What is the powerset of an ordered pair? I'm trying to prove the definition of an identity related to cartesian products, so naturally there are some ordered pairs. My prof did a crummy job on definitions though, and the text isn't great either, so:
If I have:
A = 1, 2
B = 3, 4
AxB = (1,3),(1,4),(2,3),(2,4)
What would be the powerset?
Would that be (partial example):
{(1,3),(1,4)} or could you separate these elements so you'd have items like 
{1}, {4}, {(1,3), (1,4)} as part of the powerset?
 A: Indeed $A \times B = \{(1,3),(1,4),(2,3),(2,4)\}$. This has 4 elements. 
This means its powerset will have $16 = 2^4$ elements and these are sets of pairs as in your first example. So $\{(1,3),(1,4)\}$ is one of them, the set $\{(1,3),(1,4),(2,3),(2,4)\}$ itself is also in the powerset, as is the emptyset $\emptyset$. Also singleton sets like $\{(1,3)\}$ (and three more), and sets of three elements like $\{(1,3),(1,4),(2,3)\}$ etc. 
In forming a powerset we just consider all elements of the set, and either put them in a subset or not. It does not matter if the elements of the set we start with are themselves pairs (like here), or are themselves sets, or anything. 
A: I think your confusion is that you didnt specify what the original set is. As another user mentioned, if your set is $A\times B$, then that has $4$ elements and its powerset contains $2^4$.
If your original set is $\{1,2,3,4\}$, then the powerset also happens to have $2^4$ elements, but these are different from the powerset in the first example.
Finally, the example you offered would be the powerset of $A\cup B\cup A\times B$. I don't really see why you would ever be interested in that, though, especially since you would be mixing single elements with ordered pairs.
