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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!

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    If is$A$ proper subset of $B$ then obviously $f(A) \neq f(B)$. The same if is $B$ proper subset of $A$. But what happens if $A \cap B \neq \emptyset$ ?2012-04-17

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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 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?

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    O.K Thank you for you time both!2012-04-18