Why is $|Y^{\emptyset}|=1$ but $|\emptyset^Y|=0$ where $Y\neq \emptyset$ I have a question about the set of functions from a set to another set. I am wondering about the degenerate cases. Suppose $X^Y$ denotes the set of functions from a set $Y$ to a set $X$, why is $|Y^{\emptyset}|=1$ but $|\emptyset^Y|=0$ where $Y\neq \emptyset$?

Because the empty function is the unique function from the empty set to an arbitrary set $Y$, while if $Y\neq\emptyset$, then there exists $y\in Y$, but there's no place in $\emptyset$ for a function to send $y$ to.
