Consider the set-function
$ f: \mathcal{P}(\mathbb N) \to [0 ,+\infty]$ with $\displaystyle{ f(A)= \sum_{n \in A } \frac{1}{3^n}}$ where $ A \subset \mathbb N$
(a) Is $f$ one-to-one ?
(b) Is $f$ bijective ?
Thanks in advance!
Consider the set-function
$ f: \mathcal{P}(\mathbb N) \to [0 ,+\infty]$ with $\displaystyle{ f(A)= \sum_{n \in A } \frac{1}{3^n}}$ where $ A \subset \mathbb N$
(a) Is $f$ one-to-one ?
(b) Is $f$ bijective ?
Thanks in advance!
HINT for (a): Suppose that $f(A)=f(B)$, but $A\ne B$. Let $m$ be the smallest integer that is in exactly one of $A$ and $B$; without loss of generality suppose that $m\in A\setminus B$. Then $$\sum_{k\in A\atop{k<m}}\frac1{3^k}=\sum_{k\in B\atop{k<m}}\frac1{3^k}\;.$$ Call this sum $s$. Then $$f(A)\ge s+\frac1{3^m}\;,$$ and $$f(B)\le s+\sum_{k>m}\frac1{3^k}\;;$$ can you take it from there?