Orchid Win7 v1 <voi### [at] devnull> wrote:
> On 13/07/2013 05:47 PM, Warp wrote:
> > Orchid Win7 v1<voi### [at] devnull>  wrote:
> >>> 2) If yes, that means there has to be a 1-to-1 mapping between those
> >>> points. Give a function that expresses such a mapping.
> >
> >> Given the 2D coordinates of a point on the unit square, you can
> >> interleave their decimal digits, which always yields a unique point on
> >> the unit line. For example,
> >
> >>     0.3425
> >>     0.2183 ->  0. 32 41 28 53
> >
> > That doesn't work because it's not a one-to-one mapping. Ie. the mapping
> > is not unambiguous.

> That would imply that two distinct 2D points exist which map to the same 
> 1D point. Can you provide such a counter-example?

By using a decimal notation you are equating the set of real numbers
with the set of integers, thus making the assumption that the set of
real numbers is countable. Not all real numbers can be represented with
digits, because digits can only be used to represent a countably infinite
set, which the set of reals isn't.

-- 
                                                          - Warp

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