The image of the Cantor set of functions.

Question

I am trying to show that $C=f_1(C)\cup f_2(C)$ where $f_1=\frac{1}{3}x$ and $f_2=\frac{2}{3}+\frac{1}{3}x$ but I am having trouble. I am not really sure how to start the reverse direction. If $x \in C$ then $f_1(x) \in C$ as it just changes to a different but proper ternary representation. $f_2(x) \in C$ because of the same reason. So $f_1(C)\cup f_2(C) \subseteq C$. But I am having trouble with the reverse. Hints would be appreciated.

Answer 1

I suppose the $C$ you're talking about is the Cantor set, the set of all $x \in [0,1]$ that can be represented in ternary without using the digit $1$.

What do all $y \in f_1(C)$ have in common? What do all $y \in f_2(C)$ have in common?

Now pick any $y \in C$, and look at it closely. Can you tell from its ternary representation whether $y$ is in $f_1(C)$? Can you tell whether it is in $f_2(C)$?

This should be a sufficient hint to help you solve the problem.