- #1

julypraise

- 110

- 0

## Homework Statement

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?

## Homework Equations

## 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.