Family over a set such that the union is the set

Question

I'm new to combinatorics, so this may be a common scenario with which I'm unfamiliar.

I'm trying to define a family of set $\mathbb{F}$ over a set $\mathcal{S}$, where the size of each member of $\mathbb{F}$ is not fixed and the empty set is excluded, such that the union of all members of $\mathbb{F}$ is equal to $\mathcal{S}$ itself.

First question is - what is the standard set notation for that? My guess is along the lines of

$$ \mathbb{F} := \left\{f \in \mathfrak{p}(\mathcal{S}) \mid \bigcup f = \mathcal{S} \right\} $$

Second question - what if $\mathcal{S}$ is a multiset? In my particular case, it would be the sum of the multiplicities from each member of $\mathbb{F}$ rather than the maximum of the multiplicities.

Lastly - where can I find out more about these types of cases? I expect this is a common application.

Answer 1

{ f $\subseteq$ P(S) - {empty set} : $\cup$f = S }
is what you want.