What operations can be performed on three variables $x$, $y$, and $z$ and produce one unique sum? I'm having a programming problem, but I feel this would be more relevant here. I have three unique variables, $x$, $y$, and $z$. What operations can I perform on these variables to produce a unique value? Are there any approaches I could take?

Thanks!

EDIT: To be more specific, I want some function $f(x, y, z)$ that produces a unique output for each set of inputs (assuming uniqueness in inputs, $1$, $2$, $3$ does not need to output something different than $3$, $2$, $1$). I realize that a generalized case of this is probably not possible, but I would definitely be able to co-opt similar solutions
$f(x,y,z) = x\sqrt{2} + y\sqrt{5} + z\sqrt{7}$

This function would increase by a similar order of magnitude to its inputs. Moreover it's possible to prove that for $x,y,z \in \mathbb{Z}, f(x,y,z)$ must be unique $\forall x,y,z$. 

Edit: Proof:

Suppose for contradiction that $\exists x_{1},y_{1},z_{1},x_{2},y_{2},z_{2} \in \mathbb{Z}$:

$f(x_{1},y_{1},z_{1}) = f(x_{2},y_{2},z_{2})$ with at least one of the following being true: $x_{1} \neq x_{2}, y_{1} \neq y_{2} , z_{1} \neq z_{2}$

Then we have $x_{1}\sqrt{2} + y_{1}\sqrt{5} + z_{1}\sqrt{7} = x_{2}\sqrt{2} + y_{2}\sqrt{5} + z_{2}\sqrt{7}$

$\Rightarrow (x_{1} - x_{2})\sqrt{2} + (y_{1} - y_{2})\sqrt{5} + (z_{1} - z_{2})\sqrt{7} = 0$

In the case that $x_{1} = x_{2}$ and $y_{1} \neq y_{2}$ and $z_{1} \neq z_{2}$:

$\frac{\sqrt{5}}{\sqrt{7}} = \frac{z_{2} - z_{1}}{y_{1}-y_{2}}$

This would imply that $\frac{\sqrt{5}}{\sqrt{7}} \in \mathbb{Q}$ which leads to a contradiction (distinct irrational divided by distinct irrational must also be irrational - simple to prove)

A similar proof applies to the case where $y_{1} = y_{2}, x_{1} \neq x_{2}$ and  $z_{1} \neq z_{2}$ and the case when $z_{1} = z_{2}, x_{1} \neq x_{2}$ and $y_{1} \neq y_{2}$ 

If we have $x_{1} = x_{2}$ and $y_{1} = y_{2}$, then we get $z_{1} = z_{2}$ which is a contradiction. Therefore we have proved the following conditions:

$x_{1} \neq x_{2}, y_{1} \neq y_{2} , z_{1} \neq z_{2}$

let $a = x_{1} - x_{2}, b = -(y_{1}-y_{2}), c = -(z_{1} - z_{2})$

Then we have $a\sqrt{2} = b\sqrt{5} + c\sqrt{7}$

$\Rightarrow \sqrt{2} = \frac{b\sqrt{5} + c\sqrt{7}}{a}$

let $b_{1} = \frac{b}{a}, c_{1} = \frac{c}{a}$

$\Rightarrow b_{1}, c_{1} \in \mathbb{R}$

$2 = (b_{1}\sqrt{5} + c_{1}\sqrt{7})^{2}$

$\sqrt{35} = 2 - 5b_{1}^{2} - 7c_{1}^{2}$

$\Rightarrow \sqrt{35} \in \mathbb{Q}$ which is a contradiction.

This implies that $\forall x,y,z \in \mathbb{Z}, f(x,y,z)$ is unique.
You can use the concept of binary expression.for example, (1,2,3) goes to the binary expression of 123.