## Please check my proof; how many functions exist from X to Y?

1. The problem statement, all variables and given/known data
I just need to know if it makes sense; I was told that I can't have anyone make any improvements on what I've written myself.

Question: If $X = \{ x_1 , \ldots, x_n \}$ and $Y = \{ y_1 , \ldots, y_m \}$, how many functions from $X$ to $Y$ exist?

My answer: $m^n$ functions

3. The attempt at a solution
For any element $x \in X$, there exists a unique $y \in Y$ for which $F(x) = y$.

Every $n$ element in [itex[X[/itex] will be paired with any one of the $m$ elements in $Y$.
i.e. there exist $m$ possible $F(x_1)$ in $Y$ that can be paired with $x_1$.
$x_2$ can be paired with $m$ possible $F(x_2)$
$\vdots$
$x_n$ can be paired with $m$ possible $F(x_n)$.

Because the domain $D_F = X$, every function generated through F will contain $n$ coordinate pairs. Furthermore, since there are $m$ possible values $F(x) = y$ for each element $x$, there are $n$ factors of $m$, or $m^n$, possible functions.

Thanks for any commentary (but not actual help!) you can provide.
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 Recognitions: Homework Help Science Advisor Makes perfect sense to me.