mathematical notations - for selecting from a set under some constraints

Question

I have a set $X = \{0, 1, ... , n\}$.

I want to choose a number for each $Z_i (i = 1, ..., m)$, $m \leqslant n$, from the set $X$ such that $Z_i \neq Z_j (j = {1, ... ,m}, j \neq i)$ with the exception $Z_i = Z_j \iff Z_j = 0 $.

After 'such that' in the above statement, is there a better/shorter way of stating it mathematically? Is it correct to state: $Z_i \neq Z_j (j = {1, ... ,m}, j \neq i) \iff Z_j > 0 $, or should I ask, is it the best way of stating it?

Thanks in advance.

Answer 1

How about $Z_i=Z_j\Rightarrow ((i=j) \text{ or } (Z_i=Z_j=0))$?

Or perhaps just saying: There are no repeated values among the $Z_i$s except that the value $0$ may be repeated.