Powers and Unions

Why

How does the power set relate to a union?

Notation preliminaries

Let $E$ denote a set. Let $\mathcal{A} $ denote a set of subsets of the set denoted by $E$. We define $\bigcup_{A \in \mathcal{A} } A$ to mean $\bigcup \mathcal{A} $.

Basic properties

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

$\powerset{E} \cup \powerset{F} \subset \powerset{(E \cup F)}$

$\bigcup_{X \in \mathcal{C} } \powerset{X} \subset \powerset{(\bigcup_{X \in \mathcal{C} } X)}$

$E = \bigcup \powerset{E}$

$\powerset{(\bigcup E)} \supset E$.

Typically $E \neq \powerset{(\bigcup E)}$, in which case $E$ is a proper subset.

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