I'm trying to prove that the Cantor set is equal to a certain set of 'escape points' for a mathematical feedback system.
In this proof I'm going to need the fact that every element of the Cantor set has a base-3 representation in which only 0's and 2's occur. However, I'm having a hard time with these triadic expansions. Can you help me by working out an easier problem (also using triadic expansions)?
If $x$ is in the Cantor set, then so is $1-x$.
I don't know if there is an easier way to prove this, but I'm specifically looking for a proof using the fact that x can be written as $0.a_1a_2a_3...$ where $a_i$ is either $0$ or $2$.
Let $x=0.a_1a_2a_3\ldots$, where the expansion is ternary and each $a_k$ is $0$ or $2$. For $k\in\Bbb Z^+$ let $b_k=2-a_k$; clearly $b_k\in\{0,2\}$. Let $y=0.b_1b_2b_3\ldots\;$; then $x+y=0.222\ldots=1$, so $y=1-x$ is in the Cantor set.
(I can recast this in terms of infinite series if you like, but this should be sufficient.)