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$(a)$ 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?

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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 ... ?

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  • $\begingroup$ Can you elaborate more on why a set $A$ to a set $B$ is a subset of $A\times B$ and why if either $A$ or $B$ is empty then $A\times B =\emptyset $? $\endgroup$ – Tom Aug 25 '13 at 7:58
  • $\begingroup$ @Tom: The first is just a matter of definition: a function from $A$ to $B$ is a particular kind of relation from $A$ to $B$, and by definition the relations from $A$ to $B$ are precisely the subsets of $A\times B$. The second is really also just a matter of definition: $A\times B$ is by definition the set of all ordered pairs $\langle a,b\rangle$ such that $a\in A$ and $b\in B$. If either $A$ or $B$ is empty, then there clearly are no such pairs, so $A\times B=\varnothing$ as well. $\endgroup$ – Brian M. Scott Aug 25 '13 at 8:00
  • $\begingroup$ Is there an explicit, obvious difference between $A$ to $B$ and $A\times B$? Because I can't see none. They both seem to be the same to me. $\endgroup$ – Tom Aug 25 '13 at 8:03
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    $\begingroup$ @Tom: In general you should expect that in order to prove something about a mathematical object, you’ll have to use its definition in some way. Since a function from $A$ to $B$ is by definition a certain kind of subset of $A\times B$, you should expect to have to consider what $\varnothing\times S$ and $S\times\varnothing$ are in order to answer questions about functions from $\varnothing$ to $S$ or from $S$ to $\varnothing$. I cannot emphasize this strongly enough: when you’re just beginning and therefore haven’t much of a feel for what to look for, your automatic reaction should be ... $\endgroup$ – Brian M. Scott Aug 25 '13 at 8:20
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    $\begingroup$ ... to start with the definitions of the objects involved. Here that immediately tells you that $f$ can only be the empty set, and the questions then become: Is the empty set a function from a non-empty $S$ to $\varnothing$? (No.) Is the empty set a function from $\varnothing$ to a non-empty set $S$? $\endgroup$ – Brian M. Scott Aug 25 '13 at 8:22
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HINT: Recall that a function from $A$ to $B$ is a subset of $A\times B$, whose domain equals $A$.

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