# Union of Power Sets

## Theorem

The union of the power sets of two sets $S$ and $T$ is a subset of the power set of their union:

$\powerset S \cup \powerset T \subseteq \powerset {S \cup T}$

### Union of Power Sets not always Equal to Powerset of Union

Equality does not hold in general:

The union of the power sets of two sets $S$ and $T$ is not necessarily equal to the power set of their union.

## Proof

 $\ds X$ $\in$ $\ds \powerset S \cup \powerset T$ $\ds \leadsto \ \$ $\ds X$ $\subseteq$ $\ds S \lor X \subseteq T$ Definition of Set Union and Definition of Power Set $\ds \leadsto \ \$ $\ds X$ $\subseteq$ $\ds S \cup T$ Definition of Set Union $\ds \leadsto \ \$ $\ds X$ $\in$ $\ds \powerset {S \cup T}$ Definition of Power Set

$\blacksquare$