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How to avoid rounding off in binary representation 
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#1
Mar3012, 08:46 PM

P: 110

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] onetoone? (*note: N does not have 0) If not, can you change bit so that the changed funtion becomes onetoone? 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 onetoone. (Am I right?) So can I in some way make a function that is onetoone and similar to this one by somehow avoiding the rounding off problem? Could you give me just a hint? Thanks. 


#2
Mar3012, 09:40 PM

PF Gold
P: 1,622




#3
Mar3012, 11:02 PM

P: 110




#4
Mar3112, 10:29 AM

HW Helper
P: 6,189

How to avoid rounding off in binary representation
I haven't completely figured it out, but you can make a distinction between 3 types of sets:
1. Sets X with a finite number of elements. Let's call them "0tail". 2. Sets X that contain all elements of ##\mathbb{N}_{>n}## for some number n. Let's call them "1tail". 3. Sets X that belong neither to sets of type 1 nor to sets of type 2. Let's call them "notail". Now we can for instance take your binary representation, but divide all 0tailers by 2. And divide all 1tailers by 2 and subtract from 1. That way, you'd have a bijection of ##\mathcal{P}(\mathbb{N})## to ##[0,1)##. EDIT: edited the 1tail operation. 


#5
Mar3112, 10:37 AM

PF Gold
P: 1,622

I have not checked if I Like Serena's solution works, but another (similar) way of doing this is to expand the codomain to be [0,1] and all real numbers in [1,2] with which have tails consisting completely of 1s. Then you just map the problematic elements into your expanded codomain.



#6
Mar3112, 11:08 AM

P: 110




#7
Mar3112, 11:56 AM

PF Gold
P: 1,622

Edit: It is also worthwhile to check if you can use I Like Serena's solution. His/her posts are very good. 


#8
Mar3112, 09:19 PM

P: 110

I think it's quite enough. Thanks guys.



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