# Thread: cardinality of the set of all Liouville numbers

1. ## cardinality of the set of all Liouville numbers

Hey guys, I have a problem and I don't know a good way to deal with it... so basically, it says that a number $\displaystyle x$ is called Liouville number, for every $\displaystyle N>0$ there exist integers p and q > 1 such that $\displaystyle \left |{x-\frac{p}{q}}\right |<\frac{1}{q^N}$

question: prove that the set of all Liouville numbers has the same cardinality as the $\displaystyle \mathbb{R}$

Please, how can I approach this question??? is there a way to construct a bijection from the Liouvilles to the reals???

2. Originally Posted by morito14
Hey guys, I have a problem and I don't know a good way to deal with it... so basically, it says that a number $\displaystyle x$ is called Liouville number, for every $\displaystyle N>0$ there exist integers p and q > 1 such that $\displaystyle \left |{x-\frac{p}{q}}\right |<\frac{1}{q^N}$

question: prove that the set of all Liouville numbers has the same cardinality as the $\displaystyle \mathbb{R}$

Please, how can I approach this question??? is there a way to construct a bijection from the Liouvilles to the reals???
See >>here<< (you will need to fill in a gap of two yourself)

CB

3. OK, I see the answer. Not hard at all. Thanks a lot

Hmmm... just one doubt, for the proof to work you need to assume a n to be large enough. How one can justified that?

Finally, we can prove that x is irrational, but do we need to prove that the set of all Liouville numbers is actually equal to the set of the irrationals?

4. Originally Posted by morito14
OK, I see the answer. Not hard at all. Thanks a lot

Hmmm... just one doubt, for the proof to work you need to assume a n to be large enough. How one can justified that?
I read that as for any n>0 there are a p and q with the required properties? (the paragraph on Liouville's constant can be adapted to do that)

Finally, we can prove that x is irrational, but do we need to prove that the set of all Liouville numbers is actually equal to the set of the irrationals?
There is a subset of the Liouville numbers that is equinumerous with the reals (those zero except at the n! decimal place), and that should be sufficient to prove that the Liouville numbers are equinumerous with the reals.