Is there a way to express a binary operation that takes two sets and returns a set containing each of the elements of the first set having been unioned with each element of the 2nd set?
For example, the power set of $X_1 = \{a,b\}$ is:
$\mathcal{P}_{X_1} = \{\emptyset, \{a\},\{b\},\{a,b\}\}$
The power set of $X_1 \cup \{c\}$ is:
$\mathcal{P}_{X_1 \cup \{c\}} = \{\emptyset, \{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$
but this could also be expressed as:
$\mathcal{P}_{X_1} * \mathcal{P}_{\{c\}}$
where $*$ is the operator that I am asking about.
I know I can just write $\mathcal{P}_{X_1 \cup \{c\}}$ in this instance, but what about for general sets, what if I wanted to do something like:
$X_2 = \{\{a\},\{b\}, \{a,c\}\}$ and $X_3 = \{\emptyset,\{c\},\{a,d\}\}$
and get back:
$X_2 * X_3 = \{\{a\},\{b\}, \{a,c\},\{b,c\},\{a,d\}, \{a,b,d\}, \{a,d,c\}\}$
Is there a general operator that can express this? Or, failing this, how should I go about writing it?