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  1. Michael Greinecker wrote:

    There is exactly one such function that trivialy maps all elements of the empty set to elements of (a set) S.

    How is a function with empty domain like? Why does there exist a unique one for a codomain $S$?

  2. Also

    In the usual set-theoretic way of identifying a function with its graph

    I was wondering how to characterize the subsets of the $x \sim f(x)$ plane that can be the graphs of some functions?

Thanks and regards!

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A function $f$ from $X$ to $S$ is a subset of $X\times S$ such that if $\langle x,s\rangle,\langle x,t\rangle\in f$, then $s=t$. If $X=\varnothing$, then $X\times S=\varnothing\times S=\varnothing$, and the only subset of $X\times S$ in that case is $\varnothing$: $\varnothing$ is the only function from $\varnothing$ to $S$, no matter what set $S$ is.

I’ve really already answered your second question as well: $F\subseteq X\times S$ is a function iff for each $x\in X$ there is at most one $s\in S$ such that $\langle x,s\rangle\in F$. If you replace at most one with exactly one, you ensure that the domain of $F$ is all of $X$. Another way to describe this is in terms of slices of the $X\times S$ plane. For each $x\in X$ let $S_x=\{x\}\times S$, the vertical slice at $x$. Then $F$ is a function if $|F\cap S_x|\le 1$ for each $x\in X$. (In elementary texts this is sometimes called the vertical line test.)

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