Notation for set of sequences from another set that have a unique variable.

Question

I have difficulty wrapping my head around how to write such notation.

Here's my data:

$j = 2$

$k = 10$

$X$ - combination, sequence of digits, ex: $(1, 2, 5)$

$Sx$ - sum of digits inside sequence $X$

$C$ - set of all combinations $X$, where $|X| \in {j .. k}$

$C_n$ - set of combinations $X$, where $|X| = n$

$U$ - subset of $C$, where [where Sx is unique to only one X]

$U_n$ - subset of $U$, where $|X| = n$

My main issue is to properly describe a condition 'where' for $U$ that would only fetch $X$ from $C$, that has unique $S_x$ and this I would like to request help with.

Thank you in advance :)

Answer 1

Here I use $S(X)$ to denote what you call $Sx$: $$U=\{X \in C \mid \forall X' \in C, S(X)=S(X') \Rightarrow X=X'\}.$$ This says $X$ is in $U$ if whenever $X$ has the same sum as $X'$, in fact $X$ is equal to $X'$. So this precisely picks out the elements that have a unique sum.