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:


 * $\mathcal P \left({S}\right) \cup \mathcal P \left({T}\right) \subseteq \mathcal P \left({S \cup T}\right)$

Equality does not hold in general.

Proof
Now we show by counterexample that it is not always the case that $\mathcal P \left({S}\right) \cup \mathcal P \left({T}\right) = \mathcal P \left({S \cup T}\right)$.

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

But note that $X \not \subseteq S \land X \not \subseteq T$.

Thus:

So $\mathcal P \left({S \cup T}\right) \ne \mathcal P \left({S}\right) \cup \mathcal P \left({T}\right)$.