Let $X \subseteq \mathbb{R}$. Let $A$ and $B$ be finite subsets of $X$. The statement $$\sum_{a \in A} 2^a = \sum_{b \in B}2^b \iff A = B $$ is true if $X = \mathbb{N}$ or $X = \mathbb{Z}$; this follows from the uniqueness of finite binary represntation (for naturals and dyadic rationals). However, the statement is false if $X = \mathbb{R}$ as, for example, $2^0+2^2 = 5 = 2^1 + 2^{\log_2{3}}$.

My question: what about the rationals? Is this statement true for $X = \mathbb{Q}$? In other words, is there a unique representation of the form $\sum_{a \in A}2^a$ for all $s \in S$, where $$S = \left\{\sum_{a \in A}2^a : A \subset \mathbb{Q} \text{ is finite}\right\}.$$

Bonus question: is there better, more concise notation for $S$?

Bonus answer: $ S = \mathbb{N}[2^{-1/2},2^{-1/3},...]$ (similar to how the dyadic rationals are $\mathbb{N}[2^{-1}]$.