How to define a set of sets with no two elements from the same set

Question

TL;DR I'm trying to notate the phrase in bold.

I'm attempting to define an independence system, but it's been a while since I've used mathematical notation.

I have a set $E$ of all the elements. I also have a set $V$ which contains sets $V_1, V_2,$ etc. Each $V_x$ is a subset of $E$. (That is, it just contains elements.) Note that each $V_x$ is also disjoint from each other $V_x$ (they have no elements in common), and that the union of all $V_x$ gives $E$.

I want to define my independence system to be such that a set that has only zero or one element from each $V_x$ is independent. (Thus, a set with one element from each $V_x$ will be maximally independent.) In my head, I've phrased this as "no two elements from the same $V_x$".

How do I mathematically introduce such an independence system?

Currently, I've defined it as the set $I$ such that

$${S}\in{I}\Leftrightarrow\nexists, v_x, v_y \in S \,|\, v_x \ne v_y \land \forall{V_x}, v_x,v_y\in{V_x}$$

This is supposed to read as "$I$ is a set consisting of all sets that don't have two elements from the same $V_x$".

But my notation just seems messy, unclear, and possibly incorrect. I'm especially having a hard time figuring out how to mathematically say "no two elements from the same set".

Any help writing this idea down in mathematical notation would be appreciated.

Answer 1

There is not a unique approach to this, but here's one idea. Define

$$ \mathfrak{I} := \{S\subset E : |S \cap V_x| \leq 1, \text{ for all $x$} \}. $$

Thus, a set $S \in \mathfrak{I}$ satisfies presicely that the cardinal of its elements which belong to $V_x$ is at most one, for any $x$ (by the way, one should be more precise and indicate what the elements $x$ are, i.e. to which set do they belong).

Answer 2

Here's another way.

You really ought to describe $V$ using some index set $X$, i.e. $V=\{V_x\mid x\in X\}$. You could simply take $X=\{1,2,3,\dots,n\}$ for example, then $V=\{V_1,V_2,\dots,V_n\}$. Then we can define a characteristic function $f:E\to X$ with $f(v)=x$ if $v\in V_x$. If we want to say that each $V_x$ is nonempty, then this amounts to saying $f$ is surjective.

Now $I$ is the set of subsets $S\subset E$ such that the set $g=\{(f(s),s)\mid s\in S\}$ is a partial function $g:X\to E$. That is, if both $(x,s)$ and $(x,s')$ are pairs in $g$, then $s=s'$ (since $g$ is a function). Thus, if $s,s'$ are two distinct elements of $S$, then $f(s)=x\neq x'=f(s)$, and therefore $s\in V_x$ and $s'\in V_{x'}$ are in different sets of $V$.

Such a function $g$ is a partial choice function on $V$. Hence $I$ can be defined as the set of ranges of the partial choice functions on $V$.

Alternatively, we can do without the partiality and define $I$ by considering all total choice functions on $V$, and let $I$ be the smallest set containing the range of each choice function on $V$ that is closed under subsets.