Powers and Intersections

Why

How does the power set relate to an intersection?

Notation preliminaries

First, if we have a set of sets—denote it $\mathcal{C} $—and all members are subsets of a fixed set—denote it $E$—then the set of sets is a subset of $\powerset{E}$. In this case, we can write

\[ \bigcap \Set{X \in \powerset{E}}{x \in \mathcal{C} } \]

Which is a sort of justification for the notation

\[ \bigcap_{X \in \mathcal{C} } X. \]

Basic properties

Here are some basic interactions between the powerset and intersections.¹

$\powerset{A} \cap \powerset{F} = \powerset{(A \cap F)}$

$\bigcap_{X \in \mathcal{A} } \powerset{A} = \powerset{(\bigcap_{X \in \mathcal{A} } A)}$

$\bigcap_{X \in \powerset{E}} X = \emptyset$

Future editions will expand on these propositions and provide accounts of them. ↩︎