Proving a set is uncountable?

• April 22nd 2013, 12:29 PM
gridvvk
Proving a set is uncountable?
I have to show that the set defined by {x element of (0,1) : the decimal expansion of x has only odd digits} is uncountable.

My attempt: I suppose we can show this by contradiction, but I'm sure how to go about doing so. If we were to use Cantor's diagonal argument doesn't that rely on infinite decimal expansion, doesn't the fact we are only looking at expansions that have odd digits imply finite decimal expansion?
• April 22nd 2013, 01:02 PM
Plato
Re: Proving a set is uncountable?
Quote:

Originally Posted by gridvvk
I have to show that the set defined by {x element of (0,1) : the decimal expansion of x has only odd digits} is uncountable.

My attempt: I suppose we can show this by contradiction, but I'm sure how to go about doing so. If we were to use Cantor's diagonal argument doesn't that rely on infinite decimal expansion, doesn't the fact we are only looking at expansions that have odd digits imply finite decimal expansion?

Consider $x=\sum\limits_{k = 1}^\infty {\frac{{{\delta _k}}}{{{{10}^k}}}}$ where each $\delta_n$ is an odd digit.

That is that such that $x$ has only odd digits. Can't you do a Cantor type argument on the set of all such numbers?
• April 22nd 2013, 01:07 PM
gridvvk
Re: Proving a set is uncountable?
Yes, you are correct. I misunderstood my own problem. I thought it meant that there are only an odd amount of places in the decimal expansion. Thanks for the help.