# Function uniqueness

Function "uniqueness"..

Ok, pardon the complete lack of terminology here.

I can define a function with one parameter such that no two different inputs give the same output. Example:

f(x) = x + 1

No value of x gives the same result as another value of x.

I believe that it is impossible to define a function which accepts two parameters, yet retains this property of "uniqueness". Am I correct? Is there a name for what I am trying to describe?

I'm pretty sure I am correct, but I have no idea how to prove it.

Lets say I have a function

f(a,b) = k

where k is some sort of calculation.

if k is arithmetic, i can break the uniqueness by doing the calculation outside f, and then feeding 0 as one of the parameters

f(a,b) = k = f(k,0)

if k is geometric I can do the same as above, only pass 1 instead of 0. Outside of these two classes of functions (and the trivial one where a or b is omitted in the calculation of the function) I cant seem to find a good way of proving myself.

Proving this for more and more "classes of calculations" seem futile, I'm not even sure everything can be categorized as neatly as arithmetic and geometric functions can. There might be a better way of going about this?

All feedback is welcome.

k

Last edited:

mathman

There is no "nice" way to define a two parameter function with a unique value. However, there are messy looking ways. Example consider the decimal expansions of x and y:
0.x1x2x3x4x5x6x7..... and 0.y1y2y3y4y5y6y7....
Then define z=f(x,y) by:
0.x1y1x2y2x3y4x4y4......
Note that this works from the unit square to the unit interval.
To do the whole real line, to get x and y use an arctan transformation (properly normalized) and the reverse process (tan) to go from z to the whole real line.

Normalization involves shifting the interval and dividing or multiplying by pi.

What you're looking for is called a function that is one to one. It is also called injection/injective.

A typical example is: f(x,y) = a^x * b^y where a, b are primes that don't equal each other.

Ex: let f(x,y) = 2^x * 3 ^ y

It should be easy to see that f(3,4) = 2^3 * 3^4 is a unique number or the fundumental theorem of mathematics (sometimes called of arithmetic) would be contradicted.

mathman

What you're looking for is called a function that is one to one. It is also called injection/injective.

A typical example is: f(x,y) = a^x * b^y where a, b are primes that don't equal each other.

Ex: let f(x,y) = 2^x * 3 ^ y

It should be easy to see that f(3,4) = 2^3 * 3^4 is a unique number or the fundumental theorem of mathematics (sometimes called of arithmetic) would be contradicted.
This is good as long as x and y are integers. However, for real x and y, you would lose uniqueness.

Thanks a lot to both of you, I was clearly wrong in assuming that it cannot be done.

k