Select one or zero elements from a set I am far from a mathematician. Still. I want to formally express that only 0 or 1 element of a series of sets (1...n) is selectet to form a new set.
Example:
I have three sets $S_1 = \{1,2,3\}$, $S_2 = \{4,5,6\}$ and $S_3 = \{7,8,9\}$.
Now I want to define the set $L$ as a set containing 0 or 1 elements from $S_1, S_2,S_3$. 
Example 1:
Containing one element from each set: $L = \{1,5,8\}$ 
Example 2:
Containing one element from $S_2$ and $S_3$and none from $S_3$: $L = \{5,8\}$.
How can I define $L$ formally?
 A: Suppose you have $n$ sets $S_1, S_2, \ldots, S_n$, and for simplicity let $[n] = \{1,2,\ldots,n\}$. Moreover, let $\Omega = \bigcup_{k \in [n]}S_k$ be the collection of all the elements in $S_k$. Then you can define your set $L$ as
$$L \subseteq \Omega\ \ \text{ such that }\ \ \forall k \in [n].\ |L \cap S_k| \leq 1.$$
If you are familiar with the concept of partial functions you can alternatively say that $L$ is the image of some partial function $f : [n] \rightharpoonup \Omega$ with property $f(k) \in S_k$ for any $k$ such that $f(k)$ is defined.
However, in my opinion the best solution is to define $L$ using plain words:

Let $L$ be a subset of $\Omega$ with at most one common element with any of $S_k$, for $k \in [n]$.

Complicated formulas in most cases decrease readability, while textual definitions can be just as precise. If you are unsure, use both words and a formula, e.g.

Let $L$ be a subset of $\bigcup_{k \in [n]}S_k$ such that it has at most one common element with any of $S_k$, for $k \in [n]$, that is, $$\forall k \in [n].\ |L \cap S_k| \leq 1.$$

I hope this helps $\ddot\smile$
A: A set $L$ satisfies your criterion iff
$$\#(L \cap S_a) \in \{0, 1\} \qquad \forall a \in \{1, 2, 3\}.$$
Note that this may or may not be the behavior you have in mind when a given element in $S_1 \cup S_2 \cup S_3$ occurs in more than one $S_a$.
A: $\left(L\subseteq \bigcup_iS_i\right)\ \ \land\ \ \ \forall (x,y\in L).\,\forall i.\,\left((x\in S_i)\land (y\in S_i)\implies x=y\right)$
For the second point use "there exists a unique..", $\exists!$, and the set difference, $\setminus$.
A: I would use $L\in{S}\cup{S_{12}}\cup{S_{13}}\cup{S_{23}}\cup{S_{123}}$, where:


*

*$S=\{\}$

*$S_{12}={S_1}\times{S_2}=\{\{1,4\},\{1,5\},\{1,6\},\{2,4\},\{2,5\},\{2,6\},\{3,4\},\{3,5\},\{3,6\}\}$

*$S_{13}={S_1}\times{S_3}=\{\{1,7\},\{1,8\},\{1,9\},\{2,7\},\{2,8\},\{2,9\},\{3,7\},\{3,8\},\{3,9\}\}$

*$S_{23}={S_2}\times{S_3}=\{\{4,7\},\{5,8\},\{6,9\},\{4,7\},\{5,8\},\{6,9\},\{4,7\},\{5,8\},\{6,9\}\}$

*$S_{123}={S_1}\times{S_2}\times{S_3}=\{\{1,4,7\},\dots,\{3,6,9\}\}$

Or simply $L\in\{\}\cup({S_1}\times{S_2})\cup({S_1}\times{S_3})\cup({S_2}\times{S_3})\cup({S_1}\times{S_2}\times{S_3})$
