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

## Theorem

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

## Proof

Let $S = \set {1, 2, 3}, T = \set {2, 3, 4}, X = \set {1, 2, 3, 4}$.

 $\displaystyle S \cup T$ $=$ $\displaystyle \set {1, 2, 3, 4}$ $\displaystyle \leadsto \ \$ $\displaystyle X$ $\subseteq$ $\displaystyle S \cup T$ $\displaystyle \leadsto \ \$ $\displaystyle X$ $\in$ $\displaystyle \powerset {S \cup T}$

But note that $X \nsubseteq S \land X \nsubseteq T$.

Thus:

 $\displaystyle X$ $\nsubseteq$ $\displaystyle S \land X \nsubseteq T$ $\displaystyle \leadsto \ \$ $\displaystyle X$ $\notin$ $\displaystyle \powerset S \land X \notin \powerset T$ $\displaystyle \leadsto \ \$ $\displaystyle \neg (X$ $\in$ $\displaystyle \powerset S \lor X \in \powerset T)$ $\displaystyle \leadsto \ \$ $\displaystyle X$ $\notin$ $\displaystyle \powerset S \cup \powerset T$ $\displaystyle \leadsto \ \$ $\displaystyle \powerset {S \cup T}$ $\nsubseteq$ $\displaystyle \powerset S \cup \powerset T$

So:

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

$\blacksquare$