Firstly it's important to lay down a definition to avoid ambiguity. So like in base 10 where we define so in base 3 we define . Similarly .
I'm not going to give you a tutorial on how to convert any number to and from ternary, as it's actually not entirely necessary when we're only looking at the cantor set. We only need to consider negative powers of 3, which are easy to work out. So like you pointed out, which means that
So, using this knowledge we can see that all the numbers between and are exactly those numbers in base 3 between and
which are the numbers , any number with a one in it's first decimal place.
So now looking at the cantor set in an iterative way, that is, defining
we can see that with each iteration we are removing the middle third of each interval that remains.
Now we consider exactly what it is we are removing. After the first removal, we have taken away all the numbers between one third and 2 thirds. That is, all the numbers between and . These numbers all share a common property that should be easy to see as I've already discussed this.
If you look for yourself what is happening after the next iteration you should see a similar thing happenening again. If you keep repeating this you should be able to see just exactly which elements belong in .
Hope this helps