Power set of Cartesian product I'm trying to see the differences between a Power set of a cartisian product and the cartisian product of two power sets. I have these 2 examples:
$P(\{1, 2\} \times \{3, 4\})$
$P(\{1, 2\}) \times P(\{3, 4\})$
With the first one I can arrive here, but I'm not sure how to continue to compute the power set.
$P(\{(1,3), (1,4), (2,3), (2,4)\})$
The second one I'm stuck on the cartisian product between the to result of the power set.
$\{\{\}, \{1\}, \{2\}, \{1,2\}\} \times \{\{\}, \{3\}, \{4\}, \{3,4\}\} $
How can I keep going from there? And the notation is correct?

The element $(\{1,2\},\{3\})$ is in $\mathcal{P}(\{1,2\})\times\mathcal{P}(\{3,4\})$, but it is not an element of $\mathcal{P}(\{1,2\}\times\{3,4\})$. Elements of that latter set are sets of ordered pairs, not ordered pairs of sets.
