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.
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) ).
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
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
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?
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.
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.
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.
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 …
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)?
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.
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
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
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.
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...
@@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
@@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 😭
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.
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”
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.
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.
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!
The way he says “Cantor” haunts me
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.
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
Also, the cantor set is nowhere-dense in R
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) ).
Thank you very much about the explanation in ternary digits why the Cantor set is uncountable. Love it
Next: Vitali sets.
I'm feeling very broken up by this topic ... hehe ... Visually for fun, it makes a beautiful 2D (and especially extended 3D) plot ...
Can we say that F1, the [0,1] set has zero holes but uncountably many? Since the colors are just inverted on the blackboard.
After watching the video the Cantor set is more like a 'Can Do' set for me,great video very helpful
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
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
R is uncountable: ua-cam.com/video/H_-2E6B6OrY/v-deo.html
@@drpeyam Perfection
Just read about this today in Understanding Analysis by Abbot, what the hell...
Thank you πm!
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?
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.
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.
Such a great explanation sir🛐 Thank you very much ❤ Love you Sir ❤
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.
Youre a superb mathematician dr Peyam
Nicely explained, thank you!
amazing explanation !!!
Love me some CANTOR sets. Very good!
Ok. Neat. Thank you very much.
Thank you !!
there should be something called "the lovecraft's set". the scariest of them all.
ふむふむ
難しいところがあったのでまた見に来たい
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 …
Thats insane 😳
excellent video. thank you
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)?
Can you express these ternary numbers as fractions with a factor three in the denominator?
Man you are the one
Love u sir .❤️❤️❤️
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.
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
@@Leidl.Michael Correct.
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
@@Leidl.Michael Yes, that's a number that probably isn't in C+Q. But almost all numbers are not in C+Q.
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.
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...
@@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
Wait, considering [2/3,1] , doesnt it start with 0,6xxx... ? I don't understand how it starts with 0,2?
Because we’re writing in ternary! Think binary but with 0, 1, 2
@@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 😭
But then you can’t remove the middle set!
just beautiful...
How is the set uncountable....?
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.
Thank u sir
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”
There are countably infinite rationals but uncountably infinite cantor numbers so, is it obvious to say that there exists irrational cantor numbers?
Of course
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.
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.
0.1 = 0.022222222
@@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}?
It’s more like there exists some representation without 1’s like the above
Bye
Amazing to see provocative bots on a math channel
They’re so annoying 😭
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!
There are some videos on Lebesgue integration, check them out
@@drpeyam Thank you very much!
Have a nice day.
Hello sir