# A set of algebraic operations producing unique results based on order?

1. Sep 15, 2011

### KingNothing

I'm trying to find a set of five (5) algebraic functions a(x), b(x), c(x), d(x), and e(x) that for every order they can be applied, will produce a unique result. That is, a(b(c(d(e(x))))) should be different from e(d(c(b(a(x))))) for every possible x. And every other unique ordering should produce a unique result as well.

Constraints:
x is constrained to integers where 0 <= x <= (2^16 - 1).
The result of each ordering must be constrained to 0 <= result <= (2^32 - 1).
The functions may only contain basic operations: addition, subtraction, division and multiplication.

If you could provide an example of five such functions, it would help me greatly. I would love to do some further reading after I get a working example of said functions.

EDIT: Bonus points if the five functions also produce unique results for any partial ordering. That is, a(b(d(x))) must be different than d(e(x)), and so on.

Last edited: Sep 15, 2011