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.
@nicholashayek54952 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-sw1mb2 жыл бұрын
Also, the cantor set is nowhere-dense in R
@pedrocusinato17434 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) ).
@TheRobinr2002 жыл бұрын
Thank you very much about the explanation in ternary digits why the Cantor set is uncountable. Love it
@Jim-be8sj4 жыл бұрын
Next: Vitali sets.
@umerfarooq48314 жыл бұрын
After watching the video the Cantor set is more like a 'Can Do' set for me,great video very helpful
@பேராண்டி Жыл бұрын
Such a great explanation sir🛐 Thank you very much ❤ Love you Sir ❤
@gabest44 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.
@lautaromoyano50094 жыл бұрын
Just read about this today in Understanding Analysis by Abbot, what the hell... Thank you πm!
@theproofessayist84414 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
@drpeyam4 жыл бұрын
R is uncountable: kzbin.info/www/bejne/fpCQY3hsd5uCqLs
@theproofessayist84414 жыл бұрын
@@drpeyam Perfection
@Mr_mechEngineer4 жыл бұрын
Youre a superb mathematician dr Peyam
@andreutormos72104 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.88504 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 ...
@benjaminjonen27362 жыл бұрын
Nicely explained, thank you!
@marcoardanese60132 жыл бұрын
amazing explanation !!!
@Moramany4 жыл бұрын
Love me some CANTOR sets. Very good!
@blizzard_inc4 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?
@iabervon4 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.
@MikeRosoftJH10 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.
@michalbotor4 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.
@dgrandlapinblanc2 жыл бұрын
Ok. Neat. Thank you very much.
@lazbn908 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 …
@michalbotor4 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 Жыл бұрын
Man you are the one
@AA-gl1dr3 жыл бұрын
excellent video. thank you
@michalbotor4 жыл бұрын
there should be something called "the lovecraft's set". the scariest of them all.
@humblehmathgeo4 жыл бұрын
Thank you !!
@purim_sakamoto3 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
@gavasiarobinssson51084 жыл бұрын
Can you express these ternary numbers as fractions with a factor three in the denominator?
@thenewdimension98323 жыл бұрын
Love u sir .❤️❤️❤️
@Sky-pg6xy3 жыл бұрын
Thats insane 😳
@cuneytkaymak49972 ай бұрын
Wait, considering [2/3,1] , doesnt it start with 0,6xxx... ? I don't understand how it starts with 0,2?
@drpeyam2 ай бұрын
Because we’re writing in ternary! Think binary but with 0, 1, 2
@cuneytkaymak49972 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 😭
@drpeyam2 ай бұрын
But then you can’t remove the middle set!
@md2perpe4 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.Michael4 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
@md2perpe4 жыл бұрын
@@Leidl.Michael Correct.
@Leidl.Michael4 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
@md2perpe4 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 Жыл бұрын
How is the set uncountable....?
@MikeRosoftJH10 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.
@markmajkowski95453 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.iconoclasta2 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_Ashraf2 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.iconoclasta2 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
@edgardojaviercanu47404 жыл бұрын
just beautiful...
@mohammedmadani72774 жыл бұрын
Thank u sir
@Leidl.Michael4 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.
@yashagrahari3 жыл бұрын
There are countably infinite rationals but uncountably infinite cantor numbers so, is it obvious to say that there exists irrational cantor numbers?
@drpeyam3 жыл бұрын
Of course
@edwardh3714 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.
@drpeyam4 жыл бұрын
0.1 = 0.022222222
@GuyMichaely4 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}?
@drpeyam4 жыл бұрын
It’s more like there exists some representation without 1’s like the above
@GuyMichaely4 жыл бұрын
Amazing to see provocative bots on a math channel
@drpeyam4 жыл бұрын
They’re so annoying 😭
@matematicasemplice4 жыл бұрын
Bye
@inkognito84004 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!
@drpeyam4 жыл бұрын
There are some videos on Lebesgue integration, check them out