Register to reply

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

by SithsNGiggles
Tags: check, exist, functions, proof
Share this thread:
SithsNGiggles
#1
Mar3-12, 12:50 PM
P: 187
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 [itex]X = \{ x_1 , \ldots, x_n \}[/itex] and [itex]Y = \{ y_1 , \ldots, y_m \}[/itex], how many functions from [itex]X[/itex] to [itex]Y[/itex] exist?

My answer: [itex]m^n[/itex] functions

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

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

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

Thanks for any commentary (but not actual help!) you can provide.
Phys.Org News Partner Science news on Phys.org
What lit up the universe?
Sheepdogs use just two simple rules to round up large herds of sheep
Animals first flex their muscles
Dick
#2
Mar3-12, 04:05 PM
Sci Advisor
HW Helper
Thanks
P: 25,228
Makes perfect sense to me.


Register to reply

Related Discussions
Lebesgue measurability proof - check my proof? Calculus & Beyond Homework 2
An inner product must exist on the set of all functions in Hilbert space Advanced Physics Homework 5
Proof that ghosts don't exist General Discussion 198
Proof that there exist such an element in Q Calculus & Beyond Homework 3