Is $\varnothing^0 = \varnothing^\varnothing$? As per the title, is $\varnothing^0 = \varnothing^\varnothing$, assuming $ \varnothing $ is the empty set? If this is the case, how is it so?
 A: The superscript notation in use here has a variety of different related uses. Usually it will be clear from the context which one is meant, but there's not really enough context here to be sure. That said,
$\varnothing^0$ would most commonly mean an empty product of no copies of the empty set. No matter which set it is we have no copies of, this is the set $\{\langle\rangle\}$ whose only element is the emtpy tuple.
$\varnothing^\varnothing$ would most commonly mean the set of all functions $\varnothing\to\varnothing$, that is, functions $f$ such that $f(x)$ is defined exactly if $x\in\varnothing$ (so: never) and $f(x)\in\varnothing$ whenever $f(x)$ is defined. There is exactly one function with this property: the empty function.
The empty function is represented by the empty set of pairs from $\varnothing\times\varnothing$ so $\varnothing^\varnothing=\{\varnothing\}$.
Is $\{\langle\rangle\}$ the same as $\{\varnothing\}$? They are at least morally different, in that it usually pays to be clear about whether we're working with sets of tuples or with sets of functions, and not care whether one of the things happen to be represented by the same object when everything is reduced to axiomatic set theory.
On the other hand, a fairly common formalization of tuples (when their length isn't fixed) is that a tuple is a map from the first few natural numbers to whatever the elements of the tuple is. Under this formalization, in particular the empty tuple is represented by the empty map!
So in this sense, $\varnothing^0$ and $\varnothing^\varnothing$ (and in fact $A^0$ and $A^\varnothing$ for any set $A$) do indeed denote the same mathematical object.
However, this identity depends on representational choices along the way that one really should try not to let anything substantial depend on, so if you try to use $\varnothing^0 = \varnothing^\varnothing$ for anything, that's a sign that your plan is either confused or at last confusING.
