$(a)$ How many functions are there from a nonempty set $S$ into
$\emptyset$?$(b)$ How many functions are there from $\emptyset$ into an arbitrary
set $S$?
This question seems very simplistic but I don't know the answer. I think for $(a)$ that there isn't a function that maps a set $S$ into a empty set? For $(b)$ I assume it to be all function that map the empty set to an arbitrary set since all sets contain the empty set?
Best Answer
A function from a set $A$ to a set $B$ is a subset of $A\times B$ satisfying certain conditions, one of which is that its domain is $A$. If either $A$ or $B$ is empty, $A\times B=\varnothing$, and $\varnothing$ is therefore the only subset of $A\times B$. If $A\ne\varnothing$, $\varnothing$ is not a function with domain $A$, so you’re quite right about $(a)$: there are no such functions. If $A=\varnothing$, though, it’s a different story. The domain of the function $\varnothing$ is $\{a:\langle a,b\rangle\in\varnothing\}$, which is ... ?