Tap to unmute

Cantor set

  Рет қаралды 18,605

Dr Peyam

Dr Peyam

Күн бұрын

Пікірлер: 67
@dabbinrascal7975
@dabbinrascal7975 4 жыл бұрын
The way he says “Cantor” haunts me
@lucasdepetris5896
@lucasdepetris5896 4 жыл бұрын
I've just finished the 2nd year of the math degree. In the next year, first term I'll have real analysis. This video definitely blew my mind and gave me motivation for studying.
@nicholashayek5495
@nicholashayek5495 2 жыл бұрын
I just started on my first year and currently getting slaughtered in real analysis... it was a mistake for sure, but these videos help a lot
@Andy-sw1mb
@Andy-sw1mb 2 жыл бұрын
Also, the cantor set is nowhere-dense in R
@pedrocusinato1743
@pedrocusinato1743 4 жыл бұрын
Here is another cool way to define the Cantor set Let f : [0, 1/3] U [2/3, 1] -> [0,1) with f(x) = 3x - floor(3x). The Cantor set is the set of all x in [0,1] which for all n natural, f applied n times to x is defined (the function never drops in the interval (1/3, 2/3) ).
@TheRobinr200
@TheRobinr200 2 жыл бұрын
Thank you very much about the explanation in ternary digits why the Cantor set is uncountable. Love it
@Jim-be8sj
@Jim-be8sj 4 жыл бұрын
Next: Vitali sets.
@umerfarooq4831
@umerfarooq4831 4 жыл бұрын
After watching the video the Cantor set is more like a 'Can Do' set for me,great video very helpful
@பேராண்டி
@பேராண்டி Жыл бұрын
Such a great explanation sir🛐 Thank you very much ❤ Love you Sir ❤
@gabest4
@gabest4 4 жыл бұрын
Can we say that F1, the [0,1] set has zero holes but uncountably many? Since the colors are just inverted on the blackboard.
@lautaromoyano5009
@lautaromoyano5009 4 жыл бұрын
Just read about this today in Understanding Analysis by Abbot, what the hell... Thank you πm!
@theproofessayist8441
@theproofessayist8441 4 жыл бұрын
Now waiting for the beautiful Cantor Diagonalization argument!!! :). Also, @ comment made at beginning of video Immanuel Kant and a bunch of other German intellectuals glare menacingly! ゴゴゴゴゴゴゴゴ
@drpeyam
@drpeyam 4 жыл бұрын
R is uncountable: kzbin.info/www/bejne/fpCQY3hsd5uCqLs
@theproofessayist8441
@theproofessayist8441 4 жыл бұрын
@@drpeyam Perfection
@Mr_mechEngineer
@Mr_mechEngineer 4 жыл бұрын
Youre a superb mathematician dr Peyam
@andreutormos7210
@andreutormos7210 4 жыл бұрын
Mindblowing that numbers of the form 0.2200202002220 (ternary expansion without 1) have the same cardinality than the interval (0,1) and therefore the CANTOR set is uncontable :o
@algorithminc.8850
@algorithminc.8850 4 жыл бұрын
I'm feeling very broken up by this topic ... hehe ... Visually for fun, it makes a beautiful 2D (and especially extended 3D) plot ...
@benjaminjonen2736
@benjaminjonen2736 2 жыл бұрын
Nicely explained, thank you!
@marcoardanese6013
@marcoardanese6013 2 жыл бұрын
amazing explanation !!!
@Moramany
@Moramany 4 жыл бұрын
Love me some CANTOR sets. Very good!
@blizzard_inc
@blizzard_inc 4 жыл бұрын
I enjoyed the video! However, wouldn't one third, or 0.1000... in ternary also be in the cantor set? I know this can also be expressed as 0.02222... , but it still feels icky to say that all ternary expansions of the cantor set don't contain a 1. Also, for the connection with binary, it seems to me a bit weird how 0.1000... and 0.0111... in binary are the same, yet their corresponding elements of the cantor set, 0.2000... and 0.0222... are not the same. Doesn't this mean that their cardinality is not necessarily the same, as it implies that that correspondance isn't a bijection?
@iabervon
@iabervon 4 жыл бұрын
It's better to say that the Cantor Set is all the numbers with a ternary representation without any 1s (although they may also have another representation with 1s). And it's true that the binary thing only proves that the Cantor Set's cardinality is at least that of the unit interval, but it's also obviously a subset of it, so it's also at most that of the unit interval.
@MikeRosoftJH
@MikeRosoftJH 10 ай бұрын
OK, this isn't quite a one-to-one mapping between the Cantor set and an interval. But it can be seen that on both sides there are just countably many problematic numbers; and removing a countably infinite set from a set with cardinality of the continuum doesn't change its cardinality.
@michalbotor
@michalbotor 4 жыл бұрын
