How to mathematically describe the number of Element x in a set

Question

I am trying to formulate the following.

I have a Set A={x, y, z}, I also have a Set B, C and D, which all are subsets of A. It is not exactly defined which elements are in B, C and D. I only want to define a constraint so that the sum of all elements x in B, C and D equals 1.

What would be the mathematically correct way to describe this constraint? I really have a problem finding the correct notation for something like this.

Answer 1

If I understand correctly, your constraint should read

$$ \sum_{x \in B\cup C\cup D} x = 1 $$

Answer 2

Since these are sets (as opposed to multisets), an equivalent constraint seems to be that the element $x \in A$ appears in exactly one of the sets in $\mathcal F = \{B,C,D\}$. If this is what we want, then we can formulate this as: $$ \sum_{S \in \mathcal F} |\{x\} \cap S| = 1 $$