Mathematics

6:39 PM

Hello, if anyone has a moment, I need some help with the cantor set. I've been given a problem where the cantor set is defined as $[0,1]\setminus\bigcup_{n=1}^{\infty } C_{n} $, and I'm tasked with building a bijection between it and [0,1].

Fargle

@Rithaniel Are you familiar with how the Cantor set relates to the ternary representation of its elements?

Rithaniel

Yeah, it only contains elements which can be written with 0s and 2s, but this is the complement in regards to [0,1], and I don't know that any kind of corollary holds.

Fargle

@Rithaniel Well, you can get an "almost-bijection" by taking that ternary expansion, replacing all the 2's by 1's, and thinking of it as a binary number.

Rithaniel

Yes, I'm familiar with that, the thing which is throwing me for a loop is the fact that this is the complement of Cantor, isn't it?

Fargle

Huh. What are the sets $C_n$?

Ted Shifrin

6:46 PM

Howdy @Fargle

Fargle

The complement of Cantor shouldn't be Cantor, because it has measure and contains nontrivial intervals.

Heya @Ted

Ted Shifrin

@Rithaniel: You switch to base 2 instead of base 3.

Rithaniel

$C_{n}=\bigcup_{i=0}^{3^{3n-1}-1} (\frac{3i+1}{3^{n}},\frac{3i+2}{3^{n}})$

There is a chance that I am misinterpreting what I'm reading, of course.

Fargle

Is it possible the "top" index is wrong? The way it's written, $C_1$ would include the interval $\left(\frac{25}{3}, \frac{26}{3}\right)$, which is not in $[0,1]$

Rithaniel

Also, crunching some numbers, it seems that the complement to Cantor in [0,1] contains ternary expansions that contain 1s, 2s, and 0s.

Oh, right

$C_{n}=\bigcup_{i=0}^{3^{n-1}-1} (\frac{3i+1}{3^{n}},\frac{3i+2}{3^{n}})$

Typo on my part, apologies.

Fargle

6:49 PM

No problem--just wanted to make sure.

This is the usual Cantor set then, not its complement. You're removing $(1/3, 2/3)$, and then $(1/9, 2/9)$ and $(7/9, 8/9)$, and so on.

Rithaniel

Ah, okay. In that case it was simply me misreading the content of the problem.

Leaky Nun

hi @TedShifrin

Ted Shifrin

hi Leaky

Rithaniel

Thank you for the clarification.

Fargle

No worries!

Leaky Nun

6:51 PM

@TedShifrin hast du schonmal in Deutschland gewesen?

Fargle

I'm not actually sure how you would change the "almost-bijection" I mentioned to a true bijection, however.

Rithaniel

Though, how about that for a puzzle? A bijection from the actual complement of cantor to [0,1]

Ted Shifrin

ja, einmal als ich ein Kind war, und zweimal lätzes Sommer

Fargle

For example, 1/9 = 0.0022222... and 2/9 = 0.02, but they both get mapped to 1/4 = 0.01 = 0.0011111111...

Leaky Nun

@TedShifrin wo zu?

Fargle

6:53 PM

@Rithaniel That would be a doozy, for sure.

Ted Shifrin

Lätzes Sommer in München ...

howdy @MikeM

Leaky Nun

@TedShifrin ich gehe nach Freiburg bald

fur 3 Tage

Rithaniel

Hmmm, 1/9 would = .01 as a non-repeating expansion, so that's not good. I see the issue.

Mike Miller

Hi

Leaky Nun

@Fargle @Rithaniel you would have more luck bijecting the cantor set to $2^\Bbb N$

or $P(\Bbb N)$

Ted Shifrin

6:57 PM

That's where they started, Leaky.

Leaky Nun

I see

well there are countably many overlaps with your "almost" bijection

or maybe can we use Schroeder-Berstein?

Ted Shifrin

I had typed that earlier, but I didn't see the injection the nonobvious way.

Rithaniel

So, wait, 5/9 in ternary is 0.12, correct?

Leaky Nun

you mean from [0,1] to Cantor?

Ted Shifrin

Yup.

Leaky Nun

6:59 PM

express in base 2 with no final 111.... (this guarantees uniqueness)

change 1 to 2

convert from base 3

Rithaniel

But 5/9 is in cantor?

Ted Shifrin

We already discussed that, too.

Leaky Nun

then you have injections both way

Fargle

@Rithaniel There's a 1 in it that can't be "removed" by turning it into an infinite tail of 2's, so it is not in Cantor.

Leaky Nun

and Schroeder-Berstein says you're done

Daminark

7:00 PM

Hey everyone!

Leaky Nun

hi @Daminark

Fargle

Heya @Daminark

How's things?

Rithaniel

But, by the definition set up above, the interval (4/9,5/9) is removed from cantor, which means that 5/9 is left in cantor, correct?

Leaky Nun

no, 5/9 is not in Cantor

(1/3,2/3) is already removed in the first iteration

Daminark

Everything's alright, how about you?

Rithaniel

7:03 PM

Riiiiiight, got it, I missed that bit.

Leaky Nun

proceeds to make a pun on "bit"

Ted Shifrin

proceeds to ignore Leaky's humor and rolls a mere 3.8 eyes

Fargle

@Daminark Pretty alright. I had to miss class today because of a bad headcold, but other than that things are golden.

Daminark

Hopefully it's a class you'll be able to catch up relatively soon?

Mike Miller

I'm too sleepy

Ted Shifrin

7:07 PM

gotta love allergies and sinus infections :(

Leaky Nun

let's hope they don't differentiate to become cosinus infections

Mike Miller

You're awful

Jasper Loy

I have not heard from Mr Eyeglasses for over a year. I gave up trying to contact him.

Mike Miller

I feel bad for never responding to your email - I did get if

Jasper Loy

Well, it's alright. I know you are busy. Unlike me. I have no life.

Mike Miller

7:09 PM

I did that thing j do a lot where I think of saying something meaningful but put it off because of the time investment

I don't like you picking on yourself :+(

Leaky Nun

@JasperLoy what do you do for a living?

Jasper Loy

In fact, most of the people I know in real life have abandoned me. My only friends are people I talk to online now.

Transcript for

$Mathematics$ Mathematics