If X is a set containing n elements and Y is a set containing m elements, how many functions are there from X to Y? How many of these functions are one-to-one?

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- May 29th 2009, 07:53 PMfardeen_genCombinations?
If X is a set containing n elements and Y is a set containing m elements, how many functions are there from X to Y? How many of these functions are one-to-one?

- May 31st 2009, 01:49 AMIsomorphism
For every element in X, I just need to associate an element of Y(for which I have m choices). Thus $\displaystyle m^n$ functions.

For one-one functions, For every element in X, I need to associate*only one*element of Y. Thus for the first one I have, m choices. Then for the second element, I have m-1 choices and so on. So it is $\displaystyle \frac{m!}{(m-n)!} $ - May 31st 2009, 01:56 AMMoo
And for your information, there are $\displaystyle m\times (m-1)\times \cdots\times (m-n+1)$ injective functions from X to Y.

- May 31st 2009, 02:26 AMIsomorphism
- May 31st 2009, 02:33 AMMoo
Yes, it's the same.

But that's what I had in my notes. (they're from last year, so I don't have the proofs...)

I'm wondering... If $\displaystyle m\neq n$, is it possible to have one-to-one functions from X to Y ? :D

We may consider that we're dealing with functions whose domain is X, and whose range is Y. In which case, there is no one-to-one functions (Thinking)

If $\displaystyle m=n$, the number of bijective functions is n! - May 31st 2009, 06:46 AMpankaj