What is the set of functions from $X$ into $\emptyset$.

Question

Let $X$ be a set.

What is the set of functions from $\emptyset$ into $X$ .

My answers:

It is not a function because there is not a any element in set of domain.
It is empty set because for all $x\in X$ there is no any element in the set of range.

Can check you my answer?

See http://math.stackexchange.com/questions/475613/nonempty-set-mapped-to-emptyset-and-vice-versa?rq=1 — Bert Lindenhovius, Nov 16 '16 at 01:00
Your second answer is correct. Your first, however, is not: $\varnothing$ is the unique function from $\varnothing$ to $X$. — Brian M. Scott, Nov 16 '16 at 07:07
@BrianM.Scott How can it be unique function? Can you explain? — , Nov 16 '16 at 09:12
@Kahler: A function $f$ from $\varnothing$ to $X$ is by definition a subset of $\varnothing\times X$ such that for each $a\in\varnothing$ there is exactly one $x\in X$ such that $\langle a,x\rangle\in f$. $\varnothing\times X=\varnothing$, and the only subset of $\varnothing$ is $\varnothing$, so either $\varnothing$ is the only function from $\varnothing$ to $X$, or there are no such functions. The only question, then, is whether $\varnothing$ is a function from $\varnothing$ to $X$: is it true that *if* $a\in\varnothing$, then there is exactly one $x\in X$ such that ... — Brian M. Scott, Nov 16 '16 at 17:08
... $\langle a,x\in\varnothing$? If this were not the case, either there would be an $a\in\varnothing$ with no $x\in X$ such that $\langle a,x\in\varnothing$, or there would be an $a\in\varnothing$ and $x_1,x_2\in X$ such that $x_1\ne x_2$ and $\langle a,x_1\rangle,\langle a,x_2\rangle\in\varnothing$. But neither of these is the case, *because there is no* $a\in\varnothing$ *in the first place. It is true that if* $\langle a\in\varnothing$ then there is exactly one $x\in X$ such that $\langle a,x\rangle\in\varnothing$, simply because there is no $a\in\varnothing$ that ... — Brian M. Scott, Nov 16 '16 at 17:13

What is the set of functions from $X$ into $\emptyset$.

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