Exploring the Cardinality of Cantor Set and Real Numbers

[benorin]

So the problem, and my partial solution are in the attached PDF.

I would like feedback on my proof of the first statement, if it is technically correct and if it is good. Any ideas as to how I can use/generalize/extend the present proof to proof the second statement, namely that E (the Cantor set) has the same cardinality as $$\mathbb{R}$$? Please, not the ternary expansion correspondence to the reals in [0,1] .

 

[Tide]

You might consider summarizing your arguments here since your attachment isn't visible until it gets approved.

From the words you use it sounds like you are using the standard argument that the Cantor set consists of all ternary numbers expanded as decimals that do not contain the digit "1" Obviously, that set has the same cardinality as the reals using Cantor's diagonalization.

 

[benorin]

If I post it as a JPEG or BMP or other graphics file, will I still have to wait for this pending approval stuff?

Wrong dimensions! Hate image crap, what with the "waiting for approval" stuff ? Manual content approval or what?

 

[benorin]

OK, so its no longer pending approval (and I have gotten some sleep). Please respond soon, this is due in the morning.

Thanks,
-Ben

 

[benorin]

FYI, Theorem 2.20 (e) states that to every linear transformation $$T:\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$$ there exists a real number $$\Delta \left( T\right)$$ such that $$m\left( T\left( A\right) \right) = \Delta \left( T\right) m\left( A\right)$$ for every Lebesgue measurable set A.

 

[NateTG]

How do you know that the Cantor set actually has a Lebesgue measure? Beyond that, the proof looks OK.