[EMAIL PROTECTED] wrote:
>
> Consider functions of one variable whose domain and range are both
> {0,1,2,...,n-1}. There are n^n possible functions.
n!, I'd say, since the range of any function that isn't one-to-one is
_not_ {0..n-1}. Did you mean that the range was a subset of {0..n-1}? Or
perhaps (equivalently) you meant to say "codomain" instead of "range"?
> How many of these
> are linear [i.e. F(a+b) = F(a) + F(b) + c, where c is the same for all
> a,b (if it were different, that would be trivial)]? For any one
> definition of +, there will be some number;
This strikes me as completely false. Can't be bothered to prove it,
though. Especially since the problem is currently not well-defined :-)
> I'm interested in the sum
> over all definitions of + that satisfy the usual requirements of
> associativity, commutativity, additive identity, etc.
Hmm. This is horribly inexact. Do you mean the usual requirements for a
group? A field? What?
And like anonymous says, if you are going to ask these weird questions
(some of which are quite entertaining), you could at least say why.
Cheers,
Ben.
--
http://www.apache-ssl.org/ben.html
"My grandfather once told me that there are two kinds of people: those
who work and those who take the credit. He told me to try to be in the
first group; there was less competition there."
- Indira Gandhi
