how does this help? we already know $\bigcup = \bigcap$
twilit field
I think this is still missing a few steps
grand pondBOT
rafilou2003
visual tiger
I think you would gain to write out exactly what you wanna prove at the end of your proof
something like:
pure shore
[Fyi its not inherently the fact that your "proof" is written in words that make...
another little comment related to what ΣAC was saying, you wanna prove that if x is in one of the As it's in the rest, why doesn't the proof start with, let x be in A_alpha or something like that. if you are trying to prove a statement you should start with the first given and end on what was required.
visual tiger
We suppose (Conditions)....$\$
Then... $\$
But... $\$
$(...) \$
thus (Goal)
grand pondBOT
rafilou2003
twilit field
I guess what I'm trying to say that as x is in the union, it must belong to atleast one of the sets.
visual tiger
I guess what I'm trying to say that as x is in the union, it must belong to atle...
sure, but is that what we're after?
last slate
the proof is simple, just apply aesop tactic. Just kidding
twilit field
But as this same x is in the intersection, it must belong to all the sets
visual tiger
But as this same x is in the intersection, it must belong to all the sets
what are you trying to prove
twilit field
if a given x( An arbitrary element) belongs to all the sets, the sets must be equal
twilit field
yes.
visual tiger
and btw do we need the intersection to be non-empty for "the sets must be equal" to be true?
twilit field
But all the elements in the intersection, are also in the union
visual tiger
sure
twilit field
and for an element to be in the union it needs to belong to atleast one of the sets
visual tiger
but I still don't think you understand what you want
what was the "thing" you wanted to prove
the sets must be equal right?
twilit field
yup
twilit field
yes
visual tiger
so our goal
can you write it mathematically?
so at least we know where we're supposed to go in the mathematical proof
just write the goal for now
twilit field
The goal is to prove that $A_i = A_j $ for all $i,j \in I$
visual tiger
for some?
zealous schooner
for ALL
grand pondBOT
(why am i here )= idk
twilit field
let $x$ be an arbitrary element in $A_i$
visual tiger
?
grand pondBOT
(why am i here )= idk
visual tiger
there's a step before "Ai = Aj"
zealous schooner
what is A_i?
twilit field
A_i is a set
visual tiger
any set?
twilit field
a set where i \in J
zealous schooner
an arbitrary set
zealous schooner
how do we prove a "forall" statement
grand pondBOT
(why am i here )= idk
pure shore
I think it would be enough to state that i is arbitrary though, but that's just my opinion
visual tiger
before going in the Ai=Aj territory
zealous schooner
technically you need A_i and A_j both to prove that they're equal
visual tiger
so same with j
A forall statement proof starts with "Let ... be ..."
"forall i,j in I ..."
so the proof start with
"let i,j \in I"
twilit field
Let $i,j \in I $ where both are arbitrary
visual tiger
drop " A_i, A_j be arbitrary sets such that"
they come for free with your "i,j in I" package
twilit field
Just this?
zealous schooner
let A_i A_j where i,j in I
twilit field
Let $i,j \in I$ where both are arbitrary
grand pondBOT
(why am i here )= idk
twilit field
now we know that $x\in \bigcap$
grand pondBOT
(why am i here )= idk
visual tiger
now we know that $x\in \bigcap$
who's x
twilit field
so $x \in A_i, A_j$
grand pondBOT
(why am i here )= idk
twilit field
where $x \in \bigcap$
grand pondBOT
(why am i here )= idk
visual tiger
how is x defined here?
twilit field
x is an element of the intersection
twilit field
As I've said before $x \in \bigcup$
grand pondBOT
(why am i here )= idk
twilit field
so it belongs to either A_i or A_j
zealous schooner
why?
twilit field
oh right
visual tiger
couldn't it belong to some other rando A_k
twilit field
it could yeah
visual tiger
yeah so
twilit field
well, x is an arbitrary elemnt
visual tiger
never proved a set equality before?
twilit field
Not in 2 years or so
visual tiger
ok well do you remember double inclusion at least?
visual tiger
to prove X = Y, you show every x is in Y and every y is in X
great
so let's do that
so show A_i \subseteq A_j
and then vice versa will be easy
twilit field
But why is saying that $x\in A_i$ and $x\in A_j$ means that $A_i=A_j$ where both are arbitrary worng ?
grand pondBOT
(why am i here )= idk
visual tiger
But why is saying that $x\in A_i$ and $x\in A_j$ means that $A_i=A_j$ where both...
so Ai and Aj have an element in common
so what?
twilit field
we now repeat this for all their elements
twilit field
$x_1 from A_i$
grand pondBOT
(why am i here )= idk
twilit field
then $x_2$ from A_i
grand pondBOT
(why am i here )= idk
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
twilit field
and so on until $x_n$ from $A_i$
grand pondBOT
(why am i here )= idk
twilit field
where $i \in I$
grand pondBOT
(why am i here )= idk
visual tiger
so you take a finite amount of x, and only from Ai?
twilit field
And we then repeat this process for $A_j$
grand pondBOT
(why am i here )= idk
visual tiger
ok, just something to tell you
once you took an ARBITRARY guy from the first set
and shown it's in the other set
it's done
there's no "repeating the process"
twilit field
I see
visual tiger
showing it always works once is gonna have the same result as showing it always works multiple times
twilit field
Okay, so now I prove that they are subsets of one another