Generalized Set Dualities
Why
If all sets considered in a union or intersection are subsets of a fixed set, then the union and intersection of any set of sets is well defined. We can then derive generalized version of DeMorgan's laws.1
New notation
Let $E$ denote a set.
Let $\mathcal{A} $ denote a set of subsets of
$E$.
Then define
\[
\bigcup_{A \in \mathcal{A} } A := \bigcup \mathcal{A} , \quad
\bigcap_{A \in \mathcal{A} } A := \bigcap \mathcal{A} .
\]
In this case we have
$\complement{\cup_{A \in \mathcal{A} } A} =
\cap_{A \in A} \complement{A}$.
$\complement{\cap_{A \in \mathcal{A} } A} =
\cup_{A \in A} \complement{A}$.
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