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MathGroup Archive 2006

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Re: Searching for a function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70837] Re: Searching for a function
  • From: Bill Rowe <readnewsciv at sbcglobal.net>
  • Date: Sat, 28 Oct 2006 23:38:27 -0400 (EDT)

On 10/28/06 at 5:21 AM, banerjee.28 at osu.edu (Bonny Banerjee) wrote:

>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?

Yes and no. Yes, in the sense that for any finite data set there 
is always a function that exactly maps members of A to B. If 
nothing else, a polynomial of degree Length[data]-1 will do the 
trick though not usually the best choice. For this purpose, 
cubic interpolation is usually a better choice

Cubic interpolation can be accomplished using the built in 
function Interpolation

But the general problem you are expressing, that is find an 
arbitrary function from a finite list of points at which the 
function has been sampled has no solution or perhaps more 
correctly no unique solution.
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