Is there an application form $\emptyset\to\emptyset$. What is the cardinal of $\mathcal F(\emptyset,\emptyset)$ where $\mathcal F(X,Y)$ is the set of the function from $X\to Y$ ? I would say $0$ because a function can't associated nothing at nothing, but I also know that $0^0=1$, and thus the cardinal of $\mathcal F(X,Y)$ such that $|X|=|Y|=0=|\emptyset|$ would be $1$. 
 A: A function $f:X\rightarrow Y$ is by definition (especially in set theory) a subset of $X\times Y$ such that for each $x\in X$ there is exactly one $y\in Y$ with $\langle x,y\rangle\in f$. 
If $X=\varnothing$ then $X\times Y=\varnothing$ wich has exactly one subset: $f=\varnothing$. 
The mentioned condition on $f$ for being a function is vacuously satisfied, so $f=\varnothing$ is a function.
This shows that there is exactly one function $\varnothing\rightarrow Y$. The so-called empty function.
$Y=\varnothing$ is a special case here.

Edit:
What above is called "function" is often called "graph of function". 
In e.g. the theory of categories a function $f:X\rightarrow Y$ is defined to be a triple $\langle X,G_f,Y\rangle$ where $G_f$ is its graph and denotes the subset of $X\times Y$ mentioned above. In that context the empty function $f:\varnothing\rightarrow Y$ is the triple $\langle\varnothing,\varnothing,Y\rangle$.
A: Yes; this application is known as the "empty function", or "empty graph".  In other words, $\mathcal F(\emptyset,\emptyset) = \{\emptyset\} \neq \emptyset$. 
There are several reasons that considering this to be an application makes set theory "nice".
