Re: Cantor and the binary tree

On 11 Jun 2005 06:29:14 -0700, mueckenh@xxxxxxxxxxxxxxxxx wrote:

>
>
>Randy Poe wrote:
>> mueckenh@xxxxxxxxxxxxxxxxx wrote:
>> > Is here a bijection possible between uncountable sets?
>>
>> Sometimes.
>>
>> There exist bijections between R and C, for instance. But
>> not between R and P(R).
>>
>> > Can you enumerate R <--> R?
>>
>> I can construct a bijection from R to R, if that is what
>> you are asking.
>>
>> Here's one: f(x) = x.
>
>And I can construct a bijection from nodes to paths:
>
> 0.
> / \
> 0 1
> /\ /\
> 0 1 0 1
> /\ /\ /\ /\
> ...................
>
>Everywhere a path branches off, it gets the node. The other one keeps
>that node which already was mapped on it before. I need not knwow which
>path gets which node, because all are of similar value. And so on in
>infinity.

That is not a bijection.

If you compare, '000...' with '010...', you would map '000...' to '0'
and '010...' to '01'. If you compare '000...' with '011...', you
would map '000...' to '0' and '011...' to '01'. QED.

Martin

.

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