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I hope to find a function inversible to create a set of numbers $\{a, b, c, ...N\}$ from $x$, such that all numbers are unique, it means that if $f(x)=a$, this $a$ it will never create again. Example :

$$f(x)=a, f(x)=b, f(x)=c ... f^{-1}(c)=x, f^{-1}(b)=x, f^{-1}(a)=x $$ $x, a, b, c$ cannot be created again from any numbers.

which function can do this ? I guess there is no such a function, or maybe it has another name in mathematics "procedure", "application" ?

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closed as not a real question by Jasper Loy, Hagen von Eitzen, Thomas, Norbert, J. M. Oct 5 '12 at 13:10

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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None. If $f$ is a function then from $f(x)=a$ and $f(x)=b$ we infer $a=f(x)=b$, whereas you seem to imply that $a\ne b$ should hold. –  Hagen von Eitzen Oct 3 '12 at 20:37
    
How do you mean? ..and what about the $n\mapsto (n+1)$ function $\mathbb N\to\mathbb N$? –  Berci Oct 3 '12 at 20:42
    
Actually it is not a function, I want a sort of function that maps one value to many others, from N to (N,N,N) for example –  user15992 Oct 3 '12 at 20:47
    
@HenningMakholm ok then what do we call it ? –  user15992 Oct 3 '12 at 20:49
    
I recommend looking at this article on Multi-Valued Functions. I think this is the mathematical concept that you are looking for. You could also consider vector valued functions, which may also represent what you are looking for. I'm not entirely sure that you will have a nice inverse function, but I think you could at least use these functions to describe a suitable algorithm. –  Carl Morris Oct 3 '12 at 20:53
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1 Answer 1

This is not possible, at least if your $x$ does not change between the equations you give. By definition, a "function" in mathematics always produces the same result when applied to the same argument. It has no memory.

In some computer programming languages there are things called "functions" that do not behave like that, but they are not what we call a "function" in mathematics". (The connection is that the programming-language functions were originally intended for pieces of programs that computed actual mathematical functions; the scope of the concept later broadened).

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@user995434: Even a multi-valued function gives you the same set of possible values each time you apply it to the same input. –  Henning Makholm Oct 3 '12 at 23:02
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