$\begingroup$Consider the family of sets $F = \{S_1, \dots, S_m\}$ $\forall m \in \mathbb{N}$, where $S_i$ is a set of elements $\forall i \leq m$. Let us define $C \subseteq F$.
Is there any convention for the notation of the size of $C$ (i.e. the number of sets in $C$)?
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$\begingroup$If $C$ is a family of sets, especially a finite family, then $C$ is just a set of sets. Therefore $|C|$ works just fine.
$\endgroup$0$\begingroup$I've seen $\lvert F \rvert, F^=,F\!\!\!{}^{{}^{{}^{\large{=}}}}$ and $\operatorname{card}(F)$ for the cardinality of $F.$
In some contexts (finite combinatorics?), $\#F$ or $\#(F)$ might be used.
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