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MathGroup Archive 2006

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Re: Re: Searching for a function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70886] Re: [mg70854] Re: Searching for a function
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 30 Oct 2006 05:33:32 -0500 (EST)
  • References: <ehv8vp$g6f$1@smc.vnet.net> <200610290340.XAA08628@smc.vnet.net>

This adds nothing to the information or the lack of it, in your  
previous post. If your sets are finite and they have the discrete  
topology then every function is continuous, so you actually have n^m  
functions, where n is the cardinality of the soure and m the  
cardinality of the target. One can also imagine that what you meant  
is a continuous function from an interval to another interval, which  
takes a given finite set of points to certain specified images. In  
this case you will have infinitely many functions. In fact, the same  
is true if you restrict yourself to polynomials of arbitrary degrees.  
On the other hand if you restrict yourself to polynomial of a given  
degree you can make sure that the answe r is finite. For example, in  
your case:

A=Range[2,10,2];B=Range[1,10,2]; ls = Transpose[{A, B}];


InterpolatingPolynomial[ls,x]

x-1

In general a polynomial of degree n is determined by its values at n  
points so this answer is  unique for polynomials of degree <=5. For  
example, here is how we can get another polynomial which satisfies  
all your conditions:


AppendTo[ls, {3, 9/2}];


f[x_] = Expand[InterpolatingPolynomial[ls, x]]

x^5/42 - (5*x^4)/7 + (170*x^3)/21 - (300*x^2)/7 +
   (2213*x)/21 - 647/7

You can check that this polynomial satisfies all your conditions, in  
fact:


f /@ A == B


True

Andrzej Kozlowski
Tokyo, Japan

On 29 Oct 2006, at 12:40, Bonny Banerjee wrote:

> I am looking for continuous functions only from set A to set B.  
> Sorry for
> not making it clear.
>
> --Bonny.
>
>
>
>
> "Bonny Banerjee" <banerjee.28 at osu.edu> wrote in message
> news:ehv8vp$g6f$1 at smc.vnet.net...
>> Is it possible for Mathematica to solve this problem:
>>
>> Given sets A and B, does there exist a function from A to B? If  
>> yes, what
>> is
>> the function?
>>
>>
>> Here is an example:
>>
>> Let, A = {x such that 0<x<11 and Mod[x,2]==0}
>>
>> B = {y such that 0<y<11 and Mod[y+1,2]==0}
>>
>> Then, there exists a function from A to B
>>
>> y = x - 1
>>
>>
>> Thus, is there a way to specify arbitrary sets A, B, and use  
>> Mathematica
>> to
>> figure out whether there exits a function from A to B or not?
>>
>> Thanks,
>> Bonny.
>>
>>
>
>


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