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

[SithsNGiggles]

Homework Statement

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

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 anyone 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.

 

[Dick]

Makes perfect sense to me.