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Topic: Effective method of getting a one-to-one onto function from a one-to-one
function?

Replies: 4   Last Post: Sep 15, 2013 8:23 AM

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Sandy

Posts: 40
Registered: 7/9/13
Re: Effective method of getting a one-to-one onto function from a
one-to-one function?

Posted: Sep 15, 2013 8:23 AM
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William Elliot wrote:
> On Fri, 13 Sep 2013, Sandy wrote:
>

>> Suppose X is a set with aleph_0 members and f:X-->N is a one-to-one function.
>> Is there an effective (maybe even "canonical") method of defining g:X-->N in
>> terms of f so that g is one-to-one and onto?

>
> Carefully check this.
>
> Let [n,m] = [n,m] /\ N.
>
> Set s1 = min{ n : 0 < |f^-1([1,n])| },
> s_(k+1) = min{ n : sk < |f^-1([1,n])| }.
>
> (sk)_k is an increasing sequence into N.
>
> Let g(x) = min{ k | f(x) in [1,sk] }.


Thank you. I shall look into it.




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