at uni i met a math doctor who was interested in constructing various sets, especially fractal-like one, by means of so called iterated function systems. he would obtain cantor set like so. he would define C_0 to be [0, 1], he would define two transformations: T_1(x) := x/3 and T_2(x) := (x+2)/3, and then he would define the following set recursively: C_{N+1} := T_1(C_N) union T_2(C_N), for N = 0, 1, 2, ... finally he would call C := lim_N C_N the cantor set. this iterated function systems where actually way more interesting and powerful than that, as one could have many more functions T_1, T_2,... acting on a set and/or a probability measure that was choosing which T_is will act on a C_j set in the j-th iteration.
@dgrandlapinblanc
@dgrandlapinblanc 2 жыл бұрын
Ok. Neat. Thank you very much.
@lazbn90
@lazbn90 8 ай бұрын
Not True that any metric space is a subspace of C, if that means it can be embedded into it. As an example any connected space. You have to add extra hypothesis like compactness, totally disconnected …
@michalbotor
@michalbotor 4 жыл бұрын
dr peyam: how much could we push this removal of the part of the Fj-th set, so that the resulting cantorest set F has still all the properties of the original cantor set (with some minor adjustments, such as the length of the Fj-th set)?
@MrWater2
@MrWater2 Жыл бұрын
Man you are the one
@AA-gl1dr
@AA-gl1dr 3 жыл бұрын
excellent video. thank you
@michalbotor
@michalbotor 4 жыл бұрын
there should be something called "the lovecraft's set". the scariest of them all.
@humblehmathgeo
@humblehmathgeo 4 жыл бұрын
Thank you !!
@purim_sakamoto
@purim_sakamoto 3 жыл бұрын
ふむふむ 難しいところがあったのでまた見に来たい
@gavasiarobinssson5108
@gavasiarobinssson5108 4 жыл бұрын
Can you express these ternary numbers as fractions with a factor three in the denominator?
@thenewdimension9832
@thenewdimension9832 3 жыл бұрын
Love u sir .❤️❤️❤️
@Sky-pg6xy
@Sky-pg6xy 3 жыл бұрын
Thats insane 😳
@cuneytkaymak4997
@cuneytkaymak4997 2 ай бұрын
Wait, considering [2/3,1] , doesnt it start with 0,6xxx... ? I don't understand how it starts with 0,2?
@drpeyam
@drpeyam 2 ай бұрын
Because we’re writing in ternary! Think binary but with 0, 1, 2
@cuneytkaymak4997
@cuneytkaymak4997 2 ай бұрын
@@drpeyam but then, that it consists of just 0,1,2 is not a surprise, it is not that special, because we choose to write it that way, for example we can choose to write it in base 4 so that it just consists of the numbers 0,1,2,3. I still dont get it 😭
@drpeyam
@drpeyam 2 ай бұрын
But then you can’t remove the middle set!
@md2perpe
@md2perpe 4 жыл бұрын
Consider C+Q = { c+q | c ∈ C, q ∈ Q }, where C is the Cantor set and Q the rationals. Since C is a null set (it has Lebesgue measure zero) and Q is countable, C+Q is a countable union of null sets and is itself a null set. But it is a very dense such, since every interval of the reals contains an uncountable number of points from C+Q.
@Leidl.Michael
@Leidl.Michael 4 жыл бұрын
so to interpret this a little bit clearer Q is a countable set which is dense in R and C+Q is an uncountable set which is dense in R and has lebesgue measure 0 basically uncountable dust spread along the axis C+Q≠R because of the different measure, can someone give me a a point, which lies in R and not in C+Q? thats not trivial since there are irrational numbers in C
@md2perpe
@md2perpe 4 жыл бұрын
@@Leidl.Michael Correct.
@Leidl.Michael
@Leidl.Michael 4 жыл бұрын
ok i think an example would be the number a=0.101001000100001000001...in ternacy expansion but i have no proof, only a good feeling that it cant be expressed as a=c+q, c in C and q in Q
@md2perpe
@md2perpe 4 жыл бұрын
@@Leidl.Michael Yes, that's a number that probably isn't in C+Q. But almost all numbers are not in C+Q.
@RupaliYadav-rm4sh
@RupaliYadav-rm4sh Жыл бұрын
How is the set uncountable....?
@MikeRosoftJH
@MikeRosoftJH 10 ай бұрын
Cantor set is the set of all real numbers in an interval from 0 to 1 whose base-3 expansion doesn't contain the digit 1. But these can be easily easily mapped to base-2 expansions of real numbers in the same interval: just replace the digit 2 to digit 1. This isn't quite a one-to-one function between Cantor set and the interval (can you see why?); but what we get is the set of all but countably many points in that interval. That, of course, has the same cardinality as the interval itself. Therefore, Cantor set has the cardinality of the continuum.
@markmajkowski9545
