Re: bijection of R: R <--> Rx.....xR

On Wed, 7 Sep 2005 23:14:30 +1000, "Peter Webb"
<webbfamily-diespamdie@xxxxxxxxxxxxxxx> wrote:

>
>"Timothy Golden http://www.BandTechnology.com"; <tttpppggg@xxxxxxxxx> wrote
>in message news:1126096850.166195.157500@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>> Dirk Once wrote:
>>>there is also a bijection
>>> R <--> RxRxR
>>>and in fact between R and the product set of any countable
>>>number of R's:
>>> R <--> Rx.....xR
>>
>> Could someone please expound on this?
>> How does R map to RxR?
>>
>> -Tim
>>
>
>Let a1, a2, a3, b1 ,b2, b3 etc be digits "0" .. "9"
>
>(0.a1a2a3a4... , 0.b1b2b3b4...) <--> 0.a1b1a2b2a3b3....
>
>Note that we "disallow" representation like 0.4999... for 0.5, so that every
>real has a unique representation.

You need to disallow that, but after you disallow that your map
is not quite a bijection. For example what pair a, b maps to

0.090909090909... ?

You need to take b = 0.9999... here, which is not allowed.

>This sort of thing will work for any finite R <--> Rx...xR. I can't quite
>see how to extend it to a countably infinite number of R terms though.
>
>


************************

David C. Ullrich
.

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