(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Is the fundtion defined [itex]f:P(\mathbb{N}) \to [0,1][/itex] by

[itex] f(X) = 0.a_{1} a_{2} \dots [/itex] in binary representation where [itex] a_{k}=1 [/itex] if [itex]k\in X[/itex] and otherwise [itex]0[/itex] one-to-one?

(*note: N does not have 0)

If not, can you change bit so that the changed funtion becomes one-to-one?

2. Relevant equations

3. The attempt at a solution

I know this function is surjective if I interpret the decimal expression in binary representation. But then in this case f({1}) = f(Z\{1}) as 0.100000.. = 0.011111111..., therefore not one-to-one. (Am I right?)

So can I in some way make a function that is one-to-one and similar to this one by somehow avoiding the rounding off problem? Could you give me just a hint? Thanks.

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# How to avoid rounding off in binary representation

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