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