@markmajkowski9545 3 жыл бұрын
Isn’t this list able as a subset of the integers divided by powers of 3? And Cantor found a way to describe what would be fractions of powers of three in an uncountable way? Given your “ball” can’t you place it on an integer divided by a power of 3 and have it include every Cantor Set element. Perhaps since you have a sum of fractions of 1/3^n you cannot list. It “feels” like a clever way to define a set of numbers for which some aspects of math may not be developed - and by eliminating “segments” we understand from a larger segment we are left with the numbers we don’t understand on that interval as our “set”
@gentil.iconoclasta
@gentil.iconoclasta 2 жыл бұрын
Olá, bom dia. Então, escrevi um mini artigo (2 páginas) no qual forneço uma fórmula para cada etapa de construção do Conjunto de Cantor - uma sequência que nos fornece os pontos extremos dos subintervalos. A quem possa interessar posso enviar o PDF por e-mail.
@Aqsa_Ashraf
@Aqsa_Ashraf 2 жыл бұрын
Yes please send me I'll be very grateful to u .. Actually I hv participated in poster competition at University level and the topic is to elaborate cantor set with solid example ND formula...
@gentil.iconoclasta
@gentil.iconoclasta 2 жыл бұрын
@@Aqsa_Ashraf Yes, it will be a pleasure, I leave here the link to access the formula: drive.google.com/file/d/1z0CqQal30oKyt2vxJSJmCLraygeK1y0S/view?usp=drivesdk
@edgardojaviercanu4740
@edgardojaviercanu4740 4 жыл бұрын
just beautiful...
@mohammedmadani7277
@mohammedmadani7277 4 жыл бұрын
Thank u sir
@Leidl.Michael
@Leidl.Michael 4 жыл бұрын
totally disconnected implies that you can't even draw a line in the cantor set because all subsets with more than one element in it are not connected and therefore not path-connected.
@yashagrahari
@yashagrahari 3 жыл бұрын
There are countably infinite rationals but uncountably infinite cantor numbers so, is it obvious to say that there exists irrational cantor numbers?
@drpeyam
@drpeyam 3 жыл бұрын
Of course
@edwardh371
@edwardh371 4 жыл бұрын
Wouldn't the point 1/3 be in the Cantor set? F1 is [0, 1/3] U [2/3, 1]. The point 1/3 point is never removed by successive Fn. The ternary representation of 1/3 is 0.1. Something does not seem correct.
@drpeyam
@drpeyam 4 жыл бұрын
0.1 = 0.022222222
@GuyMichaely
@GuyMichaely 4 жыл бұрын
@@drpeyam in that case wouldn't it be more accurate to say that the Cantor set is [0, 1] \ {x | x has at least one non 0 digit after a 1 in ternary}?
@drpeyam
@drpeyam 4 жыл бұрын
It’s more like there exists some representation without 1’s like the above
@GuyMichaely
@GuyMichaely 4 жыл бұрын
Amazing to see provocative bots on a math channel
@drpeyam
@drpeyam 4 жыл бұрын
They’re so annoying 😭
@matematicasemplice
@matematicasemplice 4 жыл бұрын
Bye
@inkognito8400
@inkognito8400 4 жыл бұрын
Hey, enjoy your vids for quite a while now. Just out of curiosity, do you plan to do something on ordinal numbers or measure theory?I would think that many people would find it interesting.Thanks anyways!
@drpeyam
@drpeyam 4 жыл бұрын
There are some videos on Lebesgue integration, check them out
@inkognito8400
@inkognito8400 4 жыл бұрын
@@drpeyam Thank you very much! Have a nice day.
@s2pmathematics55
@s2pmathematics55 2 жыл бұрын
Hello sir
Baire Category Theorem
16:01
Dr Peyam
Рет қаралды 14 М.
Cantor Intersection Theorem
12:59
Dr Peyam
Рет қаралды 8 М.
“Don’t stop the chances.”
00:44
ISSEI / いっせい
Рет қаралды 62 МЛН
Сестра обхитрила!
00:17
Victoria Portfolio
Рет қаралды 958 М.
99.9% IMPOSSIBLE
00:24
STORROR
Рет қаралды 31 МЛН
Class 6  Ch- 12 LIGHT, SHADOWS AND REFLECTION   One Shot Science
14:41
journey into fractals: the Cantor set and ternary expansion.
20:58
Michael Penn
Рет қаралды 20 М.
What happens at infinity? - The Cantor set
16:25
Zach Star
Рет қаралды 270 М.
Uncountable set with Lebesgue measure 0 - Cantor Set | Measure Theory
12:18
Cantor's Infinity Paradox | Set Theory
14:07
Up and Atom
Рет қаралды 396 М.
Properties of Compactness
19:32
Dr Peyam
Рет қаралды 8 М.
An Introduction to Cantor and Infinity
14:51
ThatMathThing
Рет қаралды 2,3 М.
The Devil's Staircase | Infinite Series
13:34
PBS Infinite Series
Рет қаралды 267 М.
Infinity is bigger than you think - Numberphile
8:00
Numberphile
Рет қаралды 8 МЛН
Compactness
22:38
Dr Peyam
Рет қаралды 28 М.
“Don’t stop the chances.”
00:44
ISSEI / いっせい
Рет қаралды 62 МЛН